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DESIGN FOR A BRAIN

BY THE SAME AUTHOR

An Introduction to Cybernetics

/4 s.

DESIGN FOR A BRAIN

The origin of adaptive behaviour

W. ROSS ASHBY

M.A., M.D., D.P.M.

Director, Burden Neurological Institute;
Late Director of Research, Barnuood House, Gloucester

SECOND EDITION
REVISED

NEW YORK

JOHN WILEY & SONS. Inc.

London: CHAPMAN <fe HALL. Limited
1960

First Published 1952

Reprinted (with corrections) 1954
Second Edition (revised) 1960

CATALOGUE NO. 493/4

Printed in Great Britain by Butler & Tanner Ltd., Frome and London

Preface

The book is not a treatise on all cerebral mechanisms but a pro-
posed solution of a specific problem: the origin of the nervous
system's unique ability to produce adaptive behaviour. The
work has as basis the fact that the nervous system behaves adap-
tively and the hypothesis that it is essentially mechanistic; it
proceeds on the assumption that these two data are not irrecon-
cilable. It attempts to deduce from the observed facts what sort
of a mechanism it must be that behaves so differently from any
machine made so far. Other proposed solutions have usually left
open the question whether some different theory might not fit the
facts equally well: I have attempted to deduce what is necessary,
what properties the nervous system must have if it is to behave
at once mechanistically and adaptively.

For the deduction to be rigorous, an adequately developed logic
of mechanism is essential. Until recently, discussions of mechan-
ism were carried on almost entirely in terms of some particular
embodiment — the mechanical, the electronic, the neuronic, and so
on. Those days are past. There now exists a well developed
logic of pure mechanism, rigorous as geometry, and likely to play
the same fundamental part, in our understanding of the complex
systems of biology, that geometry does in astronomy. Only by
the development of this basic logic has the work in this book been
made possible.

The conclusions reached are summarised at the end of Chapter
18, but they are likely to be unintelligible or misleading if taken
by themselves; for they are intended only to make prominent the
key points along a road that the reader has already traversed.
They may, however, be useful as he proceeds, by helping him to
distinguish the major features from the minor.

Having experienced the confusion that tends to arise whenever
we try to relate cerebral mechanisms to observed behaviour, I
made it my aim to accept nothing that could not be stated in
mathematical form, for only in this language can one be sure,
during one's progress^ that one is not unconsciously changing the

PREFACE

meaning of terms, or adding assumptions, or otherwise drifting
towards confusion. The aim proved achievable. The concepts
of organisation, behaviour, change of behaviour, part, whole,
dynamic system, co-ordination, etc. — notoriously elusive but
essential — were successfully given rigorous definition and welded
into a coherent whole. But the rigour and coherence depended
on the mathematical form, which is not read with ease by every-
body. As the basic thesis, however, rests on essentially common-
sense reasoning, I have been able to divide the account into two
parts. The main account (Chapters 1-18) is non-mathematical
and is complete in itself. The Appendix (Chapters 19-22) contains
the mathematical matter.

Since the reader will probably need cross-reference frequently,
the chapters have been divided into sections. These are indicated
thus: S. 4/5, which means Chapter ,4's fifth section. Each figure
and table is numbered within its own section: Figure 4/5/2 is the
second figure in S. 4/5. Section-numbers are given at the top of
every page, so finding a section or a figure should be as simple
and direct as finding a page.

It is a pleasure to be able to express my indebtedness to the
Governors of Barnwood House and to Dr. G. W. T. H. Fleming
for their generous support during the prosecution of the work, and
to Professor F. L. Golla and Dr. W. Grey Walter for much help-
ful criticism.

Preface to the Second Edition

At the time when this book was first written, information theory
was just beginning to be known. Since then its contribution to
our understanding of the logic of mechanism has been so great
that a separate treatment of these aspects has been given in my
Introduction to Cybernetics * (which will be referred to in this book
as /. to C). Its outlook and methods are fundamental to the
present work.

The overlap is small. I. to C. is concerned with first principles,
as they concern the topics of mechanism, communication, and
regulation; but it is concerned with the principles and does not
appreciably develop their applications. It considers mechanisms
as if they go in small discrete steps, a supposition that makes their
logical properties very easy to understand. Design for a Brain,
while based on the same principles, mentions them only so far as
is necessary for their application to the particular problem of the
origin of adaptive behaviour. It considers mechanisms that
change continuously (i.e. as the steps shrink to zero), for this
supposition makes their practical properties more evident. It has.
been written to be complete in itself, but the reader may find
/. to C. helpful in regard to the foundations.

In the eight years that have elapsed between the preparations
of the two editions, our understanding of brain-like mechanisms
has improved immeasurably. For this reason the book has been
re-arranged, and the latter two-thirds completely re-written.
The new version, I am satisfied, presents the material in an alto-
gether clearer, simpler, and more cogent form than the earlier.

The change of lay-out has unfortunately made a retention of
the previous section-numberings impossible, so there is no cor-
respondence between the numberings in the two editions. I
would have avoided this source of confusion if I could, but felt
that the claims of clarity and simplicity must be given precedence
over all else.

* Chapman & Hall, London : John Wiley & Sons, New York ; 3rd imp.
1958. Also translations in Czech, French, Polish, Russian and Spanish.

vii

Contents

CHAPTER PAGE

Preface v

Preface to the Second Edition vii

1 The Problem 1

Behaviour, reflex and learned. Relation of part to part.
Genetic control. Restrictions on the concepts. Conscious-
ness. The problem.

2 Dynamic Systems 13

Variable and system. The operational method. Phase-
space and field. The natural system. Strategy for the com-
plex system.

3 The Organism as Machine 30

The specification of behaviour. Organism and environment.
Essential variables.

4 Stability 44

Diagram of immediate effects. Feedback. Goal-seeking.
Stability and the whole.

5 Adaptation as Stability 58

Homeostasis. Generalised homeostasis. Survival. Stability
and co-ordination.

6 Parameters 71

Parameter and field. Stimuli. Joining systems. Para-
meter and stability. Equilibria of part and whole.

7 The Ultrastable System 80

The implications of adaptation. The implications of double
feedback. Step-functions. Systems containing step-
mechanisms. The ultrastable system.

8 The Homeostat 100

The Homeostat as adapter. Training. Some apparent
faults.

9 Ultrastability in the Organism 122

Step-mechanisms in the organism. A molecular basis for
memory ? Are step-mechanisms necessary ? Levels of feed-
back. Control of aim. Gene-pattern and ultrastability.
Summary.

CONTENTS
CHAPTER PAGE

10 The Recurrent Situation 138

Accumulation of adaptations.

11 The Fully- joined System 148

Adaptation-time. Cumulative adaptation.

12 Temporary Independence 158

Independence. The effects of constancy. The effects of
local stabilities.

13 The System with Local Stabilities 171

Progression to equilibrium. Dispersion. Localisation in the
poly stable system.

14 Repetitive Stimuli and Habituation 184

Habituation. Minor disturbances.

15 Adaptation in Iterated and Serial Systems 192

Iterated systems. Serial adaptation.

16 Adaptation in the Multistable System 205

The richly -joined environment. The poorly- joined environ-
ment. Retroactive inhibition.

17 Ancillary Regulations 218

Communication within the brain. Ancillary regulations.
Distribution of feedback.

18 Amplifying Adaptation 231

Selection in the state-determined system. Amplifying
adaptation. The origin of adaptive behaviour.

APPENDIX

19 The State-determined System 241

The logic of mechanism. Canonical representation. Trans-
formations.

Stability

Probability of stability.

253

Parameters

Joining systems. The state-determined system.

262

The Effects of Constancy

Ultrastability. Temporary independence. Diagrams of
effects.

272

References

281

Index

283

77554

CHAPTER 1

The Problem

1/1. How does the brain produce adaptive behaviour ? In
attempting to answer the question, scientists have discovered two
sets of facts and have had some difficulty in reconciling them.
On the one hand the physiologists have shown in a variety of ways
how closely the brain resembles a machine: in its dependence on
chemical reactions, in its dependence on the integrity of anatomical
paths, and in the precision and determinateness with which its
component parts act on one another. On the other hand, the
psychologists and biologists have confirmed with full objectivity
the layman's conviction that the living organism behaves typically
in a purposeful and adaptive way. These two characteristics of
the brain's behaviour have proved difficult to reconcile, and some
workers have gone so far as to declare them incompatible.

Such a point of view will not be taken here. I hope to show
that a system can be both mechanistic in nature and yet produce
behaviour that is adaptive. I hope to show that the essential
difference between the brain and any machine yet made is that
the brain makes extensive use of a method hitherto little used in
machines. I hope to show that by the use of this method a
machine's behaviour may be made as adaptive as we please, and
that the method may be capable of explaining even the adaptive-
ness of Man.

But first we must examine more closely the nature of the
problem, and this will be commenced in this chapter. The suc-
ceeding chapters will develop more accurate concepts, and when
we can state the problem with precision we shall not be far from
its solution.

Behaviour, reflex and learned

1/2. The activities of the nervous system may be divided more
or less distinctly into two types. The dichotomy is probably an

DESIGN FOR A BRAIN 1/3

over-simplification, but it will be sufficient until we have developed
a more elaborate technique.

The first type is reflex behaviour. It is inborn, it is genetically
determined in detail, it is a product, in the vertebrates, chiefly
of centres in the spinal cord and in the base of the brain, and it is
not appreciably modified by individual experience. The second
type is learned behaviour. It is not inborn, it is not genetically
determined in detail (more fully discussed in S. 1/9), it is a product
chiefly of the cerebral cortex, and it is modified markedly by the
organism's individual experiences.

1/3. With the first or reflex type of behaviour we shall not be
concerned. We assume that each reflex is produced by some
neural mechanism whose physico-chemical nature results inevit-
ably in the characteristic form of behaviour, that this mechanism
is developed under the control of the gene-pattern and is inborn,
and that the pattern of behaviour produced by the mechanism is
usually adapted to the animal's environment because natural
selection has long since eliminated all non-adapted variations.
For example, the complex activity of ' coughing ' is assumed to
be due to a special mechanism in the nervous system, inborn and
developed by the action of the gene-pattern, and adapted and
perfected by the fact that an animal who is less able to clear its
trachea of obstruction has a smaller chance of survival.

Although the mechanisms underlying these reflex activities are
often difficult to study physiologically, and although few are known
in all their details, yet it is widely held among physiologists that
no difficulty of principle is involved. Such behaviour and such
mechanisms will not therefore be considered further.

1/4. It is with the second type of behaviour that we are con-
cerned: the behaviour that is not inborn but learned. Examples
of such reactions exist in abundance, and any small selection
must seem paltry. Yet I must say what I mean, if only to give
the critic a definite target for attack. Several examples will
therefore be given.

A dog selected at random for an experiment with a conditioned
response can be made at will to react to the sound of a bell either
with or without salivation. Further, once trained to react in
one way it may, with little difficulty, be trained to react later in

1/5 THE PROBLEM

the opposite way. The salivary response to the sound of a bell
cannot, therefore, be due to a mechanism of fixed properties.

A rat selected at random for an experiment in maze-running
can be taught to run either to right or left by the use of an appro-
priately shaped maze. Further, once trained to turn to one side
it can be trained later to turn to the other.

Perhaps the most striking evidence that animals, after training,
can produce behaviour which cannot possibly have been inborn
is provided by the circus. A seal balances a ball on its nose for
minutes at a time; one bear rides a bicycle, and another walks
on roller skates. It would be ridiculous to suppose that these
reactions are due to mechanisms both inborn and specially per-
fected for these tricks.

Man himself provides, of course, the most abundant variety of
learned reactions: but only one example will be given here. If
one is looking down a compound microscope and finds that the
object is not central but to the right, one brings the object to
the centre by pushing the slide still farther to the right. The
relation between muscular action and consequent visual change
is the reverse of the usual. The student's initial bewilderment
and clumsiness demonstrate that there is no neural mechanism
inborn and ready for the reversed relation. But after a few days'
practice co-ordination develops.

These examples, and all the facts of which they are representa-
tive, show that the nervous system is able to develop ways of
behaving which are not inborn and are not specified in detail by
the gene-pattern.

1/5. Learned behaviour has many characteristics, but we shall
be concerned chiefly with one: when animals and children learn,
not only does their behaviour change, but it changes usually for
the better. The full meaning of ' better ' will be discussed in
Chapter 5, but in the simpler cases the improvement is obvious
enough. ' The burned child dreads the fire ' : after the experi-
ence the child's behaviour towards the fire is not only changed,
but is changed to a behaviour which gives a lessened chance of
its being burned again. We would at once recognise as abnormal
any child who used its newly acquired knowledge so as to get
to the flames more quickly.
To demonstrate that learning usually changes behaviour from a

DESIGN FOR A BRAIN 1/6

less to a more beneficial, i.e. survival-promoting, form would
need a discussion far exceeding the space available. But in this
introduction no exhaustive survey is needed. I require only
sufficient illustration to make the meaning clear. For this pur-
pose the previous examples will be examined seriatim.

When a conditioned reflex is established by the giving of food
or acid, the amount of salivation changes from less to more. And
the change benefits the animal either by providing normal lubri-
cation for chewing or by providing water to dilute and flush away
the irritant. When a rat in a maze has changed its behaviour so
that it goes directly to the food at the other end, the new behaviour
is better than the old because it leads more quickly to the animal's
hunger being satisfied. The circus animals' behaviour changes
from some random form to one determined by the trainer, who
applied punishments and rewards. The animals' later behaviour
is such as has decreased the punishments or increased the rewards.
In Man, the proposition that behaviour usually changes for the
better with learning would need extensive discussion. But in the
example of the finger movements and the compound microscope,
the later movements, which bring the desired object directly to
the centre of the field, are clearly better than the earlier move-
ments, which were ineffective for the microscopist's purpose.

Our problem may now be stated in preliminary form: what
cerebral changes occur during the learning process, and why does
the behaviour usually change for the better ? What type of
mechanistic process could show the same self-advancement ?

1/6. The nervous system is well provided with means for action.
Glucose, oxygen, and other metabolites are brought to it by the
blood so that free energy is available abundantly. The nerve
cells composing the system are not only themselves exquisitely
sensitive, but are provided, at the sense organs, with devices of
even higher sensitivity. Each nerve cell, by its ramifications,
enables a single impulse to become many impulses, each of which
is as active as the single impulse from which it originated. The
ramifications are followed by repeated stages of further ramifica-
tion, so that however small a change at any point we can put
hardly any bound to the size of the change or response that may
follow as the effect spreads. And by their control of the muscles,
the nerve cells can rouse to activity engines of high mechanical

1/7 THE PROBLEM

power. The nervous system, then, possesses almost unlimited
potentialities for action. But do these potentialities solve our
problem ? It seems not. We are concerned primarily with the
question why, during learning, behaviour changes for the better:
and this question is not answered by the fact that a given behaviour
can change to one of4 lesser or greater activity. The examples
given in S. 1/5, when examined for the energy changes before and
after learning, show that the question of the quantity of activity
is usually irrelevant.

But the evidence against regarding mere activity as sufficient
for a solution is even stronger : often an increase in the amount of
activity is not so much irrelevant as positively harmful. If a
dynamic system is allowed to proceed to vigorous action without
special precautions, the activity will usually lead to the destruction
of the system itself. A motor car with its tank full of petrol may
be set into motion, but if it is released with no driver its activity,
far from being beneficial, will probably cause the motor car to
destroy itself more quickly than if it had remained inactive. The
theme is discussed more thoroughly in S. 20/10; here it may be
noted that activity, if inco-ordinated, tends merely to the system's
destruction. How then is the brain to achieve success if its
potentialities for action are partly potentialities for self-destruction?

The relation of part to part

1/7. Our basic fact is that after the learning process the behaviour
is usually better adapted than before. We ask, therefore, what
property must be possessed by the neurons so that the manifesta-
tion by the neuron of this property shall result in the whole
organism's behaviour being improved.

A first suggestion is that if the nerve-cells are all healthy and
normal as little biological units, then the whole will appear healthy
and normal. This suggestion, .however, must be rejected as
inadequate. For the improvement in the organism's behaviour
is often an improvement in relation to entities which have no
counterpart in the life of a neuron. Thus when a dog, given food
in an experiment on conditioned responses, learns to salivate, the
behaviour improves because the saliva provides a lubricant for
chewing. But in the neuron's existence, since all its food arrives
in solution, neither ' chewing ' nor ' lubricant ' can have any direct

DESIGN FOR A BRAIN 1/8

relevance or meaning. Again, a maze-rat that has learned suc-
cessfully has learned to produce a particular pattern of move-
ment; yet the learning has involved neurons which are firmly
supported in a close mesh of glial fibres and never move in their
lives.

Finally, consider an engine-driver who has just seen a signal
and whose hand is on the throttle. If the light is red, the excita-
tion from the retina must be transmitted through the nervous
system so that the cells in the motor cortex send impulses down
to those muscles whose activity makes the throttle close. If the
light is green, the excitation from the retina must be transmitted
through the nervous system so that the cells in the motor cortex
make the throttle open. And the transmission is to be handled,
and the safety of the train guaranteed, by neurons which can
form no conception of ' red ', ' green ', ' train ', ' signal ', or
'accident ' ! Yet the system works.

Clearly, ' normality ' at the neuronic level is inadequate to
ensure normality in the behaviour of the whole organism, for the
two forms of normality stand in no definite relationship.

1/8. In the case of the engine-driver, it may be that there is- a
simple mechanism such that a red light activates a chain of nerve-
cells leading to the muscles which close the throttle while a green
light activates another chain of nerve-cells leading to the muscles
which make it open. In this way the effect of the colour of the
signal would be transmitted through the nervous system in the
appropriate way.

The simplicity of the arrangement is due to the fact that we
are supposing that the two reactions are using two independent
mechanisms. This separation may well occur in the simpler
reactions, but it is insufficient to explain the events of the more
complex reactions. In most cases the ' correct ' and the ' incor-
rect ' neural activities are alike composed of excitations, of
inhibitions, and of other processes each of which is physiological
in itself, but whose correctness is determined not by the process
itself but by the relations which it bears to other processes.

This dependence of the ' correctness ' of what is happening at
one point in the nervous system on what is happening at other
points would be shown if the engine-driver were to move over to
the other side of the cab. For if previously a flexion of the elbow

1/8 THE PROBLEM

had closed the throttle, the same action will now open it; and
what was the correct pairing of red and green to push and pull
must now be reversed. So the local action in the nervous system
can no longer be regarded as ' correct ' or ' incorrect ' in any
absolute sense, and the first simple solution breaks down.

Another example is given by the activity of chewing in so far
as it involves the tongue and teeth in movements which must
be related so that the teeth do not bite the tongue. No move-
ment of the tongue can by itself be regarded as wholly wrong, for
a movement which may be wrong when the teeth are just meeting
may be right when they are parting and food is to be driven on
to their line. Consequently the activities in the neurons which
control the movement of the tongue cannot be described as either
4 correct ' or * incorrect ': only when these activities are related to
those of the neurons which control the jaw movements can a
correctness be determined; and this property now belongs, not to
either separately, but only to the activity of the two in combination.

These considerations reveal the main peculiarity of the problem.
When the nervous system learns, its behaviour changes for the
better. When we consider its various parts, however, we find that
the value of one part's behaviour cannot be judged until the
behaviour of the other parts is known; and the values of their
behaviours cannot be known until the first part's behaviour is
known. All the valuations are thus conditional, each depending
on the others. Thus there is no criterion for ' better ' that can
be given absolutely, i.e. unconditionally. But a neuron must do
something. How then do the activities of the neurons become
co-ordinated so that the behaviour of the whole becomes better,
even though no absolute criterion exists to guide the individual

neuron

Exactly the same problem faces the designer of an artificial
brain, who wants his mechanical brain to become adaptive in its
behaviour. How can he specify the ' correct ' properties for each
part if the correctness depends not on the behaviour of each part
but on its relations to the other parts ? His problem is to get
the parts properly co-ordinated. The brain does this auto-
matically. What sort of a machine can be ^Z/-co-ordinating ?

This is our problem. It will be stated with more precision in
S. 1/17. But before this statement is reached, some minor topics
must be discussed.

DESIGN FOR A BRAIN 1/9

The genetic control of cerebral function

1/9. In rejecting the genetic control of the details of cerebral
function (in adaptation, S. 1/4) we must be careful not to reject
too much. The gene-pattern certainly plays some part in the
development of adaptive behaviour, for the various species,
differing essentially only in their gene-patterns, show character-
istic differences in their powers of developing it; the insects, for
instance, typically show little power while Man shows a great deal.

One difficulty in accounting for a new-born baby's capacity for
developing adaptations is that the gene-pattern that makes the
baby what it is has about 50,000 genes available for control of
the form, while the baby's brain has about 10,000,000,000 neurons
to be controlled (and the number of terminals may be 10 to 100
times as great). Clearly the set of genes cannot determine the
details of the set of neurons. Evidently the gene-pattern deter-
mines a relatively small number of factors, and then these factors
work actively to develop co-ordination in a much larger number
of neurons.

This formulation of how the gene-pattern comes into the picture
will perhaps suffice for the moment; it will be resumed in S. 18/6.
(/. to C, S. 14/6, also discusses the topic.)

Restrictions on the concepts to be used

1/10. Throughout the book I shall adhere to certain basic
assumptions and to certain principles of method.

I shall hold the biologist's point of view. To him, the most
fundamental facts are that the earth is over 2,000,000,000 years
old and that natural selection has been winnowing the living
organisms incessantly. As a result they are today highly special-
ised in the arts of survival, and among these arts has been the
development of a brain. Throughout this book the brain will be
treated simply as an organ that has been developed in evolution
as a specialised means to survival.

1/11. Conformably with this point of view, the nervous system,
and living matter in general, will be assumed to be essentially
similar to all other matter. So no use of any ' vital ' property
or tendency will be made, and no Deus ex machina will be invoked.

1/14 THE PROBLEM

The sole reason admitted for the behaviour of any part will be
of the form that its own state and the condition of its immediate
surroundings led, in accordance with the usual laws of matter,
to the observed behaviour.

1/12. The ' operational ' method will be followed; so no psycho-
logical concept will be used unless it can be shown in objective
form in non-living systems; and when used it will be considered
to refer solely to its objective form. Related is the restriction
that every concept used must be capable of objective demonstra-
tion. In the study of Man this restriction raises formidable
difficulties extending from the practical to the metaphysical.
But as most of the discussion will be concerned with the observed
behaviour of animals and machines, the peculiar difficulties will
seldom arise.

1/13. No teleological explanation for behaviour will be used. It
will be assumed throughout that a machine or an animal behaved
in a certain way at a certain moment because its physical and
chemical nature at that moment allowed it no other action. Never
will we use the explanation that the action is performed because
it will later be advantageous to the animal. Any such explanation
would, of course, involve a circular argument; for our purpose
is to explain the origin of behaviour which appears to be teleo-
logically directed.

1/14. It will be further assumed (except where the contrary is
stated explicitly) that the fuctioning units of the nervous system,
and of the environment, behave in a determinate way. By this
I mean that each part, if in a particular state internally and affected
by particular conditions externally, will behave in one way only,
(This is the determinacy shown, for instance, by the relays and
other parts of a telephone exchange.) It should be noticed that
we are not assuming that the ultimate units are determinate, for
these are atoms, which are known to behave in an essentially
indeterminate way; what we shall assume is that the significant
unit is determinate. The significant unit (e.g. the relay, the
current of several milliamperes, the neuron) is usually of a size
much larger than the atomic so that only the average property
of many atoms is significant. These averages are often determinate

DESIGN FOR A BRAIN 1/15

in their behaviour, and it is to these averages that our assumption
applies.

The question whether the nervous system is composed of parts
that are determinate or stochastic has not yet been answered.
In this book we shall suppose that they are determinate. That
the brain is capable of behaving in a strikingly determinate way
has been demonstrated chiefly by feats of memory. Some of the
demonstrations depend on hypnosis, and are not quite sufficiently
clear in interpretation for quotation here. Skinner, however, has
produced some striking evidence by animal experiment that the
nervous system, if the surrounding conditions can be restored
accurately, may behave in a strictly reproducible way. By
differential reinforcement with food, Skinner trained twenty
young pigeons to peck at a translucent key when it was illuminated
with a complex visual pattern. They were then transferred to the
usual living quarters where they were used for no further experi-
ments but served simply as breeders. Small groups were tested
from time to time for retention of the habit.

' The bird was fed in the dimly-lighted experimental apparatus
in the absence of the key for several days, during which
emotional responses to the apparatus disappeared. On the
day of the test the bird was placed in the darkened box. The
translucent key was present but not lighted. No responses
were made. When the pattern was projected upon the key,
all four birds responded quickly and extensively. . . . This
bird struck the key within two seconds after presentation of
a visual pattern that it had not seen for four years, and at
the precise spot upon which differential reinforcement had
previously been based.'

The assumption that the parts are determinate is thus not un-
reasonable. But we need not pre-judge the issue; the book is an
attempt to follow the assumption of determinacy wherever it leads.
When it leads to obvious error will be time to question its validity.

1/15. To be consistent with the assumptions already made, we
must suppose (and the author accepts) that a real solution of our
problem will enable an artificial system to be made that will be
able, like the living brain, to develop adaptation in its behaviour.
Thus the work, if successful, will contain (at least by implication)
a specification for building an artificial brain that will be similarly
self-co-ordinating.

1/16 THE PROBLEM

The knowledge that the proposed solution must be put to this
test will impose some discipline on the concepts used. In particular,
this requirement will help to prevent the solution from being a
mere verbalistic ' explanation ', for in the background will be the
demand that we build a machine to do these things.

Consciousness
1/16. The previous sections have demanded that we shall make
no use of the subjective elements of experience; and I can antici-
pate by saying that in fact the book makes no such use. At
times its rigid adherence to the objective point of view may
jar on the reader and may expose me to the accusation that I am
ignoring an essential factor. A few words in explanation may
save misunderstanding.

Throughout the book, consciousness and its related subjective
elements are not used for the simple reason that at no point have I
found their introduction necessary. This is not surprising, for the
book deals with only one of the properties of the brain, and with
a property — learning — that has long been recognised to have no
necessary dependence on consciousness. Here is an example to
illustrate their independence. If a cyclist wishes to turn to the
left, his first action must be to turn the front wheel to the right
(otherwise he will fall outwards by centrifugal force). Every
practised cyclist makes this movement every time he turns, yet
many cyclists, even after they have made the movement hundreds
of times, are quite unconscious of making it. The direct inter-
vention of consciousness is evidently not necessary for adaptive
learning.

Such an observation, showing that consciousness is sometimes
not necessary, gives us no right to deduce that consciousness
does not exist. The truth is quite otherwise, for the fact of the
existence of consciousness is prior to all other facts. If I perceive
— am aware of — a chair, I may later be persuaded, by other
evidence, that the appearance was produced only by a trick of
lighting; I may be persuaded that it occurred in a dream, or
even that it was an hallucination; but there is no evidence in
existence that could persuade me that my awareness itself was
mistaken — that I had not really been aware at all. This know-
ledge of personal awareness, therefore, is prior to all other forms
of knowledge.

DESIGN FOR A BRAIN 1/17

If consciousness is the most fundamental fact of all, why is it
not used in this book ? The answer, in my opinion, is that
Science deals, and can deal, only with what one man can demon-
strate to another. Vivid though consciousness may be to its
possessor, there is as yet no method known by which he can
demonstrate his experience to another. And until such a method,
or its equivalent, is found, the facts of consciousness cannot be
used in scientific method.

The problem

1/17. It is now time to state the problem. Later, when more
exact concepts have been developed, it will be possible to state
the problem more precisely (S. 5/14).

It will be convenient, throughout the discussion, to have some
well-known, practical problem to act as type-problem, so that
general statements can always be referred to it. I select the
following. When a kitten first approaches a fire its reactions are
unpredictable and usually inappropriate. It may walk almost
into the fire, or it may spit at it, or may dab at it with a paw,
or try to sniff at it, or crouch and ' stalk ' it. Later, however,
when adult, its reactions are different. It approaches the fire and
seats itself at a place where the heat is moderate. If the fire burns
low, it moves nearer. If a hot coal falls out, it jumps away. Its
behaviour towards the fire is now ' adaptive '. I might have taken
as type-problem some experiment published by a psychological
laboratory, but the present example has several advantages. It
is well known; it is representative of a wide class of important
phenomena; and it is not likely to be called in question by the
discovery of some small technical flaw.

We take as basic the assumptions that the organism is mechan-
istic in nature, that it is composed of parts, that the behaviour of
the whole is the outcome of the compounded actions of the parts,
that organisms change their behaviour by learning, and that they
change it so that the later behaviour is better adapted to their
environment than the earlier. Our problem is, first, to identify
the nature of the change which shows as learning, and secondly,
to find why such changes should tend to cause better adaptation
for the whole organism.

CHAPTER 2

Dynamic Systems

2/1. In the previous chapter we have repeatedly used the con-
cepts of a system, of parts in a whole, of the system's behaviour,
and of its changes of behaviour. These concepts are fundamental
and must be properly defined. Accurate definition at this stage
is of the highest importance, for any vagueness here will infect
all the subsequent discussion; and as we shall have to enter the
realm where the physical and the psychological meet, a realm
where the experience of centuries has found innumerable possi-
bilities of confusion, we shall have to proceed with unusual caution.

That some caution is necessary can be readily shown. We have,
for instance, repeatedly used the concept of a ' change of
behaviour ', as when the kitten stopped dabbing at the red-hot
coal and avoided it. Yet behaviour is itself a sequence of changes
(e.g. as the paw moves from point to point). Can we distinguish
clearly those changes that constitute behaviour from those changes
that are from behaviour to behaviour ? It is questions such as
these which emphasize the necessity for clarity and a secure
foundation. (The subject has been considered more extensively
in /. to C, Part I; the shorter version given here should be sufficient
for our purpose in this book.)

We start by assuming that we have before us some dynamic
system, i.e. something that may change with time. We wish to
study it. It will be referred to as the ' machine ', but the word
must be understood in the widest possible sense, for no restriction
is implied at the moment other than that it should be objective.

2/2. As we shall be more concerned in this chapter with prin-
ciples than with practice, we shall be concerned chiefly with
constructing a method for the study of this unknown machine.
When the method is constructed, it must satisfy the demands
implied by the axioms of S. 1/10-15:

(1) The method must be precisely defined, and in operational
form;

DESIGN FOR A BRAIN 2/3

(2) it must be applicable equally readily (at least in principle)

to all material 'machines', whether animate or inanimate;

(3) its procedure for obtaining information from the ' machine '

must be wholly objective (i.e. accessible or demonstrable
to all observers);

(4) it must obtain its information solely from the ' machine '

itself, no other source being permitted.

The actual form developed may appear to the practical worker
to be clumsy and inferior to methods already in use; it probably
is. But it is not intended to compete with the many specialised
methods already in use. Such methods are usually adapted to a
particular class of dynamic systems: one method is specially suited
to electronic circuits, another to rats in mazes, another to solutions
of reacting chemicals, another to automatic pilots, another to
heart-lung preparations. The method proposed here must have
the peculiarity that it is applicable to all; it must, so to speak,
specialise in generality.

Variable and system

2/3. In /. to C, Chapter 2, is shown how the basic theory can
be founded on the concept of unanalysed states, as a mother might
distinguish, and react adequately to, three expressions on her
baby's face, without analysing them into so much opening of the
mouth, so much wrinkling of the nose, etc. In this book, however,
we shall be chiefly concerned with the relations between parts, so
we will assume that the observer proceeds to record the behaviour
of the machine's individual parts. To do this he identifies any
number of suitable variables. A variable is a measurable quantity
which at every instant has a definite numerical value. A ' grand-
father ' clock, for instance, might provide the following variables:
— the angular deviation of the pendulum from the vertical; the
angular velocity with which the pendulum is moving; the angular
position of a particular cog-wheel; the height of a driving weight;
the reading of the minute-hand on the scale; and the length of
the pendulum. If there is any doubt whether a particular
quantity may be admitted as a ' variable ' I shall use the criterion
whether it can be represented by a pointer on a dial.

All the quantities used in physics, chemistry, biology, physio-
logy, and objective psychology, are variables in the defined sense.

2/4 DYNAMIC SYSTEMS

Thus, the position of a limb can be specified numerically by co-
ordinates of position, and movement of the limb can move a pointer
on a dial. Temperature at a point can be specified numerically
and can be recorded on a dial. Pressure, angle, electric potential,
volume, velocity, torque, power, mass, viscosity, humidity, sur-
face tension, osmotic pressure, specific gravity, and time itself,
to mention only a few, can all be specified numerically and
recorded on dials. Eddington's statement on the subject is
explicit: ' The whole subject matter of exact science consists of
pointer readings and similar indications.' ' Whatever quantity
we say we are " observing ", the actual procedure nearly always
ends in reading the position of some kind of indicator on a
graduated scale or its equivalent.'

Whether the restriction to dial-readings is justifiable with living
subjects will be discussed in the next chapter.

One minor point should be noticed as it will be needed later.
The absence of an entity can always be converted to a reading on
a scale simply by considering the entity to be present but in
zero degree. Thus, ' still air ' can be treated as a wind blowing at
0 m.p.h. ; 4 darkness ' can be treated as an illumination of 0 foot-
candles ; and the giving of a drug can be represented by indicating
that its concentration in the tissues has risen from its usual value
of 0 per cent.

2/4. It will be appreciated that every real ' machine ' embodies
no less than an infinite number of variables, all but a few of which
must of necessity be ignored. Thus if we were studying the swing
of a pendulum in relation to its length we would be interested in
its angular deviation at various times, but we would often ignore
the chemical composition of the bob, the reflecting power of its
surface, the electric conductivity of the suspending string, the
specific gravity of the bob, its shape, the age of the alloy, its
degree of bacterial contamination, and so on. The list of what
might be ignored could be extended indefinitely. Faced with
this infinite number of variables, the experimenter must, and of
course does, select a definite number for examination — in other
words, he defines an abstracted system. Thus, an experimenter
once drew up Table 2/4/1. He thereby selected his variables,
of time and three others, ready for testing. This experiment
being finished, he later drew up other tables which included new

DESIGN FOR A BRAIN

2/5

Time
(mins.)

Distance of

secondary
coil (cm.)

Part of skin
stimulated

Secretion of
saliva during

30 sees.

(drops)

Table 2/4/1

variables or omitted old. These new combinations were new
systems.

2/5. Because any real ' machine ' has an infinity of variables,
from which different observers (with different aims) may reason-
ably make an infinity of different selections, there must first be
given an observer (or experimenter); a system is then defined as
any set of variables that he selects from those available on the
real * machine '. It is thus a list, nominated by the observer,
and is quite different in nature from the real ' machine '. Through-
out the book, ' the system ' will always refer to this abstraction,
not to the real material i machine '.

Among the variables recorded will almost always be ' time ', so
one might think that this variable should be included in the list
that specifies the system. Nevertheless, time comes into the
theory in a way fundamentally different from that of all the
others. (The difference is shown most clearly in the canonical
equations of S. 19/9.) Experience has shown that a more con-
venient classification is to let the set of variables be divided into
4 system ' and 4 time '. Time is thus not to be included in the
variables of the system. In Table 2/4/1 for instance, ' the
system ' is defined to be the three variables on the right.

2/6. The state of a system at a given instant is the set of numerical
values which its variables have at that instant.

Thus, the six-variable system of S. 2/3 might at some instant
have the state: —4°, 0-3 radians/sec, 128°, 52 cm., 42-8 minutes,
88-4 cm.

Two states are equal if and only if the two numerical values in
each pair are equal, all pairs showing equality.

2/7 DYNAMIC SYSTEMS

The operational method

2/7. The variables being decided on, the recording apparatus
is now assumed to be connected and the experimenter ready to
start observing. We must now make clear what is assumed about
his powers of control over the system.

Throughout the book we shall consider only the case in which
he has access to all states of the system. It is postulated that the
experimenter can control any variable he pleases: that he can
make any variable take any arbitrary value at any arbitrary
time. The postulate specifies nothing about the methods: it
demands only that certain end-results are to be available. In
most cases the means to be used are obvious enough. Take the
example of S. 2/3: an arbitrary angular deviation of the pendulum
can be enforced at any time by direct manipulation; an arbitrary
angular momentum can be enforced at any time by an appropriate
impulse; the cog can be disconnected and shifted, the driving-
weight wound up, the hand moved, and the pendulum-bob lowered.

By repeating the control from instant to instant, the experi-
menter can force a variable to take any prescribed series of values.
The postulate, therefore, implies that any variable can be forced
to follow a prescribed course.

Some systems cannot be forced, for instance the astronomical,
the meteorological, and those biological systems that are accessible
to observation but not to experiment. Yet no change is neces-
sary in principle : the experimenter simply waits until the desired
set of values occurs during the natural changes of the system,
and he counts that instant as if it were the instant at which the
system were started. Thus, though he cannot create a thunder-
storm, he can observe how swallows react to one simply by
waiting till one occurs ' spontaneously '.

It will also be assumed (except where explicitly mentioned) that
he has similarly complete control over those variables that are
not in the system yet which have an effect on it. In the experi-
ment of Table 2/4/1 for instance, Pavlov had control not only of
the variables mentioned but also of the many variables that might
have affected the system's behaviour, such as the lights that
might have flashed, the odours that might have been applied, and
the noises that might have come from outside.

The assumption that the control is complete is made because,

DESIGN FOR A BRAIN 2/8

as will be seen later (and as has been shown in /. to C), it makes
possible a theory that is clear, simple, and coherent. The theories
that arise when we consider the more realistic state of affairs in
which not all states are accessible, or not all variables controllable,
are tangled and complicated, and not suitable as a basis. These
complicated variations can all be derived from the basic theory
by the addition of complications. For the moment we shall
postpone them.

2/8. The primary operation that wins new knowledge from the
* machine ' is as follows : — The experimenter uses his power of
control to determine (select, enforce) a particular state in the
system. He also determines (selects, enforces) the values of the
surrounding conditions of the system. He then allows one unit
of time to elapse and he observes to what state the system goes
as it moves under the drive of its own dynamic nature. He
observes, in other words, a transition, from a particular state,
under particular conditions.

Usually the experimenter wants to know the transitions from
many states under many conditions. Then he often saves time
by allowing the transitions to occur in chains; having found that
A is followed by B, he simply observes what comes next, and thus
discovers the transition from B, and so on.

This description may make the definition sound arbitrary and
unnatural ; in fact, it describes only what every experimenter does
when investigating an unknown dynamic system. Here are some
examples.

In chemical dynamics the variables are often the concentra-
tions of substances. Selected concentrations are brought together,
and from a definite moment are allowed to interact while the
temperature is held constant. The experimenter records the
changes which the concentrations undergo with time.

In a mechanical experiment the variables might be the positions
and momenta of certain bodies. At a definite instant the bodies,
started with selected velocities from selected positions, are allowed
to interact. The experimenter records the changes which the
velocities and positions undergo with time.

In studies of the conduction of heat, the variables are the
temperatures at various places in the heated body. A prescribed
distribution of temperatures is enforced, and, while the tempera-

2/10 DYNAMIC SYSTEMS

tures of some places are held constant, the variations of the
other temperatures are observed after the initial moment.

In physiology, the variables might be the rate of a rabbit's
heart-beat, the intensity of faradisation applied to the vagus
nerve, and the concentration of adrenaline in the circulating
bloocj. The intensity of faradisation will be continuously under
the experimenter's control. Not improbably it will be kept first
at zero and then increased. From a given instant the changes
in the variables will be recorded.

In experimental psychology, the variables might be ' the number
of mistakes made by a rat on a trial in a maze ' and 4 the amount
of cerebral cortex which has been removed surgically '. The
second variable is permanently under the experimenter's control.
The experimenter starts the experiment and observes how the
first variable changes with time while the second variable is held
constant, or caused to change in some prescribed manner.

2/9. The detailed statement just given about what the experi-
menter can do and observe is necessary because we must (as later
chapters will show) be quite clear about the sources of the experi-
menter's knowledge.

Ordinarily, when* an experimenter examines a machine he makes
full use of knowledge ' borrowed ' from past experience. If he
sees two cogs enmeshed he knows that their two rotations will not
be independent, even though he does not see them actually rotate.
This knowledge comes from previous experiences in which the
mutual relations of similar pairs have been tested and observed
directly. Such borrowed knowledge is, of course, extremely use-
ful, and every skilled experimenter brings a great store of it to
every experiment. Nevertheless it must be excluded from any
fundamental method, if only because it is not wholly reliable: the
unexpected sometimes happens; and the only way to be certain
of the relation between parts in a new machine is to test the
relation directly.

2/10. While a single primary operation may seem to yield little
information, the power of the method lies in the fact that the
experimenter can repeat it with variations, and can relate the
different responses to the different variations. Thus, after one
primary operation the next may be varied in any of three ways :

DESIGN FOR A BRAIN 2/11

the system may be changed by the inclusion of new variables
or by the omission of old; the initial state may be changed;
or the surrounding states may be changed. By applying these
variations systematically, in different patterns and groupings, the
different responses may be interrelated to yield relations.

By further orderly variations, these relations may be further
interrelated to yield secondary, or hyper-, relations ; and so on.
In this way the 'machine' may be made to yield more and more
complex information about its inner organisation.

What is fundamental about this method is that the transition
is a purely objective and demonstrable fact. By basing all our
later concepts on jthe properties of transitions we can be sure that
the more complex concepts involve no component other than the
objective and demonstrable. All our concepts will eventually be
denned in terms of this method. For example, ' environment ' is
so defined in S. 3/8, ' adaptation ' in S. 5/3, and ' stimulus ' in
S. 6/5. If any have been omitted it is by oversight; for I hold
that this procedure is sufficient for their objective definition.

Phase-space and Field

2/11. Often the experimenter, while controlling the external
conditions, allows the system to pass from state to state without
interrupting its flow, so that if he started it at state A and it went
to B, he allows it then to proceed from B to C, from C to Z),
and so on.

A line of behaviour is specified by a succession of states and the
time-intervals between them. The first state in a line of behaviour
will be called the initial state. Two lines of behaviour are equal
if all the corresponding pairs of states are equal, and if all the
corresponding pairs of time-intervals are equal.

2/12. There are several ways in which a line of behaviour may
be recorded.

The graphical method is exemplified by Figure 2/12/1. The
four variables form, by definition, the system that is being
examined. The four simultaneous values at any instant define
a state. And the succession of states at their particular intervals
constitute and specify the line of behaviour. The four traces
specify one line of behaviour.

2/12

DYNAMIC SYSTEMS

Sometimes a line of behaviour can be specified in terms of
elementary mathematical functions. Such a simplicity is con-
venient when it occurs, but is rarer in practice than an acquaintance
with elementary mathematics would suggest. With biological
material it is rare.

r*~\

imimA0~%iwMJi ■ ;

Time — *-

Figure 2/12/1 : Events during an experiment on a conditioned reflex in
a sheep. Attached to the left foreleg is an electrode by which a shock
can be administered. Line A records the position of the left forefoot.
Line B records the sheep's respiratory movements. Line C records
by a rise (E) the application of the conditional stimulus : the sound
of a buzzer. Line D records by a vertical stroke (F) the application of
the electric shock. (After Liddell et al.)

Another form is the tabular, of which an example is Table 2/12/1.
Each column defines one state; the whole table defines one line
of behaviour (other tables may contain more than one line of
behaviour). The state at 0 hours is the initial state.

Time (hours)

7-35

7-26

7-28

7-29

156-7

154-6

1541

151-5

110-3

116-7

118-3

118-5

222

153

150

14-6

Table 2/12/1 : Blood changes after a dose of ammonium chloride, w
= serum pH ; x = serum total base ; y = serum chloride ; z = serum
bicarbonate ; (the last three in m. eq. per 1.).

DESIGN FOR A BRAIN

2/13

The tabular form has one outstanding advantage: it contains
the facts and nothing more. Mathematical forms are apt to
suggest too much: continuity that has not been demonstrated,
fictitious values between the moments of observation, and an
accuracy that may not be present. Unless specially mentioned,
all lines of behaviour will be assumed to be recorded primarily
in tabular form.

2/13. The behaviour of a system can also be represented in.
phase-space. By its use simple proofs may be given of many
statements difficult to prove in the tabular form.

Figure 2/13/1.

If a system is composed of two variables, a particular state
will be specified by two numbers. By ordinary graphic methods,
the two variables can be represented by axes ; the two numbers
will then define a point in the plane, Thus the state in which
variable x has the value 5 and variable y the value 10 will be
represented by the point A in Figure 2/13/1. The representative
point of a state is the point whose co-ordinates are respectively equal
to the values of the variables. By S. 2/5 ' time ' is not to be one
of the axes.

Suppose next that a system of two variables gave the line of
behaviour shown in Table 2/13/1. The successive states will be

2/14

DYNAMIC SYSTEMS

graphed, by the method, at positions B, C, and D (Figure 2/13/1).
So the system's behaviour corresponds to a movement of the
representative point along the line in the phase-space.

By comparing the Table and the Figure, certain exact corre-
spondences can be found. Every state of the system corresponds

Time

Table 2/13/1.

uniquely to a point in the plane, and every point in the plane
(or in some portion of it) to some possible state of the system.
Further, every line of behaviour of the system corresponds
uniquely to a line in the plane. If the system has three variables,
the graph must be in three dimensions, but each state still corre-
sponds to a point, and each line of behaviour to a line in the phase-
space. If the number of variables exceeds three, this method of
graphing is no longer physically possible, but the correspondence
is maintained exactly no matter how numerous the variables.

2/14. A system's field* is the phase-space containing all the lines
of behaviour found by releasing the system from all possible initial
states in a particular set of surrounding conditions.

In practice, of course, the experimenter would test only a repre-
sentative sample of the initial states. Some of them will probably
be tested repeatedly, for the experimenter will usuallv want to
make sure that the system is giving reproducible lines of behaviour.
Thus in one experiment, in which dogs had been severely bled
and then placed on a standard diet, their body-weight x and the
concentration y of haemoglobin in their blood were recorded at
weekly intervals. This two- variable system, tested from four
initial states by thirty-six primary operations, gave the field shown
in Figure 2/14/1. Other examples occur frequently later.

It will be noticed that a field is defined, in accordance with

* Some name is necessary for such a representation as Figure 2/13/1, especially
as the concept must be used incessantly throughout the book. I hope that a
better word than 4 field ' will be found, but I have not found one yet.

DESIGN FOR A BRAIN

2/14

S. 2/9, by reference exclusively to the observed values of the
variables and to the results of primary operations on them. It
is therefore a wholly objective property of the system.

The concept of 4 field ' will be used extensively. It defines the
characteristic behaviour of the system, replacing the vague con-
cept of what a system ' does ' or how it ' behaves ' (often describ-
able only in words) by the precise construct of a ' field '. Further

5 10 15

Weight of dog (kg.)

Figure 2/14/1 : Arrow-heads show the direction of movement of the
representative point ; cross-lines show the positions of the representative
point at weekly intervals.

it presents all a system's behaviours (under constant conditions)
frozen into one unchanging entity that can be thought of as a
unit. Such entities can readily be compared and contrasted, and
so we can readily compare behaviour with behaviour, on a basis
that is as complete and rigorous as we care to make it.

The reader may at first find the method unusual. Those who
are familiar with the phase-space of mechanics will have no
difficulty, but other readers may find it helpful if at first, whenever
the word ' field ' occurs, they substitute for it some phrase like
4 typical way of behaving '.

2/15 DYNAMIC SYSTEMS

The Natural System

2/15. In S. 2/5 a system was defined as any arbitrarily selected
set of variables. The right to arbitrary selection cannot be
waived, but the time has now come to recognise that both Science
and common sense insist that if a system is to be studied with
profit its variables must have some naturalness of association.
But what is ' natural ' ? The problem has inevitably arisen after
the restriction of S. 2/9, where we repudiated all borrowed
knowledge. If we restrict our attention to the variables, we find
that as every real 4 machine ' provides an infinity of variables,
and as from them we can form another infinity of combinations,
we need some test to distinguish the natural system from the
arbitrary.

One criterion will occur to the practical experimenter at once.
He knows that if an active and relevant variable is left unobserved
or uncontrolled the system's behaviour will become capricious,
not capable of being reproduced at will. This concept may
readily be made more precise. We simply state formally the
century-old idea that a ' machine ' is something that, if its internal
state is known, and its surrounding conditions, then its behaviour
follows necessarily. That is to say, a particular surrounding
condition (or input, i.e. those variables that affect it) and a
particular state determine uniquely what transition will occur.

So the formal definition goes as follows. Take some particular
set of external conditions (or input-value) C and some particular
state S ; observe the transition that is induced by its own internal
drive and laws ; suppose it goes to state S{. Notice whether,
whenever C and S occur again, the transition is also always to
St ; if so, record that the transitions that follow C and S are
invariant. Next, vary C (or S, or both) to get another pair —
Cx and S1 say ; see similarly whether the transitions that follow
C± and S1 are also invariant. Proceed similarly till all possible
pairs have been tested. If the outcome at every pair was
4 invariant ' then the system is, by definition, a machine with
input. (This definition accords with that given in /. to C.)

In the world of biology, the concept of the machine with input
often occurs in the specially simple case in which all the events
(in one field) occur in only one set of conditions (i.e. C has the
same value for all the lines of behaviour). The field then comes

DESIGN FOR A BRAIN

2/15

from a system that is isolated. Thus, an experimenter may
subject a Protozoon to a drug at a certain concentration; he then
observes, without further experimental interference, the whole line
of behaviour (which may be long and complex) that follows. This
case occurs with sufficient frequency in biological systems and in
this book to deserve a special name; it will be referred to here as
a state-determined system.

Time (seconds)

Line

Variable

0-2

0-3

0-2

0-4

0-6

2-3

2-6

- 0-2

- 01

2-4

1-8

Table 2/15/1.

As illustration of the definition, consider Table 2/15/1, which
shows two lines of behaviour from a system that is not state-
determined. On the first line of behaviour the state x = 0,
y = 2-0 was followed after 0-1 seconds by the state x = 0-2,

Figure 2/15/1 : Field of a simple pendulum 40 cm. long swinging in a
vertical plane when g is 981 cm./sec.2. x is the angle of deviation from
the vertical and y the angular velocity of movement. Cross-strokes
mark the position of the representative point at each one-tenth second.
The clockwise direction should be noticed.

2/16 DYNAMIC SYSTEMS

y = 2-1. On line 2 the state x = 0, y = 2-0 occurred again; but
after 0-1 seconds the state became x = 0-1, */ = 1-8 and not
x = 0-2, y = 2-1. As the two states that follow the state x = 0,
y = 2-0 are not equal, the system is not state-determined.

A well-known example of a state-determined system is given
by the simple pendulum swinging in a vertical plane. It is known
that the two variables — (x) angle of deviation of the string from
vertical, (y) angular velocity (or momentum) of the bob — are
such that, all else being kept constant, their two values at a
given instant are sufficient to determine the subsequent changes
of the two variables (Figure 2/15/1).

The field of a state-determined system has a characteristic
property: through no point does more
than one line of behaviour run. This
fact may be contrasted with that of a
system that is not state-determined.
Figure 2/15/2 shows such a field (the
system is described in S. 19/13). The
system's regularity would be established
if we found that the system, started at

A, always went to A', and, started at

B, always went to B' . But such a
system is not state-determined; for to
say that the representative point is
leaving C is insufficient to define its

future line of behaviour, which may go to A' or B '. Even if the
lines from A and B always ran to A' and B', the regularity in no
way restricts what would happen if the system were started at
C: it might go to D. If the system were state-determined, the
lines CA', CB\ and CD would coincide.

Figure 2/15/2 : The field
of the system shown in
Figure 19/13/1.

2/16. We can now return to the question of what we mean when
we say that a system's variables have a ' natural ' association.
What we need is not a verbal explanation but a definition, which
must have these properties:

(1) it must be in the form of a test, separating all systems into

two classes;

(2) its application must be wholly objective;

(3) its result must agree with common sense in typical and

undisputed cases.

DESIGN FOR A BRAIN 2/17

The third property makes clear that we cannot expect a proposed
definition to be established by a few lines of verbal argument:
it must be treated as a working hypothesis and used ; only experi-
ence can show whether it is faulty or sound. (Nevertheless, in
J. to C, S. 13/5, I have given reasons suggesting that the property
of being state-determined must inevitably be of fundamental
interest to every organism that, like the human scientist, wants
to achieve mastery over its surroundings.)

Because of its importance, science searches persistently for the
state-determined. As a working guide, the scientist has for some
centuries followed the hypothesis that, given a set of variables,
he can always find a larger set that (1) includes the given variables,
and (2) is state-determined. Much research work consists of
trying to identify such a larger set, for when the set is too small,
important variables will be left out of account, and the behaviour
of the set will be capricious. The assumption that such a larger
set exists is implicit in almost all science, but, being fundamental,
it is seldom mentioned explicitly. Temple, though, refers to
4. . . the fundamental assumption of macrophysics that a com-
plete knowledge of the present state of a system furnishes sufficient
data to determine definitely its state at any future time or its
response to any external influence \ Laplace made the same
assumption about the whole universe when he stated that, given
its state at one instant, its future progress should be calculable.
The definition given above makes this assumption precise and
gives it in a form ready for use in the later chapters.

The assumption is now known to be false at the atomic level.
We, however, will seldom discuss events at this level; and as the
assumption has proved substantially true over great ranges of
macroscopic science, we shall use it extensively.

Strategy for the complex system

2/17. The discussion of this chapter may have seemed confined
to a somewhat arbitrary set of concepts, and the biologist, accus-
tomed to a great range of variety in his material, may be thinking
that the concepts and definitions are much too restricted. As this
book puts forward a theory of the origin of adaptation, it must
show how a theory, developed so narrowly, can be acceptable.
In this connexion we must note that theories are of various

2/17 DYNAMIC SYSTEMS

types. At one extreme is Newton's theory of gravitation — at once
simple, and precise, and exactly true. When such a combination
is possible, Science is indeed lucky ! Darwin's theory, on the
other hand, is not so simple, is of quite low accuracy numerically,
and is true only in a partial sense — that the simple arguments
usually used to apply it in practice (e.g. how spraying with D.D.T,
will ultimately affect the genetic constitution of the field mouse,
by altering its food supply) are gross simplifications of the complex
of events that will actually occur.

The theory attempted in this book is of the latter type. The
real facts of the brain are so complex and varied that no theory
can hope to achieve the simplicity and precision of Newton's;
what then must it do ? I suggest that it must try to be exact in
certain selected cases, these cases being selected because there we
can be exact. With these exact cases known, we can then face
the multitudinous cases that do not quite correspond, using the
rule that if we are satisfied that there is some continuity in the
systems' properties, then insofar as each is near some exact case,
so will its properties be near to those shown by the exact case.

This scientific strategy is by no means as inferior as it may
sound; in fact it is used widely in many sciences of good repute.
Thus the perfect gas, the massless spring, the completely reflecting
mirror, the leakless condenser are all used freely in the theories
of physics. These idealised cases have no real existence, but they
are none the less important because they are both simple and
exact, and are therefore key points in the general theoretical
structure.

In the same spirit this book will attend closely to certain
idealised cases, important because they can be exactly defined and
because they are manageably simple. Maybe it will be found
eventually that not a single mechanism in the brain corresponds
exactly to the types described here ; nevertheless the work will not
be wasted if a thorough knowledge of these idealised forms enables
us to understand the workings of many mechanisms that resemble
them only as approximations.

CHAPTER 3

The Organism as Machine

3/1. In accordance with S. 1/11 we shall assume at once that
the living organism in its nature and processes is not essentially
different from other matter. The truth of the assumption will
not be discussed. The chapter will therefore deal only with the
technique of applying this assumption to the complexities of
biological systems.

The specification of behaviour

3/2. If the method laid down in the previous chapter is to be
followed, we must first determine to what extent the behaviour
of an organism is capable of being specified by variables, remem-
bering that our ultimate test is whether the representation can
be by dial readings (S. 2/3).

There can be little doubt that any single quantity observable
in the living organism can be treated at least in principle as a
variable. All bodily movements can be specified by co-ordinates.
All joint movements can be specified by angles. Muscle tensions
can be specified by their pull in dynes. Muscle movements can
be specified by co-ordinates based on the bony structure or on
some fixed external point, and can therefore be recorded numeric-
ally. A gland can be specified in its activity by its rate of
secretion. Pulse-rate, blood-pressure, temperature, rate of blood-
flow, tension of smooth muscle, and a host of other variables can
be similarly recorded.

In the nervous system our attempts to observe, measure, and
record have met great technical difficulties. Nevertheless, much
has been achieved. The action potential, one of the essential
events in the activity of the nervous system, can now be measured
and recorded. The excitatory and inhibitory states of the centres
are at the moment not directly recordable, but there is no reason
to suppose that they will never become so.

3/3 THE ORGANISM AS MACHINE

3/3. Few would deny that the elementary physico-chemical
events in the living organism can be treated as variables. But
some may hesitate before accepting that readings on dials (and
the complex relations deducible from them) are adequate for the
description of all significant biological events. As the remainder
of the book will assume that they are sufficient, I must show how
the various complexities of biological experience can be reduced
to this standard form.

A simple case which may be mentioned first occurs when an
event is recorded in the form ' strychnine was injected at this
moment ', or ' a light was switched on ', or ' an electric shock was
administered '. Such a statement treats only the positive event
as having existence and ignores the other state as a nullity. It
can readily be converted to a numerical form suitable for our
purpose by using the device mentioned in S. 2/3. Such events
would then be recorded by assuming, in the first case, that the
animal always had strychnine in its tissues but that at first the
quantity present was 0 mg. per g. tissue; in the second case, that
the light was always on, but that at first it shone with a brightness
of 0 candlepower; and in the last case, that an electric potential
was applied throughout but that at first it had a value of 0 volts.
Such a method of description cannot be wrong in these cases for
it defines exactly the same set of objective facts. Its advantage
from our point of view is that it provides a method which can be
used uniformly over a wide range of phenomena: the variable is
always present, merely varying in value.

But this device does not remove all difficulties. It sometimes
happens in physiology and psychology that a variable seems to have
no numerical counter-part. Thus in one experiment two cards,
one black and one brown, were shown alternately to an animal as
stimuli. One variable would thus be ' colour ' and it would have
two values. <The simplest way to specify colour numerically is to
give the wave-length of its light ; but this method cannot be used
here, for ' black ' means ' no light ', and ' brown ' does not occur
in the spectrum. Another example would occur if an electric
heater were regularly used and if its switch indicated only the
degrees ' high ', ' medium ', and 4 low '. Another example is given
on many types of electric apparatus by a pilot light which, as a
variable, takes only the two values ' lit ' and ' unlit '. More
complex examples occur frequently in psychological experiments.

DESIGN FOR A BRAIN 3/3

Table 2/4/1 , for instance, contains a variable ' part of skin stimu-
lated ' which, in Pavlov's Table, takes only two values: 'usual
place ' and ' new place '. Even more complicated variables are
common in Pavlov's experiments. Many a Table contains a
variable ' stimulus ' which takes such values as ' bubbling water ',
1 metronome ', ' flashing light '. A similar difficulty occurs when
an experimenter tests an animal's response to injections of toxins,
so that there will be a variable ' type of toxin ' which may take
the two values 4 Diphtheria type Gravis ' and ' Diphtheria type
Medius '. And finally the change may involve an extensive
re-organisation of the whole experimental situation. Such would
occur if the experimenter, wanting to test the effect of the general
surroundings, tried the effect of the variable ' situation of the
experiment ' by giving it alternately the two* values' ' in the
animal house ' and ' in the open air '. Can such variables be
represented by number ?

In some of the examples, the variables might possibly be speci-
fied numerically by a more or less elaborate specification of their
physical nature. Thus ' part of skin stimulated ' might be
specified by reference to some system of co-ordinates marked on
the skin ; and the three intensities of the electric heater might be
specified by the three values of the watts consumed. But this
method is hardly possible in the remainder of the cases ; nor is it
necessary. For numbers can be used cardinally as well as
ordinally, that is, they may be used as mere labels without any
reference to their natural order. Such are the numberings of the
divisions of an army, and of the subscribers on a telephone system ;
for the subscriber whose number is, say, 4051 has no particular
relation to the subscriber whose number is 4052: the number
identifies him but does not relate him.

It may be shown (S. 21/6) that if a variable takes a few values
which stand in no simple relation to one another, then each value
may be allotted an arbitrary number; and provided that the
numbers are used systematically throughout the experiment, and
that their use is confined to the experiment, then no confusion
can arise. Thus the variable ' situation of the experiment '
might be allotted the arbitrary value of ' 1 ' if the experiment
occurs in the animal house, and ' 2 ' if it occurs in the open air.

Although ' situation of the experiment ' involves a great number
of physical variables, the aggregate may justifiably be treated as

3/5 THE ORGANISM AS MACHINE

a single variable provided the arrangement of the experiment is
such that the many variables are used throughout as one aggre-
gate which can take either of two forms. If, however, the
aggregate were split in the experiment, as would happen if we
recorded four classes of results:

(1) in the animal house in summer

(2) in the animal house in winter

(3) in the open air in summer

(4) in the open air in winter

then we must either allow the variable ' condition of experiment '
to take four values, or we could consider the experiment as
subject to two variables; 'site of experiment' and 'season of
year ', each of which takes two values. According to this method,
what is important is not the material structure of the technical
devices but the experiment's logical structure.

3/4. But is the method yet adequate ? Can all the living
organisms' more subtle qualities be numericised in this way ? On
this subject there has been much dispute, but we can avoid a part
of the controversy; for here we are concerned only with certain
qualities defined.

First, we shall be dealing not with qualities but with behaviour:
we shall be dealing, not with what an organism feels or thinks,
but with what it does. The omission of all subjective aspects
(S. 1/16) removes from the discussion the most subtle of the
qualities, while the restriction to overt behaviour makes the
specification by variable usually easy. Secondly, when the non-
mathematical reader thinks that there are some complex quantities
that cannot be adequately represented by number, he is apt
to think of their representation by a single variable. The use of
many variables, however, enables systems of considerable com-
plexity to be treated. Thus a complex system like ' the weather
over England ', which cannot be treated adequately by a single
variable, can, by the use of many variables, be treated as ade-
quately as we please.

3/5. To illustrate the method for specifying the behaviour of a
system by variables, two examples will be given. They are of
little intrinsic interest; more important is the fact that they

DESIGN FOR A BRAIN

3/5

demonstrate that the method is exact and that it can be extended
to any extent without loss of precision.

The first example is from a physiological experiment. A dog
was subjected to a steady loss of blood at the rate of one per cent
of its body weight per minute. Recorded are the three variables :

(x) rate of blood-flow through the inferior vena cava,
(y) » „ „ „ » muscles of a leg,

(2) „ „ „ „ „ gut.

The changes of the variables with time are shown in Figure 3/5/1.
It will be seen that the changes of the variables show a charac-
teristic pattern, for the blood-flow through leg and gut falls more
than that through the inferior vena cava, and this difference is
characteristic of the body's reaction to haemorrhage. The use

4 8

Figure 3/5/1 : Effects of haemor-
rhage on the rate of blood-flow
through : x, the inferior vena cava;
y, the muscles of a leg ; and 2, the
gut. (From Rein.)

Figure 3/5/2 : Phase-space and
line of behaviour of the data
shown in Figure 3/5/1.

of more than one variable has enabled the pattern of the reaction
to be displayed.

The changes specify a line of behaviour, shown in Figure 3/5/2.
Had the line of behaviour pointed in a different direction, the
change would have corresponded to a change in the pattern of
the body's reaction to haemorrhage.

The second example uses certain angles measured from a
cinematographic record of the activities of a man. His body
moved forward but was vertical throughout. The four variables
are:

(w) angle between the right thigh and the vertical

(x)

left

3/7

THE ORGANISM AS MACHINE

(y) angle between the right thigh and the right tibia
(z) „ „ „ left „ „ „ left

In w and x the angle is counted positively when the knee comes
forward: in y and z the angles are measured behind the knee.
The line of behaviour is specified in Table 3/5/1. The reader can
easily identify this well-known activity.

Time (seconds)

0-2

0-3

0-4

0-5

0-6

0-7

0-8

- 10

- 20

— 35

- 35

- 10

- 20

- 35

170

180

160

120

100

170

120

100

170

180

160

120

Table 3/5/1.

3/6. In a physiological experiment the nervous system is usually
considered to be state-determined. That it can be made state-
determined is assumed by every physiologist before the work
starts, for he assumes that it is subject to the fundamental assump-
tion of S. 2/15: that if every detail within it could be determined,
its subsequent behaviour would also be determined. Many of the
specialised techniques such as anaesthesia, spinal transection,
root section, and the immobilisation of body and head in clamps
are used to ensure proper isolation of the system — a necessary
condition for it to be state-determined (S. 2/15). So unless there
are special reasons to the contrary, the nervous system in a physio-
logical experiment can usually be assumed to be state-determined.

3/7. Similarly it is usually agreed that an animal undergoing
experiments on its conditioned reflexes is a physico-chemical
system such that if we knew every detail we could predict its
behaviour. Pavlov's insistence on complete isolation was intended
to ensure that this was so. So unless there are special reasons to
the contrary, the animal in an experiment with conditioned
reflexes can usually be assumed to be state-determined.

DESIGN FOR A BRAIN 3/8

3/8. These two examples, however, are mentioned only as
introduction; rather we shall be concerned with the nature of the
free-living organism within a natural environment.

Given an organism, its environment is defined as those variables
whose changes affect the organism, and those variables which are
changed by the organism's behaviour. It is thus defined in a purely
functional, not a material, sense. It will be treated uniformly
with our treatment of all variables : we assume it is representable
by dials, is explorable (by the experimenter) by primary opera-
tions, and is intrinsically state-determined.

Organism and environment

3/9. The theme of the chapter can now be stated: the free-
living organism and its environment, taken together, may be
represented with sufficient accuracy by a set of variables that
forms a state-determined system.

The concepts developed in the previous sections now enable us
to treat both organism and environment by identical methods,
for the same primary assumptions are made about each.

3/10. As example, that the organism and its environment form
a single state-determined system, consider (in so far as the activities
of balancing are concerned) a bicycle and its rider in normal
progression.

First, the forward movement may be eliminated as irrelevant,
for we could study the properties of this dynamic system equally
well if the wheels were on some backward-moving band. The
variables can be identified by considering what happens. Suppose
the rider pulls his right hand backwards: it will change the
angular position of the front wheel (taking the line of the frame as
reference). The changed angle of the front wheel will start the
two points, at which the wheels make contact with the ground,
moving to the right. (The physical reasons for this movement
are irrelevant: the fact that the relation is determined is sufficient.)
The rider's centre of gravity being at first unmoved, the line
vertically downwards from his centre of gravity will strike the
ground more and more to the left of the line joining the two
points. As a result he will start to fall to the left. This fall will
excite nerve-endings in the organs of balance in the ear, impulses

3/11 THE ORGANISM AS MACHINE

will pass to the nervous system, and will be switched through it,
if he is a trained rider, by such a route that they, or the effects
set up by them, will excite to activity those muscles which push
the right hand forwards.

We can now specify the variables which must compose the
system if it is to be state-determined. We must include: the
angular position of the handlebar, the velocity of lateral movement
of the two points of contact between wheels and road, the distance
laterally between the line joining these points and the point
vertically below the rider's centre of gravity, and the angular
deviation of the rider from the vertical. These four variables are
defined by S. 3/8 to be the ' environment ' of the rider. (Whether
the fourth variable is allotted to ' rider ' or to ' environment ' is
optional (S. 3/12). To make the system state-determined, there
must be added the variables of the nervous system, of the relevant
muscles, and of the bone and joint positions.

As a second example, consider a butterfly and a bird in the air,
the bird chasing the butterfly, and the butterfly evading the bird.
Both use the air around them. Every movement of the bird
stimulates the butterfly's eyes and this stimulation, acting through
the butterfly's nervous system, will cause changes in the butter-
fly's wing movements. These movements act on the enveloping
air and cause changes in the butterfly's position. A change of
position immediately changes the excitations in the bird's eye,
and this leads through its nervous system to changed movements
of the bird's wings. These act on the air and change the bird's
position. So the processes go on. The bird has as environment
the air and the butterfly, while the butterfly has the air and the bird.
The whole may reasonably be assumed to be state-determined.

3/11. The organism affects the environment, and the environ-
ment affects the organism : such a system is said to have i feed-
back ' (S. 4/14)-

The examples of the previous section provide illustration. The
muscles in the rider's arm move the handlebars, causing changes
in the environment; and changes in these variables will, through
the rider's sensory receptors, cause changes in his brain and
muscles. When bird and butterfly manoeuvre in the air, each
manoeuvre of one causes reactive changes to occur in the other.

The same feature is shown by the example of S. 1/17 — the

DESIGN FOR A BRAIN 3/11

type problem of the kitten and the fire. The various stimuli
from the fire, working through the nervous system, evoke some
reaction from the kitten's muscles; equally the kitten's move-
ments, by altering the position of its body in relation to the fire,
will cause changes to occur in the pattern of stimuli which falls
on the kitten's sense-organs. The receptors therefore affect the
muscles (by effects transmitted through the nervous system), and
the muscles affect the receptors (by effects transmitted through
the environment). The action is two-way and the system possesses
feedback.

The observation is not new:

4 In most cases the change which induces a reaction is brought
about by the organism's own movements. These cause a
change in the relation of the organism to the environment:
to these changes the organism reacts. The whole behaviour
of free-moving organisms is based on the principle that it
is the movements of the organism that have brought about
stimulation.'

(Jennings.)

' The good player of a quick ball game, the surgeon con-
ducting an operation, the physician arriving at a clinical
decision — in each case there is the flow from signals inter-
preted to action carried out, back to further signals and on
again to more action, up to the culminating point of the
achievement of the task '.

(Bartlett.)

1 Organism and environment form a whole and must be
viewed as such.'

(Starling.)

It is necessary to point to the existence of feedback in the
relation between the free-living organism and its environment
because most physiological experiments are deliberately arranged
to avoid feedback. Thus, in an experiment with spinal reflexes,
a stimulus is applied and the resulting movement recorded; but
the movement is not allowed to influence the nature or duration
of the stimulus. The action between stimulus and movement is
therefore one-way. A similar absence of feedback is enforced
in the Pavlovian experiments with conditioned reflexes: the
stimulus may evoke salivation, but the salivation has no effect
on the nature or duration of the stimulus.

Such an absence of feedback is, of course, useful or even essen-

3/11 THE ORGANISM AS MACHINE

tial in the analytic study of the behaviour of a mechanism,
whether animate or inanimate. But its usefulness in the labora-
tory should not obscure the fact that the free-living animal is not
subject to these constraints.

Sometimes systems which seem at first sight to be one-way
prove on closer examination to have feedback. Walking on a
smooth pavement, for instance, seems to involve so little reference
to the structures outside the body that the nervous system might
seem to be producing its actions without reference to their effects.
Tabes dorsalis, however, prevents incoming sensory impulses from
reaching the brain while leaving the outgoing motor impulses un-
affected. If walking were due simply to the outgoing motor
impulses, the disease would cause no disturbance to walking. In
fact, it upsets the action severely, and demonstrates that the
incoming sensory impulses are really playing an essential, though
hidden, part in the normal action.

Another example showing the influence of feedback occurs when
we try to place a point accurately (e.g. by trying to pass a wire
through a small hole in a board) wrhen we cannot see by how
much we are in error (e.g. when we have to pass the wire through
from the far side, towards us). The difficulty we encounter is
precisely due to the fact that while we can affect the movements
of the wire, its movements and its relations to the hole can no
longer be communicated back to us.

Sometimes the feedback can be demonstrated only with diffi-
culty. Thus, Lloyd Morgan raised some ducklings in an incubator.

The ducklings thoroughly enjoyed a dip. Each morning,

at nine o'clock, a large black tray was placed in their pen,

and on it a flat tin containing water. To this they eagerly

ran, drinking and washing in it. On the sixth morning the

tray and tin were given them in the usual way, but without

any water. They ran to it, scooped at the bottom and made

all the motions of the beak as if drinking. They squatted

in it, dipping their heads, and waggling their tails as usual.

For some ten minutes they continued to wash in non-existent

water . . .'

Their behaviour might suggest that the stimuli of tray and tin

were compelling the production of certain activities and that the

results of these activities were having no back-effect. But further

experiment showed that some effect was occurring:

'The next day the experiment was repeated with the dry tin.

DESIGN FOR A BRAIN 3/12

Again they ran to it, shovelling along the bottom with their
beaks, and squatting down in it. But they soon gave up.
On the third morning they waddled up to the dry tin, and
departed.'

Their behaviour at first suggested that there was no feedback.
But on the third day their change of behaviour showed that, in
fact, the change in the bath had had some effect on them.

The importance of feedback lies in the fact that systems which
possess it have certain properties (S. 4/16) which cannot be shown
by systems lacking it. Systems with feedback cannot adequately
be treated as if they were of one-way action, for the feedback intro-
duces properties which can be explained only by reference to the
particular feedback used. (On the other hand a one-way system
can, without error, be treated as if it contained feedback: we
assume that one of the two actions is present but at zero degree
(S. 2/3). In other words, systems without feedback are a sub-
class of the class of systems with feedback.)

3/12. As the organism and its environment are to be treated as a
single system, the dividing line between ' organism ' and ' environ-
ment ' becomes partly conceptual, and to that extent arbitrary.
Anatomically and physically, of course, there is usually a unique
and obvious distinction between the two parts of the system; but
if we view the system functionally, ignoring purely anatomical
facts as irrelevant, the division of the system into ' organism ' and
4 environment ' becomes vague. Thus, if a mechanic with an
artificial arm is trying to repair an engine, then the arm may be
regarded either as part of the organism that is struggling with
the engine, or as part of the machinery with which the man is
struggling.

Once this flexibility of division is admitted, almost no bounds
can be put to its application. The chisel in a sculptor's hand
can be regarded either as a part of the complex biophysical
mechanism that is shaping the marble, or it can be regarded as
a part of the material which the nervous system is attempting to
control. The bones in the sculptor's arm can similarly be regarded
either as part of the organism or as part of the ' environment ' of
the nervous system. Variables within the body may justifiably
be regarded as the ' environment ' of some other part. A child
has to learn not only how to grasp a piece of bread, but how to

3/14 THE ORGANISM AS MACHINE

chew without biting his own tongue ; functionally both bread and
tongue are part of the environment of the cerebral cortex. But
the environments with which the cortex has to deal are sometimes
even deeper in the body than the tongue: the child has to learn
how to play without exhausting itself utterly, and how to talk
without getting out of breath.

These remarks are not intended to confuse, but to show that
later arguments (in Chapters 15 and 16) are not unreasonable.
There it is intended to treat one group of neurons in the brain
as the environment of another group. These divisions, though
arbitrary, are justifiable because we shall always treat the system
as a whole, dividing it into parts in this unusual way merely for
verbal convenience in description.

It should be noticed that from now on ' the system ' means
not the nervous system but the whole complex of the organism
and its environment. Thus, if it should be shown that ' the
system ' has some property, it must not be assumed that this
property is attributed to the nervous system: it belongs to the
whole; and detailed examination may be necessary to ascertain
the contributions of the separate parts.

3/13. In some cases the dynamic nature of the interaction
between organism and environment can be made intuitively more
obvious by using the device, common in physics, of regarding the
animal as the centre of reference. In locomotion the animal
would then be thought of as pulling the world past itself. Pro-
vided we are concerned only with the relation between these two,
and are not considering their relations to any third and inde-
pendent body, the device will not lead to error. It was used in
the ' rider and bicycle ' example.

By the use of animal-centred co-ordinates we can see that the
animal has much more control over its environment than might at
first seem possible. Thus, while ^a frog cannot change air into
water, a frog on the bank of a stream can, with one small jump,
change its world from one ruled by the laws of mechanics to one
ruled by the laws of hydrodynamics.

Essential variables
3/14. The biologist must view the brain, not as being the seat of
the 4 mind ', nor as something that 4 thinks ', but, like every other

DESIGN FOR A BRAIN 3/14

organ in the body, as a specialised means to survival (S. 1/10).
We shall use the concept of ' survival ' repeatedly; but before we
can use it, we must, by S. 2/10, transform it to our standard
form and say what it means in terms of primary operations.

Physico-chemical systems may undergo the most extensive
transformations without showing any change obviously equivalent
to death, for matter and energy are indestructible. Yet the dis-
tinction between a live horse and a dead one is obvious enough.
Further, there can be no doubt about the objectivity of the
difference, for they fetch quite different prices in the market.
The distinction must be capable of objective definition.

It is suggested that the definition may be obtained in the
following way. That an animal should remain ' alive ', certain
variables must remain without certain ' physiological ' limits.
What these variables are, and what the limits, are fixed when
the species is fixed. In practice one does not experiment on
animals in general, one experiments on one of a particular species.
In each species the many physiological variables differ widely in
their relevance to survival. Thus, if a man's hair is shortened
from 4 inches to 1 inch, the change is trivial ; if his systolic blood-
pressure drops from 120 mm. of mercury to 30, the change will
quickly be fatal.

Every species has a number of variables which are closely
related to survival and which are closely linked dynamically so
that marked changes in any one leads sooner or later to marked
changes in the others. Thus, if we find in a rat that the pulse-
rate has dropped to zero, we can predict that the respiration rate
will soon become zero, that the body temperature will soon fall to
room temperature, and that the number of bacteria in the tissues
will soon rise from almost zero to a very high number. These
important and closely linked variables will be referred to as the
essential variables of the animal.

How are we to discover them, considering that we may not use
borrowed knowledge but must find them by the methods of
Chapter 2 ? There is no difficulty. Given a species, we observe
what follows when members of the species are started from a
variety of initial states. We shall find that large initial changes
in some variables are followed in the system by merely transient
deviations, while large initial changes in others are followed by
deviations that become ever greater till the ' machine ' changes

3/15 THE ORGANISM AS MACHINE

to something very different from what it was originally. The
results of these primary operations will thus distinguish, quite
objectively, the essential variables from the others.

3/15. The essential variables are not uniform in the closeness or
urgency of their relations to lethality. There are such variables
as the amount of oxygen in the blood, and the structural integrity
of the medulla oblongata, whose passage beyond the normal limits
is followed by death almost at once. There are others, such as
the integrity of a leg-bone, and the amount of infection in the
peritoneal cavity, whose passage beyond the limit must be regarded
as serious though not necessarily fatal. Then there are variables,
such as those of severe pressure or heat at some place on the skin,
whose passage beyond normal limits is not immediately dangerous,
but is so often correlated with some approaching threat that is
serious that the organism avoids such situations (which we call
' painful ') as if they were potentially lethal. All that we require
is the ability to arrange the animal's variables in an approximate
order of importance. Inexactness of the order is not serious, for
nowhere will we use a particular order as a basis for particular
deductions.

We can now define ' survival ' objectively and in terms of a
field : it occurs when a line of behaviour takes no essential variable
outside given limits.

CHAPTER 4

Stability

4/1. The words l stability ', ' steady state ', and ' equilibrium '
are used by a variety of authors with a variety of meanings,
though there is always the same underlying theme. As we shall
be much concerned with stability and its properties, an exact
definition must be provided.

The subject may be opened by a presentation of the three
standard elementary examples. A cube resting with one face
on a horizontal surface typifies ' stable ' equilibrium ; a sphere
resting on a horizontal surface typifies 'neutral' equilibrium;
and a cone balanced on its point typifies ' unstable ' equilibrium.
With neutral and unstable equilibria we shall have little concern,
but the concept of ' stable equilibrium ' will be used repeatedly.

These three dynamic systems are restricted in their behaviour
by the fact that each system contains a fixed quantity of energy,
so that any subsequent movement must conform to this invari-
ance. We, however, shall be considering systems which are
abundantly supplied with free energy so that no such limitation
is imposed. Here are two examples.

The first is the Watt's governor. A steam-engine rotates a pair
of weights which, as they are rotated faster, separate more widely
by centrifugal action; their separation controls mechanically
the position of the throttle; and the position of the throttle
controls the flow of steam to the engine. The connexions are
arranged so that an increase in the speed of the engine causes a
decrease in the flow of steam. The result is that if any transient
disturbance slows or accelerates the engine, the governor brings
the speed back to the usual value. By this return the system
demonstrates its stability.

The second example is the thermostat, of which many types
exist. All, however, work on the same principle: a chilling of
the main object causes a change which in its turn causes the
heating to become more intense or more effective; and vice versa.

4/3

STABILITY

The result is that if any transient disturbance cools or overheats
the main object, the thermostat brings its temperature back to
the usual value. By this return the system demonstrates its
stability.

4/2. An important feature of stability is that it does not refer
to a material body or ' machine ' but only to some aspect of it.
This statement may be proved most simply by an example showing
that a single material body can be in two different equilibrial
conditions at the same time. Consider a square card balanced
exactly on one edge; to displacements at right angles to this edge
the card is unstable; to displacements exactly parallel to this
edge it is, theoretically at least, stable.

The example supports the thesis that we do not, in general,
study physical bodies but only entities carefully abstracted from
them. The matter will become clearer when we conform to the
requirements of S. 2/10 and define stability in terms of the results
of primary operations. This may be done as follows.

4/3. Consider a corrugated surface, laid horizontally, with a ball
rolling from a ridge down towards a trough. A photograph taken
in the middle of its roll would look like
Figure 4/3/1. We might think of the
ball as being unstable because it has
rolled away from the ridge, until we
realise that we can also think of it as
stable because it is rolling towards the
trough. The duality shows we are
approaching the concept in the wrong
way. The situation can be made clearer
if we remove the ball and consider only Figure 4/3/1

the surface. The top of the ridge, as
it would affect the roll of a ball, is now recognised as a position
of unstable equilibrium, and the bottom of the trough as a position
of stability. We now see that, if friction is sufficiently marked
for us to be able to neglect momentum, the system composed of
the single variable ' distance of the ball laterally ' is state-deter-
mined, and has a definite, permanent field, which is sketched in
the Figure.

From B the lines of behaviour diverge, but to A they converge.

DESIGN FOR A BRAIN 4/4

We conclude tentatively that the concept of ' stability ' belongs
not to a material body but to a field. It is shown by a field if
the lines of behaviour converge. (An exact definition is given in

S. 4/8.)

4/4. The points A and B are such that the ball, if released on
either of them, and mathematically perfect, will stay there.
Given a field, a state of equilibrium is one from which the repre-
sentative point does not move. When the primary operation is
applied, the transition from that state can be described as ' to
itself '.

(Notice that this definition, while saying what happens at the
equilibrial state, does not restrict how the lines of behaviour may
run around it. They may converge in to it, or diverge from it,
or behave in other ways.)

Although the variables do not change value when the system
is at a state of equilibrium, this invariance does not imply that
the ' machine ' is inactive. Thus, a motionless Watt's governor
is compatible with the engine working at a non-zero rate. (The
matter has been treated more fully in /. to C, S. 11/15.)

4/5. To illustrate that the concept of stability belongs to a
field, let us examine the fields of the previous examples.

The cube resting on one face yields a state-determined system
which has two variables:

(x) the angle at which the face makes with the horizontal, and
(y) the rate at which this angle changes.

(This system allows for the momentum of the cube.) If the cube
does not bounce when the face meets the table, the field is similar
to that sketched in Figure 4/5/1. The stability of the cube
when resting on a face corresponds in the field to the convergence
of the lines of behaviour to the centre.

The square card balanced on its edge can be represented approxi-
mately by two variables which measure displacements at right
angles (a?) and parallel (y) to the lower edge. The field will
resemble that sketched in Figure 4/5/2. Displacement from the
origin 0 to A is followed by a return of the representative point
to -O, and this return corresponds to the stability. Displacement
from O to B is followed by a departure from the region under

4/5

STABILITY

o O O D o o o

Figure 4/5/1 : Field of the two-variable system described in the text.
Below is shown the cube as it would appear in elevation when its main
face, shown by a heavier line, is tilted through the angle x.

consideration, and this departure correspond^to the instability.
The uncertainty of the movements near O corresponds to the
uncertainty in the behaviour of the card when released from the
vertical position.

The Watt's governor has a more complicated field, but an
approximation may be obtained without
difficulty. The system may be specified
to an approximation sufficient for our
purpose by three variables:

(x) the speed of the engine and
governor (r.p.m.),

(y) the distance between the weights,
or the position of the throttle,
and

(z) the velocity of flow of the steam.

(y represents either of two quantities because they are rigidly
connected). If, now, a disturbance suddenly accelerates the
engine, increasing x, the increase in x will increase y; this increase
in y will be followed by a decrease of z, and then by a decrease of
x. As the changes occur not in jumps but continuously, the line

Figure 4/5/2.

DESIGN FOR A BRAIN 4/6

of behaviour must resemble that sketched in Figure 4/5/3. The

other lines of the field could be added by considering what would

^ happen after other disturbances

(lines starting from points other
than A). Although having
different initial states, all the
lines would converge towards 0.

4/6. In some of our examples,
for instance that of the cube, the
lines of behaviour terminate in a
point at which all movement
ceases. In other examples the
movement does not wholly cease ;
many a thermostat settles down,
when close to its resting state,

Figure 4/5/3 : One line of behav- to a regular small oscillation,

iour in the field of the Watt's We shall seldom be interested in

governor. For clarity, the resting .. , . ., „ , . ,

state of the system has been used the details ot what happens at

as origin. The system has been the exact centre,
displaced to A and then released.

4/7. More important is the underlying theme that in all cases
the stable system is characterised by the fact that after a displace-
ment we can assign some bound to the subsequent movement of the
representative point, whereas in the unstable system such limita-
tion is either impossible or depends on facts outside the subject of
discussion. Thus, if a thermostat is set at 37° C. and displaced
to 40°, we can predict that in the future it will not go outside
specified limits, which might be, in one apparatus, 36° and 40°.
On the other hand, if the thermostat has been assembled with a
component reversed so that it is unstable (S. 4/14) and if it is
displaced to 40°, then we can give no limits to its subsequent
temperatures; unless we introduce such new topics as the melting-
point of its solder.

4/8. These considerations bring us to the definitions. Given
the field of a state-determined system and a region in the field,
the region is stable if the lines of behaviour from all points in the
region stay within the region.

Thus, in Figure 4/3/1 make a mark on either side of A to define
a region. All representative points within are led to A, and none

4/10

STABILITY

can leave the region; so the region is stable. On the other hand,
no such region can be marked around B (unless restricted to the
single point of B itself).

The definition makes clear that change of either the field or the
region may change the result of the test. We cannot, in general,
say of a given system that it is stable (or unstable) unconditionally.
The field of Figure 4/5/1 showed this, and so does that of Figure
4/5/2. (In the latter, the regions restricted to any part of the
?/-axis with the origin are stable; all others are unstable.)

The examples above have been selected to test the definition
severely. Often the fields are simpler. In the field of the cube,
for instance, it is possible to draw many boundaries, all.oval, such
that the regions inside them are stable. The field of the Watt's
governor is also of this type.

A field will be said to be stable if the whole region it fills is
stable ; the system that provided the field can then be called stable.

4/9. Sometimes the conditions are even simpler. The system
may have only one state of equilibrium and the lines of behaviour
may all either converge in to it or all diverge from it. In such a
case the indication of which way the lines go may be given suffi-
ciently by the simple, unqualified, statement that ' it is stable '
(or not). A system can be described adequately by such an
unqualified statement (without reference to the region) only when
its field, .i.e. its behaviour, is suitably simple.

4/10. If a line of behaviour is re-entrant to itself, the system
undergoes a recurrent cycle. If
the cycle is wholly contained in a
given region, and the lines of be-
haviour lead into the cycle, the
cycle is stable.

Such a cycle is commonly shown,
by thermostats which, after correct-
ing any gross displacement, settle
down to a steady oscillation. In
such a case the field will show,
not convergence to a point but
convergence to a cycle, such as
is shown exaggerated in Figure 4/10/1,

Figure 4/10/1.

DESIGN FOR A BRAIN

4/11

4/11. This definition of stability conforms to the requirement
of S. 2/10; for the observed behaviour of the system determines
the field, and the field determines the stability.

The diagram of immediate effects

4/12. The description given in S. 4/1 of the working of the
Watt's governor showed that it is arranged in a functional circuit :
the chain of cause and effect is re-entrant. Thus if we represent
1 A has a direct effect on B ' or ' A directly disturbs B ' by the
symbol A — > B, then the construction of the Watt's governor may
be represented by the diagram:

Speed of
engine

Distance
between
weights

\ /

Velocity

of flow

of steam

■

(The number of variables named here is partly optional.)

I now want to make clear that this type of diagram, if accurately
defined, can be derived wholly from the results of primary operations.
No metaphysical or borrowed knowledge is necessary for its
construction. To show how this is done, take an actual Watt's
governor as example:

Each pair of variables is taken in turn. Suppose the relation
between ' speed of engine ' and ' distance between weights ' is
first investigated. The experimenter would fix the variable
4 velocity of flow of steam ' and all other extraneous variables that
might interfere to confuse the direct relation between speed of
engine and distance between weights. Then he would try various
speeds of the engine, and would observe how these changes affected
the behaviour of ' distance between the weights '. He would
find that changes in the speed of the engine were regularly followed
by changes in the distance between the weights. Thus the transi-
tion of the variable 4 distance between weights ' (one distance
changing to another) is affected by the value of the speed of the

4/13

STABILITY

engine. He need know nothing of the nature of the ultimate
physical linkages, but he would observe the fact. Then, still
keeping 4 velocity of flow of steam ' constant, he would try various
distances between the weights, and would observe the effect of
such changes on the speed of the engine; he would find them to
be without effect. He would thus have established that there is
an arrow from left to right but not from right to left in

Speed of
engine

Distance
between
weights

This procedure could then be applied to the two variables
' distance between weights ' and ' velocity of flow of steam ',
while the other variable ' speed of engine ' was kept constant.
And finally the relations between the third pair could be established.

The method is clearly general. To find the immediate effects
in a system with variables A, B, C, D . . . take one pair, A and
B say; hold all other variables C, D . . . constant; note B's
behaviour when A starts at Ax\ and also its behaviour when A
starts at A2. If these behaviours of B are the same, then there is
no immediate effect from A to B. But if the J5's behaviours are
unequal, and regularly depend on what value A starts from,
then there is an immediate effect, which we may symbolise by
A-+B.

By interchanging A and B in the process we can then test for
B — > A. And by using other pairs in turn we can determine all
the immediate effects. The process consists purely of primary
operations, and therefore uses no borrowed knowledge (the process
is further considered in S. 12/3). We shall frequently use this
diagram of immediate effects.

4/13. It should be noticed that this arrow, though it sometimes
corresponds to an actual material channel (a rod, a wire, a nerve
fibre, etc.), has fundamentally nothing to do with material con-
nexions but is a representation of a relation between the behaviours
at A and B. Strictly speaking, it refers to A and B only, and not
to anything between them.

That it is the functional, behaviourial, relation between A and
B that is decisive (in deciding whether we may hypothesise a
channel of communication between them) was shown clearly on

DESIGN FOR A BRAIN

4/14

that day in 1888 when Heinrich Hertz gave his famous demon-
stration. He had two pieces of apparatus (A and B, say) that
manifestly were not connected in any material way ; yet whenever
at any arbitrarily selected moment he closed a switch in A a
spark jumped in B, i.e. B's behaviour depended at any moment
on the position of A's switch. Here was a flat contradiction:
materially the two systems were not connected, yet functionally
their behaviours were connected. All scientists accepted that
the behavioural evidence was final — that some linkage was
demonstrated.

Feedback
4/14. A gas thermostat also shows a functional circuit or feed-
back ; for it is controlled by a capsule which by its swelling moves
a lever which controls the flow of gas to the heating flame, so the
diagram of immediate effects would be :

Temperature

Diameter

of capsule

«,

Size of
gas flame

Position
of lever

■>

Velocity

Position

of gas

> flow

of gas

tap

The reader should verify that each arrow represents a physical
action which can be demonstrated if all variables other than the
pair are kept constant.

Another example is provided by ' reaction ' in a radio receiver.
We can represent the action by two variables linked in two ways :

Amplitude of

oscillation of the

anode-current

Amplitude of

oscillation of the

grid-potential

The lower arrow represents the grid-potential's effect within the
valve on the anode-current. The upper arrow represents some
arrangement of the circuit by which fluctuation in the anode

4/15 STABILITY

current affects the grid-potential. The effect represented by the
lower arrow is determined by the valve-designer, that of the
upper by the circuit-designer.

Such systems whose variables affect one another in one or more
circuits possess what the radio-engineer calls ' feedback ' ; they are
also sometimes described as ' servo-mechanisms '. They are at
least as old as the Watt's governor and may be older. But only
during the last decade has it been realised that the possession of
feedback gives a machine potentialities that are not available to a
machine lacking it. The development occurred mainly during the
last war, stimulated by the demand for automatic methods of
control of searchlight, anti-aircraft guns, rockets, and torpedoes,
and facilitated by the great advances that had occurred in elec-
tronics. As a result, a host of new machines appeared which
acted with powers of self- adjustment and correction never before
achieved. Some of their main properties will be described in
S. 4/16.

The nature, degree, and polarity of the feedback usually have
a decisive effect on the stability or instability of the system. In
the Watt's governor or in the thermostat, for instance, the con-
nexion of a part in reversed position, reversing the polarity of
action of one component on the next, may, and probably will,
turn the system from stable to unstable. In the reaction circuit
of the radio set, the stability or instability is determined by the
quantitative relation between the two effects.

Instability in such systems is shown by the development of a
' runaway '. The least disturbance is magnified by its passage
round the circuit so that it is incessantly built up into a larger
and larger deviation from the central state. The phenomenon is
identical with that referred to as a ' vicious circle '.

4/15. The examples shown have only a simple circuit. But more
complex systems may have many interlacing circuits. If, for
instance, as in S. 8/2, four variables all act on each other, the
diagram of immediate effects would be that shown in Figure

4-±=*3 3- 4

2
C

Figure 4/15/1.

DESIGN FOR A BRAIN

4/16

4/15/1 (A). It is easy to verify that such a system contains
twenty interlaced circuits, two of which are shown at B and C.

The further development of the theory of systems with feed-
back cannot be made without mathematics. But here it is
sufficient to note two facts: a system which possesses feedback
is usually actively stable or actively unstable; and whether it is
stable or unstable depends on the quantitative details of the
particular arrangement.

Goal-seeking

4/16. Every stable system has the property that if displaced
from a state of equilibrium and released, the subsequent movement
is so matched to the initial displacement that the system is
brought back to the state of equilibrium. A variety of disturbances
will therefore evoke a variety of matched reactions. Reference to a
simple field such as that of Figure 4/5/1 will establish the
point.

This pairing of the line of return to the initial displacement
has sometimes been regarded as ' intelligent ' and peculiar to living
things. But a simple refutation is given by the ordinary pen-
dulum: if we displace it to the right, it develops a force which
tends to move it to the left; and if we displace it to the left, it
develops a force which tends to move it to the right. Noticing
that the pendulum reacted with forces which though varied in
direction always pointed towards the centre, the mediaeval scien-
tist would have said ' the pendulum seeks the centre '. By this
phrase he would have recognised that the behaviour of a stable
system may be described as ' goal-seeking '. Without introducing
any metaphysical implications we may recognise that this type of
behaviour does occur in the stable dynamic systems. Thus
Figure 4/16/1 shows how, as the control setting of a thermostat

Figure 4/16/1 : Tracing of the temperature (solid line), of a thermostatically
controlled bath, and of the control setting (broken line).

4/17 STABILITY

was altered, the temperature of the apparatus always followed it,
the set temperature being treated as if it were a goal.

Such a movement occurs here in only one dimension (tempera-
ture), but other goal-seeking devices may use more. The radar-
controlled searchlight, for example, uses the reflected impulses
to alter its direction of aim so as to minimise the angle between
its direction of aim and the bearing of the source of the reflected
impulses. So if the aircraft swerves, the searchlight will follow
it actively, just as the temperature followed the setting. Such
a system is goal-seeking in two dimensions.

The examples show the common feature that each is ' error-
controlled ' : each is partly controlled by the deviation of the
system's state from the state of equilibrium (which, in these
examples, can be moved by an outside operation). The thermo-
stat is affected by the difference between the actual and the set
temperatures. The searchlight is affected by the difference
between the two directions. Thus, machines with feedback are not
subject to the oft-repeated dictum that machines must act blindly and
cannot correct their errors. Such a statement is true of machines
without feedback, but not of machines in general (S. 3/11).

Once it is appreciated that feedback can be used to correct any
deviation we like, it is easy to understand that there is no limit
to the complexity of goal-seeking behaviour which may occur in
machines quite devoid of any ' vital ' factor. Thus, an automatic
anti-aircraft gun may be controlled by the radar-pulses reflected
back both from the target aeroplane and from its own bursting
shells, in such a way that it tends to minimise the distance between
shell-burst and plane. Such a system, wholly automatic, cannot
be distinguished by its behaviour from a humanly operated gun:
both will fire at the target, following it through all manoeuvres,
continually using the errors to improve the next shot. It will be
seen, therefore, that a system with feedback may be both wholly
automatic and yet actively and complexly goal-seeking. There
is no incompatibility.

4/17. It will have been noticed that stability, as defined, in no
way implies fixity or rigidity. It is true that the stable system
usually has a state of equilibrium at which it shows no change;
but the lack of change is deceptive if it suggests rigidity: if dis-
placed from the state of equilibrium it will show active, perhaps

DESIGN FOR A BRAIN 4/18

extensive and complex, movements. The stable system is re-
stricted only in that it doe£ not show the unrestricted divergencies
of instability.

Stability and the whole

4/18. An important feature of a system's stability (or instability)
is that it is a property of the whole system and can be assigned to
no part of it. The statement may be illustrated by a consideration
of the first diagram of S. 4/14 as it is related to the practical
construction of the thermostat. In order to ensure the stability
of the final assembly, the designer must consider:

(1) The effect of the temperature on the diameter of the cap-

sule, i.e. whether a rise in temperature makes the capsule
expand or shrink.

(2) Which way an expansion of the capsule moves the lever.

(3) Which way a movement of the lever moves the gas-tap.

(4) Whether a given movement of the gas -tap makes the

velocity of gas-flow increase or decrease.

(5) Whether an increase of gas-flow makes the size of the gas-

flame increase or decrease.

(6) How an increase in size of the gas-flame will affect the tem-

perature of the capsule.

Some of the answers are obvious, but they must none the less
be included. When the six answers are known, the designer can
ensure stability only by arranging the components (chiefly by
manipulating (2), (3) and (5)) so that as a whole they form an
appropriate combination. Thus five of the effects may be decided,
yet the stability will still depend on how the sixth is related to
them. The stability belongs only to the combination; it cannot be
related to the parts considered separately.

In order to emphasise that the stability of a system is inde-
pendent of any conditions which may hold over the parts which
compose the whole, some further examples will be given. (Proofs
of the statements will be found in Ss. 20/9 and 21 /12.)

(a) Two systems may be joined so that they act and interact
on one another to form a single system: to know that the two
systems when separate were both stable is to know nothing about
the stability of the system formed by their junction: it may be
stable or unstable.

4/19 STABILITY

(b) Two systems, both unstable, may join to form a whole
which is stable.

(d) In a stable system the effect of fixing a variable may be to
render the remainder unstable.

Such examples could be multiplied almost indefinitely. They
illustrate the rule that the stability (or instability) of a dynamic
system depends on the parts and their interrelations as a whole.

4/19. The fact that the stability of a system is a property of the
system as a whole is related to the fact that the presence of stability
always implies some co-ordination of the actions between the parts.
In the thermostat the necessity for co-ordination is clear, for if
the components were assembled at random there would be only
an even chance that the assembly would be stable. But as the
system and the feedbacks become more complex, so does the
achievement of stability become more difficult and the likelihood
of instability greater. Radio engineers know only too well how
readily complex systems with feedback become unstable, and how
difficult is the discovery of just that combination of parts and
linkages which will give stability.

The subject is discussed more fully in S. 20/10: here it is
sufficient to note that as the number of variables increases so
usually do the effects of variable on variable have to be co-
ordinated with more and more care if stability is to be achieved.

CHAPTER 5

Adaptation as Stability

5/1. The concept of ' adaptation ' has so far been used without
definition; this vagueness must be corrected. Not only must
the definition be precise, but it must, by S. 2/10, be given in terms
that can be reduced wholly to primary operations.

5/2. The suggestion that an animal's behaviour is ' adaptive '
if the animal 4 responds correctly to a stimulus ' may be rejected
at once. First, it presupposes an action by an experimenter and
therefore cannot be applied when the free-living organism and
its environment affect each other reciprocally. Secondly, the
definition provides no meaning for ' correctly ' unless it means
4 conforming to what the experimenter thinks the animal ought
to do '. Such a definition is useless.

Homeostasis

5/3. I propose the definition that a form of behaviour is adaptive
if it maintains the essential variables (S. 3/14) within physiological
limits. The full justification of such a definition would involve
its comparison with all the known facts — an impossibly large
task. Nevertheless it is fundamental in this subject and I must
discuss it sufficiently to show how fundamental it is and how
wide is its applicability.

First I shall outline the facts underlying Cannon's concept of
1 homeostasis '. They are not directly relevant to the problem
of learning, for the mechanisms are inborn; but the mechanisms
are so clear and well known that they provide an ideal basic
illustration. They show that:

(1) Each mechanism is c adapted ' to its end.

(2) Its end is the maintenance of the values of some essential

variables within physiological limits.

(3) Almost all the behaviour of an animal's vegetative system

is due to such mechanisms.
58

5/4 ADAPTATION AS STABILITY

5/4. As first example may be quoted the mechanisms which
tend to maintain within limits the concentration of glucose in
the blood. The concentration should not fall below about 0-06
per cent or the tissues will be starved of their chief source of
energy; and the concentration should not rise above about
0.18 per cent or other undesirable effects will occur. If the
blood-glucose falls below about 0-07 per cent the adrenal glands
secrete adrenaline, which makes the liver turn its stores of glycogen
into glucose; this passes into the blood and the fall is opposed.
In addition, a falling blood-glucose stimulates the appetite so that
food is taken, and this, after digestion, provides glucose. On
the other hand, if it rises excessively, the secretion of insulin by
the pancreas is increased, causing the liver to remove glucose
from the blood. The muscles and skin also remove it; and the
kidneys help by excreting glucose into the urine if the concentra-
tion in the blood exceeds 0-18 per cent. Here then are five
activities all of which have the same final effect. Each one
acts so as to restrict the fluctuations which might otherwise occur.
Each may justly be described as ' adaptive ', for it acts to preserve
the animal's life.

The temperature of the interior of the warm-blooded animal's
body may be disturbed by exertion, or illness, or by exposure to
the weather. If the body temperature becomes raised, the skin
flushes and more heat passes from the body to the surrounding
air; sweating commences, and the evaporation of the water
removes heat from the body ; and the metabolism of the body is
slowed, so that less heat is generated within it. If the body is
chilled, these changes are reversed. Shivering may start, and the
extra muscular activity provides heat which warms the body.
Adrenaline is secreted, raising the muscular tone and the metabolic
rate, which again supplies increased heat to the body. The hairs
or feathers are moved by small muscles in the skin so that they
stand more erect, enclosing more air in the interstices and thus
conserving the body's heat. In extreme cold the human being,
when almost unconscious, reflexly takes a posture of extreme
flexion with the arms pressed firmly against the chest and the
legs fully drawn up against the abdomen. The posture is clearly
one which exposes to the air a minimum of surface. In all these
ways, the body acts so as to maintain its temperature within
limits.

DESIGN FOR A BRAIN 5/4

The amount of carbon dioxide in the blood is important in
its effect on the blood's alkalinity. If the amount rises, the rate
and depth of respiration are increased, and carbon dioxide is
exhaled at an increased rate. If the amount falls, the reaction
is reversed. By this means the alkalinity of the blood is kept
within limits.

The retina works best at a certain intensity of illumination.
In bright light the nervous system contracts the pupil, and in
dim relaxes it. Thus the amount of light entering the eye is
maintained within limits.

If the eye is persistently exposed to bright light, as happens
when one goes to the tropics, the pigment-cells in the retina
grow forward day by day until they absorb a large portion of the
incident light before it reaches the sensitive cells. In this way
the illumination on the sensitive cells is kept within limits.

If exposed to sunshine, the pigment-bearing cells in the skin
increase in number, extent, and pigment-content. By this change
the degree of illumination of the deeper layers of the skin is kept
within limits.

When dry food is chewed, a copious supply of saliva is poured
into the mouth. Saliva lubricates the food and converts it from
a harsh and abrasive texture to one which can be chewed without
injury. The secretion therefore keeps the frictional stresses below
the destructive level.

The volume of the circulating blood may be disturbed by
haemorrhage. Immediately after a severe haemorrhage a number
of changes occur: the capillaries in limbs and muscles undergo
constriction, driving the blood from these vessels to the more
essential internal organs; thirst becomes extreme, impelling the
subject to obtain extra supplies of fluid; fluid from the tissues
passes into the blood-stream and augments its volume; and
clotting at the wound helps to stem the haemorrhage. A haemor-
rhage has a second effect in that, by reducing the number of
red corpuscles, it reduces the amount of oxygen which can be
carried to the tissues; the reduction, however, itself stimulates
the bone-marrow to an increased production of red corpuscles.
All these actions tend to keep the variables ' volume of circu-
lating blood ' and ' oxygen supplied to the tissues ' within normal
limits.

Every fast-moving animal is liable to injury by collision with

5/6 ADAPTATION AS STABILITY

hard objects. Animals, however, are provided with reflexes that
tend to minimise the chance of collision and of mechanical injury.
A mechanical stress causes injury — laceration, dislocation, or
fracture — only if the stress exceeds some definite value, depending
on the stressed tissue — skin, ligament, or bone. So these reflexes
act to keep the mechanical stresses within physiological limits.

Many more examples could be given, but all can be included
within the same formula. Some external disturbance tends to
drive an essential variable outside its normal limits; but the
commencing change itself activates a mechanism that opposes the
external disturbance. By this mechanism the essential variable
is maintained within limits much narrower than would occur if
the external disturbance were unopposed. The narrowing is the
objective manifestation of the mechanism's adaptation.

5/5. The mechanisms of the previous section act mostly within
the body, but it should be noted that some of them have acted
partly through the environment. Thus, if the body-temperature
is raised, the nervous system lessens the generation of heat within
the body and the body-temperature falls, but only because the
body is continuously losing heat to its surroundings. Flushing
of the skin cools the body only if the surrounding air is cool;
and sweating lowers the body-temperature only if the surround-
ing air is unsaturated. Increasing respiration lowers the carbon
dioxide content of the blood, but only if the atmosphere contains
less than 5 per cent. In each case the chain of cause and effect
passes partly through the environment. The mechanisms that
work wholly within the body and those that make extensive use
of the environment are thus only the extremes of a continuous
series. Thus, a thirsty animal seeks water: if it is a fish it does
no more than swallow, while if it is an antelope in the veldt it
has to go through an elaborate process of search, of travel, and
of finding a suitable way down to the river or pond. The homeo-
static mechanisms thus extend from those that work wholly
within the animal to those that involve its widest-ranging acti-
vities ; the principles are uniform throughout.

Generalised homeostasis
5/6. Just the same criterion for ' adaptation ' may be used in
judging the behaviour of the free-living animal in its learned

DESIGN FOR A BRAIN 5/6

reactions. Take the type-problem of the kitten and the fire.
When the kitten first approaches an open fire, it may paw at the
fire as if at a mouse, or it may crouch down and start to ' stalk '
the fire, or it may attempt to sniff at the fire, or it may walk un-
concernedly on to it. Every one of these actions is liable to lead
to the animal's being burned. Equally the kitten, if it is cold,
may sit far from the fire and thus stay cold. The kitten's
behaviour cannot be called adapted, for the temperature of its
skin is not kept within normal limits. The animal, in other words,
is not acting homeostatically for skin temperature. Contrast this
behaviour with that of the experienced cat: on a cold day it
approaches the fire to a distance adjusted so that the skin tempera-
ture is neither too hot nor too cold. If the fire burns fiercer, the
cat will move away until the skin is again warmed to a moderate
degree. If the fire burns low the cat will move nearer. If a red-
hot coal drops from the fire the cat takes such action as will keep
the skin temperature within normal limits. Without making any
enquiry at this stage into what has happened to the kitten's brain,
we can at least say that whereas at first the kitten's behaviour
was not hom'eostatic for skin temperature, it has now become so.
Such behaviour is ' adapted ' : it preserves the life of the animal
by keeping the essential variables within limits.

The same thesis can be applied to a great deal, if not all, of
the normal human adult's behaviour. In order to demonstrate
the wide application of this thesis, and in order to show that even
Man's civilised life is not exceptional, some of the surroundings
which he has provided for himself will be examined for their
known physical and physiological effects. It will be shown that
each item acts so as to narrow the range of variation of his
essential variables.

The first requirement of a civilised man is a house; and its
first effect is to keep the air in which he lives at a more equable
temperature. The roof keeps his skin at a more constant dryness.
The windows, if open in summer and closed in winter, assist in
the maintenance of an even temperature, and so do fires and
stoves. The glass in the windows keeps the illumination of the
rooms nearer the optimum, and artificial lighting has the same
effect. The chimneys keep the amount of irritating smoke in the
rooms near the optimum, which is zero.

Many of the other conveniences of civilisation could, with little

5/7 ADAPTATION AS STABILITY

difficulty, be shown to be similarly variation-limiting. An attempt
to demonstrate them all would be interminable. But to confirm
the argument we will examine a motor-car, part by part, in order
to show its homeostatic relation to man.

Travel in a vehicle, as contrasted with travel on foot, keeps
several essential variables within narrower limits. The fatigue
induced by walking for a long distance implies that some vari-
ables, as yet not clearly known, have exceeded limits not trans-
gressed when the subject is carried in a vehicle. The reserves
of food in the body will be less depleted, the skin on the soles of
the feet will be less chafed, the muscles will have endured less
strain, in winter the body will have been less chilled, and in
summer it will have been less heated, than would have happened
had the subject travelled on foot.

When examined in more detail, many ways are found in which
it serves us by maintaining our essential variables within narrower
limits. The roof maintains our skin at a constant dryness. The
windows protect us from a cold wind, and if open in summer,
help to cool us. The carpet on the floor acts similarly in winter,
helping to prevent the temperature of the feet from falling below
its optimal value. The jolts of the road cause, on the skin and
bone of the human frame, stresses which are much lessened by
the presence of springs. Similar in action are the shock-absorbers
and tyres. A collision would cause an extreme deceleration which
leads to very high values for the stress on the skin and bone of
the passengers. By the brakes these very high values may be
avoided, and in this way the brakes keep the variables ' stress on
bone ' within narrower limits. Good headlights keep the lumin-
osity of the road within limits narrower than would occur in their
absence.

The thesis that 4 adaptation ' means the maintenance of essential
variables within physiological limits is thus seen to hold not only
over the simpler activities of primitive animals but over the more
complex activities of the ' higher ' organisms.

5/7. Before proceeding further, it must be noted that the word
' adaptation ' is commonly used in two senses which refer to
different processes.

The distinction may best be illustrated by the inborn homeo-
static mechanisms : the reaction to cold by shivering, for instance.

DESIGN FOR A BRAIN 5/8

Such a mechanism may undergo two types of ' adaptation '. The
first occurred long ago and was the change from a species too
primitive to show such a reaction to a species which, by natural
selection, had developed the reaction as a characteristic inborn
feature. The second type of 4 adaptation ' occurs when a member
of the species, born with the mechanism, is subjected to cold and
changes from not-shivering to shivering. The first change
involved the development of the mechanism itself; the second
change occurs when the mechanism is stimulated into showing
its properties.

In the learning process, the first stage occurs when the animal
4 learns ' : when it changes from an animal not having an adapted
mechanism to one which has such a mechanism. The second
stage occurs when the developed mechanism changes from in-
activity to activity. In this chapter we are concerned with the
characteristics of the developed mechanism. The processes which
led to its development are discussed in Chapter 9.

5/8. We can now recognise that ' adaptive ' behaviour is equivalent
to the behaviour of a stable system, the region of the stability being
the region of the phase-space in which all the essential variables lie
within their normal limits.

The view is not new (though it can now be stated with more
precision):

4 Every phase of activity in a living being must be not
only a necessary sequence of some antecedent change in its
environment, but must be so adapted to this change as to
tend to its neutralisation, and so to the survival of the
organism. ... It must also apply to all the relations of
living beings. It must therefore be the guiding principle,
not only in physiology . . . but also in the other branches
of biology which treat of the relations of the living animal
to its environment and of the factors determining its survival
in the struggle for existence.'

(Starling.)

4 In an open system, such as our bodies represent, com-
pounded of unstable material and subjected continuously to
disturbing conditions, constancy is in itself evidence that
agencies are acting or ready to act, to maintain this constancy.'

(Cannon.)

4 Every material system can exist as an entity only so long
as its internal forces, attraction, cohesion, etc., balance the

5/9 ADAPTATION AS STABILITY

external forces acting upon it. This is true for an ordinary
stone just as much as for the most complex substances; and
its truth should be recognised also for the animal organism.
Being a definite circumscribed material system, it can only
continue to exist so long as it is in continuous equilibrium
with the. forces external to it: so soon as this equilibrium
is seriously disturbed the organism will cease to exist as the
entity it was.'

(Pavlov.)

McDougall never used the concept of ' stability ' explicitly, but
when describing the type of behaviour which he considered to
be most characteristic of the living organism, he wrote:

4 Take a billard ball from the pocket and place it upon the
table. It remains at rest, and would continue to remain so
for an indefinitely long time, if no forces were applied to it.
Push it in any direction, and its movement in that direction
persists until its momentum is exhausted, or until it is
deflected by the resistance of the cushion and follows a new
path mechanically determined. . . . Now contrast with this
an instance of behaviour. Take a timid animal such as a
guinea-pig from its hole or nest, and put it upon the grass
plot. Instead of remaining at rest, it runs back to its hole;
push it in any other direction, and, as soon as you withdraw
your hand, it turns back towards its hole ; place any obstacle
in its way, and it seeks to circumvent or surmount it, rest-
lessly persisting until it achieves its end or until its energy
is exhausted.'

He could hardly have chosen an example showing more clearly
the features of stability.

Survival

5/9. Are there aspects of ' adaptation ' not included within the
definition of ' stability ' ? Is ' survival ' to be the sole criterion
of adaptation ? Is it to be maintained that the Roman soldier
who killed Archimedes in Syracuse was better ' adapted ' in his
behaviour than Archimedes ?

The question is not easily answered. It is similar to that of
S. 3/4 where it was asked whether all the qualities of the living
organism could be represented by number; and the answer must
be similar. It is assumed that we are dealing primarily with
the simpler rather than with the more complex creatures, though

DESIGN FOR A BRAIN

5/10

the examples of S. 5/6 have shown that some at least of man's

activities may be judged properly by this criterion.

In order to survey rapidly the types of behaviour of the more

primitive animals, we may examine the classification of Holmes,

who intended his list to be exhaustive but constructed it with

no reference to the concept of stability. The reader will be able

to judge how far our formulation (S. 5/8) is consistent with his

scheme, which is given in Table 5/9/1.

'Useless tropistic reaction.
Misdirected instinct.
Abnormal sex behaviour.
Non-adaptive ^ Pathological behaviour.

Useless social activity.
Superfluous random
movements.

'Capture, devouring of

food.
Activities preparatory, as

making snares, stalking.
Collection of food, digging.
Migration.
Caring for food, storing,

burying, hiding.
Preparing of food.

Behaviour -

Adaptive

Self-
maintaining

Sustentative

{Against enemies — fight,
flight.
Against inanimate forces.
Reactions to heat, gravity,
chemicals.
Against inanimate objects.

Ameliorative Rest, sleep, play, basking.

^maintaining { <With thesec w,e Qaxre not concerned,

Table 5/9/1 : All forms of animal behaviour, classified by Holmes.

For the primitive organism, and excluding behaviour related to
racial survival, there seems to be little doubt that the ' adaptive-
ness ' of behaviour is properly measured by its tendency to
promote the organism's survival.

5/10. A most impressive characteristic of living organisms is
their mobility, their tendency to change. McDougall expressed
this characteristic well in the example of S. 5/8. Yet our formula-
tion transfers the centre of interest to the state of equilibrium,
to the fact that the essential variables of the adapted organism
change less than they would if it were unadapted. Which is
important : constancy or change ?

5/11 ADAPTATION AS STABILITY

The two aspects are not incompatible, for the constancy of some
variables may involve the vigorous activity of others. A good
thermostat reacts vigorously to a small change of temperature,
and the vigorous activity of some of its variables keeps the others
within narrow limits. The point of view taken here is that the
constancy of the essential variables is fundamentally important,
and that the activity of the other variables is important only in
so far as it contributes to this end. (The matter is discussed
more thoroughly in /. to C, Chapter 10.)

Stability and co-ordination

5/11. So far the discussion has traced the relation between the
concepts of ' adaptation ' and of ' stability '. It will now be
proposed that ' motor co-ordination ' also has an essential con-
nexion with stability.

4 Motor-co-ordination ' is a concept well understood in physio-
logy, where it refers to the ability of the organism to combine the
activities of several muscles so that the resulting movement
follows accurately its appropriate path. Contrasted to it are the
concepts of clumsiness, tremor, ataxia, athetosis. It is suggested
that the presence or absence of co-ordination may be decided, in
accordance with our methods, by observing whether the move-
ment does, or does not, deviate outside given limits.

Figure 5/11/1.

The formulation seems to be adequate provided that we measure
the limb's deviations from some line which is given arbitrarily,
usually by a knowledge of the line followed by the normal limb.
A first example is given by Figure 5/11/1, which shows the line
traced by the point of an expert fencer's foil during a lunge.
Any inco-ordination would be shown by a divergence from the
intended line.

A second example is given by the record of Figure 5/11/2. The
subject, a patient with a tumour in the left cerebellum, was asked

DESIGN FOR A BRAIN

5/12

to follow the dotted lines with a pen. The left- and right-hand
curves were drawn with the respective hands. The tracing shows
clearly that the co-ordination is poorer in the left hand. What
criterion reveals the fact ? The essential
distinction is that the deviations of the
lines from the dots are larger on the left
than on the right.

The degree of motor co-ordination
achieved may therefore be measured by
the smallness of the deviations from some
standard line. Later it will be suggested
that there are mechanisms which act to
maintain variables within narrow limits.
If the identification of this section is
accepted, such mechanisms could be
regarded as appropriate for the co-ordina-
tion of motor activity.

Figure 5/11/2 : Record
of the attempts of a
patient to follow the
dotted lines with the left
and right hands. (By
the courtesy of Dr.W.T.
Grant of Los Angeles.)

5/12. So far we have noticed in stable
systems only their property of keeping
variables within limits. But such sys-
tems have other properties of which we shall notice two. They
are also shown by animals, and are then sometimes considered
to provide evidence that the organism has some power of
4 intelligence ' not shared by non-living systems. In these two
instances the assumption is unnecessary.

The first property is shown by a stable system when the lines
of behaviour do not return directly, by a straight line, to the state
of equilibrium (e.g. Figure 4/5/3).
When this occurs, variables may be
observed to move away from their
values in the state of equilibrium, only
to return to them later. Thus, sup-
pose in Figure 5/12/1 that the field
is stable and that at the equilibrial
state R x and y have the values X
and Y. For clarity, only one line of
behaviour is drawn. Let the system be displaced to A and
its subsequent behaviour observed. At first, while the repre-
sentative point moves towards B, y hardly alters; but xt which

Figure 5/12/1.

5/13 ADAPTATION AS STABILITY

started at X\ moves to Ar and goes past it to X". Then x remains
almost constant and y changes until the representative point
reaches C. Then y stops changing, and x changes towards, and
reaches, its resting value X. The system has now reached a state
of equilibrium and no further changes occur. This account is just
a transcription into words of what the field defines graphically.

Now the shape and features of any field depend ultimately on
the real physical and chemical construction of the ' machine '
from which the variables are abstracted. The fact that the line
of behaviour does not run straight from A to R must be due to
some feature in the 4 machine ' such that if the machine is to
get from state A to state R, states B and C must be passed
through of necessity. Thus, if the machine contained moving
parts, their shapes might prohibit the direct route from A to R;
or if the system were chemical the prohibition might be thermo-
dynamic. But in either case, if the observer watched the machine
work, and thought it alive, he might say : ' How clever ! x
couldn't get from A to R directly because this bar was in the
way; so x went to B, which made y carry x from B to C; and
once at C, x could get straight back to R. I believe x shows
foresight.'

Both points of view are reasonable. A stable system may be
regarded both as blindly obeying the laws of its nature, and also
as showing skill in getting back to its state of equilibrium in spite
of obstacles.*

5/13. The second property is shown when an organism reacts to
a variable with which it is not directly in contact. Suppose,
for instance, that the diagram of immediate effects (S. 4/12) is
that of Figure 5/13/1; the variables have been divided by the
dotted line into ' animal ' on the right and ' environment ' on
the left, and the animal is not in direct contact with the variable
marked X. The system is assumed to be stable, i.e. to have

* I would like to acknowledge that much of what I am describing was
arrived at independently by G. Sommerhoff. I met his Analytical Biology
only when the first edition of Design for a Brain was in proof, and I could do
no more than add his title to my list of references. Since then it has become
apparent that our work was developing in parallel, for there is a deep similarity
of outlook and method in the two books. The superficial reader might notice
some differences and think we are opposed, but I am sure the distinctions are
only on minor matters of definition or emphasis. The reader who wishes to
explore these topics further should consult his book as a valuable independent
contribution.

DESIGN FOR A BRAIN

5/14

arrived at the ' adapted ' condition (S. 5/7). If disturbed, its
changes will show co-ordination of part with part (S. 5/12), and
this co-ordination will hold over the whole system (S. 4/18). It
follows that the behaviour of the ' animal '-part will be co-
ordinated with the behaviour of X although the ' animal ' has
no immediate contact with it. (Example in S. 8/7.)

In the higher organisms, and especially in Man, the power to
react correctly to something not immediately visible or tangible
has been called ' imagination ', or ' abstract thinking ', or several
other names whose precise meaning need not be discussed at
the moment. Here we should notice that the co-ordination of

-< — .

"* — I

! >

'-. >

Figure 5/13/1.

the behaviour of one part with that of another part not in direct
contact with it is simply an elementary property of the stable
system.

5/14. Let us now re-state our problem in the new vocabulary.
If, for brevity, we omit minor qualifications, we can state it thus :
A determinate ' machine ' changes from a form that produces
chaotic, unadapted behaviour to a form in which the parts are
so co-ordinated that the whole is stable, acting to maintain its
essential variables within certain limits — how can this happen ?
For example, what sort of a thermostat could, if assembled at
random, rearrange its own parts to get itself stable for temperature?
It will be noticed that the new statement involves the concept
of a machine changing its internal organisation. So far, nothing
has been said of this important concept; so it will be treated in
the next chapter.

CHAPTER 6

Parameters

6/1. So far, we have discussed the changes shown by the vari-
ables of a state-determined system, and have ignored the fact that
all its changes occur on a background, or on a foundation, of
constancies. Thus, a particular simple pendulum provides two
variables which are known (S. 2/15) to be such that, if we are
given a particular state of the system, we can predict correctly
its ensuing behaviour ; what has not been stated explicitly is that
this is true only if the length of the' string remains constant. The
background, and these constancies, must now be considered.

Every system is formed by selecting some variables out of the
totality of possible variables. ' Forming a system ' means dividing
the variables of the universe into two classes: those within the
system and those without. These two types of variable are in
no way different in their intrinsic physical nature, but they stand
in very different relations to the system.

6/2. Given a system, a variable not included in it is a parameter.
The word variable will, from now on, be reserved for one within
the system.

In general, given a system, the parameters will differ in their
closeness of relation to it. Some will have a direct relation to it :
change of their value would affect the system to a major degree;
such is the parameter ' length of pendulum ' in its relation to the
two-variable system of the previous section. Some are less closely
related to it, their changes producing only a slight effect on it;
such is the parameter 4 viscosity of the air ' in relation to the same
system. And finally, for completeness, may be mentioned the
infinite number of parameters that are without detectable effect
on the system; such are the brightness of the light shining on
the pendulum, the events in an adjacent room, and the events in
the distant nebulae. Those without detectable effect may be
ignored; but the relationship of an effective parameter to a
system must be clearly understood.

DESIGN FOR A BRAIN

6/3

Given a system, the effective parameters are usually innumer-
able, so that a list is bounded only by the imagination of the
writer. Thus, parameters whose change might affect the be-
haviour of the same system of two variables are:

( 1 ) the length of the pendulum (hitherto assumed constant),

(2) the lateral velocity of the air (hitherto assumed to be con-

stant at zero),

(3) the viscosity of the surrounding medium (hitherto assumed

constant),

(4) the movement, if any, of the point of support,

(5) the force of gravity,

(6) the magnetic field in which it swings,

(7) the elastic constant of the string of the pendulum,

(8) its electrostatic charge, and the charges on bodies nearby ;
but the list has no end.

Parameter and field

6/3. The effect on a state-determined system of a change of
parameter-value will now be shown. Table 6/3/1 shows the

Length
(cm.)

Line

Vari-
able

Time

005

010

015

0-20

0-25

0-30

147

142

129

108

129

108

- 24

- 58

147

144

135

121

101

121

101

- 6

- 36

Table 6/3/1.

results of twenty-four primary operations applied to the two-
variable system mentioned above, x is the angular deviation

6/4

PARAMETERS

from the vertical, in degrees; y is the angular velocity, in degrees
per second; the time is in seconds.

The first two Lines show that the lines of behaviour following
the state x = 14, y = 129 are equal, so the system, as far as
it has been tested, is state-determined. The line of behaviour is
shown solid in Figure 6/3/1. In these swings the length of the
pendulum was 40 cm. This parameter was then changed to 60 cm.
and two further lines of behaviour were observed. On these two,
the lines of behaviour following
the state x = 21, y = 121 are
equal, so the system is again
state-determined. The line of
behaviour is shown dotted in the
same figure. But the change of
parameter-value has caused the
line of behaviour from x = 0,
y = 147 to change.

The relationship which the para-
meter bears to the two variables
is therefore as follows :

(1) So long as the parameter is constant, the system of x and
y is state-determined, and has a definite field.

(2) After the parameter changes from one constant value to
another, the system of x and y becomes again state-determined, and
has a definite field, but this field is not the same as the previous one.

The relation is general. A change in the value of an effective
parameter changes the line of behaviour from each state. From
this follows at once : a change in the value of an effective parameter
changes the field.

From this follows the important quantitative relation : a system
can, in general, show as many fields as its parameters can show
combinations of values.

IOO

Figure 6/3/1,

6/4. The importance of distinguishing between change of a
variable and change of a parameter, that is, between change of
state and change of field, can hardly be over-estimated. It is this
distinction that will enable us to avoid the confusion threatened
in S. 2/1 between those changes that are behaviour and changes
that occur from one behaviour to another. In order to make the
distinction clear I will give some examples.

DESIGN FOR A BRAIN

6/4

In a working clock, the single variable defined by the reading
of the minute-hand on the face is state-determined as a one-
variable system; for after some observations of its behaviour, we
can predict the line of behaviour which will follow any given state.
If now the regulator (the parameter) is moved to a new position,
so that the clock runs at a different rate, and the system is re-
examined, it will be found to be still state-determined but to have
a different field.

If a healthy person drinks 100 g. of glucose dissolved in water,
the amount of glucose in his blood usually rises and falls as A
in Figure 6/4/1. The single variable ' blood-glucose ' is not state-

Figure 6/4/1 : Changes in blood-glucose after the ingestion of 100 g.
of glucose : (A) in the normal person, (B) in the diabetic.

determined, for a given state (e.g. 120 mg./lOO ml.) does not
define the subsequent behaviour, for the blood-glucose may rise or
fall. By adding a second variable, however, such as 4 rate of
change of blood-glucose ', which may be positive or negative, we
obtain a two-variable system which is sufficiently state-determined
for illustration. The field of this two-variable system will re-
semble that of A in Figure 6/4/2. But if the subject is diabetic,
the curve of the blood-glucose, even if it starts at the same initial
value, rises much higher, as B in Figure 6/4/1. When the field
of this behaviour is drawn (B, Figure 6/4/2), it is seen to be not the
same as that of the normal subject. The change of value of the
parameter ' degree of diabetes present ' has thus changed the
field.

Girden and Culler developed a conditioned response in a dog
which was under the influence of curare (a paralysing drug)-

6/5 PARAMETERS

When later the animal was not under its influence, the conditioned
response could not be elicited. But when the dog was again put
under its influence, the conditioned response returned. Thus

St .200-

/^\

blood
100 ml

(

u "" -IOC
o

t 1

(£ 100 200 100 200 300

BLOOD GLUCOSE (mg. per 100 ml J

Figure 6/4/2 : Fields of the two lines of behaviour, A and B from
Figure 6/4/1. Cross-strokes mark each quarter-hour.

two characteristic lines of behaviour (two responses to the stimulus)
existed, and one line of behaviour was shown when the parameter
* concentration of curare in the tissues ' had a high value, and
the other when the parameter had a low value.

Stimuli

6/5. Many stimuli may be represented adequately as a change
of parameter- value, so it is convenient here to relate the physio-
logical and psychological concept of a ' stimulus ' to our methods.

In all cases the diagram of immediate effects is

(experimenter) — - > stimulator — ► animal — > recorders.

In some cases the animal, at some state of equilibrium, is
subjected to a sudden change in the value of the stimulator, and
the second value is sustained throughout the observation. Thus,
the pupillary reaction to light is demonstrated by first accustoming
the eye to a low intensity of illumination, and then suddenly
raising the illumination to a high level which is maintained while
the reaction proceeds. In such cases the stimulator is parameter
to the system ' animal and recorders ' ; and the physiologist's
comparison of the previous control-behaviour with the behaviour
after stimulation is equivalent, in our method, to a comparison of
the two lines of behaviour that, starting from the same initial
state, run in the two fields provided by the two values of the
stimulator. In this type, the stimulator behaves as a step-
function (S. 7/13).

DESIGN FOE A BRAIN 6/6

Sometimes a parameter is changed sharply and is immediately
returned to its initial value, as when the experimenter applies a
a tap on a tendon. The effect of the parameter-change is a brief
change of field which, while it lasts, carries the representative
point away from its original position. When the parameter is
returned to its original value, the original field and state of equili-
brium are restored, but the representative point is now away
from the state of equilibrium; it therefore moves along a line of
behaviour, and the organism ' responds '. (Usually the point
returns to the same state of equilibrium : but if there is more than
one, it may go to some other state of equilibrium.) Such a
stimulus will be called impulsive.

It will be necessary later to be more precise about what we mean
by ' the ' stimulus. Consider, for instance, a dog developing a
conditioned reflex to the ringing of an electric bell. What is the
stimulus exactly ? Is it the closing of the contact switch ? The
intermittent striking of the hammer on the bell ? The vibrations
in the air ? The vibrations of the ear-drum, of the ossicles, of
the basilar membrane ? The impulses in the acoustic nerve, in the
temporal cortex ? If we are to be precise we must recognise that
the experimenter controls directly only the contact switch, and
that this acts as parameter to the complexly-acting system of
electric bell, middle ear, and the rest.

When the ' stimulus ' becomes more complex we must generalise.
One generalisation increases the number of parameters made to
alter, as when a conditioned dog is subjected to combinations of a
ticking metronome, a smell of camphor, a touch on the back, and
a flashing light. Here we should notice that if the parameters
are not all independent but change in groups, like the variables
in S. 3/3, we can represent each undivided group by a single
value and thus avoid using unnecessarily large numbers of
parameters.

Joining dynamic systems

6/6. We can now make clear what is meant, essentially, by the
concept of two (or more) systems being ' joined '.

This concept is of the highest importance in biology, in which it
occurs frequently and prominently. It occurs whenever we think
of one system having an effect on another, or communicating with
it, or forcing it, or signalling to it.

6/7 PARAMETERS

(The exact nature of the operation of joining is shown most
clearly in the mathematical form (S. 21/9) for there one can see
what is essential and what irrelevant. A detailed treatment has
been given in /. to C, S. 4/6; here we can discuss it less rigorously.)

To join two systems, A and B say, so that A affects B, A must
affect 2?'s conditions. In other words, the values of some of B's
parameters (perhaps one only) must become functions of (de-
pendent on) the values of A's variables. Thus, if B is a developing
egg in an incubator and A is the height of the barometer, then
A could be i joined ' so as to affect B if the temperature (or other
suitable parameter) were made sensitive to the pressure.

In this example there is no obvious way of making the develop-
ment of the egg affect the height of the barometer, so the joining
of B to A can hardly be done. In most cases, however, joinings
are possible in either direction. If both are made, then feedback
has been set up between the two systems.

In very simple cases, the behaviour of the whole formed by
joining parts can be traced step by step by logical or mathematical
deduction. Each part can be thought of as having its own phase-
space, filled by a field ; which field it is will depend on the position
of the other part's representative point. Each representative
point now undergoes a transition, guided by its own field, whose
form depends on the position of the other. So step by step, each
goes forward guided by the other and also guiding it. (The process
has been traced in detail in /. to C, S. 4/7.)

This picture is too complicated for any imaginative grasp of
how two actual systems will behave; the details must be worked
out by some other method. What is important is that the nature
of the process is conceptually quite free from vagueness or ambi-
guity; so it may properly be included in a rigorous theory of
dynamic systems.

Parameter and stability

6/7. We now reach the main point of the chapter. Because
a change of parameter-value changes the field, and because a
system's stability depends on its field, a change of parameter-
value will in general change a system's stability in some way.

A simple example is given by a mixture of hydrogen, nitrogen,
and ammonia, which combine or dissociate until the concentrations

DESIGN FOR A BRAIN 6/8

reach the state of equilibrium. If the mixture was originally
derived from pure ammonia, the single variable ' percentage
dissociated ' forms a one-variable state-determined system.
Among its parameters are temperature and pressure. As is well
known, changes in these parameters affect the position of the
state of equilibrium.

Such a system is simple and responds to the changes of the
parameters with only a simple shift of equilibrium. No such
limitation applies generally. Change of parameter-value may
result in any change which can be produced by the substitution of
one field for another: stable systems may become unstable, states
of equilibrium may be moved, single states of equilibrium may
become multiple, states of equilibrium may become cycles; and
so on. Figure 21/8/1 provides an illustration.

Here we need only the relationship, which is reciprocal: in
a state-determined system, a change of stability can only be due to
change of value of a parameter, and change of value of a parameter
causes a change in stability.

Equilibria of part and whole

6/8. In general, as S. 4/18 showed, the relation between the
stabilities of the parts and that of the whole may be complex,
and may require specialised methods for its treatment. There is,
however, one quite simple relationship that will be of the greatest
use to us and which can be readily described.

Suppose we join two parts, A with variables u and v, and B
with variables w, x and y. If A's variables have values 7 and 2,
and B's have values 3, 1 and 5, then the whole is, naturally, a
system with the five variables u, v, w, x and y; and in the cor-
responding state the variables of the whole have the values 7,
2, 3, 1 and 5 respectively.

Suppose now that this state — (7, 2, 3, 1, 5) — of the whole is a
state of equilibrium of the whole. This implies that the transi-
tion is from that state to itself (S. 4/4). This implies that A,
with the values 3, 1, 5 on its parameters, goes from (7, 2) to (7, 2);
i.e. does not change. Thus, the whole's being at a state of equili-
brium at (7, 2, 3, 1, 5) implies that A, when at (7, 2), with values
(3, 1, 5) on its parameters, must be at a state of equilibrium.
Similarly B, when its parameters are at (7, 2), must have a state

6/10 PARAMETERS

of equilibrium at (3, 1, 5). So, the whole's being at a state of
equilibrium implies that each part must be at a state of equilibrium,
in the conditions provided (at its parameters) by the other parts.

Conversely, suppose A is in equilibrium at state (7, 2) when
its parameters have the values (3, 1, 5); and that B is in equi-
librium at state (3, 1, 5) when its parameters are at (7, 2). It
follows that the whole will have a state of equilibrium at the
state (7, 2, 3, 1, 5), for at this state neither A nor B can change.

To sum up : That a whole dynamic system should be in equilibrium
at a particular state it is necessary and sufficient that each part should
be in equilibrium at that state, in the conditions given to it by the
other parts.

6/9. Suppose now that a whole, made by joining parts, is moving
along a line of behaviour. Suppose the line of behaviour meets a
state that is one of equilibrium for one part (in the conditions
given at that moment by the others) but not equilibrial for the
other parts. The part in equilibrium will stop, momentarily;
but the other parts, not in equilibrium, will change their states
and will thereby change the conditions of the part in equilibrium.
Usually the change of conditions (change of parameter- values)
will make the state no longer one of equilibrium: so the part that
stopped willl now start moving again.

Clearly, at any state of the whole, if a single part is not at
equilibrium (even though the remainder are) this part will change,
will provide new conditions for the other parts, will thus start
them moving again, and will thus prevent that state from being
one of equilibrium for the whole. As equilibrium of the whole
requires that all the parts be in equilibrium, we can say (meta-
phorically) that every part has a power of veto over the states of
equilibrium of the whole.

6/10. The importance of this fact can now be indicated. By
this fact each part acts selectively towards the set of possible
equilibria of the whole. Since Chapter 1 we have been looking
for some factor that can be both mechanistic and also selective.
The next chapter will show this factor in action.

CHAPTER 7

The Ultrastable System

7/1. We have now assembled the necessary concepts. They
are all denned as relations between primary operations, so they
are fully objective and conform to the basic requirements of
S. 2/10. We can now reconsider the basic problem of S. 5/14,
and can consider what is implied by the fact that the kitten
changes from having a cerebral mechanism that produces un-
adapted behaviour to having one which produces behaviour that
is adapted.

The implications of adaptation

7/2. In accordance with S. 3/11, the kitten and environment
are to be considered as interacting; so the diagram of immediate
effects will be of the form of Figure 7/2/1 . (The diagram resembles
t that of Figure 5/13/1, except that the fine net-

£ny* ..^ work of linkages that actually exists in environ-
ment and R has been represented by shading.)
The arrows to and from R represent, of course, the

t sensory and motor channels. The part R belongs

f to the organism,* but is here defined purely func-

tionally; at this stage any attempt to identify R
with anatomical or histological structures must be
QpJffT made with caution. R is defined as the system
Figure 7/2/1 tnat acts wnen tne kitten reacts to the fire — the
part responsible for the overt behaviour.
It was also given, in S. 5/14, that the kitten has a variety of
possible reactions, some wrong, some right. This variety of
reactions implies, by S. 6/3, that some parameters, call them S,
have a variety of values, i.e. are not fixed throughout. These
parameters, since their primary action is to affect the kitten's
behaviour (and only mediately that of the environment), evidently
have an immediate effect on R but not on the environment.

tiny.

7/3 THE ULTRASTABLE SYSTEM

Thus we get Figure 7/2/2. By S. 6/3, Enyt

the number of distinct values possible to
S must be at least as great as the num-
ber of distinct ways of behaving (both
adapted and non-adapted) possible to R.

Figure 7/2/2.

7/3. The essential variables must now be
introduced; what affects them ? Clearly
they must be affected by something, for
we are not interested in the case of the
organism that is immortal because nothing
threatens it. Possibilities are that they
are affected by the environment, by R, or by both.

The case of most interest is that in which they are immediately
affected by the environment only. This case makes the problem
for the kitten as harsh, as realistic as possible. This is the case
when a hot coal falls from the fire and rolls towards the kitten:
the environment threatens to have a direct effect on the essential
variables, for if the kitten's brain does nothing the kitten will
get burnt. This is the case when the animal in the desert is being
dried by the heat, so that if the animal does nothing it will die of
thirst.

Immediate effects from R to the essential variables would be
appropriate if the kitten's brain could act so as to change it from
an organism that must not get burnt to one that benefited by
being burnt ! (Such a change of goal may be of importance in
the higher functionings of the nervous system, when a sub-goal
may be established or changed provisionally; but the situation
does not occur at the fundamental level
that we are considering here, and we
shall not consider such possibilities
further.)

The diagram of immediate effects now
has the form of Figure 7/3/1. The
essential variables have been represented
collectively by a dial with a pointer, and
with two limit-marks, to emphasise that
what matters about the essential variables
is whether or not the value is within
Figure 7/3/1. physiological limits.

DESIGN FOR A BRAIN 7/4

7/4. Continuing to examine the case that gives the kitten the
maximal difficulty, let us consider the case in which the effects that
the various states of the environment will have on the essential
variables, though definite, is not known to the reacting part R.
This is the case of a bird, driven to a strange island and seeing a
strange berry, who does not know whether it is poisonous or not.
It is the case of the cat in Thorndike's cage, who does not know
whether the lever must be pushed to right or left for the door to
open. It is the assumption made in S.l/17, where the kitten was
confronted with a fire as an example of an organism in a situation
where its previous experience gave no reliable indication of how
the various states of the environment were paired to the states
1 within ' and c without ' the physiological limits of the essential
variables.

To be adapted, the organism, guided by information from the
environment, must control its essential variables, forcing them to
go within the proper limits, by so manipulating the environment
(through its motor control of it) that the environment then acts
on them appropriately. Thus the diagram of immediate effects
of this process is

Reacting
part R

Environment

Essential
variables

In the case we are considering, the reacting part R is not specially
related or adjusted to what is in the environment and how it is
joined to the essential variables. Thus the reacting part R can
be thought of as an organism trying to control the output of a
Black Box (the environment), the contents of which is unknown
to it.

It is axiomatic (for any Black Box when the range of its inputs
is given) that the only way in which the nature of its contents
can be elicited is by the transmission of actions through it. This
means that input-values must be given, output-values observed,
and the relationships in the paired values noticed. In the kitten's
case, this means that the kitten must do various things to the
environment and must later act in accordance with how these
actions affected the essential variables. In other words, it must
proceed by trial and error.

Adaptation by trial and error is sometimes treated in psycho-

7/5 THE ULTRASTABLE SYSTEM

logical writings as if it were merely one way of adaptation, and
an inferior way at that. The argument given above shows that
the method of trial and error holds a much more fundamental
place in the methods of adaptation. The argument shows, in
fact, that when the organism has to adapt (to get its essential
variables within physiological limits) by working through an
environment that is of the nature of a Black Box, then the process
of trial and error is necessary, for only such a process can elicit
the required information.

The process of trial and error can thus be viewed from two very
different points of view. On the one hand it can be regarded as
simply an attempt at success; so that when it fails we give zero
marks for success. From this point of view it is merely a second-
rate way of getting to success. There is, however, the other point
of view that gives it an altogether higher status, for the process
may be playing the invaluable part of gathering information,
information that is absolutely necessary if adaptation is to be
successfully achieved. It is for this reason that the process must
enter into the kitten's adaptation.

7/5. As the kitten proceeds by trial and error, its final behaviour

will depend on the outcome of the trials, on how the essential

variables have been affected. This is equivalent to saying that

the essential variables are to have an effect on which behaviour

the kitten will produce; and this is *

Env •
equivalent, to saying that in the diagram

of immediate effects there must be a

channel from the essential variables to

the parameters S; so it will resemble

Figure 7/5/1. The organism that can

adapt thus has a motor output to the

environment and two feedback loops.

The first loop was shown in Figure 7/2/1 ;

it consists of the ordinary sensory input

from eye, ear, joints, etc., giving the Fjgure 7/5/1*

organism non-affective information about

the world around it. The second feedback goes through the

essential variables (including such correlated variables as the pain

receptors, S. 3/15); it carries information about whether the

essential variables are or are not driven outside the normal limits,

DESIGN FOR A BRAIN 7/6

and it acts on the parameters S. The first feedback plays its
part within each reaction; the second determines which reaction
shall occur.

7/6. Since the argument here is crucial, let us trace it in detail,
using the basic operational concepts of S. 2/7-10.

We start with the common observation that the burned kitten
dreads the fire. Translated into full operational form, this
observation becomes:

(1) with the essential variables within their limits, the overt

behaviour (of R) is such as is consequent on the parameters
having values S^

(2) when the essential variables are sent outside the limits (i.e

if the kitten is burned), the overt behaviour is such as is
consequent on their having values 52.

That the overt behaviour is changed shows that S2 is not the same
as Sv Thus the two different values at the essential variables
have led to different values at S; there is therefore an immediate
effect from the essential variables to the parameters S.

111. The same data will now provide us with the necessary
information about what happens within the second loop, i.e.
how the essential variables affect the parameters.

The basic rule for adaptation by trial and error is: — If the
trial is unsuccessful, change the way of behaving ; when and only
when it is successful, retain the way of behaving. Now consider
the system S and how it must behave. Within this system are
the variables that are identical with the parameters to R (a mere
change of name), and to this system the essential variables are
parameters, i.e. come as input. The basic rule is equivalent to
the following formulation:

(1) When the essential variables are not all within their normal

limits (i.e. when the trial has failed), no state of S is to
be equilibrial (for the rule here is that S must go to some
other state).

(2) When the essential variables are all within normal limits

then every state of *S is to be equilibrial (i.e. S is to be in
neutral equilibrium).

7/10 THE ULTRASTABLE SYSTEM

7/8. What has been deduced so far in this Chapter is necessary.
That is to say, any system that has essential variables with given
limits, and that adapts by the process of testing various behaviours
by how each affects ultimately the essential variables, must have
a second feedback formally identical (isomorphic) with that
described here. This deduction holds equally for brains living
and mechanical.

To be quite clear in this matter, let us consider the alternative.
Suppose some new species, or some new mechanical brain, were
found to change from the non-adapted to the adapted condition
(S. 5/7), doing this consistently even when confronted with quite
new situations ; and suppose that, in spite of what was said above,
investigation showed conclusively that there was no second feed-
back of the type described — what would we say ?

There seem to be only two possibilities. We must either invoke
a hitherto unknown channel (in spite of the investigations), as
one was invoked after the demonstration by Hertz (S. 4/13);
or we must be willing to accept as natural that the system S
should go to correct values without being given an appropriate
input. This second possibility would be accepted by no one,
for the situation would be like asking an examiner to accept as
natural a candidate who gives the correct answers without being
given the questions ! If this possibility is rejected, we are left
only with the possibility that the second channel, in some form
or other, must be there.

The implications of double feedback

7/9. We may now usefully consider the relation between adaptive
behaviour and mechanism from the opposite point of view. So
far in this chapter we have taken the facts of adaptive behaviour
as given and have deduced something of the underlying mech-
anism. We will now take the mechanism and ask: Given such a
mechanism, in whatever material" form, will it necessarily show
adaptive behaviour ? The answer to this question will occupy
the remainder of this chapter and the next.

7/10. Let us get the basic assumptions clear for a completely
new start, assuming from here to the end of the chapter only
what is stated explicitly.

DESIGN FOR A BRAIN 7/11

We assume that we have before us some system that has the
diagram of immediate effects shown in Figure 7/5/1. Some
variable, or several, called ' essential ', is given to act on a system
S so that if the variable (or all of them) is within given limits,
S is unchanging; but if it is outside the limits, S changes always.
(An. adequate variety of values is assumed possible to S so that
it does not develop, for instance, simple cyclic repetitions.) A
system called * environment ' interacts with another system R.
Environment has some effect on the variable called essential, and
S has some effect on R. Given this, and nothing more, does it
follow that the system R will change from acting non-adaptively
towards the environment, to acting adaptively towards it? (At
the moment I wish to add no further assumption; in particular
I do not wish to restrict the generality by making any assumption
that R is composed of parts resembling neurons.)

7/11. Because the whole consists of two parts coupled — on the
one side the environment and reacting part R, and on the other
the essential variables and S — we can use the veto-theorem of
S. 6/9. This says that the whole can have as states of equilibrium
only such states as allow a state of equilibrium in both the essen-
tial variables and S. Now S is at equilibrium only when the
essential variables are within the given limits. It follows that all
the possible equilibria of the whole have the essential variables
within the given limits. So if the whole is started at some state
and goes along the corresponding line of behaviour, then if it
goes to an equilibrium, the equilibrium will always be found to
be an adapted one.

Thus we arrive at the solution of the problem posed at the
end of Chapter 1 ; the mechanism has been shown to be necessary
by S. 7/8, and sufficient by the present section.

7/12. This solution, however, is severely abstract and leaves
unanswered a great number of supplementary questions that are
apt to be asked on the topic. Further, it leaves one with no
vivid imaginative or intuitive conception of what is going on when
a system (one as complex as a human being, say) goes about its
business. The remainder of the book will therefore be concerned
with expanding the solution's many implications and specialisa-
tions.

7/13 THE ULTRASTABLE SYSTEM

Here, however, a difficulty arises. The attempt to follow,
conceptually and imaginatively, the actual events in the whole
system, as environment poses problems to the essential variables
(by threatening to drive them outside their normal limits), as
the values in S determine a particular way of behaving in R, as
R behaves in that way, interacting with the environment at every
moment, as the outcome falls on the essential variables, as S is
(or perhaps is not) changed, as R behaves in a new way — all this
is apt to be exceedingly complex and difficult to grasp conceptually
if the variables in environment, R, and S all change continuously,
i.e. by infinitesimal steps.

Experience has shown that the whole system, and its psycho-
logical and physiological implications, are much easier to grasp
and understand if we study the particular case in which the
variables in environment and R all vary continuously, while
those in S vary discretely (i.e. by finite jumps, occurring at finite
intervals). Evidence will be given, in S. 9/4, suggesting that
such discrete variables are in fact likely sometimes to be of real
importance in the subject; for the time, however, let us regard
them as merely selected by us for our easier apprehension.

Step-functions

7/13. Sometimes the behaviour of a variable (or parameter) can
be described without reference to the cause of the behaviour: if
we say a variable or system is a 4 simple harmonic oscillator '
the meaning of the phrase is well understood. In this book we
shall be more interested in the extent to which a variable displays
constancy. Four types may be distinguished, and are illustrated
in Figure 7/13/1. (A) The full-function has no finite interval of
constancy ; many common physical variables are of this type : the
height of the barometer, for instance. (B) The part-function has
finite intervals of change and finite intervals of constancy; it
will be considered more fully in S. 12/18. (C) The step-function
has finite intervals of constancy separated by instantaneous jumps.
And, to complete the set, we need (D) the null- function, which
shows no change over the whole period of observation. The four
types obviously include all the possibilities, except for mixed
forms. The variables of Figure 2/12/1 will be found to be part-,
full-, step-, and step-functions respectively.

DESIGN FOR A BRAIN

7/14

TIME— >

Figure 7/13/1 : Types of behaviour of a variable : A, the full-function ;
B, the part-function ; C, the step-function ; D, the null-function.

In all cases the type-property is assumed to hold only over
the period of observation: what might happen at other times
is irrelevant.

Sometimes physical entities cannot readily be allotted their
type. Thus, a steady musical note may be considered either as
unvarying in intensity, and therefore a null-function, or as
represented by particles of air which move continuously, and
therefore a full -function. In all such cases the confusion is at
once removed if one ceases to think of the real physical object
with its manifold properties, and selects that variable in which
one happens to be interested.

7/14. Step-functions occur abundantly in nature, though the
very simplicity of their properties tends to keep them incon-
spicuous. ' Things in motion sooner catch the eye than what
not stirs '. The following examples approximate to the step-
function, and show its ubiquity:

(1) The electric switch has an electrical resistance which remains

constant except when it changes by a sudden jump.

(2) The electrical resistance of a fuse similarly stays at a low

value for a time and then suddenly changes to a very
high value.

7/15

THE ULTRASTABLE SYSTEM

(3) If a piece of rubber is stretched, the pull it exerts is approxi-

mately proportional to its length. The constant of pro-
portionality has a definite constant value unless the elastic
is stretched so far that it breaks. When this happens
the constant of proportionality suddenly becomes zero,
i.e. it changes as a step-function.

(4) If strong acid is added in a steady stream to an unbuffered

alkaline solution, the pH changes in approximately step-
function form.

(5) If alcohol is added slowly with mixing to an aqueous

solution of protein, the amount of protein precipitated
changes in approximately step-function form.

(6) As the pK is changed, the amount of adsorbed substance

often changes in approximately step-function form.

(7) By quantum principles, many atomic and molecular variables

change in step-function form.

(8) Any variable which acts only in ' all or none ' degree shows

this form of behaviour if each degree is sustained over a
• finite interval.

7/15. Whether a real variable may or may not be represented
by a step-function will usually depend on the method and perhaps
instrumentation of observation. Observers and instruments do
not, in practice, record values over both very short and very long
intervals simultaneously. Thus if the honey-gathering flights of
a bee are being studied throughout a day, the observer does not

Time — >•

Figure 7/15/1 : The same change viewed : (^4) over one interval
of time, (B) over an interval twenty times as long.

DESIGN FOR A BRAIN 7/16

usually follow the bee's movements into the details of the wing
going up and down, neither does he follow the changes that
correspond to the bee's being a little older after the day's work.
The changes of wing-position are ignored as being too fast, only
an average being noticed, and the changes of age are ignored as
being too slow, the values being treated as approximately con-
stant. Thus, whether a variable of a real object behaves as a
step-function cannot in general be decided until the details of the
method of observation is specified.

The distinction is illustrated in Figure 7/15/1 in which
x = tanh t has been graphed. If observed from t = — 2 to
t = -J- 2, the graph has form A, and is obviously not of step-
function form. But if graphed from t = — 100 to t = +100,
the result is B, and the curve is approximating to the step-function
form.

7/16. As a second example, consider the Post Office type relay.
If observed from second to second the conductivity across its
contacts varies almost exactly in step-function form. If, how-
ever, the conductivity is observed over microseconds, the values
change in a much more continuous way, for the contacts can now
be seen to accelerate, decelerate, and bounce with a graceful and
continuous trajectory. And if the relay is observed over many
years and the conductivity plotted, the curve will not be flat
but will fall gradually as oxidation and wear affect the contacts.
We have here yet another example of the thesis that specifying
a real object does not uniquely specify the system or the behaviour
(Ss. 2/4 and 6/2). A question such as ' Is the behaviour of the
Post Office relay really of step-function form ? ' is improperly put,
for it asks about a real object what is determined only by the
system, which must be specified. (The matter is taken up again
in S. 9/10.)

7/17. Behaviour of step-function form is likely to be seen when-
ever we observe a ' machine ' whose component parts are fast-
acting. Thus, if we casually alter the settings of an unknown
electronic machine we are not unlikely to observe, from time to
time, sudden changes of step-function form, the suddenness being
due to the speed with which the machine changes.

A reason can be given most simply by reference to Figure 4/3/1 .

7/18 THE ULTRASTABLE SYSTEM

Suppose that the curvature of the surface is controlled by a para-
meter which makes A rise and B fall. If the ball is resting at A,
the parameter's first change will make no difference to the ball's
lateral position, for it will continue to rest at A (though with
lessened reaction if displaced). As the parameter is changed
further, the ball will continue to remain at A until A and B are
level. Still the ball will make no movement. But if the para-
meter goes on changing and A rises above B, and if gravitation is
intense and the ball fast-moving, then the ball will suddenly move
to B. And here it will remain, however high A becomes and
however low B. So, if the parameter changes steadily, the
lateral position of the ball will tend to change in step-function
form, approximating more closely as the passage of the ball for
a given degree of slope becomes swifter.

The possibility need not be examined further, for no exact
deductions will be drawn from it. The section is intended only
to show that step-functions occur not uncommonly when the
system under observation contains fast-acting components. The
subject will be referred to again in S. 9/8.

7/18. In any state-determined system, the behaviour of a variable
at any instant depends on the values which the variable and the
others have at that instant (S. 2/15). If one of the variables
behaves as a step-function the rule still applies: whether the
variable remains constant or undergoes a change is determined
both by the value of the variable and by the values of the other
variables. So, given a state-determined system with a step-
mechanism* at a particular value, all the states with the step-
mechanism at that value can be divided into two classes : those
whose occurrence does and those whose occurrence does not lead
to a change in the step-mechanism's value. The former are its
critical states: should one of them occur, the step-function will
change value. The critical state of an electric fuse is the number
of amperes which will cause it to blow. The critical state of the
1 constant of proportionality ' of an elastic strand is the length
at which it breaks.

An example from physiology is provided by the urinary bladder

* I am indebted to Dr. J. O. Wisdom for the suggestion that a mechanism
showing a step-function as its main characteristic could conveniently be called
a ' step-mechanism '.

DESIGN FOR A BRAIN

7/19

when it has developed an automatic intermittently-emptying
action after spinal section. The bladder fills steadily with urine,
while at first the spinal centres for micturition remain inactive.
When the volume of urine exceeds a certain value the centres
become active and urine is passed. When the volume falls below

— TIME— *-

Figure 7/18/1 : Diagram of the changes in x, volume of urine in the bladder,
and y, activity in the centre for micturition, when automatic action has
been established after spinal section.

a certain value, the centre becomes inactive and the bladder refills.
A graph of the two variables would resemble Figure 7/18/1. The
two-variable system is state-determined, for it has the field of
Figure 7/18/2. The variable y is approximately a step-function.

o x, x2

Figure 7/18/2 : Field of the changes shown in Figure 7/18/1.

When it is at 0, its critical state is x = X2, y = 0, for the occur-
rence of this state determines a jump from 0 to Y. When it is at
Y, its critical state is x = Xlt y = Y, for the occurrence of this
state determines a jump from Y to 0.

7/19. A common, though despised, property of every machine is
that it may ' break ', This event is in no sense unnatural, since
it must follow the basic laws of physics and chemistry and is
therefore predictable from its immediately preceding state. In
general, when a machine ' breaks ' the representative point has met
some critical state, and the corresponding step-function has
changed value.

As is well known, almost any machine or physical system will

7/20

THE ULTRA STABLE SYSTEM

break if its variables are driven far enough away from their usual
values. Thus, machines with moving parts, if driven ever faster,
will break mechanically; electrical apparatus, if subjected to
ever higher voltages or currents, will break in insulation ; machines
made too hot will melt — if made too cold they may encounter
other sudden changes, such as the condensation which stops a
steam-engine from working below 100° C; in chemical dynamics,
increasing concentrations may meet saturation, or may cause
precipitation of proteins.

Although there is no rigorous law, there is nevertheless a wide-
spread tendency for systems to show changes of step-function
form if their variables are driven far from some usual value.
Later (S. 9/7) it will be suggested that the nervous system is not
exceptional in this respect.

Systems containing step-mechanisms

7/20. When a state-determined system includes a step-function
among its variables, the whole behaviour can undergo a simplifica-
tion not possible when the variables are all full-functions.

Suppose that we have a system with three variables, A, B, S;
that it has been tested and found state-determined; that A and
B are full-functions ; and that S is a step-mechanism. (Variables
A and B, as in S. 21/7, will be referred to as main variables.)
The phase-space of this system will resemble that of Figure 7/20/1
(a possible field has been sketched in). The phase-space no longer
fills all three dimensions, but as S can take only discrete values,
here assumed for simplicity to be a pair, the phase-space is
restricted to two planes normal to S, each plane corresponding to a

Figure 7/20/1 : Field of a state-determined system of three variables, of
which S is a step-function. The states from C to C are the critical states
of the step-function for lines in the lower plane.

DESIGN FOR A BRAIN

7/20

particular value of S. A and B being full-functions, the represen-
tative point will move on curves in each plane, describing a line of
behaviour such as that drawn more heavily in the Figure. When
the line of behaviour meets the row of critical states at C — C, S
jumps to its other value, and the representative point continues
along the heavily marked line in the upper plane. In such a field
the movement of the representative point is everywhere state-
determined, for the number of lines from any point never exceeds
one.

If, still dealing with the same real ' machine ', we ignore S,
and repeatedly form the field of the system composed of A and B
(S being free to take sometimes one value and sometimes the other),
we shall find that we get sometimes a field like I in Figure 7/20/2,

1 A

B B C

Figure 7/20/2 : The two fields of the system composed of A and B.
P is in the same position in each field.

and sometimes a field like II, the one or the other appearing
according to the value that S happens to have at the time.

The behaviour of the system AB, in its apparent possession of
two fields, should be compared with that of the system described
in S. 6/3, where the use of two parameter- values also caused the
appearance of two fields. But in the earlier case the change of
the field was caused by the arbitrary action of the experimenter,
who forced the parameter to change value, while in this case the
change of the field of AB is caused by the inner mechanisms of the
' machine ' itself.

The property may now be stated in general terms. Suppose,
in a state-determined system, that some of the variables are due to
step-mechanisms, and that these are ignored while the remainder
(the main variables) are observed on many occasions by having
their field constructed. Then so long as no step-mechanism

7/23 THE ULTRASTABLE SYSTEM

changes value during the construction, the main variables will be
found to form a state-determined system, and to have a definite
field. But on different occasions different fields may be found.

7/21. These considerations throw light on an old problem in the
theory of mechanisms.

Can a ' machine ' be at once determinate and capable of spon-
taneous change ? The question would be contradictory if posed
by one person, but it exists in fact because, when talking of living
organisms, one school maintains that they are strictly determinate
while another school maintains that they are capable of spon-
taneous change. Can the schools be reconciled ?

The presence of step-mechanisms in a state-determined system
enables both schools to be right, provided that those who maintain
the determination are speaking of the system which comprises all
the variables, while those who maintain the possibility of spon-
taneous change are speaking of the main variables only. For the
whole system, which includes the step-mechanisms, has one field
only, and is completely state-determined (like Figure 7/20/1).
But the system of main variables may show as many different
forms of behaviour (like Figure 7/20/2, I and II) as the step-
mechanisms possess combinations of values. And if the step-
mechanisms are not accessible to observation, the change of the
main variables from one form of behaviour to another will seem
to be spontaneous, for no change or state in the main variables
can be assigned as its cause.

7/22. If the system had contained two step-mechanisms, each
of two values, there would have been four fields of the main
variables. In general, n step-mechanisms, each of two values,
will give 2n fields. A moderate number of step-mechanisms may
thus give a very much larger number of fields.

7/23. After this digression on step-functions we can return to the
system of S. 7/9, with its corrective feedback, and consider its
behaviour.

To bring the concepts into correspondence, we assume that the
main variables (the continuous) are in the environment, in R,
and in the essential variables. The step-functions will be in S.
It follows that their critical states will be distributed over those
regions of the main variables' phase-space at which the essential

DESIGN FOR A BRAIN 7/23

variables are outside their normal limits. Thus if the main
variables were as few as two (for graphical purposes), the dis-
tribution might be as the* dots in I of Figure 7/23/1. The dis-
tribution means that the organism is in tolerable physiological
conditions if the representative point stays within the undotted
region.

Ill

Figure 7/23/1 : Changes of field in an ultrastable system.

states are dotted.

The critical

Suppose now that the first set of values on the step-functions S
gives such a field as is shown in I, and that the representative
point is at X. The line of behaviour from X is not stable in the
region, and the representative point follows the line to the
boundary. Here (Y) it meets a critical state and a step-function
changes value ; a new field, perhaps like II, arises. The representa-
tive point is now at Y, and the line from this point is still unstable
in regard to the region. The point follows the line of behaviour,
meets a critical state at Z, and causes a change of a step-function :
a new field (III) arises. The point is at Z, and the field includes a
state of stable equilibrium, but from Z the line leads further out
of the region. So another critical state is met, another step-
function changes value, and a new field (IV) arises. In this field,

7/25 THE ULTRASTABLE SYSTEM

the line of behaviour from Z is stable with regard to the region,
so the representative point moves to the state of equilibrium and
stops there. No further critical states are met, no further step-
functions change value, and therefore no further changes of field
take place. From now on, if the field of the main variables is
examined, it will be found to be stable. The organism, if dis-
placed moderately from the state of equilibrium, will return to it,
thus demonstrating the various evidences of adaptation noticed
in Chapter 5.

7/24. This field, and this state of equilibrium, will, under con-
stant external conditions, persist indefinitely. If the system is
now subjected to occasional small impulsive disturbances (that
simply displace the representative point as in S. 6/5) the whole
will as often display its stability ; in the same action, the organismal
part will display that it now possesses an ' adapted ' mechanism
for dealing with the environmental part.

During the description of the previous section, much notice was
taken of the trials and failures, and field IV seemed to be only
the end of a succession of failures. We thus tended to lose a
sense of proportion ; for what is really important to the living and
learning organism is the great number of times on which it can
display that it has already achieved adaptation; in fact, unless
the circumstances allow this number to be fairly large and the
number of trial-failures to be fairly small, there is no gain to the
organism in having a brain that can learn.

7/25. It should be noticed that the second feedback makes, for
its success, no demands either on the construction of the reacting
part R or on the successive values that are taken by S. Another
way of saying this is to say that the mechanism is in no way put
out of order if R is initially constructed at random or if the successive
values at S occur at random. (The meaning of ' constructed at
random ' is given in S. 13/1.)

Such a construction at random probably occurs to some extent
in the nervous system, where the ultimate units (dendrons, pieds
terminaux, protein molecules perhaps) occur in numbers far too
great for their determination by the gene-pattern in detail (S. 1/9).
In the formation of the embryo brain, therefore, some of the final
details may be determined by the accidents of local minutiae —

DESIGN FOR A BRAIN 7/26

of oxygen or salt concentration perhaps, or local strains. If the
reacting part R is initially formed by such a process, then the
action of the second feedback is unaffected: it will bring the
organism to adaptation.

In the same way, nothing was supposed about the successive
values at S (except that they must not be appreciably correlated
with the events within the field). Any uncorrelated source will
therefore serve for their supply ; so they too can be, in the denned
sense, random.

The ultrastable system

7/26. In the first edition the system described in this chapter
was called ' ultrastable ', and S. 8/6 will show why the adjective
is defensible. At that time the system was thought to be unique,
but further experience (outlined in /. to C, S. 12/8-20) has shown
that this form is only one of a large class of related forms, in
which it is conspicuous only because it shows certain features, of
outstanding biological interest, with unusual clarity. The word
may usefully be retained, in accordance with the strategy of
S. 2/17, because it represents a well-defined type, useful as a fixed
type around which discussion may move without ambiguity, and
to which a multitude of approximately similar forms, occurring
mostly in the biological world, may be related.

For convenience, its definition will be stated formally. Two
systems of continuous variables (that we called ' environment '
and ' reacting part ') interact, so that a primary feedback (through
complex sensory and motor channels) exists between them.
Another feedback, working intermittently and at a much slower
order of speed, goes from the environment to certain continuous
variables which in their turn affect some step-mechanisms, the
effect being that the step-mechanisms change value when and
only when these variables pass outside given limits. The step-
mechanisms affect the reacting part; by acting as parameters to
it they determine how it shall react to the environment.

(From this basic type a multitude of variations can be made.
Their study is made much easier by a thorough grasp of the
properties of the basic form just defined.)

7/27. The basic form has many more properties of interest than
have yet been indicated. Their description in words, however, is

7/27

THE ULTRASTABLE SYSTEM

apt to be tedious and unconvincing. A better demonstration can
be given by a machine, built so that we know its nature exactly
and on which we can observe just what will happen in various
conditions. (We can describe it either as ' a machine to do our
thinking for us' or, more respectably, as ' an analogue computer \)
One was built and called the ' Homeostat '. Its construction,
and how it behaved, will be described in the next chapter.

«A*5

CHAPTER 8

The Homeostat

8/1. The ultrastable system is much richer in interesting pro-
perties than might at first be suspected. Some of these pro-
perties are of special interest to the physiologist and the psycho-
logist, but they have to be suitably displayed before their physio-
logical and psychological applications can be perceived. For
their display, a machine was built according to the definition of
the ultrastable system. What it is, and how it behaves, are the
subjects of this chapter.*

8/2. The Homeostat (Figure 8/2/1) consists of four units, each of
which carries on top a pivoted magnet (Figure 8/2/2, M in
Figure 8/2/3). The angular deviations of the four magnets from
the central positions provide the four main variables.

Its construction will be described in stages. Each unit emits
a D.C. output proportional to the deviation of its magnet from
the central position. The output is controlled in the following
way. In front of each magnet is a trough of water; electrodes
at each end provide a potential gradient. The magnet carries
a wire which dips into the water, picks up a potential depending
on the position of the magnet, and sends it to the grid of the
triode. J provides the anode-potential at 150 V., while H is at
180 V. ; so E carries a constant current. If the grid-potential allows
just this current to pass through the valve, then no current will flow
through the output. But if the valve passes more, or less, current
than this, the output circuit will carry the difference in one direc-
tion or the other. So after E is adjusted, the output is approxi-
mately proportional to M's deviation from its central position. j*

* It was given the name of ' Homeostat ' for convenience of reference, and
the noun seems to be acceptable. The derivatives ' homeostatic ' and
1 homeostatically ', however, are unfortunate, for they suggest reference to
the machine, whereas priority demands that they be used only as derivatives
of Cannon's ' homeostasis '.

t Following the original machine in principle, Mr. Earl J. Kletsky, at the
Technische Hogeschool, Delft, Holland, has designed and built a form that
replaces the magnet, coils, vane and water by Kirchhoff adding circuits and
capacitors.

100

8/2

THE HOMEOSTAT

Figure 8/2/1 : The Homeostat. Each unit carries on top a magnet and
coil such as that shown in Figure 8/2/2. Of the controls on the front
panel, those of the upper row control the potentiometers, those of the
middle row the commutators, and those of the lower row the switches S
of Figure 8/2/3.

Figure 8/2/2 : Typical magnet (just visible), coil, pivot, vane, and water
potentiometer with electrodes at each end. The coil is quadruple, con-
sisting of A, B, C and D of Figure 8/2/3.

101

DESIGN FOR A BRAIN

8/2

fi^hs.

°<3^

Figure 8/2/3 : Wiring diagram of one unit. (The letters are explained

in the text.)

Next, the units are joined together so that each sends its
output to the other three; and thereby each receives an input
from each of the other three.

These inputs act on the unit's magnet through the coils A,

B, and C, so that the torque on the magnet is approximately
proportional to the algebraic sum of the currents in A, B, and

C. (D also affects M as a self -feedback.) But before each input
current reaches its coil, it passes through a commutator (X),
which determines the polarity of entry to the coil, and through
a potentiometer (P), which determines what fraction of the input
shall reach the coil.

As soon as the system is switched on, the magnets are moved
by the currents from the other units, but these movements change
the currents, which modify the movements, and so on. It may
be shown (S. 19/11) that if there is sufficient viscosity in the
troughs, the four-variable system of the magnet-positions is
approximately state-determined. To this system the com-
mutators and potentiometers act as parameters.

When these parameters are given a definite set of values, the
magnets show some definite pattern of behaviour; for the para-
meters determine the field, and thus the lines of behaviour. If

102

8/3 THE HOMEOSTAT

the field is stable, the four magnets move to the central position,
where they actively resist any attempt to displace them. If
displaced, a co-ordinated activity brings them back to the centre.
Other parameter-settings may, however, give instability; in which
case a ' runaway ' occurs and the magnets diverge from the central
positions with increasing velocity — till they hit the ends of the
troughs.

So far, the system of four variables has been shown to be
dynamic, to have Figure 4/15/1 (A) as its diagram of immediate
effects, and to be state-determined. Its field depends on the
thirty-two parameters X and P. It is not yet ultrastable. But
the inputs, instead of being controlled by parameters set by hand,
can be sent by the switches S through similar components arranged
on a uniselector (or ' stepping-switch ') U. The values of the
components in U were deliberately randomised by taking the
actual numerical values from Fisher and Yates' Table of Random
Numbers. Once built on to the uniselectors, the values of these
parameters are determined at any moment by the positions of
the uniselectors. Twenty-five positions on each of four uni-
selectors (one to each unit) provide 390,625 combinations of
parameter- values .

F represents the essential variable of the unit. Its contacts
close when and only when the output current exceeds a certain
value. When this happens, the coils G of the uniselector can be
energised, moving the parameters to new values. The power to
G is also interrupted by a device (not shown) that allows the power
to test F's contacts only at intervals of one to ten seconds (the
operator can adjust the frequency). Thus, if set at 3-second
intervals, at every third second the uniselector will either move to
new values (if F be receiving a current exceeding the limits) or
stay where it is (if F's current be within).

8/3. That the machine described is ultrastable can be verified
by an examination of the correspondences.

There are four main variables — the positions of the four magnets.
(There can, of course, be fewer if not all the units are used.)
These four represent both the environment and the reacting part
R of Figure 7/2/1, the allotment of the four to the two subsystems
being arbitrary. The relays F correspond to the essential
variables, and the physiological limits correspond to the currents

103

DESIGN FOR A BRAIN 8/4

that flow in F when the needles are deviated to more than about
45 degrees from the central positions. The main variables are
continuous, and act and react on one another, giving the primary
feedback, which is complex, like A of Figure 4/15/1. The field
of the four main variables has only one state of equilibrium (at
the centre), which may be stable or unstable. Thus the system
is either stable and self-correcting for small impulsive displace-
ments to the needles, or unstable and self-aggravating, running
away to the limits of the troughs. Which it will be depends on
the quantitative details of the primary feedbacks, which are
dependent on the values on the step-mechanisms.

The step-mechanisms of S. 7/12 can be made to correspond to
structures on the Homeostat in several ways, which are equivalent.
Perhaps the simplest way is to identify them with the twelve
values presented by the uniselectors at any given moment (three
on each). If the needle of a unit diverges for more than a few
seconds outside the limits of ±45°, the three values of its step-
functions will be changed to three new values. These new values
have no special relation either to the previous values or to the
problem in hand — they are just the values that next follow in
Fisher and Yates' table.

It is easily seen that if any one, two, or three of the units are
used (as is often done for simplicity) this subsystem will still be
ultrastable.

The Homeostat as adapter

8/4. A remarkable property of the nervous system is its ability
to adapt itself to surgical alterations of the bodily structure.
From the first work of Marina to the recent work of Sperry, such
experiments have aroused interest and no little surprise.

Over forty years ago, Marina severed the attachments of the
internal and external recti muscles of a monkey's eyeball and
re-attached them in crossed position so that a contraction of
the external rectus would cause the eyeball to turn not outwards
but inwards. When the wound had healed, he was surprised to
discover that the two eyeballs still moved together, so that
binocular vision was preserved.

More recently Sperry severed the nerves supplying the flexor
and extensor muscles in the arm of the spider monkey, and re-
joined them in crossed position. After the nerves had regenerated,

104

8/4 THE HOMEOSTAT

the animal's arm movements were at first grossly inco-ordinated,
but improved until an essentially normal mode of progression
was re-established. The two examples are typical of a great
number of experiments, and will suffice for the discussion. Let
us see what the Homeostat will do under a similar operation.
Figure 8/4/1 shows the Homeostat simplified to two units

°1 °2

Time

Figure 8/4/1 : Two units (1 and 2) interacting. Line 1 represents the side-
to-side movements of Unit l's needle by vertical changes. Similarly
line 2 shows the behaviour of Unit 2's needle. The lowest line (U) shows
a mark whenever Unit l's uniselector advanced a step. The dotted lines
correspond to critical states. The displacements D were caused by the
operator so as to force the system to show its response.

interacting. The diagram of immediate effects was 1 ^± 2 ; the
effect 1 — >- 2 was hand-controlled, and 2 — > 1 was uniselector-
controlled. At first the step-mechanism values combined to give
stability, shown by the responses to Dv (The reader should bear
in mind, of course, that this trifling return after displacement is
representative of all the complex returns after displacement con-
sidered in Chapter 5: Adaptation as stability.) At Rv reversal
of the commutator by hand rendered the system unstable, a
runaway occurred, and the variables transgressed the critical
states. The uniselector in Unit 1 changed position and, as it
happened, gave at its first trial a' stable field. It will be noticed
that whereas before Rx the upstroke of D± in 2 caused an up-
stroke in 1, it caused a downstroke in 1 after Rv showing that the
action 2 — > 1 had been reversed by the uniselector. This reversal
compensated for the reversal of 1 — > 2 caused at Rv

At R2 the whole process was repeated. This time three uni-
selector changes were required before stability was restored. A

105

DESIGN FOR A BRAIN 8/5

comparison of the effect of D3 on 1 with that of D2 shows that
compensation has occurred again.

If the two phenomena are to be brought into correspondence,
we must notice, as in S. 3/12, that the anatomical criterion for
dividing the system into ' animal ' and i environment ' is not
the only possible: a functional criterion is also possible. Suppose
a monkey, to get food from a box, has to pull a lever towards
itself; if we sever the flexor and extensor muscles of the arm
and re-attach them in crossed position then, so far as the cerebral
cortex is concerned, the change is not essentially different from
that of dismantling the box and re-assembling it so that the
lever has to be pushed instead of pulled. Spinal cord, peripheral
nerves, muscles, bones, lever, and box — all are ' environment '
to the cerebral cortex. A reversal in the cerebral cortex will
compensate for a reversal in its environment whether in spinal
cord, muscles, or lever. It seems reasonable, therefore, to expect
that the cerebral cortex will use the same compensatory process
whatever the site of reversal.

To apply the principle of ultrastability we must add an assump-
tion that i binocular vision ' and ' normal progression ' have
neural correlates such that deviations from binocular vision or
from normal progression cause an excitation sufficient to cause
changes of step-function form in those cerebral mechanisms that
determine the actions. (The plausibility of this assumption will
be discussed in S. 9/4.) Ultrastability will then automatically
lead to the emergence of behaviour which produces binocular
vision or normal progression.

8/5. A more complex example is shown in Figure 8/5/1. The
machine was arranged so that its diagram of immediate effects
was

The effect 3 — > 1 was set permanently so that a movement of
3 made 1 move in the opposite direction. The action 1 — > 2
was uniselector-controlled, and 2 — >► 3 hand-controlled. When
the tracing commenced, the actions 1 — >• 2 and 2 — > 3 were
demonstrated by the downward movement, forced by the operator,
of 1 at S^. 2 followed 1 downward (similar movement), and 3

106

8/5 THE HOMEOSTAT

v \j — y

Time

Figure 8/5/1 : Three units interacting. At R the effect
of 2 on 3 was reversed in polarity.

followed 2 downward (similar movement). 3 then forced 1 up-
ward, opposed the original movement, and produced stability.

At R, the hand-control (2 — >■ 3) was reversed, so that 2 now
forced 3 to move in the opposite direction to itself. This change
set up a vicious circle and destroyed the stability; but uniselector
changes occurred until the stability was restored. A forced down-
ward movement of 1, at S2, demonstrated the regained stability.

The tracing, however, deserves closer study. The action 2 — > 3
was reversed at R, and the responses of 2 and 3 at S2 demonstrate
this reversal; for while at S± they moved similarly, at S2 they
moved oppositely. Again, a comparison of the uniselector-
controlled action 1 — > 2 before and after R shows that whereas
beforehand 2 moved similarly to 1, afterwards it moved oppo-
sitely. The reversal in 2 — > 3, caused by the operator, thus
evoked a reversal in 1 — > 2 controlled by the uniselector. The
second reversal is compensatory to the first.

The nervous system provides many illustrations of such a series
of events: first the established reaction, then an alteration made
in the environment by the experimenter, and finally a. reorganisa-
tion within the nervous system, compensating for the experimental
alteration. The Homeostat can thus show, in elementary form,
this power of self -reorganisation.

107

DESIGN FOR A BRAIN 8/6

8/6. We can now appreciate how different an ultrastable system
is from a simple stable system when the conditions allow the
difference to show clearly.

The difference can best be shown by an example. The auto-
matic pilot is a device which, amongst other actions, keeps the
aeroplane horizontal. It must therefore be connected to the
ailerons in such a way that when the plane rolls to the right, its
output must act on them so as to roll the plane to the left. If
properly joined, the whole system is stable and self-correcting: it
can now fly safely through turbulent air, for though it will roll
frequently, it will always come back to the level. The Homeostat,
if joined in this way, would tend to do the same. (Though not
well suited, it would, in principle, if given a gyroscope, be able to
correct roll.)

So far they show no difference; but connect the ailerons in
reverse and compare them. The automatic pilot would act, after
a small disturbance, to increase the roll, and would persist in its
wrong action to the very end. The Homeostat, however, would
persist in its wrong action only until the increasing deviation
made the step-mechanisms start changing. On the occurrence
of the first suitable new value, the Homeostat would act to stabilise
instead of to overthrow ; it would return the plane to the horizontal ;
and it would then be ordinarily self-correcting for disturbances.

There is therefore some justification for the name ' ultrastable ';
for if the main variables are assembled so as to make their field
unstable, the ultrastable system will change this field till it is
stable. The degree of stability shown is therefore of an order
higher than that of the system with a single field.

8/7. The experiments of Marina and Sperry provide an excellent
introduction because they are conceptually so simple. Some-
times a simple experiment on adaptation may need a little thought
before we can identify the essential features. Thus Mowrer put
a rat into a box with a grilled metal floor. The grill could be
electrified so as to give shocks to the rat's paws. Inside the box
was a pedal which, if depressed, at once stopped the shocks.

When a rat was put into the box and the electric stimulation
started, the rat would produce various undirected activities such
as jumping, running, squealing, biting at the grill, and random
thrashing about. Sooner or later it would depress the pedal and

108

8/7

THE HOMEOSTAT

stop the shocks. After the tenth trial, the application of the
shock would usually cause the rat to go straight to the pedal and
depress it. These, briefly, are the observed facts.

Consider the internal linkages in this system. We can suffi-
ciently specify what is happening by using six variables, or sets
of variables: those shown in the box-diagram below. By con-

Events in

sensory cortex

motor cortex

>
5

Excitation
in skin

Position of
limbs

>
4

Voltage

Position

grill

of pedal

sidering the known actions of part on part in the real system we
can construct the diagram of immediate effects. Thus, the excita-
tions in the motor cortex certainly control the rat's bodily move-
ments, and such excitations have no direct effect on any of the
other five groups of variables; so we can insert arrow 1, and
know that no other arrow leaves that box. (The single arrow, of
course, represents a complex channel.) Similarly, the other arrows
of the diagram can be inserted. Some of the arrows, e.g. 2 and 4,
represent a linkage in which there is not a positive physical action
all the time; but here, in accordance with S. 2/3, we regard them
as permanently linked though sometimes acting at zero degree.

Having completed the diagram, we notice that it forms a
functional circuit. The system is complete and isolated, and
may therefore be treated as state-determined. To apply our
thesis, we assume that the cerebral part, represented by the boxes
around arrow 6, contains step-mechanisms whose critical states
will be transgressed if stimuli of more than physiological intensity
are sent to the brain.

We now regard the system as ultrastable, and predict what
its behaviour must be. It is started, by hypothesis, from an
initial state at which the voltage is high. This being so, the
excitation at the skin and in the brain will be high. At first

109

DESIGN FOR A BRAIN 8/8

the pattern of impulses sent to the muscles does not cause that
pedal movement which would lower the voltage on the grill.
These high excitations in the brain will cause some step-mech-
anisms to change value, thus causing different patterns of body
movement to occur. The step-mechanisms act directly only at
stage 6, but changes there will (S. 12/9) affect the field of all
six groups of main variables. These changes of field will continue
to occur as long as the high excitation in the brain persists.
They will cease when, and only when, the linkages at stage 6
transform an excitation of skin receptors into such a bodily
movement as will cause, through the pedal, a reduction in the
excitation of the skin receptors; for only such linkages can stop
further encounters with critical states. The system, that is,
will change until there occurs a stable field. The stability will
be shown by an increase in the voltage on the grill leading to
changes through skin, brain, muscles, and pedal that have the
effect of opposing the increase in voltage. The stability, in
addition, has the property that it keeps the essential variables
within physiological limits; for by it the rat is protected from
electrical injury, and the nervous system from exhaustion.

It will be noted that although action 3 has no direct con-
nexion, either visually in the real apparatus or functionally in the
diagram of immediate effects, with the site of the changes at 6,
yet the latter become adapted to the nature of the action at 3.
(The subject was discussed in S. 5/13.)

This example shows, therefore, that if the rat and its environ-
ment formed an ultrastable system and acted purely automati-
cally, they would go through the same changes as were observed
by Mowrer.

Training

8/8. The process of ' training ' will now be shown in its relation
to ultrastability.

All training involves some use of ' punishment ' or ' reward ',
and we must translate these concepts into our form. i Punish-
ment ' is simple, for it means that some sensory organs or nerve
endings have been stimulated with an intensity high enough to
cause step-function changes in the nervous system (S. 7/19 and
9/7). The concept of ' reward ' is more complex. It usually
involves the supplying of some substance (e.g. food) or condition

110

8/8

THE HOMEOSTAT

(e.g. escape) whose absence would act as ' punishment '. The
chief difficulty is that the evidence suggests that the nervous
system, especially the mammalian, contains intricate and special-
ised mechanisms which give the animals properties not to be
deduced from basic principles alone. Thus it has been shown
that dogs with an oesophageal fistula, deprived of water for some
hours, would, when offered water, drink approximately the
quantity that would correct the deprivation, and would then
stop drinking; they would stop although no water had entered
stomach or system. The properties of these mechanisms have not
yet been fully elucidated; so training by reward uses mechanisms
of unknown properties. Here we shall ignore these complica-
tions. We shall assume that the training is by pain, i.e. by some
change which threatens to drive the essential variables outside
their normal limits; and we shall assume that training by reward
is not essentially dissimilar.

It should be noticed that in training-experiments the experi-
menter often plays a dual role. He first plans the experiment,
deciding what rules shall be obeyed during it. Then, when
these have been fixed, he takes part in the experiment and obeys
these rules. With the first role we are not concerned. In the
second, however, it is important to note that the experimenter
is now within the functional machinery of the experiment. The
truth of this statement can be appreciated more readily if his
place is taken by an untrained but obedient assistant who carries
out the instructions blindly; or better still if his place is taken by
an apparatus which carries out the prescribed actions automatically.

When the whole training is arranged to occur automatically
the feedback is readily demonstrated if we construct the diagram
of immediate effects. Thus, a pike in an aquarium was separated
from some minnows by a sheet of glass; every time he dashed
at the minnows he struck the glass. The following immediate
effects can be clearly distinguished:

Activities in

motor cortex

muscles

Activities in
sensory cortex

Pressure on
nose

3
111

DESIGN FOR A BRAIN

8/8

The effect 1 represents the control exerted through spinal cord
and motor nerves. Effect 2 is discontinuous but none the less
clear: the experiment implies that some activities Jed to a high
pressure on the nose while others led to a zero pressure. Effects
3 and 4 are the simple neuro-physiological results of pressures
on the nose.

Although the diagram has some freedom in the selection of
variables for naming, the system, regarded as a whole, clearly
has feedback.

In other training experiments, the regularity of action 2
(supplied above by the constant physical properties of glass) may
be supplied by an assistant who constantly obeys the rules laid
down by the experimenter. Grindley, for instance, kept a
guinea-pig in a silent room in which a buzzer was sounded from
time to time. If and only if its head turned to the right did a
tray swing out and present it with a piece of carrot; after a few
nibbles the carrot was withdrawn and the process repeated.
Feedback is demonstrably present in this system, for the diagram
of immediate effects is:

Activities in

Position of

motor cortex

head

>
4

Activities in

Amount of

sensory

cortex

carrot pi

^esented

The buzzer, omitted for clarity, comes in as parameter and serve
merely to call this dynamic system into functional existence;
for only when the buzzer sounds does the linkage 2 exist.

This type of experiment reveals its essential dynamic structure
more clearly if contrasted with elementary Pavlovian condition-
ing. In the experiments of Grindley and Pavlov, both use the
sequences '. . . buzzer, animal's response, food . . .' In Grind-
ley's experiment, the value of the variable ' food ' depended on the
animaVs response: if the head turned to the left, ' food ' was ' no
carrot ', while if the head turned to the right, ' food ' was ' carrot
given '. But in Pavlov's experiments the nature of every stimulus
throughout the session was already determined before the session
commenced. The Pavlovian experiment, therefore, allows no

112

8/9 THE HOMEOSTAT

effect from the variable ' animal's behaviour ' to ' quantity of
food given ' ; there is no functional circuit and no feedback.

It may be thought that the distinction (which corresponds to
that made by Hilgard and Marquis between ' conditioning ' and
' instrumental learning ') is purely verbal. This is not so, for
the description given above shows that the distinction may be
made objectively by examining the structure of the experiment.

It will be seen, therefore, that the ' training ' situation neces-
sarily implies that the trainer, or some similar device, is an
integral part of the whole system, which has feedback:

Trainer

Animal

We shall now suppose this system to be ultrastable, and we
shall trace its behaviour on this supposition. The step-mechanisms
are, of course, assumed to be confined to the animal; both because
the human trainer may be replaced in some experiments by a
device as simple as a sheet of glass (in the example of the pike);
and because the rules of the training are to be decided in advance
(as when we decide to punish a house-dog whenever he jumps
into a chair), and therefore to be invariant throughout the process.
Suppose then that jumping into a chair always results in the
dog's sensory receptors being excessively stimulated. As an
ultrastable system, step-function values which lead to jumps into
chairs will be followed by stimulations likely to cause them to
change value. But on the occurrence of a set of step-function
values leading to a remaining on the ground, excessive stimula-
tion will not occur, and the values will remain. (The cessation
of punishment when the right action occurs is no less important
in training than its administration after the wrong action.)

8/9. The process can be shown on the Homeostat. Figure 8/9/1
provides an example. Three units were joined:

and to this system was joined a ' trainer ', actually myself, which
acted on the rule that if the Homeostat did not respond to a

113

DESIGN FOR A BRAIN

8/9

forced movement of 1 by an opposite movement of 2, then the
trainer would force 3 over to an extreme position. The diagram
of immediate effects is therefore really

Part of the system's feedbacks, it will be noticed, pass through T.

Di Dp

1 w-

Time »-

Figure 8/9/1 : Three units interacting. The downstrokes at S are
forced by the operator. If 2 responds with a downstroke, the
trainer drives 3 past its critical surface.

At Sv 1 was moved and 2 moved similarly. This is the ' for-
bidden ' response; so at Dv 3 was forced by the trainer to an
extreme position. Step-mechanisms changed value. At S2, the
Homeostat was tested again: again it produced the forbidden
response; so at D2, 3 was again forced to an extreme position.
At S3, the Homeostat was tested again: it moved in the desired
way, so no further deviation was forced on 3. And at S± and
S5 the Homeostat continued to show the desired reaction.

From Sx onwards, T's behaviour is determinate at every instant;
so the system composed of 1, 2, 3, T, and the uniselectors, is
state-determined.

Another property of the whole system should be noticed.
When the movement-combination ' 1 and 2 moving similarly '
occurs, T is thereby impelled, under the rules of the experiment,
to force 3 outside the region bounded by the critical states. Of

114

8/10 THE HOMEOSTAT

any inanimate system which behaved in this way we would say,
simply, that the line of behaviour from the state at which 1 and 2
started moving was unstable. So, to say in psychological terms
that the ' trainer ' has ' punished ' the ' animal ' is equivalent to
saying in our terms that the system has a set of parameter-values
that make it unstable.

In general, then, we may identify the behaviour of the animal
in ' training ' with that of the ultrastable system adapting to
another system of fixed characteristics.

8/10. How will the ultrastable system behave if it has to adapt
to two environments, which alternate ? Such a situation is not
uncommon: the diving bird has to adapt to situations both on
land and in the water; British birds have to adapt both to full
foliage in the summer and to bare branches in the winter; and
the kitten has to adapt both to the mouse that tries to escape into
a hole and to the bird that tries to escape by flying upwards.

Such cases are equivalent (by Ss. 6/3 and 7/20) to the case in
which there is one environment affected by a parameter with two
values. Each value, provided it is sustained long enough for the
characteristic behaviours of adaptation to be displayed, gives one
form to the environment; and the two forms may, if we please,
be thought of as two environments. The question can therefore
be investigated by allowing the Homeostat to adapt in the pres-
ence of an alternating parameter, each value of which must be
sustained long enough so that the change does not interrupt the
process of trial and error.

Let the Homeostat be arranged so that it is partly under uni-
selector-, and partly under hand-, control. Let it be started so
that it works as an ultrastable system. Select a commutator
switch, and from time to time reverse its polarity. This reversal
provides the system with the equivalent of two environments
which alternate. We can now predict that it will be selective for
fields that give adaptation to both environments. For consider
what field can be terminal : a field that is terminal for only one of
the parameter-values will be lost when the parameter next changes ;
but the first field terminal for both will be retained. Figure
8/10/1 illustrates the process. At Rv R2, R3, and R± the hand-
controlled commutator H was reversed. At first the change of
value caused a change of field, shown at A. But the second

115

DESIGN FOR A BRAIN 8/11

-1R3

4* /\-^ 7n 5u-

Time

Figure 8/10/1 : Record of Homeostat's behaviour when a commutator H
was reversed from time to time (at the R's). The first set of uniselector
values which gave stability for both commutator positions was terminal.

uniselector position happened to provide a field which gave
stability with both values of H. So afterwards, the changes of
H no longer caused changes in the step-mechanisms. The re-
sponses to the displacements Z), forced by the operator, show that
the system is stable for both values of H. The slight but distinct
difference in the behaviour after D at the two values of H show
that the two fields are different.

The ultrastable system is, therefore, selective for step-mechanism
values which give stability for both values of an alternating
parameter.

8/11. What will happen if the ultrastable system is given an
unusual environment ? Before the question is answered we must
be clear about what is meant by ' unusual '.

In S. 6/2 it was shown that every dynamic system is acted
on by an indefinitely large number of parameters, many of which
are taken for granted, for they are always given well-understood
' obvious ' values. Thus, in mechanical systems it is taken for
granted, unless specially mentioned, that the bodies carry a
zero electrostatic charge; in physiological experiments, that the
tissues, unless specially mentioned, contain no unusual drug; in
biological experiments, that the animal, unless specially mentioned,
is in good health. All these parameters, however, are effective
in that, had their values been different, the variables would not
have followed the same line of behaviour. Clearly the field of
a state-determined system depends not only on those para-
meters which have been fixed individually and specifically, but
on all the great number which have been fixed incidentally.

116

8/11

THE HOMEOSTAT

Now the ultrastable system proceeds to a terminal field which
is stable in conjunction with all the system's parameter- values
(and it is clear that this must be so, for whether the parameters
are at their ' usual ' values or not is irrelevant). The ultrastable
system will therefore always produce a set of step-mechanism
values which is so related to the particular set of parameter- values
that, in conjunction with them, the system is stable. If the para-
meters have unusual values, the step-mechanisms will also finish
with values that are compensatingly unusual. To the casual
observer this adjustment of the step-mechanism values to the
parameter- values may be surprising; we, however, can see that
it is inevitable.

The fact is demonstrable on the Homeostat. After the machine
was completed, some ' unusual ' complications were imposed on
it (' unusual ' in the sense that they were not thought of till the
machine had been built), and the machine was then tested to see
how it would succeed in finding a stable field when affected by
the peculiar complications. One such test was made by joining

Time

Figure 8/11/1 : Three units interacting. At J, units 1 and 2 were con-
strained to move together. New step-mechanism values were found
which produced stability. These values give stability in conjunction
with the constraint, for when it is removed, at R, the system becomes
unstable.

117

DESIGN FOR A BRAIN 8/12

the front two magnets by a light glass fibre so that they had to
move together. Figure 8/11/1 shows a typical record of the
changes. Three units were joined together and were at first
stable, as shown by the response when the operator displaced
magnet 1 at Dv At J, the magnets of 1 and 2 were joined so
that they could move only together. The result of the constraint
in this case was to make the system unstable. But the instability
evoked step-mechanism changes, and a new terminal field was
found. This was, of course, stable, as was shown by its response
to the displacement, made by the operator, at D2. But it should
be noticed that the new set of step-mechanism values was adjusted
to, or ' took notice of ', the constraint and, in fact, used it in the
maintenance of stability; for when, at R, the operator gently
lifted the fibre away the system became unstable.

There are other unusual problems, of course, for which the
Homeostat's repertoire contains no solution; putting too powerful
a magnet at one side to draw the magnets over would set such a
1 problem ' ; so would a shorting of the relay F (Figure 8/2/3).
In such a situation the Homeostat, or any ultrastable system,
would have no state of equilibrium and would thus fail to adapt.
So, too, would a living organism, if set a problem for which its
total repertoire contained no solution.

Some apparent faults

8/12. It will be apparent that the principle of ultrastability, as
demonstrated by the Homeostat, does not seem to represent
adequately the great richness of adaptations developed by the
higher animals ; with some of the inadequacies we shall deal later
in the book. There are, however, some ' faults ' of the ultrastable
system that are found on closer scrutiny actually to support the
thesis that the living brain adapts by ultrastability. We will
examine them in the next few sections.

8/13. If the relation of S. 7/5 does not hold between the essential
variables and the step-mechanisms, that is, if an ultrastable
system's critical surfaces are not disposed in proper relation to
the limits of the essential variables, the system may seek an
inappropriate goal or may fail to take corrective action when the
essential variables are dangerously near their limits.

118

8/14 THE HOMEOSTAT

In animals, though we cannot yet say much about their critical
states, we can observe failures of adaptation that may well be due
to a defect of this type. Thus, though animals usually react
defensively to poisons like strychnine — for it has an intensely
bitter taste, stimulates the taste buds strongly, and is spat out
— they are characteristically defenceless against a tasteless or
odourless poison: precisely because it stimulates no nerve-fibre
excessively and causes no deviation from the routine of chewing
and swallowing.

An even more dramatic example, showing how defenceless is
the living organism if pain has not its normal effect of causing
behaviour to change, is given by those children who congenitally
lack the normal self-protective reflexes. Boyd and Nie have
described such a case: a girl, aged 7, who seemed healthy and
normal in all respects except that she was quite insensitive to
pain. Even before she was a year old her parents noticed that
she did not cry when injured. At one year of age her arm was
noticed to be crooked: X-rays showed a recent fracture-disloca-
tion. The child had made no complaint, nor did she show any
sign of pain when the fragments were re-set without an anaesthetic.
Three months later the same injury occurred to her right elbow.
At the seaside she crawled on the rocks until her hands and knees
were torn and denuded of skin. At home her mother on several
occasions smelt burning flesh and found the child leaning uncon-
cernedly against the hot stove.

It seems, then, that if an imperfectly formed ultrastable system
is, under certain conditions, defenceless, so may be an imperfectly
formed living organism.

8/14. Even if the ultrastable system is suitably arranged — if the
critical states are encountered before the essential variables reach
their limits — it usually cannot adapt to an environment ' that
behaves with sudden discontinuities. In the earlier examples of
the Homeostat's successful adaptations the actions were always
arranged to be continuous; but suppose the Homeostat had con-
trolled a relay which was usually unchanging but which, if the
Homeostat passed through some arbitrarily selected state, would
suddenly release a powerful spring that would drag the magnets
away from their 'optimal ' central positions: the Homeostat, if
it happened to approach the special state, would take no step

119

DESIGN FOR A BRAIN 8/15

to avoid it and would blindly evoke the ' lethal ' action. The
Homeostat's method for achieving adaptation is thus essentially
useless when its environment contains such 4 lethal ' discontinuities.

The living organism, however, is also apt to fail with just the
same type of environment. The pike that collided with the
glass plate while chasing minnows failed at first to avoid collision
precisely because of the suddenness of the transition from not
seeing clear glass to feeling the impact on its nose. This flaw
in the living organism's defences has, in fact, long been known
and made use of by the hunter. The stalking cat's movements
are such as will maintain as long as possible, for the prey, the
appearance of a peaceful landscape, to be changed with the
utmost possible suddenness into one of mortal threat. In the
whole process the suddenness is essential. Consider too the
essential features of any successful trap; and the necessity, in
poisoning vermin, of ensuring that the first dose is lethal.

If, then, the ultrastable system usually fails when attempting to
adapt to an environment with sudden discontinuities, so too does
the living organism.

8/15. Another weakness shown by the ultrastable system's
method is that success is dependent on the system's using a suitable
period of delay between each trial. Thus, the system shown in
Figure 7/23/1 must persist in Trial IV long enough for the repre-
sentative point to get away from the region of the critical states.
Both extremes of delay may be fatal: too hurried a change from
trial to trial may not allow time for ' success ' to declare itself;
and too prolonged a testing of a wrong trial may allow serious
damage to occur. The optimal duration of a trial is clearly the
time taken by information to travel from the step-mechanisms
that initiate the trial, through the environment, to the essential
variable that shows the outcome. If the ultrastable system re-
quires the duration to be adjusted, so does the living organism; for
there can be little doubt that on many occasions living organisms
have missed success either by abandoning a trial too quickly, or
by persisting too long with a trial that was actually useless. (The
topic is referred to again in S. 17/10.)

The same difficulty, then, confronts both ultrastable system and
living organism.

120

8/17 THE IIOMEOSTAT

8/16. If we grade an ultrastable system's environments accord-
ing to the difficulty they present, we shall find that at the ' easy '
end are those that consist of a few variables, independent of each
other, and that at the ' difficult ' end are those that contain many
variables richly cross-linked to form a complex whole. (The
topic is developed from Chapter 11 on.)

The living organism, too, would classify environments in
essentially the same way. Not only does common experience
show this, but the construction and use of ' intelligence tests '
has shown in endless ways that the easy problem is the one
whose components are few and independent, while the difficult
problem is the one with many components that form a complex
whole. So when confronted with environments of various ' diffi-
culties ', the ultrastable system and the living organism arc likely
to fail together.

8/17. The last few sections have shown, in several ways, how
several ' inadequacies ' of the ultrastable system have made us
realise, on closer scrutiny, the inadequacies not of the ultrastable
system but of the living brain. Clearly we must beware of con-
demning a proposed model for not showing a certain property
until we are sure that the living organism really shows it.

Since this book was first published, I have often had put to me
some objection of the form ' Surely an ultrastable system could
not . . . ' When one goes into the matter, it is surprising how
often the reply proves to be ' No, and a human being couldn't do
it either ! '

121

CHAPTER 9

Ultrastability in the Organism

9/1. In the early sections of Chapter 7 we considered the ele-
mentary behavioural facts of the kitten adapting, and related
them to a mechanistic theoretical construction, the ' ultrastable '
system. In the present chapter we will consider some further
elementary relations between the real organism and the theoretical
construct. We will consider, in particular, what can be said
about the simple theoretical system shown in Figure 7/5/1. How
does it correspond to the real organism and the real environment ?
We must go with caution, for experience has shown that a jump
to conclusions that are grossly in error is only too easy.

9/2. We must be particularly careful not to take for granted
that a diagram of immediate effects — that of Figure 7/5/1 for
instance — gives a picture of what is to be seen in the nervous
system. Just as one real ' machine ' can give rise to a variety
of systems, so it can give rise to a variety of diagrams of immediate
effects, if the experimenter examines the real ' machine ' with a
variety of different technical methods. An electrical network, for
instance, may give very different diagrams of functional con-
nexion if it is explored first with slowly varying potentials and
then with potentials oscillating at a high frequency. Sometimes
it happens that two techniques may give the same diagram —
exploring a metallic network firstly with direct currents and then
by the sense of touch, say. When this happens we are delighted,
for we have found a simplicity; but we must not expect this to
happen always.

Many simple bodies have one diagram that is so obtrusive that
one is apt to think of it as the specification of how the parts are
joined. This is the diagram built up by considering the parts'
positions in three-dimensional space, and studying how each part
moves when some other part is moved. In this way (using a
method just the same as that of S. 4/12) scientist and man in

122

9/4 ULTRASTABILITY IN THE ORGANISM

the street alike build up their ideas of how things are connected in
the simple material or anatomical sense. So the child learns that
when he picks up one end of a rattle the other end comes up too ;
and so the demonstrator of anatomy shows that if a certain tendon
in the forearm is pulled, the thumb moves. These operations
specify a diagram of immediate effects, a pattern of connectivity,
of great commonness and importance. But we must beware of
thinking that it is the only pattern; for there are also systems
whose parts or variables have no particular position in space
relative to one another, but are related dynamically in some
quite different way. Such occurs when a mixture of substrates,
enzymes, and other substances occur in a flask, and in which the
variables are concentrations. Then the ' system ' is a set of con-
centrations, and the diagram of immediate effects shows how the
concentrations affect one another. Such a diagram, of course,
shows nothing that can be seen in the distribution of matter in
space; it is purely functional. Nothing that has been said so
far excludes the possibility that the anatomical-looking Figure
7 1 5 /l may not be of the latter type. We must proceed warily.

9/3. The chief part of Figure 7/5/1 that calls for comment is the
feedback from environment, through essential variables and step-
mechanisms, to the reacting part R. The channel from environ-
ment to essential variables hardly concerns us, for it will depend
on the practical details for the particular circumstances in which,
at any particular time, the free-living organism finds itself.

The essential variables, being determined by the gene-pattern
(S. 3/14), will often have some simple anatomical localisation.
Some of them, for instance, are well known to be sited in the
medulla oblongata; and others, such as the signals for pain, are
known to pass through certain sites in the midbrain and optic
thalamus. More accurate identification of these variables demands
only detailed study.

Step -mechanisms in the organism

9/4. Quite otherwise is it with the step-mechanisms. We have
at present practically no idea of where to look, nor what to look
for. In these matters we must be very careful to avoid making
assumptions unwittingly, for the possibilities are very wide.

123

DESIGN FOR A BRAIN

9/5

We must beware, for instance, of asking where they are; for
this question assumes that they must be somewhere, and then the
1 where ' is apt to be interpreted anatomically, or histologically,
or in some other way that is not appropriate to the actual variable.
Some calculating machines, for instance, carry their records in
the form of a train of pulses that circulates around a cyclic path
that includes a long column of mercury. Each pulse behaves as a
step-function in that it has only two values: present or absent.
It is localised by its position in the sequence, but it has no localisa-
tion in any particular part of the column. (This is the sort of
4 localisation ' that the Fraunhofer lines have as sunlight comes to
us: they occupy a definite place in the spectrum but no unique
place in three-dimensional space.)

With these warnings in mind, a brief review will now be given
of some of the possibilities. (The list is almost certainly not
complete, and at the present time it is probably most important
that we should be alert for forms not yet considered.)

9/5. A possibility early suggested by Young was that a closed
circuit of neurons might carry a stream of impulses and be self-
maintaining in this excited state. Unexcited it would stay at
rest, excited it would be active maximally; such a system could
carry permanently the effects of some event.

Lorente de No has provided abundant histological evidence that
neurons form not only chains but circuits. Figure 9/5/1 is taken

Figure 9/5/1 :
reflex arc.

Neurons and their connexions in the trigeminal
(Semi-diagrammatic ; from Lorente de No.)

from one of his papers. Such circuits are so common that he has
enunciated a ' Law of Reciprocity of Connexions ' : * if a cell-
complex A sends fibres to cell or cell-complex B, then B also sends
fibres to A, either direct or by means of one internuncial neuron '.

124

9/6 ULTRASTABILITY IN THE ORGANISM

A simple circuit, if excited, would tend either to sink back to
zero excitation, if the amplification-factor was less than unity,
or to rise to maximal excitation if it was greater than unity.
Such a circuit tends to maintain only two degrees of activity:
the inactive and the maximal. Its activity will therefore be of
step-function form if the time taken by the chain to build up to
maximal excitation can be neglected. Its critical states would
be the smallest excitation capable of raising it to full activity,
and the smallest inhibition capable of stopping it. McCulloch
has referred to such circuits as ' endromes ' and has studied some
of their properties.

9/6. Another source of step-functions would be provided if
neurons were amoeboid, so that their processes could make or
break contact with other cells.

That nerve-cells are amoeboid in tissue-culture has been known
since the first observations of Harrison. When nerve-tissue from
chick-embryo is grown in clotted plasma, filaments grow outwards
at about 0-05 mm. per hour. The filament terminates in an
expanded end, about 15 X 25/^ in size, which is actively amoeboid,
continually throwing out processes as though exploring the
medium around. Levi studied tissue-cultures by micro-dissection,
so that individual cells could be stimulated. He found that a
nerve-cell, touched with the needle-point, would sometimes throw
out processes by amoeboid movement.

The conditions of tissue-culture are somewhat abnormal, and
artefacts are common; but this objection cannot be raised against
the work of Speidel, who observed nerve-fibres growing into the
living tadpole's tail. The ends of the fibres, like those in the
tissue-culture, were actively amoeboid. Later he observed the
effects of metrazol in the same way: there occurred an active
retraction and, later, re-extension. More recently Carey and
others have studied the motor end-plate. They found that it,
too, is amoeboid, for it contracted to a ball after physical injury.

To react to a stimulus by amoeboid movement is perhaps the
most ancient of reactions. So the hypothesis that neurons are
amoeboid assumes only that they have never lost their original
property. It is possible, therefore, that step-functions are pro-
vided in this way.

125

DESIGN FOR A BRAIN 9/7

9/7. Every cell contains many variables that might change in
a way approximating to the step-function form, especially if the
time of observation is long compared with the average time of
cellular events. Monomolecular films, protein solutions, enzyme
systems, concentrations of hydrogen and other ions, oxidation-
reduction potentials, adsorbed layers, and many other constituents
or processes might act as step-mechanisms.

If the cell is sufficiently sensitive to be affected by changes of
atomic size, then such changes might be of step-function form,
for they could change only by a quantum jump. But this source
of step-functions is probably unavailable, for changes of this
size may be too indeterminate for the production of the regular
and reproducible behaviour considered in this book (S. 1/14).

Round the neuron, and especially round its dendrons and axons,
there is a sensitive membrane that might provide step-functions,
though the membrane is probably wholly employed in the trans-
mission of the action potential. Nerve ' fibrils ' have been des-
cribed for many years, though the possibility that they are an arte-
fact cannot yet be excluded. If they are real their extreme
delicacy of structure suggests that they might behave as step-
functions.

The delicacy everywhere evident in the nervous system has
often been remarked. This delicacy must surely imply the
existence of step-functions ; for the property of being ' delicate '
can mean little other than ' easily broken ' ; and it was observed
in S. 7/19 that the phenomenon of something ' breaking ' is the
expression of a step-function changing value. Though the argu-
ment is largely verbal, it gives some justification for the opinion
that step-mechanisms are by no means unlikely in the nervous
system.

' The idea of a steady, continuous development ', said
Jacques Loeb, ' is inconsistent- with the general physical
qualities of protoplasm or colloidal material. The colloidal
substances in our protoplasm possess critical points. . .
The colloids change their state very easily, and a number of
conditions . . . are able to bring about a change in their
state. Such material lends itself very readily to a discon-
tinuous series of changes. '

126

9/9 ULTRA STABILITY IN THE ORGANISM

A molecular basis for memory?

9/8. What is necessary that any material entity should serve as a
step-mechanism in ultrastability ? Only that it should be a
step-mechanism, that it should be able to be changed by the
essentia] variables, and that it should have an effective action on
the reacting part R.

As a dynamic system, the brain is so extremely sensitive, and
is such a powerful amplifier, that we can hardly put a limit to the
smallness of a physical change that still has a major effect in
behaviour. Even a change on a single molecule cannot be dis-
missed as ineffective, for many events in the nervous system
depend critically on how some variable is related to a threshold;
and near the threshold a small change may have great con-
sequences.

It is therefore not impossible that molecular events should
provide the step-functions. There are plenty of events that
might provide such forms: whether a molecule is in the dextro-
or the laevo- state, in the cis- or trans- state; whether a hydrogen
bond does or does not exist; whether a double bond between
carbon atoms lies in this plane or that; and so on. Such bases
would have the advantage in the living organism of providing the
necessary function at a minimal cost in weight and bulk, matters
of considerable importance to the free-living organism.

Pauling has discussed these possibilities and has suggested
limits that would narrow the field of search. If the molecular
entity is too small, thermal agitation will prevent it from showing
the constancy which it must have if it is to act as basis for such
behaviour as that of Skinner's pigeons (S. 1/14). If too large, it
will be unsuitable for the miracle of ' miniaturisation ' that has
actually been achieved in the mammalian brain.

9/9. With all these possible forms of step-mechanism in mind,
it is difficult for us to say much that is definite about the feedback
channels from essential variable through step-mechanisms to the
reacting part R of Figure 7/5/1. Clearly there is not the least
necessity for the channels to consist of anatomical or histological
tracts ; for if the step-mechanisms were molecular, the channel to
them might be biochemical or hormonic in nature; while the
channel from them to R might be extremely short, and of almost

127

DESIGN FOR A BRAIN 9/10

any nature, if they were sited inside the neurons in which they
exerted their action.

Evidently much more knowledge will be necessary before we
can identify accurately this part of the second-order feedback.
What is most important at the present time is that we should
avoid unwittingly taking for granted what has yet to be demon-
strated.

Are step-mechanisms necessary ?

9/10. Since S. 7/12 we have been considering adaptation when
the variables affecting the reacting part R of Figure 7/5/1 behave
as step-functions. The justification given then was that this case
is of central theoretic importance because of its peculiar simplicity
and clarity: even if the nervous system contained no step-
mechanisms, the student of the subject would still find considera-
tion of this form helpful to get a clear grasp of how a nervous
system can adapt. But is there no justification stronger than
this ? Does the evidence, perhaps, prove that the process of
adaptation implies the existence of step-mechanisms ?

9/11. We have already seen (S. 7/16) that even so typical a
mechanism as a Post Office relay cannot be said to be (uncondi-
tionally) a step-mechanism, for some ways of observing it (e.g.
over microseconds or over years) do not show this form; and no
particular mode of observation can claim absolute priority. Thus
even if some object in a Black Box gave convincing evidence that
it was a Post Office relay, the observer still could not claim that
the real object was a step-mechanism unconditionally.

9/12. On the other hand, contrasted with a full-function the
step-function has a remarkable simplicity of behaviour, and not
every real object can be made to show such a simplicity. To
say of the object in the Black Box that it can be made to show
behaviour of step-function type is thus to say something un-
conditionally true about it.

Again, if a system of three variables is studied and found to
produce such a field as that of Figure 7/20/1 (in which three
possible dimensions are reduced to two two-dimensional planes),
then again the observer can claim that he has demonstrated some-

128

9/13 ULTRASTABILITY IN THE ORGANISM

thing special, in the sense that the fully three-dimensional fields
(e.g. that of Figure 3/5/2) cannot be reduced to the simple form.
Thus certain behaviours, though they do not permit the deduc-
tion that they are due to something that is a step-mechanism,
may none the less permit the deduction that they are produced
by some entity with the special property that its behaviour can
be reduced to step-function form; and the latter is a meaningful
statement, for not all entities produce behaviour that can be so
reduced.

9/13. What of the nervous system ? If the variables in S (of
Figure 7/5/1) were to vary as full-functions, the observer would
see only one very complex system moving by a complex continuous
trajectory to an eventual equilibrium. Often, however, he
observes that the organism ' makes a trial ', i.e. produces some
recognisable form of behaviour and then persists in this way of
behaving for some appreciable time. Then the trial is aban-
doned, a new way of behaving occurs, and this too is persisted in
for an appreciable time; and so on.

When this happens, the observer may justly claim that the
system is showing less than its full complexity; for, through the
duration of the trial, the fact that it is persisting as this particular
form of trial means that some redundancy is occurring. The
redundancy is similar to that of the sequence of letters that goes,
e.g.

FFFFFLLLLLLLTTTTTJJJJJJJ

rather than with full variety:

EJYMSNASGCGHLAAPEYPJVRQJ

Within each trial, if it shows a characteristic way of behaving,
there will be defined a field ; and these fields will follow one another
in a discrete succession, as in Figure 7/23/1. When this occurs,
if the observer is willing to assume that the changes of field are
occurring in a state-determined whole, he may legitimately deduce
that the parameters responsible for the changes from field to field
are of such a nature, and so joined to the main variables, that
they may be represented by step-functions (for full-functions
could not give the discrete movement from distinct trial to distinct
trial). That they may be so represented is a meaningful restriction
on their nature.

If now we couple this deduction with what has been called

129

DESIGN FOR A BRAIN 9/14

Dancoff s principle — that systems made efficient by natural selec-
tion will not use variety or channel capacity much in excess of
the minimum — then we can deduce that when organisms regularly
use the method of trials there is strong presumptive (though not
conclusive) evidence that their trials will be controlled by material
entities having (relative to the rest of the system) not much more
than the minimal variety. There is therefore strong presumptive
evidence that the significant variables in S (of Figure 7/5/1) are
step-functions, and that the material entities embodying them are
of such a nature as will easily show such functional forms.

Levels of feedback

9/14. We can next consider how the formulation of Chapter 7,
and Figure 7/5/1, compares with the real organism's organisation
in respect of the division of the feedbacks into two clearly dis-
tinguishable orders: between organism and environment by the
usual sensory and motor channels, and that passing through the
essential variables and step-mechanisms to the reacting part R.

Chapter 7 followed the strategy described in S. 2/17: we
attempted to get a type-system perfectly clear, so that it would
act as a suitable reference for many real systems that do not
correspond to it exactly. To get a clear case we assumed that
the system (organism and environment joined) was subject to
just two types of disturbance from outside. Of one type is the
impulsive disturbance to the system's main variables ; by this their
state is displaced to some non-equilibrial position; this happens if
a Homeostat's needle is pushed away from the centre, as at Dx
in Figure 8/4/1 ; or if the fire by the kitten suddenly blazes up.
Then the organism, if adapted, demonstrates its adaptation by
taking the action appropriate to the new state. A number of
such impulsive disturbances, each with an interval for reaction
to occur, are necessary if the organism's adaptation is to be tested
and demonstrated.

Of the other type is the disturbance in which some parameter
to the whole system is changed (from the value it had over the
many impulsive disturbances, to some new value). This change
stands in quite a different relation to the system from the change
implied by the impulse. Whereas the impulse made the system
demonstrate its stability, the change at the parameter made the

130

9/16 ULTRASTABILITY IN THE ORGANISM

system demonstrate (if possible) its ultrastability. Whereas the
system demonstrates, after the impulse, its power of returning to
the state of equilibrium, it demonstrates, after the change of
parameter-value, its power of returning the field (of its main
variables) to a stable form. The ultrastable system is thus the
appropriate' form for the organism if the disturbances that come
to it from the world around fall into two clearly defined classes:

(1) Frequent (or even continuous) small impulsive disturbances

to the main variables.

(2) Occasional changes, of step-function form, to its parameters.

The ultrastable system is thus not merely a didactic device;
it may, in some cases, actually be the optimal mechanism by which
an organism can ensure its survival. When the disturbances that
threaten the organism have, over many generations, had the bi-modal
form just described, we may expect to find that the organism will,
under natural selection, have developed a form fairly close to the
ultrastable, in that it will have developed two readily distinguishable
feedbacks.

9/15. It is not for a moment suggested that all natural stimuli,
disturbances, and problems come to kittens in the tidily dichoto-
mous way in which we have brought them to the Homeostat.
Neither is it suggested that the real brain can always be viewed as
ultrastable, if only we can find the right way of approach. On
the contrary, it is only when we scientists are fortunate that we
will find that a complex system can be reduced conceptually into
manageable subsystems, as the Homeostat is reducible into its
continuous part with feedbacks between the needles, and its
stepwise varying part around the second feedback. If there are
many feedback loops, and there is no convenient way of indi-
vidualising them, then simplicity is not to be had, and there is
nothing for it but to treat the system as one whole, of high com-
plexity. (The subject is discussed from Chapter 11 through the
remainder of the book.)

The control of aim

9/16. The ultrastable systems discussed so far, though develop-
ing a variety of fields, have sought a constant goal. The Homeo-
stat sought central positions and the rat sought zero grill-potential.

131

DESIGN FOR A BRAIN

9/16

In this section will be described some methods by which the
goal may be varied. Variations in the goal will be important in
those cases in which the goal is only a sub-goal, sought temporarily
or provisionally for the achievement of some other goal that is
permanent (S. 3/15).

If the critical states' distribution in the main-variables' phase-
space is altered by any means whatever, the ultrastable system
will be altered in the goal it seeks. For the ultrastable system
will always develop a field which keeps the representative point
within the region of the critical states (S. 7/23). Thus if (Figure
9/16/1) for some reason the critical states moved to surround B

JO-

Figure 9/16/1.

instead of A, then the terminal field would change from one which
kept x between 0 and 5 to one which kept x between 15 and 20.
A related method is illustrated by Figure 9/16/2. An ultra-
stable system U interacts with a variable A.
E and R represent the immediate effects which
U and A have on each other; they may be
thought of as C/'s effectors and receptors. If A
should have a marked effect on the ultrastable
system, the latter will, of course, develop a field
stabilising A ; at what value will depend markedly
on the action of R. Suppose, for instance, that
U has its critical states all at values 0 and 10, so
that it always selects a field stabilising all its
main variables between these values. If R is such that, if A
has some value «, R transmits to U the value 5a — 20, then
it is easy to see that U will develop a field holding A within
one unit of the value 5; for if the field makes A go outside the
range 4 to 6, it will make U go outside the range 0 to 10, and

132

Figure 9/16/2.

9/16 ULTRASTABILITY IN THE ORGANISM

this will destroy the field. So U becomes ' 5-seeking '. If the
action of R is now changed to transmitting, not 5a — 20 but
5a + 5, then U will change fields until it holds A within one
unit of 0; and U is now ' 0-seeking'. So anything that controls
the b in R = 5a + b controls the ' goal ' sought by U.

As a more practical example, suppose U is mobile and is
ultrastable, with its critical states set so that it seeks situations
of high illumination; such would occur if its critical states
resembled, in Figure 9/16/1, B rather than A. Suppose too that
R is a ray of light. If in the path of R we place a red colour-
filter, then green light will count as ' no light ' and the system
will actively seek the red places and avoid the green. If now
we merely replace the red filter by a green, the whole aim of
its movements will be altered, for it will now seek the green
places and avoid the red.

Next, suppose R is a transducer that converts a temperature
at A into an illumination which it transmits to U. If R is
arranged so that a high temperature at A is converted into a high
illumination, then U will become actively goal-seeking for hot
places. And if the relation within R is reversed, U will seek
for cold places. Clearly, whatever controls R controls C/'s goal.

There is therefore in general no difficulty in accounting for
the fact that a system may seek one goal at one time and another
goal at another time.

Sometimes the change, of critical states or of the transducer
R. may be under the control of a single parameter. When this
happens we must distinguish two complexities. Suppose the
parameter can take only two values and the system U is very
complicated. Then the system is simple in the sense that it
will seek one of only two goals, and is complicated in the sense
that the behaviour with which it gets to the goal is complicated.
That the behaviour is complicated is no proof, or even sugges-
tion, that the parameter's relations to the system must be com-
plicated; for, as was shown in S. 6/3, the number of fields is
equal to the number of values the parameter can take, and has
nothing to do with the number of main variables. It is this
latter that determines, in general, the complexity of the goal-
seeking behaviour.

These considerations may clarify the relations between the
change of concentration of a sex-hormone in the blood of a

133

DESIGN FOR A BRAIN 9/17

mammal and its consequent sexual goal-seeking behaviour. A
simple alternation between ' present ' and ' absent ', or between
two levels with a threshold, would be sufficient to account for
any degree of complexity in the two behaviours, for the com-
plexity is not to be related to the hormone-parameter but to the
nervous system that is affected by it. Since the mammalian
nervous system is extremely complex, and since it is, at almost
every point, sensitive to both physical and chemical influences,
there seems to be no reason to suppose that the directiveness of
the sex-hormones on the brain's behaviour is essentially different
from that of any parameter on the system it controls. (That
the sex-hormones evoke specifically sexual behaviour is, of course,
explicable by the fact that evolution, through natural selection,
has constructed specific mechanisms that react to the hormone
in the specific way.)

The gene-pattern and ultrastability

9/17. We can now return to the questions of S. 1/9 and ask
what part is played by the gene-pattern in the determination of
the process of adaptation.

Taking the diagram of immediate effects (Figure 7/5/1) as
basis, the question is answerable without difficulty if we take the
system part by part and channel by channel.

The environment is, of course, assumed to be given arbitrarily;
so is the channel by which the environment affects the essential
variables (S. 7/3). The essential variables and their limits are
determined by the gene-pattern (S. 3/14); for these are species'
characteristics.

In the living organism, the reacting part R has, in effect, three
types of ' input ' : there is the sensory input from the environment,
there are the values of its parameters in S, and there are those
parameters that were determined genetically during embryonic
development. (That all three may be regarded as ' input ' has
been shown in /. to C, S. 13/11). These three sets of parameters
vary on very different time-scales: the genetic parameters, those
that make this a dog-brain and that a bird-brain, are in evidence
only at one period in a lifetime ; the parameters in S, if the adapta-
tion proceeds by clearly marked trials, change only between trial
and trial; and the parameters at the sensory input vary more or

134

9/18 ULTRA STABILITY IN THE ORGANISM

less continuously. The influence of the gene-pattern can thus be
traced in R, giving it certain anatomical tracts, biochemical pro-
cesses, histological structures, and thus determining whether it
shall adapt as a dog does or as a starfish does.

The nature of the parameters in S is wholly under genetic
control, for their physical .embodiment has probably been selected
for suitability by natural selection. (Here the nature of the
parameters — whether they are reverberating circuits, or molecular
configurations, etc. — must be clearly distinguished from the values
that any one parameter may take.)

Finally there is the relation between the essential variables and
those in S — that the essential variables must force those in S to
change when the essential variables are outside their physiological
limits, and not to change otherwise (S. 7/7). As this relation is
entirely ad hoc, it must be determined by the gene-pattern, for
there is no other source for its selection.

These are the ways, then, in which the gene-pattern must act
as determinant to the living organism's mechanism for adaptation.

9/18. A question that must be answered is whether ultrasta-
bility, as described here, can reasonably be supposed to have been
developed by natural selection; for the ad hoc features of the
previous section have no other determinant adequate for their
selection and adjustment.

For ultrastability to have been developed by natural selection,
it is necessary and sufficient that there should exist a sequence of
forms, from the simplest to the most complex, such that each form
has better survival-value than that before it. In other words,
ultrastability must not become of value to the organism only
when some complex form has all its parts and relations correct
simultaneously, for such an event occurs only rarely.

Suppose the original organism had no step-mechanisms; such an
organism would have a permanent, invariable set of reactions.
If a mutation should lead to the formation of a single step-mech-
anism whose critical states were such that, when the organism
became distressed, it changed value before the essential variables
transgressed their limits, and if the step-mechanism affected in any
way the reaction between the organism and the environment,
then such a step-mechanism might increase the organism's chance
of survival. A single mutation causing a single step-mechanism

135

DESIGN FOR A BRAIN 9/19

might therefore prove advantageous; and this advantage, though
slight, might be sufficient to establish the mutation as a species
characteristic. Then a second mutation might continue the pro-
cess. The change from the original system to the ultras table
can therefore be made by a long series of small changes, each
of which improves the chance of survival. The change is thus
possible under the action of natural selection.

Summary

9/19. The solution of the problem of Chapter 1 is now completed
in its essentials. It may be summarised as follows:

In the type-problem of S. 1/17 the disturbances that come to
the organism are of two widely different types (the distribution
is bi-modal). One type is small, frequent, impulsive, and acts on
the main variables. The other is large, infrequent, and induces a
change of step-function form on the parameters to the reacting
part. Included in the latter type is the major disturbance of
embryogenesis, which first sends the organism into the world with
a brain sufficiently disorganised to require correction (in this
respect, learning and adaptation are related, for the same solution
is valid for both).

To such a distribution of disturbances the appropriate regulator
(to keep the essential variables within physiological limits) is one
whose total feedbacks fall into a correspondingly bi-modal form.
There will be feedbacks to give stability against the frequent
impulsive disturbances to the main variables, and there will be a
slower-acting feedback giving changes of step-function form to
give stability against the infrequent disturbances of step-function
form.

Such a whole can be regarded simply as one complex regulator
that is stable against a complex (bi-modal) set of disturbances.
Or it can equivalently be regarded as a first-order regulator
(against the small impulsive disturbances) that can reorganise
itself to achieve this stability after the disturbance of embryo-
genesis or after a major change in its conditions has destroyed this
stability. When the biologist regards the system in this second
way, he says that the organism has ' learned ', and he notices that
the learning always tends towards the better way of behaving

136

9/20 ULTRASTABILITY IN THE ORGANISM

9/20. Such is the solution in outline. The reader, however, may
well feel that the amount of information given by the solution is
small.

To some extent, the generality of the ultrastable system, the
degree to which it does not specify details, is correct. Adaptation
can be shown by systems far wider in extent than the mammalian
and cerebral, and any proposed solution would manifestly be
wrong if it stated that, say, myelin was necessary, when the
Homeostat obviously contains none. Thus the generality, or if
you will the vagueness, of the ultrastable system is, from that point
of view, as it should be.

However, the attempt to apply this general formulation to the
real nervous system soon encounters major difficulties. What
these are, and how they are to be treated, will occupy the remainder
of the book.

137

CHAPTER 10

The Recurrent Situation

10/1. With the previous chapter we came to the end of our
study of how the organism changes from the unadapted to the
adapted condition. But this simple problem and solution is only
a first step towards our understanding of the living, and especially
of the human, brain. To the simple ultrastable system we must
obviously add further complications. Thus the living organism
not only becomes adapted, but it does so by a process that shows
some evidence of efficiency, in the sense that the adaptation is
reached by a path that is not grossly far from the path that would
involve the least time, and energy, and risk. Though ' efficiency '
is not yet accurately defined in this context, few would deny that
the Homeostat's performance suggests something of inefficiency.
But before we rush in to make 4 improvements ' we must be clear
about what we are assuming.

10/2. Let us return to first principles. ' Success \ or ' adapta-
tion ', means to an organism that, in spite of the world doing its
worst, the organism so responded that it survived for the duration
necessary for reproduction.

Now i what the world did ' can be regarded as a single, life-
long, and very complex Grand Disturbance (7. to C, Chapter 10),
to which the organism produces a single, life-long, and very
complex Grand Response; how they are related determines the
Grand Outcome — success or failure. In the most general case,
the partial disturbances that make up this Grand Disturbance,
and the partial responses that make up the organism's Grand
Response (I. to C, S. 13/8) may be interrelated to any degree,
from zero to complete. (The interrelation is ' complete ' when the
Grand Outcome is a function of all the partial responses; it would
correspond to an extremely complex relation between partial
responses and final outcome.)

The case of the complete interrelation, though fundamental

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10/4 THE RECURRENT SITUATION

theoretically (because of its complete generality), is of little
importance in practice, for its occurrence in the terrestrial world
is rare (though it may occur more commonly in models or in pro-
cesses of adaptation set up in large computers). Were it common,
a brain would be useless (/. to C, S. 13/5). In fact, brains have
been developed because the terrestrial environment usually con-
fronts the organism with a Grand Disturbance that has a major
degree of constraint within its component parts, of which the
organism can take advantage. Thus the organism commonly
faces a world that repeats itself, that is consistent to some degree
in obeying laws, that is not wholly chaotic. The greater the
degree of constraint, the more can the adapting organism specialise
against the particular forms of environment thsft do occur. As
it specialises so will its efficiency against the particular form of
environment increase. If the reader feels the ultrastable system,
as described so far, to be extremely low in efficiency, this is because
it is as yet quite unspecialised ; and the reader is evidently uncon-
sciously pitting it against a set of environments that he has
restricted in some way not yet stated explicitly in this book.

10/3. The chapters that follow will consider several constraints
of outstanding commonness and will show how the appropriate
specialisations exemplify the above propositions in several ways.
They will consider certain ways in which the ordinary terrestrial
environments fail to show the full range ; and we will see how these
restrictions indicate ways in which the living organism can
specialise so as to take advantage of them.

The recurrent situation

10/4. In this chapter we will consider the case, of great importance
in real life, in which the occasional disturbances (class" 2 of S. 9/14)
are sometimes repetitive, and in which a response, if adaptive on
the disturbance's first appearance, is also adaptive when the same
disturbance appears for the second, third, and later times.

We must not take for granted that one response will be adaptive
to all occurrences of the disturbance, for there are cases in which
what is appropriate to a disturbance depends on how many times
it has appeared before. An outstanding example is given by the
rat facing that environment (a natural one by S. 3/1) in which food

139

DESIGN FOR A BRAIN 10/5

will appear on two successive nights at the same place, followed,
on the third night, by a lethal mixture of the same food and poison
(the method of ' pre-baiting '). Environments such as this are
intrinsically complex. Complete adaptation here (under the
assumptions made) demands the reaction-pattern: eat, eat,
abstain. This reaction-pattern is more complex than the simple
reaction-pattern of eating, or of abstaining: for the three parts
must be related, and the triple organised holistically.

In this chapter we shall consider the other case, of frequent
occurrence, in which what is appropriate to the disturbance is
conditional on which disturbance it is, but not on when it occurs
in the sequence of disturbances.

So far, the ultrastable system (represented, say, by the Homeo-
stat) has been presented (e.g. in the Figures throughout Chapter 8)
with changes of parameter-value such that the later value is
merely different from the earlier; now we consider the case in
which the parameter takes a sequence of values, e.g.

in which repetitions occur at irregular intervals, and in which a
response to P2, say, if adaptive on P2s first occurrence, is also
adaptive to P2 on its later occurrences.

When this is the case, the opportunity exists for advantage to
be taken of the fact that P2 can be responded to at once on its
later occurrences, without the necessity of a second exploratory
series of trials and errors.

This case is particularly important because (S. 8/10) it includes
the case in which the changes of P-value correspond to changes
from one environment to another. Suppose, for instance, that a
wild rat learns first to adapt to conditions in a stable {P2), then
to conditions in a nearby barn (P3), and so on. Haying adapted
first to the stable and then to the barn, its survival value would
obviously be enhanced if it could return to the stable and at once
resume the adaptations that it had previously developed there.
An organism with such a power can accumulate adaptations.

10/5. To see what is necessary, let us see what happens in the
Homeostat. A little reflection, or an actual test, soon shows that
the present model is totally devoid of such power of accumulation.
Thus in Figure 8/4/1 the reversal at R2 restores the external

140

10/7 THE RECURRENT SITUATION

conditions to which it was already adapted at Dx; yet after the
events following the first reversal (at R^), the first adaptation (at
Dj) is totally lost; and the Homeostat treats the situation after
R2 as if the situation had occurred for the first time.

In general, if the Homeostat is given a problem A, then a prob-
lem B, and then A again, it treats A as if it had never encountered
A before; the activities during the adaptation to B have totally
destroyed the previous adaptation to A. (The psychologist would
say that retroactive inhibition was complete, S. 16/12.)

This way of adapting to A on its second presentation cannot be
improved upon if the environment is such that there is no implica-
tion that the second reaction to A should be the same as the first.
The Homeostat's behaviour might then be described as that of a
system that ' does not jump to conclusions ' and that ' treats every
new situation on its merits '. In a world in which pre-baiting was
the rule, the Homeostat would be better than the rat ! When,
however, the environment does show the constraint assumed in
this chapter, the Homeostat fails to take advantage of it. How
should it be modified to make this possible ?

10/6. The Homeostat has, in fact, a small resource for dealing
with recurrent situations, but the method is of small practical use.
In S. 8/10 we saw that the Homeostat's ultimate field is one that
is stable to all the situations, so that a change from one to another
demands no new trials.

10/7. This method, however, cannot be used extensively in the
adaptations of real life, for two reasons. The first is that when
the number of values is increased beyond a few, the time taken
for a suitable set of step-function values to be found is likely to
increase beyond anything ordinarily available, a topic that will
be treated more thoroughly in Chapter 11. The second is that
the adaptation, even if established, is secure only if the set of
parameter-values is closed, i.e. so long as no new value occurs.
Should a new value occur, everything goes back into the melting-
pot, and adaptation to the new set of values (the old set increased
by one new member) has to start from scratch. Common observa-
tion shows, of course, that each new adaptation does not destroy
all the old; evidently the method of S. 8/10 is of little practical
importance.

141

DESIGN FOR A BRAIN 10/8

The accumulator of adaptations

10/8. To see what is necessary, let us take for granted that
organisms are usually able to add new adaptations without destroy-
ing the old. Let us take this as given, and deduce what modifica-
tions it enforces on the formulation of S. 7/5. Suppose, then,
that an organism has adapted to a value P1% has then adapted to
P2 by trial and error as in S. 7/23, and that when P1 is restored
the organism is found to be adapted at once, without further
trials. What can we deduce ?

(The arguments that culminated in S. 7/8 apply here without
alteration, so we can take for granted that the adaptation to each
individual value of P takes place through the second feedback,
with essential variables controlling step-functions as in S. 7/7.
The modification to be made can be found by a direct application
of the method of S. 4/12, seeing whether variation at one variable
leads to variation at another.)

To follow the argument through, let us define two sub-sets of
the step-mechanisms in S that affect R (Figure 7/5/1):

Sx : those step-mechanisms whose change, with P at Pv would
cause a loss of the adaptation to Px (i.e. those step-
mechanisms that are effective towards R when P is at

Pi);

S2: those step-mechanisms that were permanently changed in
value after the trials that led to the adaptation to P2.

First it follows that the sets Sx and S2 are disjunct, i.e. have no
common member. For if there were such a common member it
would (as a member of S2) be changed in value when Px was
applied for the second time, and therefore (as a member of Sx)
would force the behaviour at Px to be changed on Px's second
presentation, contrary to hypothesis. Thus, for the retention of
adaptation to Plf in spite of that to P2, the step-mechanisms must
fall into separate classes.

(That the step-mechanisms must be split into classes can be
made plausible by thinking of the step-mechanisms, in any ultra-
stable system, as carrying information about how the essential
variables have behaved in the past. When Px is presented for the
second time, for the behaviour to be at once adaptive, information
must be available somewhere about how the essential variables

142

10/9 THE RECURRENT SITUATION

behaved in the past (for by hypothesis they are to give none now,
and they are the only source). Thus somewhere in the system
there must be this information stored; and these stores must not
be accessible while P2 is acting, or they will be affected by the
events and the stored information over-written. Thus there must
be separate stores for P± and P2, and provision for their separate
use.)

Next, consider the channel from the essential variables. In
condition P2, the channel from them to the step-mechanisms in
#2 was evidently open, for events at the essential variables (whether
within physiological limits or not) affected what happened in
S2 (by the ordinary processes of adaptation). On the other hand,
during this time the channel from the essential variables to the
step-mechanisms in Sx was evidently closed, for changes in the
essential variables were followed by no changes in the step-
mechanisms of Sv Thus the channel from the essential variables
to the step-mechanisms S must be divisible into sections, so that
some can conduct while the others do not; and the determination
of which is to conduct must be, at least partly, under the control
of the conditions P, varying as P varies between P± and P2.

Finally, consider the channels from S± and S2 to the reacting
part R. When Px is applied for the second time, the channel
from S2 to R is evidently closed, for though the parameters in S2
are changed (before and after P2), yet no change occurs in R's
behaviour (by hypothesis). On the other hand, that from Sx is
evidently open, for it is S^s values that determine the behaviour
under Pv and it is the adapted form that is made to appear.

10/9. To summarise: — Let it be given that the organism has
adapted to Px by trial and error, then it adapted similarly to P2,
and that when P1 was given for the second time the organism
was adapted at once, without further trials. From this we may
deduce that the step-mechanisms must be divisible into non-
overlapping sets, that the reactions to Px and P2 must each be
due to their particular sets, and that the presentation of the
problem (i.e. the value of P) must determine which set is to
be brought into functional connexion, the remainder being left in
functional isolation.

Thus if the diagram of Figure 7/5/1 is taken as basic, it must
be modified so that the step-mechanisms are split into sets, there

143

DESIGN FOR A BRAIN

10/10

must be some gating mechanism r to determine which set shall
be on the feedback circuit, and the gating mechanism r must be
controlled (usually through R, as this is the organism's structure)
by the value of P.

Envt

Figure 10/9/1.

Figure 10/9/1 presents the diagram of immediate effects, but
the Figure is best thought of as a mere mnemonic for the functional
relations, lest it suggest some anatomical form too strongly. The
parameter P can be set at various values, Plt P2, . . . The step-
mechanisms are divided into sets, and there is a gating mechanism
r, controlled by P through the environment and the reacting
part R, that determines which of the sets shall be effective in
the second feedback via the essential variables.

10/10. The diagram of Figure 10/9/1 and the behaviour of the
gating mechanism may seem somewhat complex, but we must
beware of seeing into it more complexity than is necessary. All
that is necessary is that the step-mechanisms involved in any
particular problem P{ be distinct from those involved in the
others, that if S{ be affected by the essential variables then St shall
be the mechanisms that affect R, and that there shall be a corre-
spondence between the problems and the sets of step-mechanisms.
This latter correspondence need not be orderly or ' rational ' ; it
may be perfectly well set up at random (i.e. determined by factors

144

10/11 THE RECURRENT SITUATION

outside our present view) provided only that if the presentation
of a particular problem Pt got through to some set St, then always
when Pt is presented again the actions shall again go through to
S{. Such a case would occur if the connexions were, say, electrical
and made by plugging connexions at random into a plug-board.
Once made* they would ensure that recurrence of Pt would give
the same pattern for the selection of S{ ; and the change from Pt
to some other problem, Pi say, by involving some change in the
sensory input to R, would cause some change in the distribution
over the step-mechanisms.

In the same way, if nerve-cells were to grow at random (i.e.
determined in their growth by local temporary details of oxygen
supply, mechanical forces, etc.) until their histological details were
established, and if the paths taken by impulses depended on the
concatentation of stimuli coming in, then the recurrence of Pt
would always give access to Si3 and a change from Pz to P„ by
changing the sensory stimuli, would change the distribution.

An easy method by which J7 may be provided is given in S. 16/13.

These details need not detain us. They are mentioned only to
show that the basic requirements are easily met, and that the
mechanism meeting them may look far less tidy than Figure
10/9/1 might suggest. In this sense the Figure, though helpful
in some ways, is apt to be seriously misleading. In S. 16/12 we
return to the matter.

10/11. In the previous sections, the various situations Pl9 P2,
P3, . . . were arbitrary, and not assumed to have any particular
relation between them. A special case, common enough to be of
interest, occurs when the situations usually occur in a particular
sequence. Thus a young child, reaching across the table for a
biscuit, may have first to get his hand past the edge of the table
without striking it, then the hand past his cup without spilling
it, then past the jam without his sleeve wiping it, and so on:
a sequence of actions, each of Which calls for some adaptation.
Much of life consists of just such sequences.

The system of Figure 10/9/1 can readily give such sequences
in which every part is adapted to its own little problem. The
situation of ' hand coming past the edge of the table and in danger
of striking it ' is Pv say. Adaptation to this situation can occur
in the usual way, by the basic method of the ultrastable system.

145

DESIGN FOR A BRAIN 10/12

The sleeve passing near open jam is another situation, P2; adapta-
tion to this, too, can occur. And the alterations necessary in
adaptation to P2 will not, in our present system, cause loss of
the adaptation to Pv

Whether the whole situation can be adapted to by such a
sequence of sub-adaptations (to Pv to P2, etc.) depends on the
environment: only certain types of environment will allow such
fragmentation. If such types of environment are frequent in an
organism's life, then there will be advantage in evolution if the
species changes so that each organism is provided, genetically,
with a mechanism similar to that of Figure 10/9/1.

10/12. To amplify the point, we may consider the case of an
organism that lives in an environment that consists of many
sensorily-different situations, and such that each situation is
adequately met by one of two reactions, eat or flee say, so that
the organism's problem in life is to allot one of the two reactions
to each of many situations. In such an environment, the reacting
part R of Figures 7/5/1 or 10/9/1 could be quite small, for it
requires only mechanism capable of performing two reflexes. The
stores of step-mechanisms, however, would have to be large, and
the gating mechanism r perhaps elaborate; for here would have
to be as many records as there are sensory situations. Each
would require its own locus of storage, and the gating mechanism
would have to be able to ensure that each situation led to its
particular locus. In such a world we would, therefore, expect the
organism to have a differently proportioned brain from one that
lived in a world such as was presented to the Homeostat.

It should not be beyond the biologist's powers to identify a
species with such an environment. Examination of the organism's
nervous system might then enable some fundamental identifica-
tions to be made.

10/13. The objection may be raised that the specification deduced
in this chapter is too vague to be of use to the worker who wants
to find the corresponding mechanism in, say, the human brain.
The reply must be that the specification is right to be vague; for
what is given — that adaptation occurs with accumulation —
specifies an extremely broad class of mechanisms, so that a very
great diversity of actual machines could all show adaptation with

146

10/13 THE RECURRENT SITUATION

accumulation. If they can all show it, a deduction would be
patently wrong if, without further data, it proceeded to indicate
some one of the class.

What the deduction has shown is that we must give up our
naive conviction that the outstanding behavioural properties of
adaptation indicate some unique cerebral mechanism, or that they
will provide the unique explanation of the features of the living
brain. 'Many of these features cannot be related uniquely to the
processes of adaptation, for these processes can go on in systems
that lack those neurophysiological features, systems very different
from the living brain, such as the modern computer. Only further
information, beyond that assumed in this chapter, can take the
identification further.

147

CHAPTER 11

The Fully-joined System

11/1. The Homeostat is, of course, grossly different from the
brain in many respects, one of the most obvious being that while
the brain has a very great number of component parts the Homeo-
stat has, effectively, only four. This difference does not make
the theory of the ultrastable system totally inapplicable, for much
of the theory is true regardless of the number of parts, which is
simply irrelevant. Nevertheless, we are in danger, after spending
so much time getting to know a system of four parts, of developing
a set of images, a set of working mental concepts, that is seriously
out of proportion if considered as a set of working concepts with
which to think about the real brain. Let us therefore consider
specifically the properties of the ultrastable system that has a very
great number of parts. We shall find that one difficulty becomes
outstanding, and we shall spend the remainder of the book dealing
with it, for it is the main problem in the adaptation of large
systems.

Adaptation-time

11/2. Suppose the Homeostat were of a thousand units. Such
a size goes only a small fraction of the way to brain-size, but it
will serve our purpose. Such a system will have a thousand relays
(F, Figure 8/2/3); let us suppose that all but a hundred are
shorted out, so that the system is left with one hundred essential
variables. This number may be of the same order of size as the
number of essential variables in a living organism.

Since these variables are essential, adaptation implies that all
are to be kept within their proper limits. Let us try a rough
preliminary calculation. Simplifying the situation further, let us
suppose that the step-mechanisms, as they change, give to each
essential variable a 50-50 chance of going within, or outside of,
its limits, and that the chances are independent. (The case of
independence should be considered, by the strategy of S. 2/17,

148

11/3 THE FULLY-JOINED SYSTEM

as the case is central.) We ask, how many trials will, on the average,
be required for adaptation ?

Each variable has a probability J of being kept within its limits.
At any one trial the probability that all of 100 will be kept in is
(J)100, and so the average number of trials will be 2100, by S. 22/7.
How long will this take, at, say, one per second ? The answer is:
about 1022 years, a time unimaginably vaster than all astronomic-
ally meaningful time ! For practical purposes this is equivalent
to never, and so we arrive at our major problem : the brain, though
having many components, does adapt in a fairly short time — the
1,000-unit Homeostat, though of vastly fewer components, does
not. What is wrong ?

It can hardly be that the brain does not use the basic process of
ultrast ability, for the arguments of S. 7/8 show that any system
made of parts that obey the ordinary laws of cause and effect
must use this method. Further, there is no reason to suppose
that the function 2N, where N is the number of essential variables,
is seriously in error, even though it is somewhat uncertain: other
lines of reasoning (given below) also lead to the same order of
size, a size far too large to be compatible with the known facts.
There seems to be little doubt that a 1,000-unit Homeostat would
quite fail, by its slowness in getting adapted, to resemble the
mammalian brain. Wherein, then, does this system not resemble
(in an essential way) the system of brain and environment ?

11/3. In the previous section the dynamic nature of the brain and
environment was really ignored, for the calculation was based on
a direct relation between step-mechanisms and essential variables,
while what connected them was ignored. Let us now ask the
question again, ignoring the essential variables but observing the
dynamic systems of environment and reacting part (Figure 7/2/1).
Two type-cases are worth consideration (by S. 2/17).

The first case occurs when the system is linear, like the Homeo-
stat, so that it has only one state of equilibrium, which can be
stable or unstable. In this case, unstable fields are of no use to
the organism, for they do not persist; only the stable can be of
enduring use, for only they persist. Let us ask then: if a Homeo-
stat had a thousand units, how many trials would be necessary
for a stable field to be found ? Though the answer to this ques-
tion is not known, for the mathematical problem has not yet

149

DESIGN FOR A BRAIN 11/4

been solved, there is evidence (S. 20/10) suggesting that in some
typical cases the number will be of the order of 2-^, where N is
the number of variables. There seems to be little doubt that if
a Homeostat were made with a thousand units, practically every
field would be unstable, and the chance of one occurring in a
lifetime would be practically zero. We thus arrive at much the
same conclusion as before.

The second case to be considered is that of the system that has
the transformation on its states formed at random, so that every
state goes equiprobably to every state. Such systems have been
studied by Rubin and Sitgreaves. Among their results they find
that the modal length of trajectory is \/n, where n is the number
of states. Now if the whole is made of N parts, each of which
can take any one of a states, then the whole can take any one of
aN. This is n; so the modal length of trajectory is VoN, which
can be written as (\Zo)N. Again, if we fill in some plausible
numbers, with N = 1,000, we find that the length of trajectory,
and therefore the time before the system settles to some equili-
brium, takes a time utterly beyond that ordinarily observed to
be taken by the living brain.

11/4. The three functions given by the three calculations are all
of the exponential type, in that the number of trials is proportional
to some number raised to the power of the number of essential
variables or parts. Exponential functions have a fundamental
peculiarity: they increase with deceptive slowness when the
exponent is small, and then develop with breath-taking speed as
the exponent gets larger. Thus, so long as the Homeostat has
only a few units, the number of trials it requires is not large.
Let its size undergo the moderate increase to a thousand parts,
however, and the number of trials rushes up to numbers that make
even the astronomical look insignificant. In the presence of this
exponential form, a mere speeding up of the individual trials, or
similar modification, will not bring the numbers down to an
ordinary size.

11/5. What had made the processes of the last two sections so
excessively time-consuming is that partial successes go for nothing.
To see how potent is this fact, consider a simple calculation which
will illustrate the point.

150

11/5 THR FULLY -JOINED SYSTEM

Suppose N events each have a probability p of success, and
the probabilities are independent. An example would occur if N
wheels bore letters A and B on the rim, with A's occupying the
fraction p of the circumference and 2?'s the remainder. All are
spun and allowed to come to rest; those that stop at an A count
as successes. Let us compare three ways of compounding these
minor successes to a Grand Success,* which, we assume, occurs
only when every wheel is stopped at an A.

Case 1: All N wheels are spun; if all show an A, Success is
recorded and the trials ended; otherwise all are spun again, and
so on till ' all A's ' comes up at one spin.

Case 2: The first wheel is spun; if it stops at an A it is left
there; otherwise it is spun again. When it eventually stops at an
A the second wheel is spun similarly; and so on down the line of
N wheels, one at a time, till all show^'s.

Case 3: All N wheels are spun; those that show an A are left
to continue showing it, and those that show a B are spun again.
When further A's occur they also are left alone. So the number
spun gets fewer and fewer, until all are at A's.

Regard each spin (regardless of the number of wheels turned)
as one trial. We now ask, how many trials, on the average, will
the three cases require ?

Case 1 will require ( - )N, as in S. 11/2. Case 2 will require, on

the average, l/'p for the first wheel, then \/p for the second, and
so on; and thus N/p for them all. Case 3 is difficult to calculate
but will be the average of the longest of a sample of N drawn from
the distribution of length of run for one wheel ; it will be somewhat
larger than I /p.

The calculations are of interest not for their quantitative exact-
ness but because when N gets large they tend to widely differing
values. Suppose, for instance, that p is \, that spins occur at
one a second, and that N is 1,000. Then if Tv T2, and T3 are
the average times to reach Success in Cases 1, 2, and 3 respectively,
Ti = 2iooo seconds,

T2 = — — — seconds,

T3 = rather more than \ second.

* As we shall have to consider several compoundings of minor events to
major events, I shall use the convention of I. to C, S. 13/8, and distinguish
them respectively by a lower case, or a capital, initial.

151

DESIGN FOR A BRAIN 11/6

When these values are converted to ordinary quantities, Tx is
about 10293 years, T2 is about 8 minutes, and T3 is a few seconds.
Thus, while getting Success under the rules of Case 1 (all simul-
taneously) is practically impossible, getting it under Cases 2 and 3
is easy.

The final conclusion — that Case 1 is very different from Cases
2 and 3 — does not depend closely on the particular values of p
ahd N. It illustrates the general fact that the exponential
function (Case 1) tends to become large at an altogether faster
rate than the linear. If the reader likes to try other numbers
he is likely to arrive at results showing much the same features.

11/6. Comparison of the three Cases soon shows why Cases 2
and 3 can arrive at Success so much sooner than Case 1 : they can
benefit by partial successes, which 1 cannot. Suppose, for
instance, that, under Case 1, a spin gave 999 A's and 1 B. This is
very near complete Success; yet it counts for nothing, and all the
A's have to be thrown back into the melting-pot. In Case 3,
however, only one wheel would remain to be spun; while Case 2
would perhaps get a good run of A's at the left-hand end and could
thus benefit somewhat.

The examples thus show the great, the very great, reduction
in time taken that occurs when the final Success can be reached
by stages, in which partial successes can be conserved and
accumulated.

11/7. It is difficult to find a real example which shows in one
system the three ways of progression to Success, for few systems
are constructed so flexibly. It is, however, possible to construct,
by the theory of probability, examples which show the differences
referred to. Thus suppose that, as the traffic passes, we note the
final digit on each car's number-plate, and decide that we want
to see cars go past with the final digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, in
that order. If we insist that the ten cars shall pass consecutively,
as in Case 1, then on the average we shall have to wait till about
10,000,000,000 cars have passed: for practical purposes such an
event is impossible. But if we allow success to be achieved by
first finding a ' 0 ', then finding a ' 1 ', and so on until a ' 9 ' is
seen, as in Case 2, then the number of cars which must pass will
be about fifty, and this number makes ' Success ' easily achievable.

152

11/9 THE FULLY -JOIN ED SYSTEM

11/8. A well-known physical example illustrating the difference
is the crystallisation of a solid from solution. When in solution,
the molecules of the solute move at random so that in any given
interval of time there is a definite probability that a given molecule
will possess a motion and position suitable for its adherence to
the crystal. Now the smallest visible crystal contains billions of
molecules : if a visible crystal could form only when all its mole-
cules happened simultaneously to be properly related in position
and motion to one another (Case 1), then crystallisation could
never occur: it would be too improbable. But in fact crystallisa-
tion can occur by succession (Case 2), for once a crystal has begun
to form, a single molecule which happens to possess the right
position and motion can join the crystal regardless of the positions
and motions of the other molecules in the solution. So the crystal-
lisation can proceed by stages, and the time taken resembles T2
rather than 7\.

We may draw, then, the following conclusion. A compound
event that is impossible if the components have to occur simul-
taneously may be readily achievable if they can occur in sequence
or independently.

11/9. It is now becoming apparent in what way the 1,000-unit
Homeostat, which takes such an excessively long time to adapt,
differs from the living organism, which usually gets adapted in a
fraction of its generation-time. The organism, of course, does not
reach its full adult adaptation by making trial after trial, all of
which count for nothing until suddenly everything comes right !
On the contrary, it conforms more to the rules of Cases 2 or 3,
achieving partial successes and then retaining them while improving
what is still unsatisfactory.

However, before we turn to consider the latter cases we should
notice that the 1,000-unit Homeostat is not wholly atypical, for
environments do exist, though they are rare, that demand that
all successes occur simultaneously and against which partial
successes count for nothing. When they occur, it is notorious
that they give the living organism great difficulty. A trifling
example occurs when a trout would take a fly on the surface; the
trout must both break the surface at the correct place and must
also close its jaws at the right moment; two variables here are
essential in the sense that both their values must be within proper

153

DESIGN FOR A BRAIN

11/10

limits for success to be achieved; failure in either respect means
failure totally.

The example par excellence occurs when the burglar, homeo-
statically trying to earn his daily bread by stealing, faces that
particular environment known as the combination lock. This
environment has, of course, been selected to be as difficult for
him as possible; and its peculiar difficulty lies precisely in the fact
that' partial successes — getting, say, six letters right out of seven
— count for nothing. Thus there can be no progression towards
the solution. Thus, confronted with an environment that does
not permit use to be made of partial adaptation, human and
Homeostat fail alike.

Cumulative adaptation

11/10. In terrestrial environments, however, such problems are
not common. Usually the organism has many essential variables,
and also it manages to reach an adaptation of all fairly quickly.
Let us take these apparently contradictory facts as data, and see
what can be deduced from them (following essentially the method
of S. 4/12).

Quite a simple example will suffice to show the point. Let us
suppose that an organism has three essential variables (marked
1, 2 and 3) all affected by the environment, and all able to veto
the stability of the step-mechanisms S, as in Figure 11/10/1.

Let us suppose that the organism has reached the state of being

Figure 11/10/1.
154

11/10 THR FULLY -JOINED SYSTEM

adapted at essential variables 1 and 2. (More precisely: the
disturbance that comes to the environment finds such a set of
values on the step-mechanisms S that in the reaction to it, essential
variables 1 and 2 never go outside their proper limits.) The
reaction does, however, send essential variable 3 outside its limit.

We now take as given that the system will go through a process
like that of S. 7/23, and that after it is over the step-mechanisms
will be changed to new values such that now essential variable 3
is kept within limits, and also that the other two still react in the
same way as before (this being necessary if we are to assume that
adaptations once formed are held, as we wish to).

To see what this implies, let S3 be the set of those step-mechan-
isms that ended with changed values after the trials due to 3's
adaptation (some such there must be, or the change of 3's response
to the disturbance is an effect without a cause). Now apply the
operational test of S. 4/12: as the step-mechanisms in S3 are
changed in value, but the behaviours showing at 1 and 2 are not,
it follows that, whatever is shown in Figure 11/10/1, there can
be no effective channel of communication from the step-
mechanisms S3 (in S) through R and the environment to 1 and 2.

Next, let M12 represent all those parts, in R and the environment,
that play a part in determining how the disturbance eventually
affects 1 and 2. By M3 represent similarly the parts on the channel
from S3 through R and the environment to 3. (Nothing is assumed
about how M12 and M3 are related.) Now M3 cannot be void
(or S3 would have no channel to affect 3, and the final adaptation
of 3 could not have occurred). Similarly, neither can M12 be
void. Finally, the earlier deduction that there is no channel from
S3 (through R and environment) to 1 or 2 implies that there are
no common variables to M12 and M3, nor any channel from M3 to
M12 (for otherwise changes at S3 would show at 1 and 2, contrary
to hypothesis).

It has thus been shown that, for adaptations to accumulate,
there must not be channels from some step-mechanisms (e.g. S3)
to some variables (e.g. M12), nor from some variables (e.g. M3)
to others (e.g. M12). Thus, for the accumulation of adaptations
to be possible the system must not be fully joined. The idea so
often implicit in physiological writings, that all will be well if only
sufficient cross-connexions are available, is, in this .context, quite
wrong.

155

DESIGN FOR A BRAIN 11/11

This is the point. If the method of ultrastability is to succeed
within a reasonably short time, then partial successes must be
retained. For this to be possible it is necessary that certain parts
should not communicate to, or have an effect on, certain other parts.

11/11. Now we can see where the Homeostat was misleading.
From the beginning of the book (from S. 1/7) we have treated the
brain and the environment as richly joined, each within its own
parts and each to the other. Thus the Homeostat was built so
that every unit did in fact interact with every other unit. In this
way we developed the theory of the system that is integrated
totally.

By working throughout with systems which were always assumed
to be as richly connected as the reader cared to think them, we
avoided the possibility of talking much about integration, and
then discussing a mechanism that, in fact, really worked in
separate parts. The reacting parts and the environments that we
have discussed have so far been integrated in the extreme.

11/12. Nevertheless, it is now clear that one can go too far in
this direction. The Homeostat goes too far; it is too well inte-
grated, cannot accumulate adaptations, and thereby takes a quite
un-brainlike time in reaching adaptation. What we must discuss
now is a system similar to the Homeostat in its ultrastability,
but not so richly cross-joined. To what degree, then, should it
be cross- joined if it is to resemble the nervoils system ?

The views held about the amount of internal connexion in the
nervous system — its degree of ' wholeness ' — have tended to range
from one extreme to the other. The ' reflexologists ' from Bell
onwards recognised that in some of its activities the nervous
system could be treated as a collection of independent parts. They
pointed to the fact, for instance, that the pupillary reflex to light
and the patellar reflex occur in their usual forms whether the
other reflex is being elicited or not. The coughing reflex follows
the same pattern whether the subject is standing or sitting. And
the acquirement of a new conditioned response might leave a pre-
viously established response largely unaffected. On the other
hand, the Gestalt school recognised that many activities of the
nervous system were characterised by wholeness, so that what
happened at one point was conditional on what was happening at

156

11/13 THE FULLY-JOINED SYSTEM

other points. The two sets of facts were sometimes treated as
irreconcilable.

Yet Sherrington in 1906 had shown by the spinal reflexes that
the nervous system was neither divided into permanently separated
parts nor so wholly joined that every event always influenced
every other. Rather, it showed a richer, and a more intricate
picture— one in which interactions and independencies fluctuated.
'Thus, a weak reflex may be excited from the tail of the spinal
dog without interference with the stepping-reflex '. . . . ' Two
reflexes may be neutral to each other when both are weak, but
may interfere when either or both are strong '. . . . * But to
show that reflexes may be neutral to each other in a spinal dog
is not evidence that they will be neutral in the animal with its
whole nervous system intact and unmutilated.' The separation
into many parts and the union into a single whole are simply the
two extremes on the scale of ' degree of connectedness '.

Being chiefly concerned with the origin of adaptation and co-
ordination, I have tended so far to stress the connectedness of
the nervous system. Yet it must not be overlooked that adapta-
tion may demand independence as well as interaction. The
learner-driver of a motor-car, for instance, who can only just
keep the car in the centre of the road, may find that any attempt
at changing gear results in the car, apparently, trying to leave
the road. Later, when he is more skilled, the act of changing
gear will have no effect on the direction of the car's travel. Adap-
tation thus demands not only the integration of related activities
but the independence of unrelated activities.

11/13. From now on, therefore, we'will no longer hold the implicit
assumption that all parts are richly joined. This freedom makes
possible various modifications that have not so far been explicitly
considered, and that give new forms of ultrastable system. These
forms are still ultrastable, for they conform to the definition
(which does not involve explicitly the amount of connexion), but
they will not be so excessively slow in becoming adapted as the
simple form of S. 11/2. They do this by developing partial,
fluctuating, and temporary independencies within the whole, so
that the whole becomes an assembly of subsystems within which
communication is rich and between which it is more restricted.
The study of such systems will occupy the remainder of the book.

157

CHAPTER 12

Temporary Independence

12/1. Several times we have used, without definition, the con-
cept of one variable or system being ' independent ' of another.
It was stated that a system, to be state-determined, must be
' properly isolated ' ; and some parameters in S. 6/2 were described
as ' ineffective '. So far the simple method of S. 4/12 has been
adequate, but as it is now intended to treat of systems that are
neither wholly joined nor wholly separated, a more rigorous
method is necessary.

The concept of the ' independence ' of two dynamic systems
might at first seem simple: is not a lack of material connexion
sufficient ? Examples soon show that this criterion is unreliable.
Two electrical parts may be in firm mechanical union, yet if
the bond is an insulator the two parts may be functionally inde-
pendent. And two reflex mechanisms in the spinal cord may be
inextricably interwoven, and yet be functionally independent.

On the other hand, one system may have no material con-
nexion with another and yet be affected by it markedly: the
radio receiver, for instance, in its relation to the transmitter.
Even the widest separation we can conceive — the distance between
our planet and the most distant nebulae — is no guarantee of
functional separation ; for the light emitted by those nebulae is yet
capable of stirring the astronomers of this planet into controversy.
The criterion of physical connexion or separation is thus useless.

12/2. Can we make the test for independence depend on whether
one variable (or system) gives energy or matter to the other ?
The suggestion is plausible, but experience with simple mechanisms
is misleading. When my finger strikes the key of a typewriter,
the movement of my finger determines the movement of the
type; and the finger also supplies the energy necessary for the
type's movement. The diagram

158

12/3 TEMPORARY INDEPENDENCE

would state, in this case, both that energy, measurable in ergs,
is transmitted from A to B, and also that the behaviour of B
is determined by, or predictable from, that of A. If, however,
power is freely available to B, the transmission of energy from
A to B becomes irrelevant to the question of the control exerted.
It is easy, in fact, to devise a mechanism in which the flow of
both energy and matter is from B to A and yet the control is
exerted by A over B. Thus, suppose B contains a compressor
which pumps air at a constant rate into a cylinder, creating a
pressure that is shown on a dial. From the cylinder a pipe goes
to A, where there is a tap which allows air to escape and can
thus control the pressure in the cylinder. Now suppose a stranger
comes along; he knows nothing of the internal mechanism, but
tests the relations between the two variables: A, the position of
the tap, and B, the reading on the dial. By direct testing he soon
finds that A controls B, but that B has no effect on A. The
direction of control has thus no necessary relation to the direction
of flow of either energy or matter when the system is such that
all parts are supplied freely with energy.

Independence

12/3. The test for independence can, in fact, be built up from
the results of primary operations (S. 2/10), without any reference
to other concepts or to knowledge of the system borrowed from
any other source.

The basic definition simply makes formal what was used
intuitively in S. 4/12. To test whether a variable X has an effect
on a variable Y, the observer sets the system at a state, allows
one transition to occur, and notices the value of Y that follows.
(The new value of X does not matter.) He then sets the system
at a state that differs from the first only in the value of X (in
particular, Y must be returned to its original initial state). Again
he allows a transition to occur, and he notices again the value
of Y that results. (He thus obtains two transitions of Y from
two states that differ only in the value of X.) If these two values
of Y are the same, then Y is defined to be independent of X so
far as the particular initial states and other conditions are
concerned.

By dependent we shall mean simply ' not independent '.

159

DESIGN FOR A BRAIN 12/4

This operational test provides the ' atom ' of independence.
Two transitions are needed: the concept of 'independence' is
meaningless with less.

12/4. In general, what happens when the test is applied to one
pair of initial states docs not restrict what may happen if it is
applied to other pairs. The possibility cannot be excluded that
the test may give results varying arbitrarily over the possible
pairs. Often, however, it happens that, for some given value of
all other variables or parameters, Z, Wf . . . , Y is independent
of X for all pairs of initial states that differ only in the value of X.
In this case, for that particular field and for that particular value
of the other variables and parameters, Y is independent of X in a
more extended sense. Provided the field and the initial values
of Y, Z, W, etc., do not change, Y's transition is unaffected by
X's initial value. In this case, Y is independent of X over a
region (in the phase space) represented by a line parallel to the
.Y-axis, ' independent ' in the sense that whenever the representa-
tive point, moving on a line of behaviour, leaves this region, Y
will undergo the same transition.

Sometimes it may happen that Y is independent of X not only
for all values of X but also for all values of the other variables
and parameters — Z, W, etc. In the previous paragraph a change
of Z's value might have changed the field or region so that Y
was no longer independent of X. In the present case, F's transi-
tion (from a uniform Y- value) is the same regardless of the initial
values of X, Z, W, etc. Y is then independent of X unconditionally.

It will be seen that two variables may be ' independent ' to
varying degrees : at two points, over a line, over a region, over the
whole phase-space, over a set of fields. The word is thus capable
of many degrees of application. The definitions given above are
not intended to answer the question (of doubtful validity) ' what
is independence really ? ' but simply to show how this word must
be used if a speaker is to convey an unambiguous message to his
audience. Clearly, the word often needs supplementary specifica-
tion (e.g. does i Y independent of X ' mean l over this field ' or
1 over all fields ' ?); the supplementary specification must then be
given, either by the context or explicitly.

The word ' independent ' is thus similar to the word ' stable ' :
both words are often useful in that they can convey information

160

12/8 TEMPORARY INDEPENDENCE

about a system quickly and easily when the system has a suitable
simplicity and when it is known that the listener will interpret
them suitably. But always the speaker must be prepared, if the
system is not simple, to add supplementary details or even to go
back to a description of the transitions themselves ; here there is
always security, for here the information is complete.

12/5. Because there are various degrees of independence, so that
Y may be independent of X over a small region of the field but
not independent if the same region is extended, it follows that
one system can give a variety of diagrams of immediate effects —
as many as there are ranges and conditions of independence
considered. This implication is unpleasant for us; but we cannot
evade the fact. (Fortunately it commonly happens that the inde-
pendencies in which we are interested give much the same diagram,
so often one diagram will represent all the significant aspects of
independence.)

12/6. So far we have discussed F's independence of X. What-
ever this is, it in no way restricts, in general, whether X is or is
not independent of Y. If X is independent of Y, but Y is not
independent of X, then X dominates F.

12/7. The definition given so far refers to independence between
two variables. It may happen that every variable in a system A
is independent of every variable in a system B, all possible pairs
being considered. We then say that system A is independent of
system B.

Again, such independence does not, in general, restrict the
possibilities whether B is or is not independent of A; A may
dominate B.

12/8. To illustrate the definition's use, and to show that its
answers accord with common experience, here are some examples.
If a bacteriologist wishes to test whether the growth of a micro-
organism is affected by a chemical substance, he prepares two
tubes of nutrient medium containing the chemical in different
concentrations (X) but with all other constituents equal; he seeds
them with equal numbers of organisms ; and he observes how the
numbers (Y) change as time goes on. Then he compares the two

161

DESIGN FOR A BRAIN 12/8

later numbers of organisms after two initial states that differed
only in the concentrations of chemical.

To test whether a state-determined system is dependent on a
parameter, i.e. to test whether the parameter is ' effective ', the
observer records the system's behaviour on two occasions when
the parameter has different values. Thus, to test whether a
thermostat is really affected by its regulator he sets the regulator
at some value, checks that the temperature is at its usual value,
and records the subsequent behaviour of the temperature; then
he returns the temperature to its previous value, changes the
position of the regulator, and observes again. A change of
behaviour implies an effective regulator.

Finally, an example from animal behaviour. Parker tested the
sea-anemone to see whether the behaviour of a tentacle was
independent of its connexion with the body.

4 When small fragments of meat are placed on the tentacles
of a sea-anemone, these organs wind around the bits of food
and, by bending in the appropriate direction, deliver them
to the mouth.'

(He has established that the behaviour is regular, and that the
system of tentacle-position and food-position is approximately
state-determined. He has described the line of behaviour follow-
ing the initial state: tentacle extended, food on tentacle.)

'If, now, a distending tentacle on a quiet and expanded
sea-anemone is suddenly seized at its base by forceps, cut
off and held in position so that its original relations to the
animal as a whole can be kept clearly in mind, the tentacle
will still be found to respond to food brought in contact
with it and will eventually turn toward that side which was
originally toward the mouth.'

(He has now described the line of behaviour that follows an initial
state identical with the first except that the parameter ' con-
nexion with the body ' has a different value. He observed that
the two behaviours of the variable ' tentacle-position ' are identi-
cal.) He draws the deduction that the tentacle-system is, in this
aspect, independent of the body-system:

' Thus the tentacle has within itself a complete neuro-
muscular mechanism for its own responses.'

The definition, then, agrees with what is usually accepted*
Though clumsy in simple cases, it has the advantage in complex

162

12/9

TEMPORARY INDEPENDENCE

cases of providing a clear and precise foundation. By its use
the independencies within a system can be proved by primary
operations only.

12/9. The definition makes 'independence ' depend on how the
system behaves over a single unit of time (over a single step if
changing in steps, or over an infinitesimal time if changing con-
tinuously). The dependencies so defined between all pairs of
variables give, as defined in S. 4/12, the diagram of immediate
effects.

In general, this diagram is not restricted: all geometrically
drawable forms may occur in a wide enough variety of machines.
This freedom, however, is not always possible if we consider the
relation between two variables over an extended period of time.
Thus, suppose Z is dependent on F, and Y dependent on X, so
that the diagram of immediate effects contains arrows:

X may have no immediate effect on Z, but over two steps the
relation is not free; for two different initial values of X will lead,
one step later, to two different values of Y; and these two different
values of Y will lead (as Z is dependent on Y) to two different
values of Z. Thus after two steps, whether X has an immediate
effect on Z or not, changes at X will give changes at Z; and thus
X does have an effect on Z, though delayed.

Another sort of independence is thus possible : whether changes
at X are followed at any time by changes at Z. These relations
can be represented by a diagram of ultimate effects. It must be
carefully distinguished from the diagram of immediate effects. It
is related to the latter in that it can be formed by taking the
diagram of immediate effects and adding further arrows by the
rule that if any two arrows are joined head to tail,

Y s

163

DESIGN FOR A BRAIN

a third arrow is added from tail to head, thus

12/10

X > z

The rule is applied repeatedly till no further addition of arrows
is possible. Thus the diagram of immediate effects I in Figure
12/9/1 would yield the diagram of ultimate effects II.

>- 2

=^3

Figure 12/9/1.

The diagram of ultimate effects shows at once the dependencies
in the case when we allow time for the effects to work round the
system. Thus from II of the Figure we see that variable 1 is
permanently independent of 2, 3, and 4, and that the latter three
are all ultimately dependent on each other.

The effects of constancy

12/10. Suppose eight variables have been joined, by the method
of S. 6/6, to give the diagram of immediate effects shown in
Figure 12/10/1. We now ask: what behaviour at the three

ABC
Figure 12/10/1.

variables in B will make A and C independent, in the ultimate
sense, and also leave both A and C state-determined ? That is,
what behaviour at B will sever the whole into independent parts,
giving the diagram of immediate effects of Figure 12/10/2:

164

12/11 TEMPORARY INDEPENDENCE

A C

Figure 12/10/2.

The question has not only theoretical but practical importance.
Many experiments require that one system be shielded from effects
coming from others. Thus, a system using magnets may have to
be shielded from the effects of the earth's magnetism ; or a thermal
system may have to be shielded from the effects of changes in
the atmospheric temperature; or the pressure which drives blood
through the kidneys may have to be kept independent of changes
in the pulse-rate.

A first suggestion might be that the three variables B should
be removed. But this conceptual removal corresponds to no
physical reality: the earth's magnetic field, the atmospheric
temperature, the pulse-rate cannot be 4 removed '. In fact the
answer is capable of proof (S. 22/14): that A and C should be
independent and state-determined it is necessary and sufficient that
the variables B should be null-functions. In other words, A and C
must be separated by a wall of constancies.

It also follows that if the variables B can be sometimes fluctu-
ating and sometimes constant (i.e. if they behave as part-functions),
then A and C can be sometimes functionally joined and sometimes
independent, according to B's behaviour.

12/11. Here are some illustrations to show that the theorem
accords with common experience.

(a) If A (of Figure 12/10/1) is a system in which heat-changes
are being studied, B the temperatures of the parts of the con-
tainer, and C the temperatures of the surroundings, then for A
to be isolated from C and state-determined, it is necessary and
sufficient for the .B's to be kept constant, (b) Two electrical
systems joined by an insulator are independent, if varying slowly,
because electrically the insulator is unvarying, (c) The centres
in the spinal cord are often made independent of the activities in
the brain by a transection of the cord; but a break in physical
continuity is not necessary: a segment may be poisoned, or

165

DESIGN FOR A BRAIN

12/12

anaesthetised, or frozen; what is necessary is that the segment
should be unvarying.

Physical separation, already noticed to give no certain inde-
pendence, is sometimes effective because it sometimes creates an
intervening region of constancy.

12/12. The example of Figure 12/10/1 showed one way in which
the behaviour of a set of variables, by sometimes fluctuating and
sometimes being constant, could affect the independencies within
a system. The range of ways is, however, much greater.

To demonstrate the variety we need a rule by which we can
make the appropriate modifications in the diagram of ultimate
effects when one or more of the variables is held constant. The
rule is: — Take the diagram of immediate effects. If a variable V
is constant, remove all arrows whose heads are at V; then,
treating this modified diagram as one of immediate effects, com-
plete the diagram of ultimate effects, using the rule of S. 12/9.
The resulting diagram will be that of the ultimate effects when

lixiixu/i

« 4-f==r3 4 + 3 4

Figure 12/12/1 : If a four- variable system has the diagram of immediate
effects A, and if 1 and 2 are part-functions, then its diagram of ultimate
effects will be B, C, D or E as none, 1, 2, or both 1 and 2 become inactive,
respectively.

V is constant. (It will be noticed that the effect of making V
constant cannot be deduced from the original diagram of ultimate
effects alone.) Thus, if the system of Figure 12/12/1 has the
diagram of immediate effects A, then the diagram of ultimate
effects will be B, C, D or E according as none, 1, 2, or both 1 and
2 are constant, respectively.

It can be seen that with only four variables, and with only
two of the four possibly becoming constant, the patterns of inde-

166

12/14 TEMPORARY INDEPENDENCE

pendence show a remarkable variety. Thus, in C, 1 dominates
3; but in D, 3 dominates 1. As the variables become more
numerous so does the variety increase rapidly.

12/13. The multiplicity of inter-connexions possible in a tele-
phone exchange is due primarily to the widespread use of
temporary constancies. The example serves to remind us that
' switching ' is merely one of the changes producible by a re-
distribution of constancies. For suppose a system has the

->•

>.D

^«

Figure 12/13/1.

diagram of immediate effects shown in Figure 12/13/1. If an
effect coming from C goes down the branch AD only, then, for
the branch BE to be independent, B must be constant. How the
constancy is obtained is here irrelevant. When the effect from
C is to be ' switched ' to the BE branch, B must be freed and A
must become constant. Any system with a i switching ' process
must use, therefore, an alterable distribution of constancies.
Conversely, a system whose variables can be sometimes fluctuating
and sometimes constant is adequately equipped for switching.

The effects of local stabilities

12/14. The last few sections have shown how important, in any
system that is to have temporary independencies, are variables
that temporarily go constant. As such- variables play a funda-
mental part in what follows, let us examine them more closely.

Any subsystem (including the case of the single variable) that
stays constant is, by definition, at a state of equilibrium. If the
subsystem's surrounding conditions (parameters) are constant, the
subsystem evidently has a state of equilibrium in the corresponding
field ; if it stays constant while its parameters are changing, then
that state is evidently one of equilibrium in all the fields occurring.
Thus, constancy in a subsystem's state implies that the state is

167

DESIGN FOR A BRAIN 12/15

one of equilibrium; and constancy in the presence of small impul-
sive disturbances implies stability.

The converse is also > true. If a subsystem is at a state of
equilibrium, then it will stay at that state, i.e. hold a constant
value (so long as its parameters do not change value).

Constancy, equilibrium, and stability are thus closely related.

12/15. Are such variables (or subsystems) common? Later
(S. 15/2) it will be suggested that they are extremely common,
and examples will be given. Here we can notice two types that
are specially worth notice.

One form, uncommon perhaps in the real world but of basic
importance as a type-form in the strategy of S. 2/17, is that in
which the subsystem has a definite probability p that any particu-
lar state, selected at random, is equilibrial. We shall be con-
cerned with this form in S. 13/2. (In explanation, it should be
mentioned that the sample space for the probabilities is that given
by a set of subsystems, each a machine with input and therefore
determinate in whether a given state, with given input-value, is
or is not equilibrial.) The case would arise when the observer
faced a subsystem that was known (or might reasonably be
assumed) to be a determinate machine with input, but did not
know which subsystem, out of a possible set, was before him ; the
sample space being provided by the set suitably weighted, the
observer could legitimately speak of the probability that this
system, at this state, and with this input, should be in equilibrium.

The other form, very much commoner, is that which shows
8 threshold ', so that all states are equilibrial when some para-
metric function is less than a certain value, and few or none are
equilibrial when it exceeds that value. Well-known examples are
that a weight on the ground will not rise until the lifting force
exceeds a certain value, and a nerve will not respond with an
impulse until the electric intensity, in some form, rises above a
certain value.

What is important for us here is to notice that threshold, by
readily giving constancy, can readily give what is necessary for
the connexions between variable and variable to be temporary.
Thus the changes in the diagram of Figure 12/12/1 could readily
be produced by parts showing the phenomenon of threshold.

168

12/18

TEMPORARY INDEPENDENCE

12/16. These deductions can now be joined to those of S. 12/10.
If three subsystems are joined so that their diagram of immediate
effects is

and if B is at a state that is equilibrial for all values coming from
A and C, then A and C are (unconditionally) independent. Thus,
Z?'s being at a state of equilibrium severs the functional connexion
between A and C.

Suppose now that 2?'s states are equilibrial for some states of
A and C, but not for others. As A and C, on some line of behaviour
of the wrhole system, pass through various values, so will they
(according to whether 2?\s state at the moment is equilibrial or
not) be sometimes dependent and sometimes independent.

Thus we have achieved the first aim of this chapter: to make
rigorously clear, and demonstrable by primary operations, what
is meant by ' temporary functional connexions ', when the control
comes from factors within the system, and not imposed arbitrarily
from outside.

12/17. The same ideas can be extended to cover any system as
large and as richly connected as we please. Let the system consist
of many parts, or subsystems, joined as in S. 6/6, and thus pro-,
vid'ed with basic connexions. If some of the variables or sub-
systems are constant for a time, then during that time the con-
nexions through them are reduced functionally to zero, and the
effect is as if the connexions had been severed in some material
way during that time.

If a high proportion of the variables go constant, the severings
may reach an intensity that cuts the whole system into subsystems
that are (temporarily) quite independent of one another. Thus a
whole, connected system may, if a sufficient proportion of its
variables go constant, be temporarily equivalent to a set of un-
connected subsystems. Constancies, in other words, can cut a
system to pieces. (I. to C, S. 4/20, gives an illustration of the fact.)

12/18. The field of a state-determined system whose variables
often go constant has only the peculiarity that the lines of
behaviour often run in a sub-space orthogonal to the axes. Thus,

169

DESIGN FOR A BRAIN

12/18

over an interval in which all variables but one are constant, the
corresponding line of behaviour must run as a straight line parallel
to the axis of the variable that is changing. If all but two are

inactive (along some line of behaviour),
that line in the phase-space may curve
but it must remain in the two-dimen-
sional plane parallel to the two corre-
sponding axes; and so on. If all the
variables are constant, the line
naturally becomes a point — at the
state of equilibrium. Thus a three-
variable system might give the line of
behaviour shown in Figure 12/18/1.

In the interval before they reach
equilibrium, such variables will, of
course, behave as part-functions.
Through the remaining chapters they
will show their importance. For convenience of description, a
part-function (described in time by a variable) will be said to be
active or inactive (at a given point on a line of behaviour)
according to whether the variable is changing or remaining
constant.

Figure 12/18/1. In the dif-
ferent stages the active
variables are : A, y ; B, y
and 2 ; C,z; D x ; E y ;
F x and z.

170

CHAPTER 13

The System with Local Stabilities

13/1. Having examined what is meant by a system that has
' partial, fluctuating, and temporary independencies within the
whole ' we can now consider some of the properties that a system
of such a type will show in its behaviour.

In saying 4 a system of such a type ' we have not, of course,
defined a system with precision: we have only defined a set or
class of systems. How shall we achieve precision ? Two ways
are open to us.

One way is to add further details until we have defined a parti-
cular system with full precision, so that its behaviour is deter-
minate and uniquely defined; we then follow the behaviour in all
detail. Such a study would give us an exact conclusion, but it
would give us far more detail than we require, or can conveniently
handle, in the remaining chapters.

Another way is to talk about such systems ' in general '. Here
nothing is easier than to relax our grasp and to talk vaguely about
what will ' usually ' happen, regardless of the fact that whether
particular properties (such as linearity, or the presence of thres-
hold) are ' usually ' present differs widely in the systems of the
sociologist, the neurophysiologist, and the physicist. Rigour and
precision are possible while speaking of systems ' in general '
provided two requirements are met: the set of systems under
discussion must be defined precisely, and statements made must
be precise statements about the properties of the set. In other
words, we give up the aim of being precise about the individual
system, and accept the responsibility of being precise about the
set. This second way is the method we shall largely follow in the
remaining chapters.

Having changed to the new aim, we shall often find that the
argument about the set is conducted most readily in terms of
some individual system that is followed in detail; when this
happens, the individual system must be understood to have

171

DESIGN FOR A BRAIN 13/2

importance only as a typical, generic, or ' random ' element of the
set it belongs to. Though the argument will often appear verbally
to be focused on an individual system, it is directed really at the
properties of the set, the individual system being introduced only
as a means to an end.

We shall have much to do, in what follows, with systems con-
structed in some 4 random ' way. The word will always mean that
we are discussing some generic system so as to find its typical
properties, and thus to arrive at some precise deduction about the
defined set of systems.

13/2. A set of systems of special importance for the later
chapters is the set of those systems that are made of parts that
have a high proportion of their states equilibrial, and are made
by the parts being joined at random.

More precisely, assume that we have before us a very great
number of parts, assumed to be fairly homogeneous, so that there
is a defined ' universe ', or distribution, of them. Each is assumed
to be state-determined, and thus to have in it no randomness
whatever. As a little machine with input, if it is at a certain state
and in certain conditions it will do a certain thing; and it will do
this thing whenever the state and conditions recur.

We now take a sample of these parts by some clearly defined
sampling process and thus arrive at some particular set of parts.
(It is not assumed that all parts have an equal probability of being
taken.) Again we take a sample from the possible ways of
joining them, taking it by a clearly defined sampling process, and
thus arrive at some one way of joining them.

The particular set of parts, joined in the particular way, now
gives the final system.

This particular final system, be it noticed, is state-determined.
It is not stochastic in the sense of being able, from a given state
and in given conditions, to undergo various transitions with
various probabilities. Thus the particular system is not random
at all. The randomness enters with the observer or experi-
menter; he is little interested in the particular system taken by
the sampling, but is much interested in the population from
which the particular system has come, as the neurophysiologist is
interested in the set of mammalian brains. The ' randomness '
comes in because the observer faces a system that interests him

172

13/4

THE SYSTEM WITH LOCAL STABILITIES

only because it is typical of the set. With the population as his
sample space (derived from the two primary sample spaces) he
may then legitimately speak of the probability of the system
showing a certain event, or having a certain property.

If to this specification we add the restriction that the original
parts are rich in states of equilibrium (e.g. as in S. 12/15), we get a
type of system that will be referred to frequently in what follows.
For lack of a better name I shall call it a polystable system.
Briefly, it is any system whose parts have many equilibria and
that has been formed by taking parts at random and joining them
at random (provided that these words are understood in the exact
sense given above).

Definitions can only be justified ultimately, however, by their
works. The remainder of the book will demonstrate something
of the properties of this interesting type of system, a key-system
in the strategy of S. 2/17.

13/3. In such demonstrations we shall not be discussing one
particular system, specified in all detail: we shall be discussing a
set. When a set is discussed we must be careful to keep an
important distinction in mind, and we must make the distinction
arbitrarily: (1) are we discussing what can happen? — a question
which focuses attention on the extreme possibilities, and therefore
on the rare and exceptional; or (2) are we discussing what usually
happens? — which focuses attention on the central mass of cases,
and therefore on the common and ordinary. Both questions have
their uses ; but as the answers are often quite different, we must be
careful not to confuse them.

13/4. A property shown by all state-determined systems, and
one that will be important later is the following. In a state-
determined system, if a subsystem has been constant and then
commences to show changes in its variables, we can deduce that

A B

173

DESIGN FOR A BRAIN 13/5

among its parameters must have been, when it started changing,
at least one that was itself changing. Picturesquely one might
say that change can come only from change. The reason is not
difficult to see. If variable or subsystem C is affected immediately
only by parameters A and B, and if A and B are constant over
some interval, and if, within this interval, C has gone from a state
c to the same state (i.e. if c is a state of equilibrium), then for
C to be consistent in its behaviour it must continue to repeat the
transition ' c to c ' so long as A and B retain their values, i.e. so
long as A and B remain constant. If C is state-determined, a
transition from c to some other state can occur only after A or B,
for whatever reason, has changed its value.

Thus a state-determined subsystem that is at a state of equili-
brium and is surrounded by constant parameters (variables of
other subsystems perhaps) is, as it were, trapped in equilibrium.
Once at the state of equilibrium it cannot escape from it until an
external source of change allows it to change too. The sparks
that wander in charred paper give a vivid example of this property,
for each portion, even though combustible, is stable when cold;
one spark can become two, and various events can happen, but a
cold portion cannot develop a spark unless at least one adjacent
point has a spark. So long as one spark is left we cannot put
bounds to what may happen ; but if the whole should reach a state
of l no sparks ', then from that time on it is unchanging.

Progression to equilibrium

13/5. Let us now consider how a polystable system will move
towards its final state of equilibrium. From one point of view
there is nothing to discuss, for if the parts are state-determined
and the joining defined, the whole is a state-determined system
that, if released from an initial state, will go to a terminal cycle or
equilibrium by a line of behaviour exactly as in any other case.
The fact, however, that the polystable system has parts with
many equilibria, which will often stay constant for a time, adds
special features that deserve attention; for, as will be seen later,
they have interesting implications in the behaviours of living
organisms.

13/6. A useful device for following the behaviour of these some-
what complex wholes is to find the value of the following index.

174

13/8 THE SYSTEM WITH LOCAL STABILITIES

At any given moment, the whole system is at a definite state, and
therefore so is each variable ; the state of each variable either is or
is not a state of equilibrium for that variable (in the conditions
given by the other variables). The number of variables that are
at a state of equilibrium will be represented by i. If the whole is
of n variables then obviously i must lie in the range of 0 to n.
If i equals n, then every variable is at a state of equilibrium
in the conditions given by the others, so the whole is at a state of
equilibrium (S. 6/8). If i is not equal to n, the other variables,
n — i in number, will change value at the next step in time. A
new state of the whole will then occur, and i will have a new value.
Thus, as the whole moves along a line of behaviour, i will change
in value ; and we can get a useful insight into the behaviour of the
whole by considering how i will behave as time progresses.

13/7. The behaviour of i is strictly determinate once the system
and its initial state have been given. In a set of systems, how-
ever, the behaviour of i is difficult to characterise except at the
two extremes, where its behaviour is simple and clear. Com-
parison of what happens at the extremes will give us an insight
that will be invaluable in the later chapters, for it will go a long
way towards answering the fundamental problem of Chapter 11.
(By establishing what happens in the two specially simple and
clear cases we are following the strategy of S. 2/17.)

13/8. At one extreme is the polystable system that has been
joined very richly, so that almost every variable is joined to almost
every other. (Such a system's diagram of immediate effects
would show that almost all of the n(n — 1) arrows were present.)
Let us consider the case in which, as in S. 12/15, every subsystem
has a high probability p of being at a state of equilibrium, and in
which the probabilities are all independent. How will * behave ?
(Here we want to know what will usually happen; what can
happen is of little interest.)

The probability of each part being at a state of equilibrium is
p, and so, if independence (of probability) holds, the probability
that the whole (of n variables) is at a state of equilibrium will be
pn (by S. 6/8). If p is not very close to 1, and n is large, this
quantity will be extremely small (S. 11/4). i will usually have
a value not far from np (i.e. about a fraction p of the total will be

175

DESIGN FOR A BRAIN 13/9

at equilibrium at any moment). Then the line of behaviour
will perform a sort of random walk around this value, the whole
reaching a state of equilibrium if and only if i should chance on
the extreme value of n. Thus we get essentially the same picture
as we got in S. 11/3: a system whose lines of behaviour are long
and complex, and whose chance of reaching an equilibrium in a
fairly short time is, if n is large, extremely small. In this case
the time taken by the whole to arrive at a state of equilibrium
will be extremely long, like 2\ of S. 11/5.

13/9. Particularly worth noting is what happens if i should
happen to be large but not quite equal to n. Suppose, for in-
stance, i were 999 in a 1,000-variable system of the type now being
considered. The whole is now near to equilibrium, but what will
happen ? One variable is not at equilibrium and will change.
As the system is richly connected, most of the 999 other variables
will, at the next instant (or step), find themselves in changed
conditions; whether the state each is at is now equilibrial will
depend on factors such that (by hypothesis) 999p will still be
equilibrial, and thus i is likely to drop back simply to its average
value. Thus the richly-joined form of the polystable system,
even if it should get very near to equilibrium (in the sense that
most of its parts are so) will be unable to retain this nearness but
will almost certainly fall back to an average state. Such a system
is thus typically unable to retain partial or local successes.

13/10. With the number n still large, and the probabilities p
still independent, contrast the behaviour of the previous section
(in which the system was assumed to be richly or completely
joined) with that of the polystable system in which the primary
joins between variables are scanty. (A similar system also occurs
if p is made very near to 1 ; for, by S. 12/17, as most of the variables
will be at states of equilibrium, and thus constant for most of the
time, the functional connexions will also be scanty.) How will i
behave in this case, especially as the scantiness approaches its
limit ?

Consider the case in which the scantiness has actually reached
its limit. The system is now identical with one of n variables
that has no connexions between any of them; it is a ' system '
only in the nominal sense. In it, any part that comes to a state

170

13/12 THE SYSTEM WITH LOCAL STABILITIES

of equilibrium must remain there, for no disturbance can come
to it. So if two states of the whole, earlier and later, are com-
pared, all parts contributing to i in the earlier will contribute in
the later; so the value of i cannot fall with time. It will, of course,
usually increase. Thus, this type of system goes to its final state
of equilibrium progressively. Its progression, in fact, is like that
of Case 3 of S. 11/5; for the final equilibrium has only to wait for
the part that takes longest. The time that the whole takes will
therefore be like T3, and thus not excessively long.

13/11. The two types of polystable system are at opposite poles,
and systems in the real world will seldom be found to correspond
precisely with either. Nevertheless, the two types are important
by the strategy of S. 2/17, for they provide clear-cut types with
clear-cut properties; if a real system is similar to either, we may
legitimately argue that its properties will approximate to those
of the nearer.

Polystable systems midway between the two will show a some-
what confused picture. Subsystems will be formed (e.g. as in
S. 12/17) with kaleidoscopic variety and will persist only for short
times ; some will hold stable for a brief interval, only to be changed
and to disintegrate as delimitable subsystems. The number of
variables stable, i, will keep tending to climb up, as a few sub-
systems hold stable, only to fall back by a larger or smaller
amount as they become unstable. Oscillations will be large, until
one swing happens to land i at the value n, where it will stick.

More interesting to us will be the systems nearer the limit of
disconnexion, when i's tendency to increase cumulatively is better
marked, so that i, although oscillating somewhat and often slipping
back a little, shows a recognisable tendency to move to the value
n. This is the sort of system that, after the experimenter has
seen i repeatedly return to n after displacement, is apt to make
him feel that i is ' trying ' to get to n. '

13/12. So far we have discussed only the first case of S. 12/15;
what if the polystable system were composed of parts that all had
their states of equilibrium characterised by a threshold ? This
question will specially interest the neurophysiologist, though it
will be of less interest to those who are intending to work with
adapting systems of other types.

177

DESIGN FOR A BRAIN 13/13

The presence of threshold precludes the previous assumption of
independence in the probabilities; for now a variable's chance of
being at a state of equilibrium will vary in some correspondence
with the values of the variable's parameters. In the case of two
or more neurons, the correspondence will be one way if the effect
is excitatory, and inversely if it is inhibitory. (If there is a mixture
of excitatory and inhibitory modes, the outcome may be an
approximation to the independent form.) To follow the subject
further would lead us into more detail than is appropriate in
this survey; and at the present time little can be said on the
matter.

13/13. To sum up: The polystable system, if composed of
parts whose states of equilibrium are distributed independently
of the states of their inputs, goes to a final equilibrium in a way
that depends much on the amount of functional connexion.

When the connexion is rich, the line of behaviour tends to be
complex and, if n is large, exceedingly long; so the whole tends to
take an exceedingly long time to come to equilibrium. When
the line meets a state at which an unusually large number of the
variables are stable, it cannot retain the excess over the average.

When the connexion is poor (either by few primary joins or by
many constancies in the parts), the line of behaviour tends to be
short, so that the whole arrives at a state of equilibrium soon.
When the line meets a state at which an unusually large number
of the variables are stable, it tends to retain the excess for a time,
and thus to progress to total equilibrium by an accumulation of
local equilibria.

Dispersion
13/14. The polystable system shows another property that
deserves special notice.

Take a portion of any line of behaviour of such a system. On
it we can notice, for every variable, whether it did or did not
change value along the given portion. Thus, in Figure 12/18/1,
in the portion indicated by the letters B and C both y and z
change but x does not. In the portion indicated by F, x and ~
change but y does not. By dispersion I shall refer to the fact that
the active variables (y and z) in the first portion are not identical
with those (x and z) of the second. (In the example the portions

178

13/16 THE SYSTEM WITH LOCAL STABILITIES

come from the same line, but the two portions may also come
from different lines.) As the example shows, it is not implied
that the two sets shall contain no common element, only that the
two sets are not identical.

The importance of dispersion will be indicated in S. 13/17.
Here we should notice the essential feature: though the two por-
tions may start from points that differ only in one, or a few,
variables (as in S. 12/3) the changes that result may distribute
the activations (S. 12/18) to different sets of variables, i.e. to
different places in the system. Thus, the important phenomenon
of different patterns (or values) at one place leading to activations
in different places in the system demands no special mechanism :
any polystable system tends to show it.

13/15. If the two places are to have minimal overlap, and if the
system is not to be specially designed for the separation of parti-
cular patterns of input, then all that is necessary is that the parts
should have almost all their states equilibrial. Then the number
active will be few; if the fraction of the total number is usually
about r, and if the active variables are distributed independently,
the fraction that will be common to the two sets (i.e. the overlap)
will be about r2. This quantity can be as small as one pleases by
a sufficient reduction in the value of r, which can be done by
making the parts such that the proportion of states equilibrial is
almost 1. Thus the polystable system may respond, to two
different input states, with two responses on two sets of variables
that have only small overlap.

13/16. It will be proposed later that dispersion is used widely
in the nervous system. First we should notice that it is used
widely in the sense-organs.

The fact that the sense-organs are not identical enforces an
initial dispersion. Thus if a beam of radiation of wave-length
0*5 ii is directed to the face, the eye will be stimulated but not
the skin ; so the optic nerve will be excited but not the trigeminal.
But if the wave-length is increased beyond 0-8 yu, the excitation
changes from the optic nerve to the trigeminal. Dispersion has
occurred because a change in the stimulus has moved the excita-
tion (activity) from one set of anatomical elements (variables)
to another.

179

DESIGN FOR A BRAIN 13/17

In the skin are histologically-distinguishable receptors sensitive
to touch, pain, heat, and cold. If a needle on the skin is changed
from lightly touching it to piercing it, the excitation is shifted
from the ' touch ' to the ' pain ' type of receptor; i.e. dispersion
occurs.

Whether a change in colour of a stimulating light changes
the excitation from one set of elements in the retina to another
is at present uncertain. But dispersion clearly occurs when the
light changes its position in space; for, if the eyeball does not
move, the excitation is changed from one set of elements to
another. The lens is, in fact, a device for ensuring that disper-
sion occurs: from the primitive light-spot of a Protozoon dis-
persion cannot occur.

It will be seen therefore that a considerable amount of dis-
persion is enforced before the effects of stimuli reach the central
nervous system: the different stimuli not only arrive at the
central nervous system different in their qualities but they often
arrive by different paths, and excite different groups of cells.

13/17. The sense organs evidently have as an important function
the achievement of dispersion. That it occurs or is maintained
in the nervous system is supported by two pieces of evidence.

The fact that neuronic processes frequently show threshold,
and the fact that this property implies that the functioning
elements will often be constant (S. 12/15) suggest that dispersion
is bound to occur, by S. 12/16.

More direct evidence is provided by the fact that, in such cases
as are known, the tracts from sense-organ to cortex at least
maintain such dispersion as has occurred in the sense organ.
The point-to-point representation of the retina on the visual
cortex, for instance, ensures that the dispersion achieved in the
retina will at least not be lost. Similarly the point-to-point
representation now known to be made by the projection of the
auditory nerve on the temporal cortex ensures that the dispersion
due to pitch will also not be lost. There are therefore good
reasons for believing that dispersion plays an important part in
the nervous system.

180

13/19 THE SYSTEM WITH LOCAL STABILITIES

Localisation in the polystable system

13/18. How will responses to a stimulus be localised in a poly-
stable system? — how will the set of the active variables be dis-
tributed over the whole set ?

In such a system, the reaction to a given stimulus, from a given
state, will be regular and reproducible, for the whole is state-
determined. To this extent its behaviour is lawful. But when
the observer notices which variables have shown the activity it
will probably seem lawless, for the details of where the activation
spreads to have been determined by the sampling process, and
the activated variables will probably be scattered over the system
apparently haphazardly (subject to S. 13/4). Thus the question
1 Is the reaction localised ? ' is ambiguous, for two different
answers can be given. In the sense that the activity is restricted
to certain variables of the whole system, the answer is ' yes ' ; but
in the sense that these variables occur in no simply describable
way, the answer is ' no '.

An illustration that may be helpful is given by the distribution
over a town of the chimneys that ' smoke ' (suffer from a forced
down-draught) when the wind blows from a particular direction.
The smoking or not of a particular chimney will be locally deter-
minate ; for a wind of a particular force and direction, striking the
adjacent roofs at a particular angle, will regularly produce the
same eddies, which will determine the smoking or not of the
chimney. But geographically the smoking chimneys are not
distributed with any simple regularity ; for if a plan of the town is
marked with a black dot for every chimney that smokes in an
east wind, and a red dot for every one that smokes in a west wind,
the black and red dots will probably be mixed irregularly. The
phenomenon of ' smoking ' is thus determined in detail yet
distributed geographically at random.

13/19. Such is the ' localisation ' shown by a polystable system.
In so far as the brain, and especially the cerebral cortex, cor-
responds to the polystable, we may expect it to show ' localisation '
of the same type. On this hypothesis we would expect the brain
to behave as follows.

The events in the environment will provide a continuous stream
of information which will pour through the sense organs into the

181

DESIGN FOR A BRAIN 13/19

nervous system. The set of variables activated at one moment
will usually differ from the set activated at a later moment;
and the activity will spread and wander with as little apparent
orderliness as the drops of rain that run, joining and separating,
down a window-pane. But though the wanderings seem dis-
orderly, the whole is reproducible and state-determined ; so that if
the same reaction is started again later, the same initial stimuli
will meet the same local details, will develop into the same patterns,
which will interact with the later stimuli as they did before, and
the behaviour will consequently proceed as it did before.

This type of system would be affected by removals of material
in a way not unlike that demonstrated by many workers on the
cerebral cortex. The works of Pavlov and of Lashley are typical.
Pavlov established various conditioned responses in dogs, removed
various parts of the cerebral cortex, and observed the effects on
the conditioned responses. Lashley taught rats to run through
mazes and to jump to marked holes, and observed the effects of
similar operations on their learned habits. The results were
complicated, but certain general tendencies showed clearly.
Operations involving a sensory organ or a part of the nervous
system first traversed by the incoming impulses are usually
severely destructive to reactions that use that sensory organ.
Thus, a conditioned response to the sound of a bell is usually
abolished by destruction of the cochleae, by section of the auditory
nerves, or by ablation of the temporal lobes. Equally, reactions
involving some type of motor activity are apt to be severely upset
if the centre for this type of motor activity is damaged. But
removal of cerebral cortex from other parts of the brain gave
vague results. Removal of almost any part caused some dis-
turbance, no matter from where it was removed or what type of
response or habit was being tested; and no part could be found
whose removal would destroy the response or habit specifically.

These results have offered great difficulties to many theories of
cerebral mechanisms, but are not incompatible with the theory
put forward here. For in a large polystable system the whole
reaction will be based on activations that are both numerous and
widely scattered. And, while any exact statement would have
to be carefully qualified, we can see that, just as England's paper-
making industry is not to be stopped by the devastation of any
single county, so a reaction based on numerous and widely

182

13/20 THE SYSTEM WITH LOCAL STABILITIES

scattered elements will tend to have more immunity to localised
injury than one whose elements are few and compact.

13/20. Lashley had noticed this possibility in 1929, remarking
that the memory-traces might be localised individually without
conflicting with the main facts, provided there were many traces
and that they were scattered widely over the cerebral cortex,
unified functionally but not anatomically. He did not, how-
ever, develop the possibility further; and the reason is not far
to seek when one considers its implications.

Such a localisation would, of course, be untidy; but mere un-
tidiness as such matters little. Thus, in a car factory the spare
parts might be kept so that rear lamps were stored next to radia-
tors, and ash-trays next to grease guns; but the lack of obvious
order would not matter if in some way every item could be
produced when wanted. More serious in the cortex are the effects
of adding a second reaction; for merely random dispersion provides
no means for relating their locations. It not only allows related
reactions to activate widely separated variables, but it has no
means of keeping unrelated reactions apart; it even allows them
to use common variables. We cannot assume that unrelated
reactions will always differ sufficiently in their sensory forms to
ensure that the resulting activations stay always apart, for two
stimuli may be unrelated yet closely similar. Nor is the differen-
tiation trivial, for it includes the problem of deciding whether a
few vertical stripes in a jungle belong to some reeds or to a tiger.

Not only does random dispersion lead to the intermingling of sub-
systems, with abundant chances of random interaction and con-
fusion, but even more confusion is added with every fresh act of
learning. Even if some order has been established among the
previous reactions, each addition of a new reaction is preceded
by a period of random trial and error which will necessarily cause
the changing of step-mechanisms which were already adjusted to
previous reactions, which will be thereby upset. At first sight,
then, such a system might well seem doomed to fall into chaos.
Nevertheless, I hope to show, from S. 15/4 on, that there are
good reasons for believing that its tendency will actually be
towards ever-increasing adaptation.

183

CHAPTER 14

Repetitive Stimuli and Habituation

14/1. This chapter continues the study of the polystable system
but is something of a digression; and the reader who proceeds
directly to Chapter 15 will lose nothing of the logical thread.
Nevertheless, it has been included for two reasons.

The first is that it will give us practice in understanding the
polystable system, and will show how systems of that type can
be discussed in terms that are both general and precise.

A second reason is that it gives another example of the thesis
that pervades the book : — -When a system ' runs to equilibrium '
one's first impression is that what is interesting has now come to
an end — an impression that is often valid when the system is
simple, and the equilibrium that of a run-down clock. What has
been largely overlooked, and what this book attempts to display,
is that when a complex system runs to equilibrium, the equilibrium
necessarily implies a complex relationship between the states of
the various parts. When the relationship between the states of
the parts is examined, they will show unusual and striking
features, features that are of peculiar interest to the student of
behaviour. Thus Chapter 8, on the Homeostat, showed ' only ' a
system running to equilibrium (e.g. Figure 8/5/1); yet because the
system and conditions were structured, interesting relations could
be traced between the actions and interactions of the various
parts at and around the terminal equilibrium. These relations
are what we identified in Chapter 5 as ' adaptation '. The present
chapter will give another example of how a system, ' merely '
running to equilibrium under a complex repetitive input, also
produces behaviour of psychological and physiological interest.

14/2. First a definition. When there are many states of equili-
brium in a field, then as every line of behaviour must end at some
state of equilibrium (or cycle), the lines of behaviour collect into
sets, such that the lines in one set come to one common termina-

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14/3 REPETITIVE STIMULI AND HABITUATION

tion (cycle or state of equilibrium). The whole field is thus
divisible into regions (each a confluent) such that each region
contains one and only one state of equilibrium or cycle, to which
every line of behaviour in it eventually comes. The chief pro-
perty of a confluent is that the representative point, if released
from any point within it, (a) cannot leave the confluent, (b) will
go to the state of equilibrium or cycle, where it will remain so
long as the parametric conditions persist.

The division of the whole field into confluents is not peculiar to
machines of special type, but is common to all systems that are
state-determined and that have more than one state of equilibrium
or cycle.

Habituation

14/3. Consider now what will happen if a polystable system be
subjected to an impulsive (S. 6/5) stimulus S repetitively, the
stimulus being unvarying, and with intervals between its applica-
tions sufficiently long for the system to come to equilibrium
before- the next application is made.

By S. 6/5, the stimulus S, being impulsive, will displace the
representative point from any given state to some definite state.
Thus the effect of S (acting on the representative point at a state
of equilibrium by the previous paragraph) is to transfer it to some

Figure 14/3/1 : Field of system with twelve confluents, each containing a
state of equilibrium (shown as a dot), or a cycle (X at the left). The
arrows show the displacements caused by S when it is applied to the
representative point at any state of equilibrium or on X.

185

DESIGN FOR A BRAIN 14/4

definite state in the field and there to release it. The possibilities
sketched in Figure 14/3/1 will illustrate the process sufficiently.

Suppose the system is in equilibrium at A. S is applied; its
effect is to move the representative point to the end of the arrow,
in this example moving it into another confluent. The system is
now, by hypothesis, left alone until it has settled : this means that
the basic field operates, carrying it, in this example, to the state
of equilibrium B. Here it will remain until the next application
of S, which in this example, again moves it to a new confluent ;
here the basic field takes it to the state of equilibrium C. So does
the alternation of S and the basic field take it from equilibrium to
equilibrium till it arrives at E. From this state, S moves it only
to within the same confluent and the ' leaving alone ' results in its
coming back to E. S (having by hypothesis a unique effect) now
takes it to the arrow head, and again it comes back to E. This
state of affairs is now terminal, and the representative point is
trapped within the i?-confluent.

It can now be seen that the process is selective; the representa-
tive point ends in a confluent such that the ^-displacement carries
it to some point within the confluent. Confluents such as A, C,
and D, with the S- displacement going outside, cannot hold the
representative point under the process considered; confluents
such as E, J, and L can trap it.

14/4. The diagram shows two complications that must be con-
sidered for completeness' sake.

The first is that the events of the right-hand side may occur,
where the process considered takes the representative point
cyclically through confluents P, Q, R, P, Q, R, P, . . .

The second is shown on the left at X, where the confluent comes
to a cycle. From this cycle a variety of displacements may be
caused by S, depending on the precise moment at which S is
applied (i.e. on just where the representative point happens to be).

Whether such cycles (between or within confluents) are com-
mon in the nervous system is a question to be settled by experi-
ment. The cycle within the confluent, as at X, will hardly dis-
turb the conclusions below; for either all ^-displacements fall
within the confluent (in which case it is trapped as stated above)
or it will sooner or later leave the confluent (and we are no longer
concerned with the cycle). Thus in the Figure, unless the period

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14/5 REPETITIVE STIMULI AND HABITUATION

of the cycle bears some exact simple relation to that of the applica-
tions of S (an event of zero probability if they can vary con-
tinuously), the representative point will in fact leave X and
eventually be trapped at E. (The cycle around P, Q, R adds
complications that can be dealt with in a more detailed discussion.)

14/5. The description given is not rigorous, but can easily be
made so. It is intended only to illustrate the thesis that under
repetitive applications of a stimulus (with sufficient delay between
the applications for the system to come to equilibrium) the
polystable system is selective, for it sticks sooner or later at a state
from whose confluent the stimulus cannot shift it. And, if there is a
metric and continuity over the phase space, this distance that the
stimulus S finally moves the point will be less than the average
distance, for short arrows are favoured. Thus the amounts of
change caused by the successive applications of S change from
average to less than average.

We need not attempt here to formulate calculations about the
exact amount: they can be left to those specially interested.
What we should notice is that the outcome of the process is not
symmetric. When we think of a randomly assembled system of
random parts we are apt to deduce that its response to repeti-
tive stimulation will be equally likely to decrease or to increase.
The argument shows that this is not so: there is a fundamental
tendency for the response to get smaller.

There is a line of argument, much weaker, which may help to
make the conclusion more evident. We may take it as axiomatic
that large responses tend to cause more change (or are associated
with more change) within the system than small. If the responses
have any action back on their own causes, then large responses
tend to cause a large change in what made them large; but the
small only act to small degree on the factors that made them
small. Thus factors making for smallness have a fundamentally
better chance of surviving than those that make for largeness.
Hence the tendency to smallness.

(If the point requires illustration, we could consider the question :
Of two boys making their own fireworks, who has the better
chance of survival ? — the boy who is trying to produce the biggest
firework ever, or the boy who is trying to produce the tiniest !)

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DESIGN FOR A BRAIN

14/6

14/6. The process can readily be demonstrated, almost in
truistic form, on the Homeostat (which is here treated as a system
whose variables include 4 position of uniselector ' so that it has
many states of equilibrium, differing from one another according
to the uniselector's position).

The process is shown in Figure 14/6/1. Two units were joined

Time

Figure 14/6/1 : Homeostat tracing. At each D, l's magnet is displaced
by the operator through a fixed angle. 2 receives this action through
its uniselector. When the uniselector's value makes 2's magnet meet
the critical state (shown dotted) the value is changed. After the
fourth change the value causes only a small movement of 2, so the value
is retained permanently.

1 — > 2. The effect of 1 on 2 was determined by 2's uniselector,
which changed position if 2 exceeded its critical states. The
operator then repeatedly disturbed 2 by moving 1, at D. As
often as the uniselector transmitted a large effect to 2, so often
did 2 shift its uniselector. But as soon as the uniselector arrived
at a position that gave a transmission insufficient to bring 2 to its
critical states, that position was retained. So under constant
stimulation by D the amplitude of 2's response changed from
larger to smaller.

The same process in a more complex form is shown in Figure
14/6/2. Two units are interacting: 1 ^± 2. Both effects go

Time
U

Figure 14/6/2 : Homeostat arranged as ultrastable system with two units
interacting. At each D the operator moved l's magnet through a fixed
angle. The first field such that D does not cause a critical state to be
met is retained permanently.

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14/7 REPETITIVE STIMULI AND HABITUATION

through the uniselectors, so the whole is ultrastable. At each D
the operator displaced l's magnet through a constant distance.
On the first ' stimulation ', 2's response brought the system to
its critical states, so the ultrastability found a new terminal field.
The second stimulation again evoked the process. But the new
terminal field was such that the displacement D no longer caused
2 to reach its critical states; so this field was retained. Again
under constant stimulation the response had diminished.

14/7. In animal behaviour the phenomenon of ' habituation '
is met with frequently: if an animal is subjected to repeated
stimuli, the response evoked tends to diminish. The change has
been considered by some to be the simplest form of learning.
Neuronic mechanisms are not necessary, for the Protozoa show it
clearly :

'Amoebae react negatively to tap water or to water from a
foreign culture, but after transference to such water they
behave normally.'

4 If Paramecium is dropped into \°/0 sodium chloride it
at once gives the avoiding reaction ... If the stimulating
agent is not so powerful as to be directly destructive, the
reaction ceases after a time, and the Paramecia swim about
within the solution as they did before in water.' (Jennings.)

Fatigue has sometimes been suggested as the cause of the
phenomenon, but in Humphrey's experiments it could be excluded.
He worked with the snail, and used the fact that if its support is
tapped the snail withdraws into its shell. If the taps are repeated
at short intervals the snail no longer reacts. He found that when
the taps were light, habituation appeared early; but when they
were heavy, it was postponed indefinitely. This is the opposite
of what would be expected from fatigue, which should follow more
rapidly when the heavier taps caused more vigorous withdrawals.

A variety of special explanations have been put forward to
explain its origin, but the almost universal distribution of its
occurrence in living organisms should warn us that the basis
must be some factor much more widely spread than the neuro-
physiological. The argument of this chapter suggests that it is to
be expected to some degree in all polystable systems when they
are subjected to a repetitive stimulus or disturbance.

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DESIGN FOR A BRAIN 14/8

Minor disturbances

14/8. Exactly the same type of argument — of looking for what
can be terminal — can be used when S is not an accurately repeated
stimulus but is, on each application, a sample from a set of dis-
turbances having some definite distribution. In this case, Figure
14/3/1, for example, would remain unchanged in its confluents
and states of equilibrium, but the arrow going from each state of
equilibrium would lose its uniqueness and become a cluster or
distribution of arrows from which, at each disturbance, some one
would be selected by some process of sampling.

The outcome is similar. The equilibrium whose arrows all go
far away to other confluents is soon left by the representative
point; while the equilibrium whose arrows end wholly within its
own confluent acts as a trap for it. Thus the polystable system
(if free from cycles) goes selectively to such equilibria as are
immune to the action of small irregular disturbances.

14/9. The fields (of the main variables) selected by the ultrastable
system are subject to this fact. Thus, consider the three fields
of Figure 14/9/1 as they might have occurred as terminal fields
in Figure 7/23/1. In fields A and C the undisturbed representa-

Figure 14/9/1 : Three fields of an ultrastable system, differing in their
liability to change when the system is subjected to small random dis-
turbances. (The critical states are shown by the dots.)

tive points will go to, and remain at, the states of equilibrium.
When they are there, a leftwards displacement sufficient to
cause the representative point of A to encounter the critical states
may be insufficient if applied to C; so C's field may survive
a displacement that destroyed A's. Similarly a displacement ap-
plied to the representative point on the cycle in B is more likely

190

14/10 REPETITIVE STIMULI AND HABITUATION

to change the field than if applied to C. A field like C, therefore,
with its state of equilibrium near the centre of the region, tends
to have a higher immunity to displacement than fields whose
states of equilibrium or cycles go near the edge of the region.

14/10. How would this tendency show itself in the behaviour of
the living organism ?

The processes of S. 7/23 allow a field to be terminal and yet
to show all sorts of bizarre features : cycles, states of equilibrium
near the edge of the region, stable and unstable lines mixed,
multiple states of equilibrium, multiple cycles, and so on. These
possibilities obscured the relation between a field's being terminal
and its being suitable for keeping essential variables within normal
limits. But a detailed study was not necessary; for we have
just seen that all such bizarre fields tend selectively to be destroyed
when the system is subjected to small, occasional, and random dis-
turbances. Since such disturbances are inseparable from practical
existence, the process of ' roughing it ' tends to cause their replace-
ment by fields that look like C of Figure 14/9/1 and act simply
to keep the representative point well away from the critical states.

191

CHAPTER 15

Adaptation in Iterated and
Serial Systems

15/1. The last three chapters have been concerned primarily
with technique, with the logic of mechanism, when the mechanism
shows partial, fluctuating and temporary divisions into sub-
systems within the whole ; they have considered specially the case
when the subsystems are rich in states of equilibrium. We can
now take up again the thread left at S. 11/13, and can go on to
consider the problem of how a large and complex organism can
adapt to a large and complex environment without taking the
almost infinite time suggested by S. 11/5.

The remaining chapters will offer evidence that the facts are
as follows:

(1) The ordinary terrestrial environment has a distribution of
properties very different from the distribution assumed when the
estimate of S. 11/2 came out so high.

(2) Against the actual distribution of terrestrial environments
the process of ultrastability can often give adaptation in a reason-
ably short time.

(3) When particular environments do get more complex, the
time of adaptation goes up, not only in theoretical ultrastable
systems but in real living ones.

(4) When the environment is excessively complex and close-
knit, the theoretical ultrastable system and the real living fail
alike.

In this chapter and the next we will examine environments of
gradually increasing complexity. (What is meant by 4 com-
plexity ' will appear as we proceed.)

15/2. In S. 11/11 it was suggested that the Homeostat (i.e. the
two units or so marked off to represent ' environment ') is not
typical of the terrestrial environment because in the Homeostat
every variable is joined directly to every other variable, so that

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15/2 ADAPTATION IN ITERATED AND SERIAL SYSTEMS

what happens at each variable is conditional, at that moment,
on the values of all the other variables in the system. What,
then, does characterise the ordinary terrestrial environments from
this point of view ?

Common observation shows that the ordinary terrestrial
environment usually shows several features, which are closely
related :

(1) Many of the variables, often the majority, are constant over
appreciable intervals of time, and thus behave as part-functions.
Thus, the mammal stands on ground that is almost always im-
mobile; tree-trunks keep their positions; a cup placed on a table
will stay there till a force of more than a certain amount arrives.
If one looks around one, only in the most chaotic surroundings
will all the variables be changing. This constancy, this common-
ness of part-functions, must, by S. 12/14, be due to commonness
of states of equilibrium in the parts that compose the terrestrial
environment. Thus the environment of the living organism tends
typically to consist of parts that are rich in states of equilibrium.

(2) Associated with this constancy (naturally enough by S.
12/17) is the fact that most variables of the environment have an
immediate effect on only a few of the totality of variables. At
the moment, for instance, if I dip my pen in the ink-well, hardly
a single other variable in the room is affected. Opening of the
door may disturb the positions of a few sheets of paper, but will
not affect the chairs, the electric light, the books on the shelves,
and a host of others.

We are, in fact, led again to consider the properties of a system
whose connexions are fluctuating and conditional — the type
encountered before in S. 11/12, and therefore treatable by the
same method. I suggest, therefore, that most of the environ-
ments encountered on this earth by living organisms contain many
part-functions. Conversely, a system of part-functions adequately
represents a very wide class of commonly occurring environ-
ments.

As a confirmatory example, here is Jennings' description of an
hour in the life of Paramecium, with the part-functions indicated
as they occur.

(It swims upwards and) '. . . thus reaches the surface film.'

The effects of the surface, being constant at zero throughout the

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DESIGN FOR A BRAIN 15/2

depths of the pond, will vary as part-functions. A discontinuity
like a surface will generate part-functions in a variety of ways.

' Now there is a strong mechanical jar — someone throws a
stone into the water perhaps.'

Intermittent variations of this type will cause variations of part-
function form in many variables.

(The Paramecium dives) '. . . this soon brings it into water
that is notably lacking in oxygen.'

The content of oxygen will vary sometimes as part-, sometimes as
full-, function, depending on what range is considered. Jennings,
by not mentioning the oxygen content before, was evidently
assuming its constancy.

'. . . it approaches a region where the sun has been . . .
heating the water.'

Temperature of the water will behave sometimes as part-, some-
times as full-, function.

(It wanders on) '. . . into the region of a fresh plant stem
which has lately been crushed. The plant -juice, oozing out,
alters markedly the chemical constitution of the water.'

Elsewhere the concentration (at zero) of these substances is
constant.

8 Other Paramecia . . . often strike together ' (collide).

The pressure on the Parameciwri's anterior end varies as a part-
function.

' The animal may strike against stones.'
Similar part-functions.

' Our animal comes against a decayed, softened, leaf.'

More part-functions.

'. . . till it comes to a region containing more carbon dioxide
than usual.'

Concentration of carbon dioxide, being generally uniform with
local increases, will vary in space as a part-function.

1 Finally it comes to the source of the carbon dioxide — a large
mass of bacteria, embedded in zoogloea.'

Another part-function due to contact.

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15/2 ADAPTATION IN ITERATED AND SERIAL SYSTEMS

It is clear that the ecological world of Paramecium contains
many part-functions, and so too do the worlds of most living
organisms.

A total environment, or universe, that contains many part-
functions, will show dispersion, in that the set of variables active
at one moment will often be different from the set active at another.
The pattern of activity within the environment will therefore tend,
as in S. 13/18, to be fluctuating and conditional rather than
invariant. As an animal interacts with its environment, the
observer will see that the activity in the environment is limited
now to this set, now to that. If one set persists active for a long
time and the rest remains inactive and inconspicuous, the observer
may, if he pleases, call the first set ' the ' environment. And if
later the activity changes to another set he may, if he pleases,
call it a ' second ' environment. It is the presence of part-
functions and dispersion that makes this change of view reasonable.

An organism that tries to adapt to an environment composed
largely of part-functions will find that the environment is composed
of subsystems which sometimes are independent but which from
time to time show linkage. The alternation is shown clearly
when one learns to drive a car. The beginner has to struggle
with several subsystems: he has to learn to control the steering-
wheel and the car's relation to road and pedestrian; he has to
learn to control the accelerator and its relation to engine-speed,
learning neither to, race the engine nor to stall it; and he has to
learn to change gear, neither burning the clutch nor stripping the
cogs. On an open, level, empty road he can ignore accelerator
and gear and can study steering as if the other two systems did
not exist ; and at the bench he can learn to change gear as if steer-
ing did not exist. But on an ordinary journey the relations vary.
For much of the time the three systems

driver + steering wheel + . . .
driver + accelerator + . . .
driver -f- gear lever -J- • • •

could be regarded as independent, each complete in itself. But
from time to time they interact. Not only may any two use
common variables in the driver (in arms, legs, brain) but some
linkage is provided by the machine and the world around. Thus,
any attempt to change gear must involve the position of the
accelerator and the speed of the engine; and turning sharply

195

DESIGN FOR A BRAIN 15/3

round a corner should be preceded both by a slowing down and
by a change of gear. The whole system thus shows that temporary
and conditional division into subsystems that is typical of the
whole that is composed largely of part-functions.

Thus the terrestrial environment conforms largely to the poly-
stable type.

15/3. To study how ultrastability will act when the environment
is not fully joined, we shall have to use the strategy of S. 2/17 and
pick out certain cases as type-forms. We will therefore consider
environments with four degrees of connectedness.

First we will consider (in S. 15/4-7) the ' whole ' in which the
connexion between the parts is actually zero — the limiting case
as the connexions get less and less.

In S. 15/8-11 we will consider the case in which actual con-
nexions exist, but in which the subsystems are connected in a
chain, without feedback between subsystems. These two cases
will suffice to demonstrate certain basic properties.

In the next chapter we will consider the more realistic case in
which the subsystems are joined unrestrictedly in direction, so
that feedback occurs between the subsystems. This case will be
considered in two stages: first, in S. 16/2-4 we will dispose of the
case in which the connexions are rich; and then, from S. 16/5
onwards, we will consider the most interesting case, that in which
the connexions are in all directions, so that feedback occurs
between the subsystems, but in which the connexions are not rich
so that the whole can be regarded as formed from subsystems
each of which is richly connected internally, joined by connexions
(between the subsystems) that are much poorer — the case, in fact,
of the system that is neither richly joined nor unjoined.

Adaptation in iterated systems

15/4. The first case to be considered is that in which the whole
system, of organism and environment, is actually divided into
subsystems that (at least during the time of observation) do not
have any effective action on one another. Thus instead of A
in Figure 15/4/1 we are considering B. (For simplicity, the
diagram shows lines instead of arrows.) If the whole system
consists of organism and environment, the actual division between

196

15/5 ADAPTATION IN ITERATED AND SERIAL SYSTEMS

• •

Figure 15/4/1.

the two might be that shown in Figure 15/4/2. Such an arrange-
ment would be shown functionally by any organism that deals
with its environment by several independent reactions. Such a
whole will be said to consist of iterated systems.

Environment.

Animal

Figure 15/4/2 : Diagrammatic representation of an animalof eight variables
interacting with its environment as five independent systems.

S. 13/10 exemplified the argument applicable to such a ' whole \
If i is the number of subsystems that are at a state of equilibrium
at any particular moment, then in an iterated set i cannot fall,
and will usually rise. As subsystem after subsystem reaches
equilibrium so will each stay there ; and thus the whole will change
cumulatively towards total equilibrium.

15/5. Whether the feedbacks in Figure 15/4/2 are first order or
second (S. 7/5) is here irrelevant: the whole still moves to equili-
brium progressively. Thus, if each subsystem has essential
variables and step-mechanisms as in Figure 7/5/1, the stability
of the second order will develop as in S. 7/23; and thus the
adaptation of the whole to this environment will also develop
cumulatively and progressively.

In this case, the processes of learning by trial and error will
go on in one subsystem independently of what is going on in the
others. That such independent, localised learning can occur
within one animal was shown by Parker in the following experi-
ment:

4 If a sea-anemone is fed from one side of its mouth, it will

197

DESIGN FOR A BRAIN 15/6

take in, by means of the tentacles on that side, one fragment
of food after another. If now bits of food be alternated with
bits of filter paper soaked in meat juice, the two materials
will be accepted indiscriminately for some eight or ten trials,
after which only the meat will be taken and the filter paper
will be discharged into the sea water without being brought
to the mouth. If, after having developed this state of affairs
on one side of the mouth, the experiment is now transferred
to the opposite side, both the filter paper and the meat will
again be taken in till this side has also been brought to a state
of discriminating.'

15/6. What of the time taken by the iterated set to become
adapted ? T3 (of S. 11/5) is applicable here; so the extremeness
of Tx is not to be feared. Thus, however large the whole, if it
should actually consist of iterated subsystems, then the time it
takes to get adapted may be expected to be of the same order as
that taken by one of its subsystems. If this time is fairly short,
the whole may be very large and yet become adapted in a fairly
short time.

15/7. If Figure 15/4/2 is re-drawn so as to show explicitly its
relation to the system of Figure 7/5/1 the result is that shown
in Figure 15/7/1 (where the subsystems have been reduced to
three for simplicity in the diagram).

At once the reader may be struck by the fact that the three
reacting parts in the organism (in its brain usually) are represented
as having no connexion between them: is this not a fatal flaw ?

The subject is discussed more thoroughly in S. 17/2; here a
partial answer can be given. Let us compare the course of
adaptation as it would proceed (1) with the two left-hand sub-
systems wholly unconnected as shown, and (2) with the reacting
part of subsystem A having some immediate effect on subsystem B.

The first case is straightforward: each subsystem is a little
ultrastable system, homologous with that of S. 7/5/1, and each
would proceed to adaptation in the usual way.

When B is joined so as to be affected by A, however, the whole
course is somewhat changed. A is unaffected, so it will proceed
to adaptation as before; but B, previously isolated, is now affected
by one or more parameters that need no longer be constant. The
effect on B will depend on whether the effect comes to B from A's
reacting part or from A's step-mechanisms. If from the step-

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15/7 ADAPTATION IN ITERATED AND SERIAL SYSTEMS

Environment

Organism

Figure 15/7/1 : Sketch of the diagram of immediate effects of an organism
adapting to an environment as three separate subsystems. (Compare
Figures 15/8/1 and 16/6/1.)

mechanisms, B can achieve no permanent adaptation until A has
reached adaptation, for their values will keep changing. However,
once A's step-mechanisms have reached their terminal values, B's
parameters will be constant, and B can then commence profitably
its own search, undisturbed by further changes. Thus the join
from A's step-mechanisms will about double the time taken for the
whole to reach adaptation.

If, however, the effect comes to B from A's reacting part, then
even after A has reached adaptation, every time that A shows its
adaptation (by responding appropriately to a disturbance to its
environment), the lines of behaviour that A's reacting part follow
will provide B with a varying set of values at its parameters.
B is thus in a situation homologous to that of the Homeostat
in S. 8/10, except that B may be subject to parameter-values
many more than two. The time that B will take to reach adapta-
tion under all these values is thus apt to resemble Tx (S. 11/5), and
thus to be excessively long. Thus a joining from the reacting
part of A to that of B may have the effect of postponing the whole's
adaptation almost indefinitely.

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DESIGN FOR A BRAIN

15/8

These remaiks are probably sufficient for the moment to show
that the absence of connexions between organismal subsystems
in Figure 15/7/1 does not condemn the representation off-hand.
There is more to this matter of joining than is immediately
evident. (The topic is resumed in S. 17/2.)

Serial adaptation

15/8. By S. 15/3, our second stage of connectedness in the system
occurs when the parts of the environment are joined as a chain.
Figure 15/8/1 illustrates the case.

Figure 15/8/1.

Without enquiring at the moment into exactly what will
happen, it is obvious, by analogy with the previous section, that
adaptation must occur in the sequence — A first, then B, then C.
Thus we are considering the case of the organism that faces an
environment whose parts are so related that the environment can
be adapted to only by a process that respects its natural articulation.

15/9. Such environments are of common occurrence. A puppy
can learn rabbit-catching only after it has learned how to run:
the environment does not allow the two reactions to be learned in
the opposite order. A great deal of learning occurs in this way.
Mathematics, for instance, though too vast and intricate for one
all-comprehending flash, can be mastered by stages. The stages
have a natural articulation which must be respected if mastery is

200

15/10 ADAPTATION IN ITERATED AND SERIAL SYSTEMS

to be achieved. Thus, the learner can proceed in the order
' Addition, long multiplication, . . .' but not in the order ' Long
multiplication, addition, . . .' Our present knowledge of mathe-
matics has in fact been reached only because the subject contains
such stage-by-stage routes.

As a clear illustration of such a process, here is Lloyd Morgan
on the training of a falcon:

1 She is trained to the lure — a dead pigeon . . . — at first with
the leash. Later a light string is attached to the leash, and
the falcon is unhooded by an assistant, while the falconer,
standing at a distance of five to ten yards, calls her by shout-
ing and casting out the lure. Gradually day after day the
distance is increased, till the hawk will come thirty yards or
so without hesitation; then she may be trusted to fly to the
lure at liberty, and by degrees from any distance, say a
thousand yards. This accomplished, she should learn to
stoop to the lure. . . . This should be done at first only
once, and then progressively until she will stoop backwards
and forwards at the lure as often as desired. Next she should
be entered at her quarry . . .'

The same process has also been demonstrated more formally.
Wolfe and Cowles, for instance, taught chimpanzees that tokens
could be exchanged for fruit: the chimpanzees would then learn
to open problem boxes to get tokens; but this way of getting fruit
(the ' adaptive ' reaction) was learned only if the procedure for
the exchange of tokens had been well learned first. In other
words, the environment was beyond their power of adaptation
if presented as a complex whole — they could not get the fruit —
but if taken as two stages in a particular order, could be adapted to.

\ . . the growing child fashions day by day, year by year, a
complex concatenation of acquired knowledge and skills, adding
one unit to another in endless sequence ', said Culler. I need not
further emphasise the importance of serial adaptation.

15/10. To see the process in more detail, consider the following
example. A young animal has already learned how to move about
the world without colliding with objects. (Though this learning
is itself complex, it will serve for illustration, and has the advantage
of making the example more vivid.) This learning process was
due to ultrastability : it has established a set of values on the
step-mechanisms which give a field such that the system composed

201

DESIGN FOR A BRAIN

15/10

of eyes, muscles, skin-receptors, some parts of the brain, and hard
external objects is stable and always acts so as to keep within
limits the mechanical stresses and pressures caused by objects in
contact with the skin-receptors (S. 5/4). The diagram of immedi-
ate effects will therefore resemble Figure 15/10/1. This system
will be referred to as part A, the ' avoiding ' system.

BRAIN -«-

EYES

SKIN

MUSCLE S

^-OBJECTS

Figure 15/10/1 :
system.

Diagram of immediate effects of the ' avoiding '
Each word represents many variables.

As the animal must now get its own food, the brain must
develop a set of values on step-me'chanisms that will give a field
in which the brain and the food-supply occur as variables, and
which is stable so that it holds the blood-glucose concentration
within normal limits. (This system will be referred to as part B,
the ' feeding ' system.) This development will also occur by
ultrastability ; but while this is happening the two systems will
interact.

The interaction will occur because, while the animal is making
trial-and-error attempts to get food, it will repeatedly meet
objects with which it might collide. The interaction is very
obvious when a dog chases a rabbit through a wood. Further,
there is the possibility that the processes of dispersion within brain
and environment may allow the two reactions to use common
variables. When the systems interact, the diagram of immediate
effects will resemble Figure 15/10/2.

BLOOD

glucose'

-^- BRAIN -<-

FOOD .
SUPPLY "*"

EYES

SKIN

MUSC LES

— >OBJECTS

B A

Figure 15/10/2.

Let us assume at this point (simply to get a clear discussable
case) that the step-mechanisms affecting A are, for whatever
reason, not changeable while the adaptation to B is occurring
(compare S. 10/8). As the ' avoiding ' system A is not subject

202

15/11 ADAPTATION IN ITERATED AND SERIAL SYSTEMS

to further step-function changes, its field will not alter, and it
will at all times react in its characteristic way. So the whole
system is equivalent to an ultrastable system B interacting with
an ' environment ' A. It would also be equivalent to an ultra-
stable system interacting with an inborn reflex, as in S. 3/12. B
will therefore change its step-mechanism values until the whole
has a field which is stable and which holds within limits the
variable (blood-glucose concentration) whose extreme deviations
cause the step-mechanisms to change. We know from S. 8/11
that, whatever the peculiarities of A, B's terminal field will be
adapted to them.

It should be noticed that the seven sets of variables (Figure
15/10/2) are grouped in one way when viewed anatomically and
in a very different way when viewed functionally. The anato-
mical point of view sees five sets in the animal's body and two sets
in the outside world. The functional point of view sees the whole
as composed of two parts : an ' adapting ' part B, to which A
is ' environment '.

It is now possible to predict how the system will behave after
the above processes have occurred. Because part A, the ' avoid-
ing ' system, is unchanged, the behaviour of the whole will still
be such that collisions do not occur; and the reactions to the
food supply will maintain the blood-glucose within normal limits.
But, in addition, because B became adapted to A, the getting of
food will be modified so that it does not involve collisions, for all
such variations will have been eliminated.

15/11. What of the time required for adaptation of all the essen-
tial variables when the environment is so joined in a chain ?

The dominating subsystem A will, of course, proceed to adapta-
tion in the ordinary way. B, however, even when A is adapted
may still be disturbed to some degree by changes coming to it
from A, changes that come ultimately from the disturbances to
A that A must adapt against. C -also may get upset by some of
these disturbances, transmitted through B, and so on. Thus each
subsystem down the chain is likely to be disturbed by all the
disturbances that come to the subsystems that dominate it, and
also by the reactive, adaptive changes made by the same domin-
ating subsystems.

It is now clear how important is the channel-capacity of the

203

DESIGN FOR A BRAIN 15/11

connexions that transmit disturbances down the chain. If their
capacity is high, so much disturbance may be transmitted to the
lower members of the chain that their adaptations may be post-
poned indefinitely. If their capacity is low, the attenuation may
be so rapid that C, though affected by B, may be practically
unaffected by what happens at A; and a further subsystem D
may be practically unaffected by those at B; and so on. Thus,
as the connexions between the subsystems get weaker, so does
adaptation tend to the sequential — first A, then B, then C, and
so on. (The limit, of course, is the iterated set.)

If the adaptation is sequential, the behaviour corresponds to
that of Case 2 (S. 11/5). The time of adaptation will then be
that of the moderate T2 rather than that of the excessively long
2\. Thus adaptation, even with a large organism facing a large
environment, may be achievable in a moderate time if the environ-
ment consists of subsystems in a chain, with only channels of
small capacity between them.

204

CHAPTER 16

Adaptation in the Multistable System

16/1. Continuing our study of types of environment we next
consider, after Figures 15/7/1 and 15/8/1, the case in which the
subsystems of the environment are connected unrestrictedly in
direction, so that feedbacks occur between them. This type of
environment may vary according to the amounts of communica-
tion (variety) that are transmitted between subsystem and sub-
system. Two degrees are of special interest as types:

(1) those in which it is near the maximum — the richly joined

environment. (The exposition is more convenient if we
consider this case first, as it can be dismissed briefly.)

(2) those in which the amount is small.

The richly joined environment
16/2. When a set of subsystems is richly joined, each variable
is as much affected by variables in other subsystems as by those
in its own. When this occurs, the division of the whole into
subsystems ceases to have any natural basis.

The case of the richly joined environment thus leads us back to
the case discussed in Chapter 11.

16/3. Examples of environments that are both large and richly
connected are not common, for our terrestrial environment is
widely characterised by being highly subdivided (S. 15/2). A
richly connected environment would therefore intuitively be per-
ceived as something unusual, or even unnatural. The examples
given below are somewhat recherche, but they will suffice to make
clear what is to be expected in this case.

The combination lock was mentioned in S. 11/9. Though not
vigorous dynamically, its parts, so far as they affect the output
at the bolt, are connected in that the relations between them
are highly conditional. Thus, if there are seven dials that allow
the bolt to move only when set at RHOMBUS, then the effect

205

DESIGN FOR A BRAIN 16/3

of the first dial going to R on the movement of the bolt is con-
ditional on the positions of all the other six; and similarly for
the second and remaining letters.

A second example is given by a set of simultaneous equations,
which can legitimately be regarded as the temporary environment
of a professional computer if he is paid simply to get correct
answers. Sometimes they come in the simplest form, e.g.

2x = 8 ^|
3y = -7\

iz = S J

Then they correspond to the iterated form ; and each line can
be treated without reference to the others, as in S. 15/4.
Sometimes they are rather more complex, e.g.

2x = 3^

3x — 2y = 2

x + y — z = 0.

This form can be solved serially, as in S. 15/8; for the first line
can be treated without reference to the other two; then when
the first process has been successful the second line can be treated
without reference to the last; and so to the end. The peculiarity
of this form is that the value of x is not conditional on the values
of the coefficients in the second and third lines.
Sometimes the forms are more complex, e.g.

2x + y — 3z = 2^

x — y + 2z = 0

-x _ sy + 2 = 1.

Now the value of x is conditional on the values of all the coeffi-
cients; and in finding x, no coefficient can be ignored. The same
is true of y and z. Thus if we regard the coefficients as the
environment and the values of x, y and z as output, correctness
in the answer demands that, in getting any part of the answer
(any one of the three values), all the environment must be taken
into account.

A third, and more practical, example of a richly connected
environment (now, thank goodness, no more) faced the experi-
menter in the early days of the cathode-ray oscilloscope. Adjust-
ing the first experimental models was a matter of considerable
complexity. An attempt to improve the brightness of the spot

200

16/4 ADAPTATION IN THE MULTISTABLE SYSTEM

might make the spot also move off the screen. The attempt to
bring it back might alter its rate of sweep and start it oscillating
vertically. An attempt to correct this might make its line of
sweep leave the horizontal; and so on. This system's variables
(brightness of spot, rate of sweep, etc.) were dynamically linked
in a rich and complex manner. Attempts to control it through
the available parameters were difficult precisely because the
variables were richly joined.

16/4. How long will an ultrastable system that includes such
an environment take to get adapted ?

This is the question of S. 11/2. Unless a large fraction of the
outcomes are acceptable, the time taken tends to be like Tx of
S. 11/5. As the system is made larger, so does the time of
adaptation tend to increase beyond all bounds of what is practical;
in other words, the ultrastable system probably fails. But this
failure does not discredit the ultrastable system as a model of
the brain (S. 8/17), for such an environment is one that is also
likely to defeat the living brain. That the living organism is
notoriously apt to find such environments difficult or impossible
for adaptation is precisely the reason why the combination lock
is relied on for protection.

Even when a skilled thief defeats the combination lock he
supports, rather than refutes, the thesis. Thus if he can hear, as
each dial moves, a tumbler fall into position, then the environ-
ment is to him a serial one (S. 15/8); for he can get the first dial-
setting right without reference to the others, then the second, and
so on. The time of its opening is thus made vastly shorter.
Thus the skilled thief does not really adapt successfully to the
richly joined environment — he demonstrates that what to others
is richly joined is to him joined serially.

Thus the first answer to the question : how does the ultrastable
system, or the brain, adapt to a richly joined environment ? is — it
doesn't. After the reasonableness of this answer has been made
clear, we may then notice that sometimes there are ways in
which an environment, apparently too complex for adaptation,
may eventually be adapted to; perhaps by the discovery of
ways of getting through the necessary trials much faster, or
perhaps by the discovery that the environment is not really as
complex as it looks. (/. to C, S. 13/4, discusses the matter.)

207

DESIGN FOR A BRAIN 16/5

The poorly joined environment

16/5. We will finally consider the case in which the environment
consists of subsystems joined so that they affect one another only
weakly, or occasionally, or only through other subsystems. It
was suggested in S. 15/2 that this is the common case in almost
all natural terrestrial environments.

If the degree of interaction between the subsystems varies, its
limits are: at the lower end, the iterated systems of S. 15/4 (as
the communication between subsystems falls to zero), and at the
upper end, the richly connected systems of S. 16/2 (as the com-
munication rises to its maximum).

When the communication between subsystems falls much below
that within subsystems, the subsystems will show naturally and
prominently (S. 12/17).

If such an environment acts within an ultrastable system, what
will happen ? Will adaptation occur ? As the discussion below
will show, the number of cases is so many, and the forms so
various, that no detailed and exhaustive account is possible. We
must therefore use the strategy of S. 2/17, getting certain type-
forms quite clear, and then covering the remainder by some appeal
to continuity: that so far as other forms resemble the type-forms
in their construction, to a somewhat similar degree will they
resemble the type-forms in their behaviour.

16/6. To obtain a secure basis for the discussion of this most
important case, let us state explicitly what is now assumed:

(1) The environment is assumed to be as described in S. 15/2,
so that it consists of large numbers of subsystems that have many
states of equilibrium. The environment is thus assumed to be
polystable.

(2) Whether because the primary joins between the subsystems
are few, or because equilibria in the subsystems are common, the
interaction between subsystems is assumed to be weak.

(3) The organism coupled to this environment will adapt by
the basic method of ultrastability, i.e. by providing second-order
feedbacks that veto all states of equilibrium except those that
leave each essential variable within its proper limits.

(4) The organism's reacting part is itself divided into sub-
systems between which there is no direct connexion. Each sub-

208

16/7 ADAPTATION IN THE MULTISTABLE SYSTEM

system is assumed to have its own essential variables and second
order feedback. Figure 16/6/1 illustrates the connexions, but
somewhat inadequately, for it shows only three subsystems. (It
should be compared with Figures 15/7/1 and 15/8/1.)

> r

I >

r 1

1 v

Figure 16/6/1.

Such a system is essentially similar to the multistable system
denned in the first edition. (The system denned there allowed
more freedom in the connexions between the main variables, e.g.
from reacting part to reacting part, and between reacting part and
an environmental subsystem other than that chiefly joined to it;
these minor variations are a nuisance and of little importance —
in the next chapter we shall be considering such variations.)

16/7. To trace the behaviour of the multistable system, suppose
that we are observing two of the subsystems, e.g. A and B of
Figure 16/6/1, that their main variables are directly linked so
that changes of either immediately affects the other, and that for
some reason all the other subsystems are inactive.

The first point to notice is that, as the other subsystems are
inactive, their presence may be ignored; for they become like the
4 background ' of S. 6/1. Even if some are active, they can still
be ignored if the two observed subsystems are separated from
them by a wall of inactive subsystems (S. 12/10).

The next point to notice is that the two subsystems, regarded
as a unit, form a whole which is ultrastable. This whole will
therefore proceed, through the usual series of events, to a terminal

209

DESIGN FOR A BRAIN 16/8

field. Its behaviour will not be essentially different from that
recorded in Figure 8/4/1. If, however, we regard the same
series of events as occurring, not within one ultrastable whole,
but as interactions between a minor environment and a minor
organism, each of two subsystems, then we shall observe
behaviours homologous with those observed when interaction
occurs between 4 organism ' and ' environment '. In other words,
within a multistable system, subsystem adapts to subsystem in exactly
the same way as animal adapts to environment. Trial and error
will appear to be used; and, when the process is completed, the
activities of the two parts will show co-ordination to the common
end of maintaining the essential variables of the double system
within their proper limits.

Exactly the same principle governs the interactions between three
subsystems. If the three are in continuous interaction, they form
a single ultrastable system which will have the usual properties.

As illustration we can take the interesting case in which two
of them, A and C say, while having no immediate connexion
with each other, are joined to an intervening system B, inter-
mittently but not simultaneously. Suppose B interacts first with
A: by their ultrastability they will arrive at a terminal field.
Next let B and C interact. If B's step-mechanisms, together with
those of C, give a stable field to the main variables of B and C,
then that set of B's step-mechanism values will persist indefinitely;
for when B rejoins A the original stable field will be re-formed.
But if B's set with C's does not give stability, then it will be
changed to another set. It follows that 2?'s step-mechanisms will
stop changing when, and only when, they have a set of values
which forms fields stable with both A and C. (The identity in
principle with the process described in S. 8/10 should be noted.)

16/8. The process can be illustrated on the Homeostat. Three
units were connected so that the diagram of immediate effects was
2 ^± 1 ^± 3 (corresponding to A, B, and C respectively). To
separate the effects of 2 and 3 on 1, bars were placed across the
potentiometer dishes (Figure 8/2/2) of 2 and 3 so that they could
move only in the direction recorded as downwards in Figure
16/8/1, while 1 could move either upwards or downwards. If
1 was above the central line (shown broken), 1 and 2 interacted,
and 3 was independent; but if 1 was below the central line, then

210

16/8 ADAPTATION IN THE MULTISTABLE SYSTEM

u j'k'l»m»-T7

-\j — \r

Tim<z

Figure 16/8/1 : Three units of the Homeostat interacting. Bars in the
central positions prevent 2 and 3 from moving in the direction corre-
sponding here to upwards. Vertical strokes on U record changes of
uniselector position in unit 1. Disturbance D, made by the operator,
demonstrates the whole's stability.

1 and 3 interacted, and 2 was independent. 1 was set to act on

2 negatively and on 3 positively, while the effects 2 — > 1 and

3 — > 1 were uniselector-controlled.

When switched on, at J, 1 and 2 formed an unstable system
and the critical state was transgressed. The next uniselector
connexions (K) made 1 and 2 stable, but 1 and 3 were unstable.
This led to the next position (L) where 1 and 3 were stable but
1 and 2 became again unstable. The next position (M) did
not remedy this; but the following position (N) happened to
provide connexions which made both systems stable. The values
of the step-mechanisms are now permanent; 1 can interact re-
peatedly with both 2 and 3 without loss of stability.

It has already been noticed that if A, B and C should form
from time to time a triple combination, then the step-mechanisms
of all three parts will stop changing when, and only when, the
triple combination has a stable field. But we can go further
than that. If A, B and C should join intermittently in various
ways, sometimes joining as pairs, sometimes as a triple, and
sometimes remaining independent, then their step-mechanisms
will stop changing when, and only when, they arrive at a set of
values which gives stability to all the arrangements.

Clearly the same line of reasoning will apply no matter how
many subsystems interact or in what groups or patterns they

211

DESIGN FOR A BRAIN 16/9

join. Always we can predict that their step -mechanisms will stop
changing when, and only when, the combinations are all stable.
Ultrastable systems, whether isolated or joined in multistable
systems, act always selectively towards those step-mechanism
values which provide stability.

16/9. At the beginning of the preceding section it was assumed,
for simplicity, that the process of dispersion was suspended, for
we assumed that the two subsystems interacting remained the
same two (e.g. A and B of Figure 16/6/1) during the whole pro-
cess. What modifications must be made when we allow for the
fact that in a multistable system the number and distribution of
subsystems active at each moment may fluctuate by dispersion ?

The progression to equilibrium of the whole, to a terminal field,
and thus to adaptation of the whole, will occur whether dispersion
occurs or not. The effect of dispersion is to destroy the indi-
viduality of the subsystems considered in the previous section.
There two subsystems were pictured as going through the complex
processes of ultrastability, their main variables being repeatedly
active while those of the surrounding subsystems remained
inactive. This permanence of individuality can hardly occur
when dispersion occurs. Thus, suppose that a multistable
system's field of all its main variables is stable, and that its repre-
sentative point is at a state of equilibrium R. If the representa-
tive point is displaced to a point P, the lines from this point will
lead it back to R. As the point travels back from P to R, sub-
systems will come into action, perhaps singly, perhaps in com-
bination, becoming active and inactive in kaleidoscopic variety
and apparent confusion. Travel along another line to R will
also activate various combinations of subsystems; and the set
made active in the second line may be very different from that
made active by the first.

In such conditions it is no longer profitable to observe par-
ticular sybsystems when a multistable system adapts. What
will happen is that so long as some essential variables are outside
their limits, so long will change at step-mechanisms cause com-
bination after combination of subsystems to become active. But
when a stable field arises not causing step-mechanisms to change,
it will, as usual, be retained. If now the multistable system's
adaptation be tested by displacements of its representative point,

212

16/11 ADAPTATION IN THE MULTISTABLE SYSTEM

the system will be found to respond by various activities of various
subsystems, all co-ordinated to the common end. But though
co-ordinated in this way, there will, in general, be no simple
relation between the actions of subsystem on subsystem : knowing
which subsystems were activated on one line of behaviour, and
how they interacted, gives no certainty about which will be
activated on some other line of behaviour, or how they will interact.
Later I shall pefer again to ' subsystem A adapting to, or
interacting with, subsystem B ', but this will be only a form of
words, convenient for description: it is to be understood that
what is A and what is B may change from moment to moment.

16/10. This new picture answers the objection to Figure 16/6/1
(and the others) that it shows a tidiness nowhere evident when the
organism (or the environment) is examined anatomically or
histologically. The Figures are diagrams of immediate effects,
and are intended purely as an aid to easier thinking about func-
tional and behavioural relationships. They must be regarded
as showing only functional connexions, and of these only those
between variables that are active over some small interval of
time. Figure 16/6/1 is thus apt to mislead both by suggesting
a permanence of structure that does not exist when dispersion
occurs, and by suggesting an actual two-dimensional form that
may well have no anatomical or histological existence. Neverthe-
less, the functional relationships are indisputable, and the Figure
represents them. How they are related to variables physically
identifiable in the brain has yet to be discovered.

16/11. Though the multistable system may look chaotic in
action, as the activity fluctuates over the subsystems with the
same apparent lack of order as that shown by the smoking
chimneys of S. 13/18, yet the tendency is always towards ultimate
equilibrium and adaptation. So the next question to ask is that
of Chapter 11 : will the adaptation take an excessively long time ?

Clearly, following the arguments of the previous chapter, much
will depend on the richness of connexion between the subsystems
— on how much disturbance comes to each subsystem from the
others.

At the limit, when the transfers of disturbance are all zero,
the whole system becomes identical with the iterated systems of

213

DESIGN FOR A BRAIN 16/12

S. 15/6, and the whole will progress to adaptation similarly. In
this case the time taken to reach adaptation will be the moderate
time of T3, rather than the excessive time of Tv

As the connexions become richer, whether by more basic joins
or by the subsystems having fewer states of equilibrium, so will
the system move towards the richly-connected type of S. 16/4;
and so will the time required for adaptation increase towards
that of Tv %

Summary. We are now in a position to summarise the answer,
given by the intervening chapters, to the objection, raised in
S. 11/2, that ultrastability cannot be the mode of adaptation used
by living organisms because it would take too long. We can now
appreciate that the objection was unwittingly using the assumption
that the organism and the environment were richly joined both
within themselves and to each other. Evidence has been given,
in S. 15/2, that the actual richness is by no means high. Then
Chapters 15 and 16 have shown that when it is not high, adaptation
by ultrastability can occur in a time that is no longer impossibly
long. Thus the objection has been answered, at least in outline.
There we must leave the matter, for a closer examination would
have to depend on measurements of actual brains adapting to
actual environments. The study of the matter should not be
beyond the powers of the present-day experimenter.

Retroactive inhibition

16/12. The suggestion now before the reader is that the system
of Figure 7/5/1, when looked at more closely in the forms in
which it occurs in actual organisms and environments, will be
found to break up into parts more like those of Figure 16/6/1 —
the multistable system. Let us trace out some of the properties
of this system — extra to those it possesses by being basically
ultrastable and only a particular form of Figure 7/5/1 — and see
how they accord with what is known of the living organism.

A first question to be asked about the multistable system is
whether it can take advantage of the recurrent situation, a matter
considered earlier in Chapter 10. Thus, after a multistable system
has adapted to a parameter's taking the value P2> then to its
value P8, will it, when given P2 again, retain anything of its first
adaptation ?

214

16/12 ADAPTATION IN THE MULTISTABLE SYSTEM

Before attempting the answer, let us recall that, in any poly-
stable system, any two different lines of behaviour will give
changes in two sets of variables (S. 13/14) which may or may not
overlap. Each set will be distributed over the system somewhat
as the smoking chimneys of S. 13/18 were distributed over the
town. Two disturbances (D1 and D2), to a polystable system,
will give two sets of active variables, as two winds (W1 and W2),
to a town, would ?give two sets of smoking chimneys.

Of the chimneys in the town, what fraction will smoke with
both the winds ? The precise answer would depend on precise
conditions; but we can see as a first approximation (as in S. 13/15)
that if only a small fraction smoke under Wv and a small fraction
under W2, then if the two fractions are independent, the fraction
smoking under both will be the product of the single fractions, and
thus much smaller than either. Thus if a random 1 per cent
smoke under W1 and another random 1 per cent under W2, those
that smoke under both will be only y^- of 1 per cent.

The independence, and the smallness of the overlap, can occur
only if Wx and W2 are well separated in direction. If W2 should
be very close to W^s direction, it will probably cause smoking in
many of I^'s chimneys (in the limit, of course, as it matches
W^s direction, it will make all W^s chimneys smoke).

Thus a polystable system (subject to certain conditions of
statistical independence which would require detailed examina-
tion) will respond to two parameter-values (or disturbances, or
stimuli) with two sets of variables whose overlap depends on: —
(1) the amount of activation that each causes, and (2) the resem-
blance between the parameter- values.

Suppose now that the parameter values correspond, as in
S. 10/8, to environments that have to be adapted to (or to problems
that have to be solved). Since the multistable system is also
polystable, what has just been said will be true of the multistable
system. Here the two lines of behaviour will include trials and
will cause changes in the step-mechanisms as well as in the main
variables. The degree to which the two sets of activated step-
mechanisms overlap will again depend on what fraction of all
step-mechanisms are activated and on the degree of resemblance
of the parameter- values (or environments). In particular, if the
lines of behaviour overlap on only a few step-mechanisms, the
second set of trials may cause little change in the step-mechanisms

215

DESIGN FOR A BRAIN 16/13

that have an effect on the first reaction, and thus little loss in the
first adaptation. Thus the multistable system, without further ad
hoc modification, will tend to take advantage of the recurrent situation.

16/13. It is of interest to notice that when two stimuli (or
parameter- values) are widely different, the multistable system
will tend to direct the activations to widely different sets of
step-mechanisms. It thus provides, without further ad hoc
modification, a functional equivalent of the gating mechanism

r of s. 10/9.

16/14. Conversely, as the two disturbances (or stimuli, or para-
meter-values) tend to equality, so will the overlap of the two
activated sets tend to increase. A large overlap in the step-
mechanisms will mean that the second set of trials will be severely
destructive to the first adaptation. Now the tendency for new
learning to upset old is by no means unknown in psychology ; and
an examination of the facts shows that the details are strikingly
similar to those that would be expected to occur if the nervous
system and its environment were multistable. In experimental psy-
chology 4 retroactive inhibition ' has long been recognised. The
evidence is well known and too extensive to be discussed here, so
I will give simply a typical example. Muller and Pilzecker found
that if a lesson were learned and then tested after a half -hour
interval, those who passed the half-hour idle recalled 56 per cent
of what they had learned, while those who filled the half-hour
with new learning recalled only 26 per cent. Hilgard and Marquis,
in fact, after reviewing the evidence, consider that the phenomenon
is sufficiently ubiquitous to justify its elevation to a 4 principle of
interference \ There can therefore be no doubt that the pheno-
menon is of common occurrence. New learning does tend to
destroy old.

In a multistable system, the more the stimuli used in new learn-
ing resemble those used in previous learning, the more will the
new tend to upset the old; for, by the method of dispersion
assumed here, the more similar are two stimuli the greater is the
chance that the dispersion will lead them to common variables
and to common step-mechanisms. In psychological experiments
it has repeatedly been found that the more the new learning
resembled the old the more marked was the interference. Thus

216

16/15 ADAPTATION IN THE MULTISTABLE SYSTEM

Robinson made subjects learn four-figure numbers, perform a
second task, and then attempt to recall the numbers; he found
that maximal interference occurred when the second task consisted
of learning more four-figure numbers. Similarly Skaggs found
that after learning five-men positions on the chessboard, the
maximal failure of memory was caused by learning other such
arrangements. The multistable system's tendency to be dis-
organised by new reactions is thus matched by a similar tendency
in the nervous system.

16/15. It should be noticed that the demands that a brain
model should show both retroactive inhibition and the ability
to accumulate adaptations are opposed; for retroactive inhibition
demands that later adaptations shall be destructive to earlier
adaptations, while the power to accumulate adaptations demands
that the later shall not be destructive to the earlier. The Homeo-
stat showed retroactive inhibition at maximal intensity (S. 10/5),
for any later adaptation destroyed the earlier totally. A set of
iterated systems, with some suitable gating-mechanism, shows
the maximal power of accumulating adaptations. A multistable
system of some intermediate degree can show both features
partially, and will thus resemble the living organism.

217

CHAPTER 17

Ancillary Regulations

17/1. Our study of adaptation has led us to the ultrastable
system, and then to some difficulties, in S. 11/2, about how long
an ultrastable system would take to get adapted. These diffi-
culties have been largely resolved by our identification of the
multistable system. (This is not to say that the topic of adapta-
tion is exhausted, for it extends to innumerable special cases that
deserve particular study.) In this chapter and the next we will
consider some other objections that may be raised to the thesis
that the brain is to a major degree multistable. In dealing with
them we shall encounter some new aspects of the subject that are
worthy of attention.

Communication within the brain

17/2. If it is accepted from here onwards that the formulation
of S. 16/6 and its Figure (the multistable system) solves, at least
in its major features, the problem posed in Chapter 1, there arises
the question why Figure 16/6/1 shows, in the lower part (the
organism), no joins between subsystem and subsystem. Does not
this absence make the representation a travesty of the facts ? —
a brain with no communication between its parts !

17/3. In this matter let us dispose once for all of the idea,
fostered in almost every book on the brain written in the last
century, that the more communication there is within the brain
the better. It will suffice if we remember the three following ways
in which we have already seen that some function can be success-
ful only if certain pairs of variables are not allowed to communicate,
or between which the communication must not be allowed to
increase beyond a certain degree.

(1) In S. 8/15 we saw that when an organism is adapting by
discrete trials, the essential variables must" change the step-
mechanisms at a rate much slower than the rate at which the

218

17/4 ANCILLARY REGULATIONS

main variables change. Too rapid a change at the step-
mechanisms means that the appropriateness (or not) of a set of
values does not have time to be communicated round, through
the brain and environment as they carry out the trial, to the
essential variables, which would thus be acting before the arrival
of their necessary information. If it takes ten seconds for the
goodness of a trial to be tested, then alterations should obviously
not be made more frequently than at about eleven-second intervals.
And if it takes ten years to observe adequately the effect of a
profound re-organisation of a Civil Service, then such re-organisa-
tions ought not to occur more frequently than at eleven-year
intervals. The amount of communication from essential variables
to step-functions can thus become harmful if excessive.

(2) In Chapter 10 we considered how the organism could take
advantage of the recurrent situation, so that if, having adapted
first to A and then to B, A were presented again, it could produce
the behaviour appropriate to A at once. It was shown in S. 10/8
that during the adaptation to B, the step-mechanisms concerned
with the adaptation to A must not be affected by what happens
at the essential variables. The allowing of such communication
would thus be harmful.

(3) In S. 16/11 it was shown that a multistable system's
chance of getting adapted in a reasonably short time is closely
related to its approximation to the iterated form. Thus every
addition of channels of communication takes the system further
from the iterated form and, whatever else it may do, increases
the time taken to arrive at adaptation.

Thus, in adapting systems, there are occasions when an increase
in the amount of communication can be harmful.

17/4. It may still be objected that Figure 16/6/1 should show
connexions directly between the reacting parts, because such
connexions are necessary for co-ordination to be achieved between
part and part. The objection in fact is mistaken; connexions are
not necessary. Let me explain.

First we can dismiss at once the case in which the parts of the
environment (as in Figure 15/7/1) are not joined; for then the
threats to the various essential variables come independently and
can be responded to independently. In this case the necessity
for co-ordination between parts does not arise.

219

DESIGN FOR A BRAIN 17/4

What of the case of Figure 16/6/1, when the parts of the environ-
ment are joined, and when what is done by, say, the reacting part
of A may affect, through the part B of the environment, what
happens at the second (B) essential variable ? In this case
co-ordination between the actions of the two reacting parts is
certainly necessary, for the desirable state of all essential variables
being kept within limits can be achieved only by each part's
actions being properly related to what the others are doing; for
all actions meet in the common environment.

Given, then, that co-ordination between the reacting parts is
demanded, does this imply that the reacting parts must be in direct
communication ? It does not ; for communication between them is
already available (in the case considered) through the environment

The anatomist may be excused for thinking that communication
between part and part in the brain can take place only through
some anatomically or histologically demonstrable tract or fibres.
The student of function will, however, be aware that channels are
also possible through the environment. An elementary example
occurs when the brain monitors the acts of the vocal cords by
a feedback that passes, partly at least, through the air before
reaching the brain.

As the matter is of considerable importance in the general
theory of how organism and environment interact, and as it has
hitherto received little attention (though S. 5/13 touched on it),
let us consider an example that shows how functioning parts of
the brain may sometimes be co-ordinated by a channel of com-
munication that passes through the environment.

Consider the player serving at tennis. His left arm makes a
movement that projects the ball into the air; a moment later,
his right arm makes a movement that, we will assume, strikes the
ball correctly into the opposite court. We will also assume that
the movements of the left arm are (for whatever reason) not
invariable but are subject to small random variations between
service and service. We assume that these variations are appreci-
able, so that unless the movements of the right arm are also varied,
and properly paired to those of the left, the ball is likely to go
out. Nevertheless we are assuming that the right arm's move-
ments are so paired that the ball arrives safely in the proper place
(' position of the ball's arrival ' is the essential variable, and its
normal limits are the bounds of the opposite court).

220

17/4

ANCILLARY REGULATIONS

For the co-ordination to occur, there must be some channel
from the source of the left arm's variations to the right arm's
movements (/. to C, S. 11/11; the pairing proves as much by
S. 4/13 above). Our question now is: must this channel lie within
the brain ?

Not only it need not, it usually does not; as the following
argument will show. Consider the situation at the moment when
the ball is in mid-air: is the right arm's developing movement
now guided by messages from the left arm's centre* or from the
position of the ball in the air ? The operational test (of S. 4/12
and 12/3) is decisive : let the left arm's movements remain unaltered
but assume now that the position of the ball be altered, by a sharp
gust say; is the right arm's movement altered? The normal
player, if the ball should be affected by a gust, will at once modify
his right-arm movements accordingly. These modifications, by
the basic operational test, show that the right arm is immediately

ORGANISM

Figure 17/4/1.

* We must avoid the tangles caused by the fact that the right arm is
controlled by the left motor cortex, and vice versa.

221

DESIGN FOR A BRAIN

17/5

affected, in part at least, by the position of the ball in the air.
Thus the server at tennis normally co-ordinates his left and right
arms' movements by the method: Left arm throws up the ball
with imperfect accuracy, then the position of the ball in the air
(through vision) guides the right arm. The diagram of immediate
effects is (to show the correspondence with Figure 16/6/1) as
shown in Figure 17/4/1.

Thus, within the assumptions bounding this example, co-
ordination between parts can take place through the environment;
communication within the nervous system is not always necessary.

17/5. After these observations, one may begin to wonder why
the brain should have connexions between its parts at all. There
are at least two reasons.

The first comes from the fact that, in the organism's life-long
struggle to defend its essential variables against disturbance, there
is a fundamental advantage in getting information about the
disturbance early. (The fact can either be accepted as obvious,
or proved more formally, as in /. to C, S. 12/5.) Now while
many of the disturbances that threaten an essential variable come
ultimately from the environment, some of them may come from
other parts of the same organism. Thus every child that is
learning to feed itself discovers that its lip may be hurt both by
environmental objects and also by its own attempt to pass a
spoonful of food into its closed mouth. If the lip is not to be
struck, the mouth must be opened in advance of the spoon's
arrival ; for this to be possible, information that the spoon is
approaching must get to the ' mouth centre ' before the spoon
arrives. Sometimes the information may come through the
environment (by the child watching the spoon), with the diagram
of immediate effects:

Centre for

hand
movements

— >

Position

spoon

— >

Pattern

on
retina

— >

Centre for

mouth
movements

Muscles

of
mouth

But if, for whatever reason, communication from hand to mouth
is not possible through the environment, then communication
within the brain, from hand-centre to mouth-centre, is necessary
if the mouth is to be opened before the spoon arrives. Thus

222

17/7 ANCILLARY REGULATIONS

communication within the brain can clearly be necessary or
advantageous.

17/6. A second reason why communication within the brain may
be desirable can be discussed rigorously only in the concepts of
/. to C, S. 7/7, but the reason can be sketched here.

When a system is described, it starts by being a member of a
large class of possible forms; as each specification is added, so
does the class that it may belong to shrink. Start with a system
restricted only by having the states possible to it fixed at a certain
number. If now is added the further specification ' its diagram
of immediate effects contains all possible arrows ', the possibilities
in its fields are restricted only slightly. But had it been added
that the diagram contained few arrows, the possibilities in the
fields would have been restricted severely.

Thus, other things being equal, the fewer the joins, the fewer
are the modes of behaviour available to the system. From this
point of view, extra connexions within the brain can be advan-
tageous, for they make possible a greater repertoire of behaviours.

Another way by which the same fact can be seen is to consider
the reacting parts before they wrere joined. The parameters used
in the joining must, before the joining, have had fixed values (for
otherwise the parts would not have been state-determined). Thus
before the joining each parameter must have been fixed at some
one of its possible values; after the joining the parameter would be
capable of variation as it was affected by the other part. With
the variation would have come, to the part, a corresponding
variety in its fields, and ways of behaving (S. 6/3). Thus joining,
by mobilising parameters that would otherwise be fixed, adds to
the variety of possible behaviours.

It can now be admitted, without misunderstanding, that Figure
16/6/1 would have been more realistic with some connexions
drawn between the reacting parts. The presentation and dis-
cussion at S. 16/6, however, was simpler without them.

17/7. If increased connexions between the reacting parts in the
organism bring in the two advantages just described, they also
bring in, as S. 16/4 showed, the disadvantage of lengthening,
perhaps to a very great degree, the time required for adaptation.
Doubtless there are even more factors to be reckoned in the

223

DESIGN FOR A BRAIN 17/8

balance, but what we have seen is sufficient to show that richness
of corwcr ion between the parts in the brain has both advantages and
disadvantages. Clearly the organism must develop so that its
brain finds, in this respect, an optimum.

It is not suggested that what is wanted is the optimum in the
strict sense. Finding an optimum is a much more complex
operation than finding a value that is acceptable (according to a
given criterion). Thus, suppose a man comes to a foreign market
containing a hundred kinds of fruit that are quite new to him.
To find the optimum for his palate he must (1) taste all the hundred,
(2) make at least ninety-nine comparisons, and (3) remember the
results so that he can finally go back to the optimal form. On
the other hand, to find a fruit that is acceptable he need merely
try them in succession or at random (taking no trouble to remember
the past), stopping only at the first that passes the test. To
demand the optimum, then, may be excessive; all that is required
in biological systems is that the organism finds a state or value
between given limits.

Thus, for the organism to adapt with some efficiency against
the terrestrial environment, it is necessary that the degree of
connexion between the reacting parts lie between certain limits.

Ancillary regulations
17/8. ' Between certain limits ' — we have heard that phrase
before ! Are we arguing in a circle ? Not really, for two different
adaptations are involved, of two types or levels or orders.

To see the two adaptations and their relation, recall that we
started (S. 3/14) by assuming that certain essential variables were
to be kept within limits. Call them Ev E2, E3, and 224; in Figure
8/2/1 they are clearly evident; keeping them within limits is one
adaptation. In Chapter 11 we added another essential variable
F: the time taken by the four E's to get stable within their limits;
keeping it within limits is another adaptation. This F is quite
distinct from a fifth E, which would enter the system in quite a
different way. Yet F does come to the whole as an essential
variable, for from S. 11/2 onwards we have consistently discussed
the case in which it has certain limits which we do not want it
to exceed. (The possibility of various classes of essential variables
was mentioned in S. 3/15.)

The 25' s — the four relays on the Homeostat say — are clearly

224

17/9 ANCILLARY REGULATIONS

homologous and equivalent; but F comes into the whole in a
different way. To see how, suppose that it is most desirable (for
some major essential variable S) that success, on some lesser
essential variable E, be achieved in fewer than a hundred trials
(i.e. F is to be less than 100). E, making trials, will cause change
after change to occur on its corresponding step-mechanisms; at
the same time F (increasing exhaustion perhaps) is steadily
mounting to its limit. What is to happen if F passes its limit
of 100 ? If & is such that the organism dies, nothing remains
to be said ; but if & is not totally essential, the organism is in the
condition of having made many trials in some way that has failed
to bring success quickly (the situation discussed in Chapter 11).
What is to be done ? By the method of ultrastability, F's passing
beyond its limit must induce changes, but clearly these changes
should not be simply in the same step-mechanisms that E has
been working on, or the action by F is no different from a hundred-
and-first trial by E. For F to have an appropriately effective
action, its passage beyond the limit must induce changes in those
conditions that have continued unchanged throughout E's hundred
trials. jE's trials must not consist of further samples from the
same set, but must change to samples from a new set. Thus if
the organism is a cat in a box, and if it has made 100 trials of
manipulating the levers and objects without success, now is the
time for it to make trials from a new statistical population — to
change perhaps to various forms of mewing and calling.

Thus the improvement of the E's speed of adaptation by the
selection of an appropriate value for the step-mechanisms under
F's control is not the same as making a selection on the step-
mechanisms that E itself should make. Providing an examinee
with pen, paper, and a quiet room may be called ' helping the
examinee ', but it is clearly quite distinct from the ' help ' that
would show him how to answer the individual question. F
4 helps ' the E's only in the first sense, not the second.

Thus the conclusion of S. 17/7 — that if an organism is to adapt
with reasonable speed, certain parameters will have to be brought
within certain limits — does not involve a circular appeal, for the
two selections are working at different levels, i.e. on different sets.

17/9. It is not for a moment suggested that all naturally occurring
organisms have essential variables that divide neatly into distinct

225

DESIGN FOR A BRAIN 17/9

levels: 2£'s, F's, and so on. Would that it were so ! When it
occurs, the whole act of adaptation (really life-long as we saw in
S. 10/2) can be divided into portions; then the practical scientist
can study the system portion by portion, level by level, and can
thus greatly simplify its study. The Homeostat was designed
partly so as to enable two levels to be obviously distinguishable:
(1) the four continuous variables at the magnets and (2) the
discontinuous variables on the uniselectors. When a system has
this natural internal division, the observer can take advantage
of the fact to describe the somewhat complex whole in three
stages, each considerably simpler: the continuous system and its
properties, the discontinuous system and its properties, and the
interaction between them. But when the whole system is not so
divisible it remains merely a fearfully complex whole, not capable
of reduction, and therefore as intractable to the scientist as the
examples in S. 16/3.

This book inevitably concerns itself with the case in which the
essential variables are divisible clearly into levels: the primary
levels (of Ev E2, E3, EA) in Chapters 7 to 10, and then a sharply
differentiated F in Chapters 11 to the present. In this it was
again following the strategy of S. 2/17, getting a clear grasp of
the manageable cases so that they could serve as a basis for at
least a distant survey of the unmanageable. The reader will now
appreciate that the simplicity of the earlier chapters was essentially
a didactic device, not resembling the actual complexity of actual
organisms. In fact, their real complexity is greater, by many
orders of size, than that considered here. Thus, the reacting part
R of Figure 7/5/1, which looks so simple, may not only contain
the complexities of the multistable system (Figure 16/6/1) but
also, in the higher organisms, many subsystems of the form of
Figure 7/5/1 itself, each with its own little sub-essential variables
and sub-adaptations; for much adaptation to long- term goals is
achieved by finding suitable sets of sub-goals, perhaps in complex
sequences of timing and conditionality. Thus once we have used
the carefully simplified forms of Figures 7/5/1 and 16/6/1 to
establish our understanding, we must be prepared to admit that
in the real brain the same principles work in a complexity that is
of an altogether higher order, one that may well prove to be for ever
beyond the detailed comprehension of the human scientist, who has
an I.Q. limited, for all practical purposes, to something below 200.

226

17/10 ANCILLARY REGULATIONS

With this admitted, let us continue to examine those cases in
which some division into levels, and some easy understanding, are
possible.

17/10. In Chapter 7 it was shown that the simple ultrastable
system would solve the basic problem of getting the primary
essential variables stable within their limits. But in Chapter 11
we recognised that adaptation, though it occurs in a purely logical
sense, may occur at such a low degree of efficiency as to be useless
for practical purposes. If we are to find the mechanism that
resembles the living, and especially the human, brain we must
find one that adapts, not merely in a nominal sense but with really
high efficiency. In S. 17/7 we found that such efficiency implies
adjustment of the degree of intra-cerebral connectivity to within
certain limits.

We can now notice explicitly that there are other parameters
that will also have to be adjusted if the degree of adaptation is to
be more than merely nominal. Several of these have already
been noticed in passing:

(1) In S. 8/15 we noticed that the duration of trial demanded
adjustment. In that section, the adjustment was, of course, made
by the operator before the tracings of the Homeostat's behaviour
were taken; but nothing has yet been said about how this adjust-
ment is to be made automatically in the organism.

(2) In S. 7/7 it was demanded that the essential variables
should act on the step-mechanisms in the particular way: hunt
at ' bad ' and stick at ' good '. Nothing was said about how this
particular relation was to be provided in the organism.

(3) In S. 10/8 it was shown that if an ultrastable system was
to adapt efficiently to a recurrent situation, a certain gating-
mechanism was necessary; but nothing was said about how the
organism should acquire one.

(4) S. 13/11 showed how important is the value of the para-
meter : richness of equilibria among the states of the parts.
Nothing was said about how this parameter should be adjusted
to within satisfactory limits.

Doubtless there are others that we have not yet noticed. One
other of outstanding importance deserves a section to itself.

227

DESIGN FOR A BRAIN 17/11

Distribution of feedback

17/11. Another adjustment that is necessary, if the adaptation
is to be more than merely nominal, has already been made in
Figure 16/6/1, which thereby begged an important question. In
the Figure, if we start in the environment at any subsystem and
trace a route through the essential variable that it affects, on
through the corresponding step-mechanism, reacting part, and so
back to the environment, we arrive at the same subsystem as the
one we started at. The Figure thus implies that if an essential
variable, Ex say, is being upset by a part of the environment,
E^s actions will eventually affect the very part of the environment
that is the cause of the trouble.

The correspondence undoubtedly favours efficiency in adapta-
tion, as may be seen by tracing explicitly what would happen
otherwise. (The argument is clearest when the systems are
iterated, Figure 15/7/1.) Suppose, in it, that the second-order
loops were severed and then re-connected in some random way:
so that the essential variable of A affected the reacting part of J?,
say. A disturbance to A that A is not adapted to would now
result in changes at B's step-mechanisms, though the set of values
here might be perfectly adapted to dealing with whatever disturb-
ance came to B. Thus without proper distribution of the second-
order feedbacks the effects from the essential variables would
only change at random, destroying in the process minor adapta-
tions already established. Thus without appropriate distribution
of the second-order feedbacks there cannot be that conservation
of correct adaptations in the subsystems, and the cumulative
progression to adaptation that Chapter 10 treated as of major
importance. The system would still adapt as a Homeostat does,
but it would take the excessive time of T1 rather than the moderate
time of T3 (S. 11/5).

The distribution of second-order feedbacks cannot be settled
once for all, for a part of each circuit is determined by, or supplied
by, the environment, and is thus subject to change. To this the
organism must make counter-adjustments, if the distribution is
to remain appropriate.

A well-known example that illustrates the necessity for finding
where to apply a correction is given by the aspiring chess-player
who has just -lost a game and who is considering how his strategy

228

17/13 ANCILLARY REGULATIONS

should be altered for the future. Often he is acutely aware of
the fact that he is not sure where to apply the correction. Should
he examine the last few moves and alter his tactics ? Should he,
in future, avoid that sort of middle-game ? Or, maybe, should
he stop opening with P — Q4 and change to P — K4 ? The young
chess-player has not only to solve the problems of what move to
make next but also that of where to feed back the corrections.
Thus, there may well be players today who are weak simply
because, when they lose a game, they change their opening rather
than their end-game.

(In this example the ' parts ' to be modified are strung out in
time : the modification has to find the right place in the sequence.
The example serves to remind us that a diagram of immediate
effects (such as Figure 16/6/1) represents functional, not structural
or anatomical relations.)

17/12. The last two sections have shown that at least five ancillary
regulations have to be made if the basic process of ultrastability
is to bring adaptation with reasonable efficiency and speed. The
next question thus is: how are these ancillary regulations to be
achieved ?

17/13. The answer can be given with some assurance, for all
processes of regulation are dominated by the law of requisite
variety. (It has been described in i". to C, Chapter 11 ; here will
be given only such details as are necessary.)

This law (of which Shannon's theorem 10 relating to the sup-
pression of noise is a special case) says that if a certain quantity
of disturbance is prevented by a regulator from reaching some
essential variables, then that regulator must be capable of exerting
at least that quantity of selection. (Were the law to be broken,
we would have a case of appropriate effects without appropriate
causes, such as an examinee giving correct answers before he has
been given the questions (S. 7/8). ^Scientists work on the assump-
tion that such things do not happen; and so far they have found
no fact that would make them question the assumption.) The
provision of the ancillary regulations thus demands that a process
of selection, of appropriate intensity, exist. Where shall we find
this process ?

The biologist, of course, can answer the question at once; for

229

DESIGN FOR A BRAIN 17/14

the work of the last century, and especially of the last thirty
years, has demonstrated beyond dispute that natural, Darwinian,
selection is responsible for all the selections shown so abundantly
in the biological world. Ultimately, therefore, these ancillary
regulations are to be attributed to natural selection. They will,
therefore, come to the individual (to our kitten perhaps) either
by the individual's gene-pattern or they develop under an ultra-
stability of their own. There is no other source.

17/14. The subject of adaptation in brain-like mechanisms, how-
ever, today interests an audience much wider than the biological.
I will, therefore, give a brief account of these processes of selection
so that the reader whose training has not been biological can see
just how the ancillary regulations must be developed in brains
other than the living.

The account will also serve a second purpose. So far, the book
has followed the method of starting, in Chapter 1, with the fact
of adaptation as an effect, and has argued back to its causes.
This is not the natural direction for argument, which goes alto-
gether more simply and clearly if we just take an initial state and
then ask: what will happen from now on ? I propose, therefore,
to sketch the process in its natural direction, showing that, given
a certain very general starting point, adaptation as an outcome
is inevitable.

230

CHAPTER 18

Amplifying Adaptation

Selection in the state -determined system

18/1. The origin of selections ceases to be a problem as soon as
it is realised that selection, far from being a rarity, is performed
to greater or less degree by every isolated state-determined
system (/. to C, S. 13/19). In such a system, as two lines of
behaviour may become one, but one line cannot become two, so
the number of states that it can be in can only decrease.

This selection is well known, but in simple systems it shows only
in trivial form. The spring-driven clock, for instance, is selective
for the run-down state: start it at any state of partial winding
and it will make its way to the run-down state, where it will
remain. The often-made observation that machines run to an
equilibrium expresses the same property.

In simple systems the property seems trivial, but as the system
becomes more complex so does this property become richer and
more interesting. The Homeostat, for instance, can be regarded
simply as a system, with magnets and uniselectors, that runs to
a partial equilibrium, where it sticks. But the equilibrium is only
partial, and therefore richer in content than that of the run-down
clock. The uniselectors are motionless but the magnets may still
move, and the partial equilibrium manifests a dynamic homeostasis
that has been selected by the uniselector's process of running to
equilibrium. Thus the Homeostat begins to show something of
the richness of properties that emerge when the system is complex
enough, or large enough, to show: (1) a high intensity of selection
by running to equilibrium, and also (2) that this selected set of
states, though only a small fraction of the whole, is still large
enough in itself to give room for a wide range of dynamic activities.
Thus, selection for complex equilibria, within which the observer can
trace the phenomenon of adaptation, must not be regarded as an
exceptional and remarkable event: it is the rule. The chief reason
why we have failed to see this fact in the past is that our terrestrial

231

DESIGN FOR A BRAIN 18/2

world is grossly bi-modal in its forms: either the forms in it are
extremely simple, like the run-down clock, so that we dismiss
them contemptuously, or they are extremely complex, so that we
think of them as being quite different, and say they have Life.

18/2. Today we can see that the two forms are simply at the
extremes of a single scale. The Homeostat made a start at the
provision of intermediate forms, and modern machinery, especially
the digital computers, will doubtless enable further forms to be
interpolated, until we can see the essential unity of the whole range.

Further examples of intermediate forms are not difficult to
invent. Here is one that shows how, in any state-determined
dynamic system, some properties will have a greater tendency to
persist, or ' survive ', than others. Suppose a computer has a
hundred stores, labelled 00 to 99, each of which initially holds one
decimal digit, i.e. one of 0, 1, 2, . . . , 9, chosen at random, inde-
pendently and equiprobably. It also has a source of random
numbers (drawn, preferably, from molecular, thermal, agitation).
It now repeatedly performs the following operation:

Take two random numbers, each of two digits; suppose 82
and 07 come up. In this case multiply together the numbers
in stores 82 and 07, and replace the digit in the first store
(no. 82) by the right-hand digit of the product.

Now Even x Even gives Even, and Odd x Odd gives Odd;
but Odd X Even gives Even, so the number in the first store can
change from Odd to Even, but not from Even to Odd. As a
result, the stores, which originally contained Odds and Evens in
about equal numbers, will change to containing more and more
Evens, the Odds gradually disappearing. The biologist might say
that in the ' struggle ' to occupy the stores and survive the Evens
have an advantage and will inevitably exterminate the Odds.

In fact, among the Evens themselves there are degrees of
ability to survive. For the Zeros have a much better chance
than the other Evens, and, as the process goes on, so will the
observer see the Zeros spread over the stores. In the end they
will exterminate their competitors completely.

18/3. This example is easily followed, but is uncomfortably close
to the trivial. More complex examples could easily be set up,
but they would tell us nothing of the principles at work (though

232

18/3 AMPLIFYING ADAPTATION

they would provide most valuable and convincing examples).
What all would show is that when a single-valued operation is
performed repeatedly on a set of states (this operation being the
4 laws ' of the system), the system tends to such states as are not
affected by the operation, or are affected to less than usual degree.
In other words, every single-valued operation tends to select forms
that are peculiarly able to resist its change-inducing action. In
simple systems this fact is almost truistic, in complex systems
anything but. And when it occurs on the really grand scale, on a
system with millions of variables and over millions of years, then
the states selected are likely to be truly remarkable and to show,
among their parts, a marked co-ordination tending to make them
immune to the operation.

The development of life on earth must thus not be seen as
something remarkable. On the contrary, it was inevitable. It
was inevitable in the sense that if a system as large as the surface
of the earth, basically polystable, is kept gently simmering
dynamically for five thousand million years, then nothing short
of a miracle could keep the system away from those states in
which the variables are aggregated into intensely self-preserving
forms. The amount of selection performed by this system, of
which we know only one example, is of an order of size so vastly
greater than anything that we experience as individuals, that we
not unnaturally have some difficulty in grasping that the process
is really the same as that seen so trivially in our everyday systems.
Nevertheless it is so; the greater extension in space enables a
vastly greater number of forms to be tested, and the greater
extension in time enables the forms to be worked up to a vastly
greater degree of intricate co-ordination.

We can thus trace, from a perfectly natural origin, the gene-
patterns that today inhabit the earth; we are not surprised that
the earth has developed forms that show, in conjunction with their
environments, the most remarkable power of being resistant to the
change-inducing actions of the world around them. They are
resistant, not in the static and uninteresting way that a piece of
granite, or a run-down clock, is resistant, but in the dynamic and
much more interesting way of forming intricate dynamic systems
around themselves (their so-called ' bodies ', with extensions such
as nests and tools) so that the whole is homeostatic and self-
preserving by active defences.

233

DESIGN FOR A BRAIN 18/4

18/4. What concerns us in this book is the fact that the active
defences can be direct or indirect. The direct were considered only
in S. 1/3. They include all the regulatory mechanisms that are
specified in detail by the gene-pattern. They are adapted because
the conditions that insisted on them have been constant over many
generations.

The earlier forms of gene-pattern adapted in this way only. The
later forms, however, have developed a specialisation that can
give them a defence against a class of disturbances to which the
earlier were vulnerable. This class consists of those disturbances
that, though not constant over a span of many generations (and
thus not adaptable to by the gene-pattern, for the change is too
rapid) are none the less constant over a span of a single generation.
When disturbances of this class are frequent, there is advantage
in the development of an adapting mechanism that is (1) controlled
in its outlines by the gene-pattern (for the same outlines are
wanted over many generations), and (2) controlled in details by the
details applicable to that particular generation.

This is the learning mechanism. Its peculiarity is that the
gene-pattern delegates part of its control over the organism to
the environment. Thus, it does not specify in detail how a kitten
shall catch a mouse, but provides a learning mechanism and a
tendency to play, so that it is the mouse which teaches the kitten
the finer points of how to catch mice.

This is regulation, or adaptation, by the indirect method. The
gene-pattern does not, as it were, dictate, but puts the kitten into
the way of being able to form its own adaptation, guided in detail
by the environment.

18/5. We can now answer the question raised in S. 17/12, and
can see how the law of requisite variety is to be applied to the
question of how the ancillary regulations are to be achieved, i.e.
how the necessary parameters are to be brought to their appro-
priate values.

Some may be adjusted by the direct action of the gene-pattern,
so that the organism is born with the correct values. For this
to be possible, the environmental conditions must have been
constant for a sufficiently long time, and the processes of natural
selection must have been intense enough and endured long enough
for the total selection exerted to satisfy the law.

234

18/6 AMPLIFYING ADAPTATION

Some ancillary regulations may be adjusted by the gene-pattern
at one remove. In this case the gene-pattern would establish
values that would result in the appearance of a mechanism,
actually a regulator, that would then proceed, by its own action,
to bring the parameters to appropriate values.

Other ancillary regulators might be adjusted by the gene-
pattern at two removes ; but we need not trace the matter further,
as real systems will seldom be arranged neatly in distinct levels
(S. 17/9). All we need notice here is that adaptation can be
achieved by the gene-pattern either directly or indirectly.

Amplifying adaptation

18/6. The method of adaptation by learning is the only way of
achieving adaptation when what is adaptive is constant for too
short a time for adaptation of the gene-pattern to be achieved.
For this reason alone we would expect the more advanced organ-
isms to show it. The method, however, has also a peculiar
advantage that is worth notice, particularly when we consider
the limitation implied by the law of requisite variety, and ask
how much regulation the gene-pattern can achieve in the two cases.

Direct and indirect regulation occur as follows. Suppose an
essential variable X has to be kept between limits x' and x" .
Whatever acts directly on X to keep it within the limits is regu-
lating directly. It may happen, however, that there is a mechan-
ism M available that affects X, and that will act as a regulator
to keep X within the limits x' and x" provided that a certain
parameter P (parameter to M) is kept within the limits p' andp".
If, now, any selective agent acts on P so as to keep it between
p' and p", the end result, after M has acted, will be that X is
kept between x' and x" .

Now, in general, the quantities of regulation required to keep
P in p' and p" and to keep X in x' to x" are independent. The
law of requisite variety does not link them. Thus it may happen
that a small amount of regulation supplied to P may result in a
much larger amount of regulation being shown by X.

When the regulation is direct, the amount of regulation that
can be shown by X is absolutely limited to what can be supplied
to it (by the law of requisite variety) ; when it is indirect, however,
more regulation may be shown by X than is supplied to P. Indirect

235

DESIGN FOR A BRAIN 18/7

regulation thus permits the possibility of amplifying the amount
of regulation; hence its importance.

18/7. Living organisms came across this possibility aeons ago,
for the gene-pattern is a channel of communication from parent
to offspring: 4 Grow a pair of eyes,' it says, ' they'll probably
come in useful; and better put haemoglobin into your veins —
carbon monoxide is rare and oxygen common.' As a channel of
communication it has a definite, finite capacity, Q say. If this
capacity is used directly, then, by the law of requisite variety,
the amount of regulation that the organism can use as defence
against the environment cannot exceed Q. To this limit, the
non-learning organisms must conform. If, however, the regula-
tion is done indirectly, then the quantity Q, used appropriately,
may enable the organism to achieve, against its environment, an
amount of regulation much greater than Q. Thus the learning
organisms are no longer restricted by the limit.

The possibility of such ' amplification ' is well known in other
ways. If a child wanted to discover the meanings of English
words, and his father had only ten minutes available for instruc-
tion, the father would have two possible modes of action. One is
to use the ten minutes in telling the child the meanings of as many
words as can be described in that time. Clearly there is a limit to
the number of words that can be so explained. This is the direct
method. The indirect method is for the father to spend the ten
minutes showing the child how to use a dictionary. At the end
of the ten minutes the child is, in one sense, no better off ; for not
a single word has been added to his vocabulary. Nevertheless
the second method has a fundamental advantage ; for in the future
the number of words that the child can understand is no longer
bounded by the limit imposed by the ten minutes. The reason
is that if the information about meanings has to come through
the father directly, it is limited to ten-minutes' worth; in the
indirect method the information comes partly through the father
and partly through another channel (the dictionary) that the
father's ten-minute act has made available.

In the same way the gene-pattern, when it determines the growth
of a learning animal, expends part of its resources in forming a
brain that is adapted not only by details in the gene-pattern but
also by details in the environment. The environment acts as the

236

18/7 AMPLIFYING ADAPTATION

dictionary. While the hunting wasp, as it attacks its prey, is
guided in detail by its genetic inheritance, the kitten is taught
how to catch mice by the mice themselves. Thus in the learning
organism the information that comes to it by the gene-pattern
is much supplemented by information supplied by the environ-
ment; so the total adaptation possible, after learning, can exceed
the quantity transmitted directly through the gene-pattern.

237

Summary

The primary fact is that all isolated state-determined
dynamic systems are selective : from whatever state they
have initially, they go towards states of equilibrium. The
states of equilibrium are always characterised, in their rela-
tion to the change-inducing laws of the system, by being
exceptionally resistant.

(Specially resistant are those forms whose occurrence leads,
by whatever method, to the occurrence of further replicates
of the same form — the so-called c reproducing ' forms.)

If the system permits the formation of local equilibria,
these will take the form of dynamic subsystems, exception-
ally resistant to the disruptive effects of events occurring
locally.

When such a stable dynamic subsystem is examined intern-
ally, it will be found to have parts that are co-ordinated in
their defence against disturbance.

If the class of disturbance changes from generation to
generation but is constant within each generation, even more
resistant are those forms that are born with a mechanism
such that the environment will make it act in a regulatory
way against the particular environment — the ' learning '
organisms.

This book has been largely concerned with the last stage
of the process. It has shown, by consideration of specially
clear and simple cases, how the gene-pattern can provide a
mechanism (with both basic and ancillary parts) that, when
acted on by any given environment, will inevitably tend to
adapt to that particular environment.

238

APPENDIX

CHAPTER 19

The State- determined System

19/1. The mathematics necessary for the study of adaptation
does not consist simply of the solution of a particular mathe-
matical problem. The problem, to the bio-mathematician, ranges
from the identification of the basic logic necessary for the repre-
sentation of the basic concept of mechanism, through its develop-
ment into various branches (such as from the discrete to the
continuous and from the non -metric to the metric), to the eventual
use of specialised techniques for special particular problems.

Since the problems that interest the biologist usually come
from systems of very great complexity, in which treatment of all
the facts is not possible, special importance must be given
to methods, such as that of topology, that allow simple answers
to be given to simple questions, even though the basic facts are
complex. The mathematical basis should therefore be sufficiently
general to allow specialisation into the methods of topology.
Here we have been greatly aided by the magnificent work of the
French school that writes, collectively, under the pseudonym of
N. Bourbaki. In their great Elements de, Mathematiques this
school has shown how the theory of sets, in a simple basic form,
can be gradually extended and developed, without the least loss
of precision or the least change in the fundamental concepts, into
the realms of topology, algebra, geometry, theory of functions,
differential equations, and all the various branches of mathematics.

How the theory of sets, essentially in the form used by Bourbaki,
gives a secure basis for the logic of mechanism, has already been
displayed in Part I of J. to C. (That book does not use Bourbaki's
symbols explicitly, but his concepts are used throughout and in
exactly his form; so the reader who wishes to correlate /. to C.
with Bourbaki's work will find that the correlation is in most
places obvious.)

241

DESIGN FOR A BRAIN 19/2

The logic of mechanism

19/2. Our starting-point is the idea, much more than a century
old, that a ' machine ' is that which, whenever it is in given
conditions and at a given internal state, goes always to a parti-
cular state (i.e. not to different states on different occasions).
This definition at once shows its formal correspondence with
Bourbaki's ' algebraic law of external composition '. For if the
external conditions can be at any one of a set Q, and the internal
states of the machine at any one of a set E, then the machine
defines, by its behaviour, a mapping (Bourbaki's ' application')
of Q x E into E. The concept of ' machine ' thus corresponds
exactly to one of the most basic concepts in mathematics.

After this basic identification many others follow at once.
The mapping of E into E given by holding the value of Q constant
corresponds to the machine when isolated. An element of E
that is invariant in the algebra (for some value of Q) corresponds
to a state of equilibrium of the machine when the input (or
surrounding conditions) is held constant (i.e. for a given field).
The compatibility (or not) of an equivalence relation with an
external law of composition corresponds to whether or not a
proposed simplification of a state-determined system leaves the
new system still state-determined. If it does, then the algebraic
quotient-law corresponds to the new, simplified, canonical repre-
sentation. And so on, in a manner that deserves extensive
treatment.

It is not my intention here to develop the subject ab initio
and extensively. As this book is concerned primarily with the
brain and with systems in which continuity is common, we need
only notice that Bourbaki has shown how the basic concepts,
stated in discrete form, can be specialised to the continuous forms
and to those with a metric. In this Appendix we will deal only
with such forms as are continuous and provided with a metric.

(N.B. Throughout this chapter the emphasis is on the system
that is isolated and left alone to show what it will do, apart from
occasional interferences from the experimenter. The statements
made should be interpreted accordingly. Chapter 21 deals
explicitly with the system that is being subjected to changes in
its conditions, or at its input.)

242

19/7 THE STATE-DETERMINED SYSTEM

19/3. A variable is a function of the time. A system of n
variables will usually be represented by xl9 x2, . . ., xn, or some-
times more briefly by x. The case where n = 1 is not excluded.
It will be assumed throughout that n is finite; a system with an
infinite number of variables (e.g. that of S. 19/17) will be replaced
by a system in which i is discontinuous and n finite, and which
differs from the original system by some amount that is negligible.
Each variable xt is a function of the time t; it will sometimes be
written as x^t) for emphasis. It must be single-valued, but need
not be continuous. A constant may be regarded as a variable
which undergoes zero change.

19/4. The state of a system at a time t is the set of numerical
values of x-^t), . . . , xn(t). Two states (xv . . . , xn) and
(Vv • • • > Vn) are e<Iual if x% = Vi for a11 *•

19/5. A transition can be specified only after an interval of time,
finite and represented by At or infinitesimal and represented by
dt, has been specified. It is represented by the pair of states,
one at time t and one at the specified time later.

A line of behaviour is specified by a succession of states and the
time-intervals between them. Two lines of behaviour are equal
if all the corresponding states and time-intervals along the suc-
cession are equal. (So two lines of behaviour that differ only in
the absolute times of their origin are equal.)

19/6. A primary operation is a physical event, not a mathe-
matical, requiring a real machine and a real operator or experi-
menter. He selects an initial state (x\, . . . , a?J), and then
records the transition that occurs as the system changes in
accordance with its own internal drives and laws.

19/7. If, on repeatedly applying primary operations, he finds
that all the lines of behaviour that follow an initial state S are
equal, and if a similar equality occurs after every other state
S',S", . . . , then the system is regular.

Such a system can be represented by equations of form
xx = Fx(xl . . . , xl ; t)

Xn — Fn\x\9 ' • • » xn i 0

243

DESIGN FOR A BRAIN 19/8

in which the F's are single-valued functions of their arguments
but are otherwise quite unrestricted. Obviously, if the initial
state is at t = 0, we must have

F,(«J, . . . , 4 1 0) = a$ (i = 1, . . . , n)

19/8. Theorem : The lines of behaviour of a state-determined system
define a group.

Let the initial state of the variables be x°, where the single
symbol represents all n, and let time t' elapse so that x° changes
to x' . With x' as initial state let time t" elapse so that x' changes
to x" . As the system is state-determined, the same total line of
behaviour will be followed if the system starts at x° and goes on
for time t' + t". So

x[ = Ftih . . . , xn ; t") = F,(xl . . . , xl ; f + t")

{i = 1, . . . , n)
But

x\ = F,(4 . . . , x°n; t') {i = 1,. . . , n)

giving

Ft{F^; t'), . . . ,Fn(x°; f); t") = F,(4 . . . ,x°n; V + t")

(i = 1, . . . . , n)

for all values of x°, t\ and t" over some given region; and this is
one way of defining a one-parameter finite continuous group.

The converse is not true. Thus x — (1 -f- t)x° defines a group
(with n = 1); but the times do not combine by addition, and the
system is not state-determined.

Example : The system with lines of behaviour given by
xx = x\ + x?zt + t2
x2 = x% + 2i
is state-determined, but that with lines given by

X-t ^= X-t ~\~ X'jl \~ i

x2 = X?z + t

is not.

Canonical representation

19/9. Theorem: That a system xv . . . , xn should be state-
determined it is necessary and sufficient that the x's, as functions oft,
should satisfy equations

244

19/9 THE STATE-DETERMINED SYSTEM

(1)

-^ -M*v • • • > *»)

w/^rtf iheps are single-valued, but not necessarily continuous, func-
tions of their arguments; in other words, the fluxions of the set
x±, . . . , xn can be specified as functions of that set and of no
other functions of the time, explicit or implicit. The equations, in
this form, are said to be the canonical representation of the system.
(The equations will sometimes be written

dxjdt =f(x1, . . . , xn) (i = 1, . . . , n) . (2)

and may be abbreviated even to x =f(x) if the context makes the
meaning clear.)

(1) Let the system be state-determined. Start it at x\, . . . , x„
at time t = 0 and let it change to xv . . . , xn at time t, and then
on to xx + dxv . . . , xn + dxn at time t + dt. Also start it at
xv . . . , xn at time t = 0 and let time dt elapse. By the group
property (S. 19/8) the final states must be the same. Using
the same notation as S. 19/8, and starting from x\, xt changes
to Fi (x°; t + dt and starting at xt it gets to F^x; dt). Therefore

F^x0; t + dt) = Fix; dt) (i = 1, . . . , n).

Expand by Taylor's theorem and write -xrF^a; b) as F'^a; b).

Then

Ft.(*°; t) + dt.FfaO; t\ = Fi(x; 0) + dt.F&x; 0)

(i = 1, . . . , n)

But both F{{x°; t) and F{(x; 0) equal xt.

. , n) . . (3)
. , n)

Therefore

JF>°; t) = F'lxi 0)

(i = 1,

But

xi = F^x0; t)

(* = 1,

dt dtl{x ' l)

= n+i t)

dx
so by (3), -^ = F'i(x; 0) (i = 1, . . . , n)

which proves the theorem, since Fi(x; 0) contains t only in
xv . . . , xn and not in any other form, either explicit or implicit.

245

DESIGN FOR A BRAIN

19/10

Example 1: The state-determined system of S. 19/8, treated in
this way, yields the differential equations, in canonical form:

da\
lit
dx2 _ |
dt -2J

The second system may not be treated in this way as it is not
state-determined and the group property does not hold.
Corollary:

fi(xv

xn)

diFi(Xli

««; 0

(t = if

*=o

= 2

(2) Given the differential equations, they may be written

dxt =fi{xv . . . , xn).dt (i = 1, . . . , n)

and this shows that a given set of values of xv . . . , xn, i.e.
a given state of the system, specifies completely what change,
dxiy will occur in each variable, x{, during the next time-interval,
dt. By integration this defines the line of behaviour from that
state. The system is therefore state-determined.
Example 2 : By integrating

dxx
~dt
dx2
~dt

the group equations of the example of S. 19/8 are regained.

19/10. Definition. The system is linear when the functions
fv • • • 9 fn are aN nnear functions of the arguments xv . . . , xn.

19/11. Example 3: The equations of the Homeostat may be
obtained thus : — If xt is the angle of deviation of the i-th magnet
from its central position, the forces acting on xt are the momentum,
proportional to xi3 the friction, also proportional to x{, and the
four currents in the coil, proportional to xv x2, xz and xA. If
linearity is assumed, and if all four units are construct ionally
identical, we have

(mxt)

kx{ + l(p - qftoiM + . . . + auxA)

{i = 1, 2, 3, 4)

246

19/12 THE STATE-DETERMINED SYSTEM

where p and q are the potentials at the ends of the trough, I
depends on the valve, k depends on the friction at the vane,
and m depends on the moment of inertia of the magnet. If
h = l(p — q)/m and j = k/m, the equations may be written
dxjdt — x{ \ {•

dxjdt = &(««*i + . . . + fl,4a4) -jxt J {l = l> 2' 3' 4)

which shows the 8-variable system to be state-determined and
linear.
They may also be written

dati

dt ~Xi

dxt k (Up — q). , , ,

> (i = 1, 2, 3, 4)

Let m — > 0. dxi/dt becomes very large, but not dx{/dt. So
±i tends rapidly towards

k q (gfl^l + • • • + ai*Zi)

while the x's, changing slowly, cannot alter rapidly the value
towards which it is tending. In the limit,

^ = it = ^^(-Bft + • • • + aiiXl) (i = 1, 2, 3, 4)

Change the time-scale by t = ^ t, and

dxjdr = anxx + . . . + aux4 (i == 1, 2, 3, 4)

showing the system xl9 x2, x3, x± to be state-determined and linear.
The a's are now the values set by the input controls of Figure

8/2/3.

19/12. The theorems of the preceding sections show that the
following properties are equivalent, in that the possession of any
one implies the possession of the remainder.

(1) The system is state-determined.

(2) From any point of the field departs only one line of be-

haviour.

(3) The lines of behaviour are specifiable by equations of form:

dxi/dt ='fi(xl, . . . , xn) (i = 1, . . . , n)

247

DESIGN FOR A BRAIN 19/13

in which the right-hand side contains no functions of t except those
whose fluxions are given on the left.

19/13. A simple example of a system which is regular but not
state-determined is given by the following apparatus. A table
top is altered so that instead of being flat, it undulates irregularly
but gently like a putting-green (Figure 19/13/1). Looking down

Figure 19/13/1.

on it from above, we can mark across it a grid of lines to act
as co-ordinates. If we place a ball at any point and then release
it, the ball will roll, and by marking its position at, say, every
one-tenth second we can determine the lines of behaviour of the
two-variable system provided by the two co-ordinates.

If the table is well made, the lines of behaviour will be accur-
ately reproducible and the system will be regular. Yet the
experimenter, if he knew nothing of forces, gravity, or momenta,
would find this two- variable system unsatisfactory. He would
establish that the ball, started at A, always went to A' \ and
started at B it always went to B'. He would find its behaviour at
C difficult to explain. And if he tried to clarify the situation by
starting the ball at C itself, he would find it went to D ! He would
say that he could make nothing of the system ; for although each
line of behaviour is accurately reproducible, the different lines
of behaviour have no simple relation to one another. He will,
therefore reject this two-variable system and will not rest till
he has discovered, either for himself or by following Newton,

248

19/16 THE STATE -DETERMINED SYSTEM

a system that is state-determined. In my theory I insist on the
systems being state-determined because I agree with the experi-
menter who, in his practical work, is similarly insistent.

Transformations of the canonical representation

19/14. Sometimes systems that are known to be isolated and
complete are treated by some method not identical with that used
here. In those cases some manipulation may be necessary to
convert the other form into ours. Some of the possible manipula-
tions will be shown in the next few sections.

19/15. Systems can sometimes be described better after a change
of co-ordinates. This means changing from the original variables
xv . . . , xn to a new set yl9 . . . , yn, equal in number to the
old and related by single-valued functions <j>{:

Vi = <f>i(xv .-.,#«) (i = 1, . . . , n)

If we think of the variables as being represented by dials, the
change means changing to a new set of dials each of which indicates
some function of the old. If the functions </>t- are unchanging in
time (as functions of their arguments), the new system will remain
state-determined.

19/16. In the ' Homeostat ' example of S. 19/11 a fluxion was
treated as an independent variable. I have found this treat-
ment to be generally advantageous: it leads to no difficulty or
inconsistency, and gives a beautiful uniformity of method.

For example, if we have the equations of a state-determined
system we can write them as

*< -fii^v ...,#») = 0 (i = 1, . . . , n)

treating them as n equations in 2n algebraically independent
variables xl9 . . . , xn, xv . . . , xn. Now differentiate all the
equations q times, getting (q -f\l)w equations with (q + 2)n
variables and derivatives. We can then select n of these vari-
ables arbitrarily, and noticing that we also want the next higher
derivatives of these w, we can eliminate the other qn variables,
using up qn equations. If the variables selected were zv . . . , zn
we now have n equations, in 2n variables, of type

0((zv . . . , zni zl9 . . . 3 zn) = 0 (i =* 1, . . . , n)

249

DESIGN FOR A BRAIN 19/17

where the z's are the selected x's, and z's the corresponding as's.
These have only to be solved for %,..., zn in terms of
»!, . . . , Zn and the equations are in canonical form. So the
new system is also state-determined (by S. 19/9).

This transformation implies that in a state- determined system we
can avoid direct reference to some of the variables provided we use
derivatives of the remaining variables to replace them.

Example: xx = xx — x2 \

x2 = oXi -\- x2J

can be changed to omit direct reference to x2 by using xx as a new
independent variable. It is easily converted to

dx1/dt = ij

dxjdt = — 4>x± + 2a?j

which is in canonical form in the variables x, and x*

19/17. Systems which are isolated but in which effects are
transmitted from one variable to another with some finite delay
may be rendered state-determined by adding derivatives as
variables. Thus, if the effect of x1 takes 2 units of time to reach
x2, while x2s effect takes 1 unit of time to reach xv and if we write
x(t) to show the functional dependence,

then dxx(t)/dt =/iK(0, oc2(t - 2)Y

dx2(t)/dt=f2{x1(t- 1), x2(t)}

This is not in canonical form; but by expanding xx(t — 1) and
x2{t — 2) in Taylor's series and then adding to the system as
many derivatives as are necessary to give the accuracy required,
we can obtain a state-determined system which resembles it as
closely as we please.

19/18. If a variable depends on some cumulative effect so that,
say, x1=f< (f>(x2)dt I, then if we put <j>(x2)dt = y, we get the

equivalent form

dxjdt=f(y)
dy/dt = <f>(x2)
dxjdt = . . . etc.

which is in canonical form.

250

19/21 THE STATE-DETERMINED SYSTEM

19/19. If a variable depends on velocity effects so that, for
instance

1 ~f\dt' Xv xy

dx
It

dX2 _ ft \

fa — J2\XV X2)

dx
then if we substitute for -— inf^. . .) we get the canonical form

dx
dx

i/dt ==/i{/2(a>i,0a), xv x2}^
Jdt = f2(x1, x2) ]

19/20. If one variable changes either instantaneously or fast
enough to be so considered without serious error, then its value
can be given as a function of those of the other variables; and
it can therefore be eliminated from the system.

19/21. Explicit solutions of the canonical equations

dxjdt =fi(xv . . . , xn) (i = 1, . . . , n)

will seldom be needed in our discussion, but some methods will
be given as they will be required for the examples.

(1) A simple symbolic solution, giving the first few terms of
xt as a power series in t, is given by

e, = e**x\ (i = 1, . . . , n) . . (1)

where X. is the operator

and etx = 1jrtX+tlx> +^X* + . . . . (3)

It has the important property that any function 0(xv . . . , xn)
can be shown as a function of t, if the x's start from x®, . . . , x„,

by 0(x19 . . . ,xn) = etx${xl . . . , x») . . (4)

(2) If the functions ft are linear so that

dxjdt = anxx -f a12x2 -f . . . + alnxn + bx

dxjdt = anlxx -f an2x2 -f • • • + annxn + bn
251

(5)

DESIGN FOR A BRAIN 19/22

then if the fr's are zero (as ean be arranged by a change of
origin) the equations may be written in matrix form as

x = Ax . . . (6)

where x and x are column vectors and A is the square matrix
[aij]. In matrix notation the solution may be written

x = etAx° .... (7)

(3) Most convenient for actual solution of the linear form is
the recently developed method of the Laplace transform. The
standard text-books should be consulted for details.

19/22. Any comparison of a state-determined system with the
other types of system treated in physics and thermodynamics
must be made with caution. Thus, it should be noticed that the
concept of the state-determined system makes no reference to
energy or its conservation, treating it as inelevant. It will also be
noticed that the state-determined system, whatever the ' machine '
providing it, is essentially irreversible. This can be established
either by examining the behaviour representation of S. 19/7, the
canonical representation of S. 19/9, or, in a particular case, by
examining the field of the common pendulum in Figure 2/15/1.

252

CHAPTER 20

Stability

20/1. As will be seen in S. 21/14, the canonical representation
contains all the information that the real ' machine ' can give
relative to the selected system. By selecting a particular system
the experimenter has already acknowledged that he can obtain
only a finite amount of information from the infinite amount that
exists in the real ' machine ' ; yet even this reduction is often
insufficient, for the canonical representation of the behavioural
properties of xv ... , xn may still convey an unmanageably
large amount of information. Take the case, for instance, of the
cluster of 20,000 stars, about which the astronomer asks: will
the cluster condense to a ball, or will it disperse ? The canonical
representation can be set up (it has 120,000 variables), and it
contains the answer; but the labour of extracting it is so pro-
hibitively great that astronomers, and others in like position,
have looked for methods that do not use all the information
available in the canonical representation. Hence the introduc-
tion into science of statistical and topological methods, and the
use of concepts such as independence (S. 12/4) which may, if the
case is suitable, enable us to get a simple answer to a simple
question without the necessity for our going into every detail.

Prominent among such concepts is that of stability. Its basic
elements have been given in /. to C, Chapter 5. Here we shall
treat it only in the form suitable for continuous systems, and only
with such rigour as is necessary for our main purpose.

20/2. Given a state-determined system in unvarying conditions,
so that it has one field, and given a region in the field and a point in
the region, a line of behaviour from the point is stable, with respect
to that field and region and point, if it never leaves the region.

20/3. If all the lines within a given region are stable from all
points within the region, and if all the lines meet at one point,
the system has normal stability.

253

DESIGN FOR A BRAIN 20/4

20/4. A state of equilibrium can be denned in several ways. In
the field it is a terminating point of a line of behaviour. In the
equations of S. 19/7 the state of equilibrium Xv . . . , Xn is
given by the equations

Xt = Lim F^x0; t) {i = 1, . . . , n) . (1)

t— >00

if the n limits exist. In the canonical equations the values satisfy
f(Xv . . . , Xn) = 0 (i = 1, . . . , n) . (2)

A state of equilibrium is an invariant of the group, for a change
of t does not alter its value.

3fi

If the Jacobian of the/'s, i.e. the determinant

dxj

which will

be symbolised by J, is not identically zero, then there will be
isolated states of equilibrium. If, J =0, but not all its first minors
are zero, then the equations define a curve, every point of which
is a state of equilibrium. If J = 0 and all first minors but not all
second minors are zero, then a two-way surface exists composed
of states of equilibrium; and so on.

20/5. Theorem: If the f's are continuous and differ entiable, a
state- determined system tends to the linear form (S. 19/10) in the
neighbourhood of a state of equilibrium.
Let the system, specified by

dxjdt = fi(x1, . . . , xn) (i == 1, . . . , n)

have a state of equilibrium Xlf . . . , Xn, so that

fi(Xli . . . , Xn) = 0 (i = 1, . . . , n>.

Put xt = Xt ; -f £,. (i = t, . . . , n) so that xt is measured as a
deviation £t from its equilibrial value. Then

^(A-< + I,.) =fi(X1 + ft, . . . , Xn + £,) (i = 1, . . . , n)

Expanding the right-hand side by Taylor's theorem, noting that
dXJdt = 0 and tYi&tf^X) = 0, we find, if the £'s are infinitesimal,
that

W = ^ + • • • + WJn (t - 1, . . . , »).

The partial. derivatives, taken at the point Xv . . . , Xn, are
numerical constants. So the system is linear.

254

20/6 STABILITY

20/6. In general the only test for stability is to observe or
compute the given line of behaviour and to see what happens
as t — ► oo. For the linear system, however, there are tests that
do not involve the line of behaviour explicitly. Since, by the
previous section, many systems approximate to the linear within
the region in which we are interested, the methods to be described
are often applicable.

Let the linear system be
dxjdt = aixxx + ai2x2 + . . . + ainxn (i = 1, . . . , n) (1)

or, in the concise matrix notation (S. 19/21)

x = Ax . . . . (2)

Constant terms on the right-hand side make no difference to
the stability and can be ignored. If the determinant of A is not
zero, there is a single state of equilibrium. The determinant
an — A a12 • • • dm

#21 #22 A • • • a2n

anl an2 ... ann—X

when expanded, gives a polynomial in X of degree n which, when
equated to 0, and, if necessary, multiplied by —1, gives the
characteristic equation of the matrix A :

Xn + mj?-1 + m2Xn~2 + . . . + ™>n = 0.
Each coefficient rat is the sum of all i-rowed principal (co-axial)
minors of A, multiplied by (— 1)*. Thus,

mi = — (%1 + «22 + • • • + ann)\ ™n = (~ !)" \ Al

Example: The linear system

dxjdt = — 5x1 + 4cT2 — 6^3"j
dxjdt = 7x± — 6x2 + 8#3 V
dxjdt = — 2x-l + 4#2 — 4#3J

has the characteristic equation

P + 15A2 + 2A + 8 = 0.
Of this equation, the roots Al9 . . . , Xn are the latent roots
of A. The integral of the canonical representation gives each
x{ as a linear function of the exponentials e\*, . . . , exn. For
the sum to be convergent, every real part of Xx, . . . , An must be
negative, and this criterion provides a test for the stability of
the system.

255

DESIGN FOR A BRAIN 20/7

Example: The equation A3 + 15 A2 + 2 A + 8 = 0 has roots
— 14-902 and — 0-049 ± 0-729 V — 1, so the system is stable.

20/7. A test which avoids finding the latent roots is Hurwitz' :
a necessary and sufficient condition that the linear system is
stable is that the series of determinants

etc.

mly

m1 1

m1 1 0

m1 1 0 0

ra3 m2

m3 m2 m1

ra3 m2 mx 1

ra5 ra4 m3

m5 ra4 m3 ra«

m7 m6 m5 m^

(where, i

f q > n,

= 0), are al

positive.

Example: The system with characteristic equation
A3 + 1522 + 2A + 8 = 0
yields the series

+ 15,

0
15

8
0

These have the values —15, + 22, and +176. So the system
is stable, agreeing with the previous test.

20/8. Another test, related to Nyquist's, states that a linear
system is stable if, and only if, the polynomial

ln + m^""1 + m2An"2 + . . . + mn
changes in amplitude by nn when A, a complex variable
(X = a + hi where i = V — 1), goes from — i oo to + i oo along
the 6-axis in the complex A-plane.

Nyquist's criterion of stability is widely used in the theory
of electric circuits and of servo-mechanisms. It, however, uses
data obtained from the response of the system to persistent
harmonic disturbance. Such disturbance is of little use in the
theory of adapting systems, and will not be discussed here.

20/9. Some examples will illustrate various facts relating to
stability in linear systems.

Example 1: The diagonal terms au represent the intrinsic
stabilities of the variables; for if all variables other than xt are
held constant, the linear system's i-th equation becomes

dx{/dt = aHXi + c,
256

20/9

STABILITY

where c is a constant, showing that under these conditions xt
will converge to — c/aH if au be negative, and will diverge without
limit if au be positive.

If the diagonal terms au are much larger in absolute magnitude
than the others, the latent roots tend to the values of au. It
follows that if the diagonal terms take extreme values they
determine the stability.

Example 2: If the terms au in the first n — 1 rows (or columns)
are given, the remaining n terms can be adjusted to make the
latent roots take any assigned values.

Example 3: The matrix of the Homeostat equations of S. 19/11
is

fliJi a-iJn a,Ji

a2Ji
azlh
aAJi

12'

a22h
a»9h

13'

a23h
azzh
ai3h

a24
a34

If j = 0, the system must be unstable, for the eight latent roots
are the four latent roots of [atJ], each taken with both positive and
negative signs. If the matrix has latent roots pl9 . . . , //8, and
if A1? . . . , A4 are the latent roots of the matrix [a^h], and if
j ^ 0, then the A's and ^'s are related by Xv = ju^ + j/^a- As
j — > ± °o the 8-variable and the 4-variable systems are stable
or unstable together.

Example 4: In a stable system, fixing a variable may make
the system of the remainder unstable. For instance, the system
with matrix

— 10

— 3

— 1

— 6

is stable. But if the third variable is fixed, the system of the
first two variables has matrix

6 5"

4 - 3

and is unstable.

257

[_: _g

DESIGN FOR A BRAIN

20/10

Example 5: Making one variable more stable intrinsically
(Example 1 of this section) may make the whole unstable. For
instance, the system with matrix

— 4 —31
3 2 J

is stable. But if an becomes more negative, the system becomes

unstable when an becomes more negative than
Example 6: In the n x n matrix

a ! b

in partitioned form, let the order of [a] be k x k. If the k diagonal
elements au become much larger in absolute value than the rest,
the latent roots of the matrix tend to the k values au and the
n — k latent roots of [d]. Thus the matrix, corresponding to [d],

1 — 3^

1 2

has latent roots + 1-5 ± l-658t,
- 100 — 1

and the matrix
2 0"

2—100—1 2

0 — 3 1—3

2-1 1 2

has latent roots — 101-39, — 98-62, and + 1-506 ± l-720t.

Corollary: If system [d] is unstable but the whole 4-variable
system is stable, then making xx and x2 more stable intrinsically
will eventually make the whole unstable.

Example 7: The holistic nature of stability is well shown by
the system with matrix

— 3 — 2 2^

- 5

5 6

2 — 4

in which each variable individually, and every pair, is stable;
yet the whole is unstable.

The probability of stability

20/10. The probability that a system should be stable can be
made precise only after the system has been defined, ' stability '

258

20/10

STABILITY

defined for it, and then a proper sample space denned. In general,
the number of possible meanings of ' probability of stability ' is
too large for extensive treatment here. Each case must be
considered individually when such consideration is called for.

A case of some interest because of its central position in the
theory is the probability that a linear system shall be stable,
when its matrix is filled by random sampling from given distribu-
tions. The problem then becomes:

A matrix of order n x n has elements which are real and are
random samples from given distributions. Find the probability
that all the latent roots have negative real parts.

This problem seems to be still unsolved even in the special
cases in which all the elements have the same distributions,
selected to be simple, as the ' normal ' type e~x\ or the ' rect-
angular ' type, constant between — a and + a. Nevertheless,
as I required some indication of how the probability changed with
increasing n, the rectangular distribution (integers evenly dis-
tributed between — 9 and + 9) was tested empirically. Matrices
were formed from Fisher and Yates' Table of Random Numbers,
and each matrix was then tested for stability by Hurwitz' rule
(S. 20/7). Thus a typical 3x3 matrix was

- 1 — 3 — 8^

- 5 4-2
-4 — 4 — 9

In this case the second determinant is — 86 ; so it need not be
tested further. The testing becomes very time-consuming when
the matrices exceed 3x3, for the time taken increases approxi-
mately as ?i5. The results are summarised in Table 20/10/1.

Order of
matrix

Number
tested

Number
found
stable

Per cent
stable

2x2
3x3

4x4

320
100
100

77
12

Table 20/10/1.

The main feature is the rapidity with which the probability
tends to zero. -The figures given are compatible (#2 = 4-53,

259

DESIGN FOR A BRAIN

20/10

P = 0-10) with the hypothesis that the probability for a matrix
of order n x n is 1/2". That this may be the correct expression
for this particular case is suggested partly by the fact that it
may be proved so when n = 1 and n = 2, and partly by the
fact that, for stability, the matrix has to pass all of n tests.
And in fact about a half of the matrices failed at each test.
If the signs of the determinants in Hurwitz' test are statistically
independent, then l/2n would be the probability in this case.

In these tests, the intrinsic stabilities of the variables, as
judged by the signs of the terms in the main diagonal, were
equally likely to be stable or unstable. An interesting variation,
therefore, is to consider the case where the variables are all
intrinsically stable (all terms in the main diagonal distributed
uniformly between 0 and — 9).

The effect is to increase their probability of stability. Thus
when n is 1 the probability is 1 (instead of J); and when n is
2 the probability is § (instead of J). Some empirical tests gave
the results of Table 20/10/2.

Order of
matrix

Number
tested

Number
found
stable

Per cent
stable

2x2
3x3

120
100

87
55

72
55

Table 20/10/2.

The probability is higher, but it still falls as n is increased.

A similar series of tests was made with the Homeostat. Units
were allowed to interact with settings determined by the uni-
selectors, which were set at one position for one test, the usual
ultrastable feedback being severed. The percentage of stable
combinations was found when the number of units was two.
Then the percentage was found for the same general conditions
except that three units interacted; and then four. The general
conditions were then changed and a new triple of percentages
found. And this was repeated six times altogether. As the
general conditions sometimes encouraged, sometimes discouraged,
stability, some of the triples were all high, some all low; but in
every case the per cent stable fell as the number of interacting

260

20/10

STABILITY

iOOt

50-

2 3

NUMBER OF VARIABLES

Figure 20/1Q/1.

units was increased. The results are given in Figure 20/10/1,
in which each triple lies on one line.

These results prove little; but they suggest that the proba-
bility of stability is small in large systems assembled at random.
It seems, therefore, that large linear systems should be assumed
to be unstable unless evidence to the contrary is produced.

261

CHAPTER 21

Parameters

21/1. In the previous two chapters we have considered the
state-determiried system when it was isolated, with constant
condition? around it, or when no change came to its input. We
now turn to consider the state-determined system when it is
affected by changes in the conditions around it, when it is no
longer isolated, or when changes come to its input. We turn, in
other words, to consider the ' machine with input ' of /. to C,
Chapter 4.

Experience has shown that this change corresponds to the
introduction of parameters into the canonical representations so
that they become of the form

dxi/dt =fi{xv . . . , xn\ als a2, . . .) (i = 1, . . . , n)

21/2. If the «'s are fixed at particular values the result is to make
the/'s a particular set of functions of the #'s and thus to specify
a particular state-determined system. From this it follows that
each particular set of values at the a's specifies a particular field.
In other words, the two sets: (1) the values at the a's and (2) the
fields that the system can show can be set in correspondence
— perhaps the most fundamental fact in the whole of this book.
(Figure 21/8/1 will illustrate it.)

It should be noticed that the correspondence is not one-one but

Figure 21/2/1.
262

21/6

PARAMETERS

may be many-one, for while one value of the vector (av a2, . . .)
will indicate one and only one field, several such vectors may
indicate the same field. Thus the relation is a mapping of
Bourbaki's type. The possibilities are sufficiently indicated in
Fig. 21/2/1; in the upper line P, each dot represents one value
of the vector of parameter- values (av o2, . . .); in the lower line
F, each dot represents one field. Notice that (1) every vector
value indicates a field; (2) no vector value indicates more than
one field; (3) a field may be indicated by more than one vector
value; (4) some fields may be unindicatcd.

21/3. If the a's can take m combinations of values then m fields
are possible. The m fields will often be distinct, but the possibility
is not excluded that the m may include repetitions, and thus not
be all different.

21/4. If a parameter changes continuously (i.e. by steps that
may be as small as we please), then it will often happen that the
corresponding changes in the field will be small; but nothing here
excludes the possibility that an arbitrarily small change in a
parameter may give an arbitrarily large change in the field.
Thus the fields will often be, but need not necessarily be, a con-
tinuous function of the parameters.

21/5. If a parameter affects immediately only certain variables,
it will appear only in the corresponding /'s. Thus the canonical
representation (of a machine with input a)

dxjdt =f1(x1, x2;
dxjdt =f2(xlf x2)

corresponds to a diagram of immediate effects

21/6. Change of parameters can represent every alteration which
can be made on a state-determined system, and therefore on any
physical or biological 4 machine '. It includes every possibility of
experimental interference. Thus if a set of variables that are
joined to form the system x =f{x) are changed in their relations

263

DESIGN FOR A BRAIN 21/7

so that they form the system x = <j>(x), then the change can equally
well be represented as a change in the single system x = ip(x; a).
For if a can take two values, 1 and 2 say, and if

f{x) = y>(x; 1)
<j>{x) = y)(x; 2)

then the two representations are identical.

As example of its method, the action of S. 8/11, where the two
front magnets of the Homeostat were joined by a light glass fibre
and so forced to move from side to side together, will be shown
so that the joining and releasing are equivalent in the canonical
equations to a single parameter taking one of two values.

Suppose that units xv x2 and xz were used, and that the magnets
of 1 and 2 were joined. Before joining, the equations were
(S. 19/11)

dxjdt = anxx + a12x2 + a13aO

2/ — ttoi^i ~~ | Wo? 2 i™ ^S3*»l i

dX3/at = Cl31X1 -f~ a32<%2 ~T~ a33X3J

After joining, x2 can be ignored as a variable since xx and x2 are
effectively only a single variable. But x2s output still affects the
others, and its force still acts on the fibre. The equations there-
fore become

dxjdt = (an + a12 + a21 + a22)xx + (a13 + a23)x,
dxjdt = (a31 + aZ2)xx + a33a\

It is easy to verify that if the full equations, including the para-
meter 6, were:

dx1/dt = {an + b(a12 + a21 + a22)}x1 + (1 — b)a12x2

+ (a13 + ba23)x3

dX2/dt = ^21^1 I ^22*^2 l ^23^3

dxjdt = (a31 + ba^Xj^ + (1 — b)aZ2x2 + a33x,

then the joining and releasing are identical in their effects with
giving b the values 1 and 0 respectively. (These equations are
sufficient but not, of course, necessary.)

21/7. A variable Xk behaves as a null-function if it has the
following properties, which are easily shown to be necessary and
sufficient for each other:

(1) As a function of the time, it remains at its initial value xQk-

264

21/7 PARAMETERS

(2) In the canonical equations, fk{x^ . . . , xn) is identically

zero.

(3) In the equations of S. 19/7, Fk(x\, . . . , a>°; f) = #j>.

(Some region of the phase-space is assumed given.)

In a state-determined system, the variables other than the step-
and null-functions will be referred to as main variables.

Theorem: In a state- determined system, the subsystem of the
main -variables forms a state -determined system provided no step-
function changes from its initial value.

Suppose xv .... xk are null- and step-functions and the main-
variables are Xk+i, . . . , xn. The canonical equations of the
whole system are

dxjdt — 0

dxje/dt = 0
dxk+i/dt =fk+1(xi, . . . , X* xt+i, . . . , xn)\

dxjdt =fn(xl9 . . . , xk, xk+i, . . . , xn)

The first k equations can be integrated at once to give xx = x®y
. . . , Xk = x%. Substituting these in the remaining equations
we get:

dxt+i/dt =fk+i(x%, . . . , x\, xk+i, . . . , xn))

dxjdt = fn(x°v . . . , x°k, Xk+i, . . . , xn)j

The terms x\, . . . , a?£ are now constants, not effectively functions
of t at all. The equations are therefore in canonical form; so the
system is state-determined over any interval not containing a
change in x®, . . . , x°k.

Usually the selection of variables to form a state-determined
system is determined by the real, natural relationships existing
in the real ' machine ', and the observer has no power to alter
them without making alterations in the c machine ' itself. The
theorem, however, shows that without affecting whether it is
state-determined the observer may take null-functions into the
system or remove them from it as he pleases.

It also follows that the statements : ' parameter a was held con-
stant at a0 ', and ' the system was re-defined to include a, which,

265

DESIGN FOR A BRAIN

21/8

as a null-function, remained at its initial value of a0 ' are merely
two ways of describing the same facts.

21/8. The fact that the field is changed by a change of parameter
implies that the stabilities of the lines of behaviour may be
changed. For instance, consider the system

dxjdt = — xt -f- «#2> dxjdt = x± — x2 + 1.

When a = 0, 1, and 2 respectively, the system has the three
fields shown in Figure 21/8/1.

Figure 21 /8/1 : Three fields of xx and x2 when a has the values (left to
right) 0, 1, and 2.

When a = 0 there is a stable state of equilibrium at x1 = 0,
x2 = 1 ; when a = 1 there is no state of equilibrium ; when a = 2
there is an unstable state of equilibrium at x1 = — 2, x2 = — 1.
The system has as many fields as there are values to a.

Joining systems

21/9. (Again the basic concepts have been described in /. to C,
S. 4/7; here we will describe the theory in continuous systems.)

The simple physical act of joining two machines has, of course,
a counterpart in the equations, shown more simply in the canonical
than in the equations of S. 19/7.

One could, of course, simply write down equations in all the
variables and then simply let some parameter a have one value
when the parts are joined and another when they are separated.
This method, however, gives no insight into the real events in
4 joining ' two systems. A better method is to make the para-
meters of one system into defined functions of the variables of the
other. When this is done, the second dominates the first. If
parameters in each are made functions of variables in the other,

266

21/10 PARAMETERS

then a two-way interaction occurs. For instance, suppose we
start with the 2-variable system

dx/dt = f1(x, y; a)'
dy/dt = f2{x, y)

then the diagram of immediate effects is

and the 1 -variable system dz/dt = <f>(z; b)

> z

If we make a some function of z, a = z say for simplicity, the new
system has the equations

dx/dt =f1(x, y; zf\
dy/dt =f2(x, y) I
dz/dt = <f>(z; b) J

and the diagram of immediate effects becomes

If a further join is made by putting b = y, the equations become

dx/dt =f1(x, y; z)
dy/dt =f2{x, y)
dz/dt = 4(z; y)

and the diagram of immediate effects becomes

< y

In this method each linkage uses up one parameter. This is
reasonable; for the parameter used by the other system might
have been used by the experimenter for arbitrary control. So
the method simply exchanges the experimenter for another system.

This method of joining does no violence to each system's
internal activities: these proceed as before except as modified by
the actions coming in through the variables which were once
parameters.

21/10. Theorem: The whole made by joining parts is richer in
ways of behaving than the system obtained by leaving the parts
isolated.

267

DESIGN FOR A BRAIN 21/11

(The argument is simple and clear if it is supposed that each
part has a finite number of states possible, and if the number of
input states is also finite. The result for the infinite case, being
the limit of the finite case, is the same as that stated, but would
need a special technique for its discussion.)

Suppose the system consists of p parts, each capable of being
in any one of s states, with p and s assumed finite. Then, whether
joined or not, the set of all the parts has sv states possible. (Put
sv = k, for convenience.)

If the whole is richly joined, each of these k states may go, in a
transition, to any of the k states; for the transition of each part
is not restricted (since it is allowed to be conditional on, and to
vary with, the states of the other parts). The number of trans-
formations is thus kk.

If, however, the parts are not joined, the transformations of
each part cannot vary with the states of the others ; so the trans-
formation of the whole must be built up by taking a single trans-
formation from each part. Each part, with s states, has ss
transformations; so the whole will have (ss)r transformations
possible. This equals ks.

As s is less than k, ks is less than kk; whence the theorem.

21/11. If Xv . . . , Xn is a state of equilibrium in a system

dxi/dt =ft(xv . . . , xn\ 14, . . .) {i = 1 n)

for certain a- values, and the system is then joined to some ?/'s by
making the a's functions of the y's, then Xv . . . , Xn will still
be a state of equilibrium (of the a>system) when the y's make the
a's take their original values. Thus the zeros of the/'s, and the
states of equilibrium of the ^-system, are not altered by the
operation of joining.

21/12. On the other hand, the stabilities may be altered grossly.

In the general case, when the/'s are unrestricted, this proposi-
tion is not easily given a meaning. But in the linear case (to
which all continuous systems approximate, S. 20/5) the meaning
is clear. Three examples will be given.

Example 1: Two systems may give a stable whole if joined one
way, but an unstable whole if joined another way. Consider the
1 -variable systems dx/dt = x + ^V\ + 2han^ dy/dt = — 2r — 3y.

268

21/13 PARAMETERS

If they are joined by putting r = x, px = y, the system becomes
dx/dt = x + 2y + p

dy/dt = — 2x — 3y

The latent roots of its matrix are — 1, — 1; so it is stable. But
if they are joined by r = x, p2 = y, the roots become + 0«414
and — 2-414; and it is unstable.

Example 2 : Stable systems may form an unstable whole when
wined. Join the three systems

dx/dt = — x — 2q — 2r
dy/dt = - 2p — y + r
dz/dt = p + q — z

all of which are stable, by putting p = x, q — y, r = z. The
resulting system has latent roots +1, — 2, — 2.

Example 3 : Unstable systems may form a stable whole when
joined. Join the 2-variable system

dx/dt = 3x — Sy — 3p>
dy/dt = Sx — 9y -

which is unstable, to dz/dt = 2lq + Sr + 3z, which is also un-
stable, by q = x, r = y, p = z. The whole is stable.

The state-determined system

21/13. It is now clear that there are, in general, two ways of
getting to know a complex dynamic system (i.e. one made of
many parts).

One way is to know the parts (ultimately the individual vari-
ables) in isolation, and how they are joined. ' Knowing ' each
part, or variable, means being able to write down the correspond-
ing lines of the canonical representation (if not in mathematical
symbolism then in any other way that gives an unambiguous
statement of the same facts). Knowing how they are joined
means that certain parameters to the parts can be eliminated (for
they are functions of the variables). In this way the canonical
representation of the whole is obtained. Integration will then
give the lines of behaviour of the whole. Thus we can work
from an empirical knowledge of the parts and their joining to a
deduced knowledge of the whole.

The other way is to observe the whole and its lines of behaviour.

269

DESIGN FOR A BRAIN 21/14

These observations give the functions of S. 19/7. Differentiation
of these (as in the Corollary of S. 19/9) will give the canonical
representation, and thus those of the parts, to which the other
variables now come as parameters. Thus we can also work from
an empirical knowledge of the whole to a deduced knowledge of
the parts and their joining.

21/14. It is now becoming clear why the state-determined
system, and its associated canonical representation, is so central
in the theory of mechanism. If a set of variables is state-deter-
mined, and we elicit its canonical representation by primary
operations, then our knowledge of that system is complete. It is
certainly not a complete knowledge of the real ' machine ' that
provides the system, for this is probably inexhaustible; but it is
complete knowledge of the system abstracted — complete in the
sense that as our predictions are now single-valued and verified,
they have reached (a local) finality. If a tipster names a single
horse for each race, and if his horses always win, then though he
may be an ignorant man in other respects we would have to admit
that his knowledge in this one respect was complete.

The state-determined system must therefore hold a key place
in the theory of mechanism, by the strategy of S. 2/17. Because
knowledge in this form is complete and maximal, all the other
branches of the theory, which treat of what happens in other
cases, must be obtainable from this central case as variations
on the question: what if my knowledge is incomplete in the
following way . . . ?

So we arrive at the systems that actually occur so commonly
in the biological world — systems whose variables are not all
accessible to direct observation, systems that must be observed
in some way that cannot distinguish all states, systems that can
be observed only at certain intervals of time, and so on.

21/15. Identical with the state-determined system is the ' noise-
less transducer ' defined by Shannon. This he defines as one that,
having states a and an input x, will, if in state a„ and given input
xn, change to a new state an+1 that is a function only of xn and an:

«n+i = g(*n> a«)-

Though expressed in a superficially different form, this equation

270

21/15 PARAMETERS

is identical with a canonical representation; for it says simply
that if the parameters x and the state of the system are given,
then the system's next state is determined. Thus the communica-
tion engineer, if he were to observe the biologist and the psycho-
logist for the first time, would say that they seem to prefer to
work with noiseless systems. His remark would not be as trite
as it seems, for from it flow the possibilities of rigorous deduction.

271

CHAPTER 22

The Effects of Constancy

22/1. A variable behaves as a step-function over some given
period of observation if it changes value at only a finite number of
discrete instants, at which it changes value instantaneously and
by a finite jump.

22/2. An example of a step-function in a system will be given
to establish the main properties.

Suppose a mass m hangs downwards suspended on a massless
strand of elastic. If the elastic is stretched too far it will break
and the mass will fall. Let the elastic pull with a force of k
dynes for each centimeter increase from its unstretched length,
and, for simplicity, assume that it exerts an opposite force when
compressed. Let x, the position of the mass, be measured verti-
cally downwards, taking as zero the position of the elastic when
there is no mass.

If the mass is started from a position vertically above or below
the point of rest, the movement will be given by the equation

jt{mT)=gm-kx • ■ • (1)

where g is the acceleration due to gravity. This equation is not
in canonical form, but may be made so by writing x = xv
dx/dt = x2, when it becomes
dx±

dx» k

dt 6 m 1

(2)

If the elastic breaks, k becomes 0, and the equations become

dxx _

dXo

Assume that the elastic breaks if it is pulled longer than X.

272

22/2 THE EFFECTS OF CONSTANCY

The events may be viewed in two ways, which are equivalent.

We may treat the change of k as a change of parameter to the
2-variable system xv cc2, changing their equations from (2)
above to (3) (S. 21/1). The field of the 2-variable system will
change from A to B in Figure 22/2/1, where the dotted line at X

A B

Figure 22/2/1 : Two fields of the system (xt and x2) of S. 22/2.
unbroken elastic the system behaves as A, with broken as B.
the strand is stretched to position X it breaks.

With
When

shows that the field to its right may not be used (for at X the
elastic will break).

Equivalent to this is the view which treats them as a 3- variable
system: xv x2> and k. This system is state-determined, and has
one field, shown in Figure 22/2/2.

Figure 22/2/2

Field of the 3-variable system.
273

DESIGN FOR A BRAIN 22/3

In this form, the step-function must be brought into the
canonical equations. A possible form is:

dk JK , K

= ?(f + f tanh {q(X -*)}-*) . (4)

where K is the initial value of the variable k, and q is large and
positive. As q — > oo, the behaviour of & tends to the step-
function form.

Another method is to use Dirac's ^-function, defined by d(u) = 0
if w y£ o, while if u = 0, (5(w) tends to infinity in such a way that

6(u)du = 1.

Then if du/dt = 6{(f>{u, v, . . .)}, du/dt will be usually zero; but
if the changes of u, v, . . . take <f> through zero, then d(u) becomes
momentarily infinite and u will change by a finite jump. These
representations are of little practical use, but they are important
theoretically in showing that a step-function can occur in the
canonical representation of a system.

22/3. In a state-determined system, a step-function will change
value if, and only if, the system arrives at certain states: the
critical. In Figure 22/2/2, for instance, all the points in the plane
k = K (the upper plane) and to the right of the line xx = X are
critical states for the step-function k when it has the initial
value K.

The critical states may, of course, be distributed arbitrarily.
More commonly, however, the distribution is continuous. In this
case there will be a critical surface

<j>(k, a?2 a?n) = 0

which, given k, divides the critical from the non-critical states.
In Figure 22/2/2, for instance, the surface intersects the plane
k = K at the line xx = X. (The plane k = 0 is not intersected
by it, for there are no states in this system whose occurrence will
result in k changing from 0.)

Commonly </> is a function of only a few of the variables of the
system. Thus, whether a Post Office type relay opens or shuts
depends only on the two variables: the current in the coil, and
whether the relay is already open or shut.

Such relays and critical states occur in the Homeostat. When

274

22/5 THE EFFECTS OF CONSTANCY

two, three or four units are in use, the critical surfaces will form
(to a first approximation) a square, cube, or tesseract respectively
in the phase-space around the origin. The critical states will fill
the space outside this surface. As there is some backlash in the
relays, the critical surfaces for opening are not identical with
those for closing.

Systems with multiple fields

22/4. If, in the previous example, someone unknown to us were
occasionally to break and sometimes to replace the elastic, and if
we were to test the behaviour of the system cclt x2 over a prolonged
time including many such actions, we would find that the system
was often state-determined with a field like A of Figure 22/2/1,
and often state-determined with a field like B; and that from time
to time the field changed suddenly from the one form to the other.
Such a system could be said without ambiguity to have two
fields. Similarly, if parameters capable of taking r combinations
of values were subject to occasional change by some other,
unobserved system, a system might be found to have r fields.

22/5. The argument can be used to some degree in the converse
direction; for the correspondence of S. 21/2 may be used, with
caution, conversely; for though the number of fields does not
prescribe the number of parameter-values it does prescribe their
minimal number with precision. Thus fields that change like the
first row of letters in S. 9/13 demand a minimum of 4 parameter-
values, while those that change like the second row demand a
minimum of 15.

If the observer should find that one field persists, the minimal
number of parameter- values is, of course, one. If the field should
change suddenly to a new field, which persists, he may deduce
that the parameter- value must have changed (for no single value
could give two fields), and that -the minimal number of values
over the new persistence is again one. Thus he may legitimately
deduce that the minimal variety attributable to the parameter-
values is, on the scale of S. 7/13, that of the step-function — the
null-function provides too little, and the part-function an un-
necessary excess. (Compare S. 9/10-13.)

275

DESIGN FOR A BRAIN 22/6

The ultrastable system

22/6. The definition and description already given in S. 7/26
have established the elementary properties of the ultrastable
system. A restatement in mathematical form, however, has the
advantage of rendering a misunderstanding less likely, and of
providing a base for quantitative studies.

If a system is ultrastable, it is composed of main variables xt
and of step-functions ait so that the whole is state-determined:

dxi/dt =fi(x\ a) (i = 1, . . . , n)

dajdt = gi(x; a) {i = 1, 2, . . .)

The functions gt must be given some form like that of S. 22/2.
The system is started with the representative point within the
critical surface <f>(x) = 0, contact with which makes the step-
functions change value. When they change, the new values of
a{ are to be random samples from some distribution, assumed
given.

Thus in the Homeostat, the equations of the main variables
are (S. 19/11):

dXi/dt = aaxx + ai2x2 + ai3x3 + a^ (i = 1, 2, 3, 4)
The a's are step-functions, coming from a distribution of ' rect-
angular ' form, lying evenly between — 1 and -f- 1. The critical

surfaces of the a's are specified approximately by | x | ± - = 0.

Each individual step-function a^ depends only on whether x}
crosses the critical surface.

As the a's change discontinuously, an analytic integration of
the differential equations is not, so far as I am aware, possible.
But the equations, the description, and the schedule of the
uniselector- wirings (the random samples) define uniquely the
behaviour of the x's and the a's. So the behaviour could be
computed to any degree of accuracy by a numerical method.

22/7. How many trials will be necessary, on the average, for a
terminal field to be found ? If an ultrastable system has a
probability p that a new field of the main variables will be stable,
and if the fields' probabilities are independent, then the number
of fields occurring (including the terminal) will be, on the average,
\/p.

276

22/9 THE EFFECTS OF CONSTANCY

For at the first field, a proportion p will be terminal, and
q (= 1 — p) will not. Of the latter, at the second field, the pro-
portion p will be terminal and q not ; so the total proportion stable
at the second field will be pq, and the number still unstable q2.
Similarly the proportion becoming terminal at the u-th field will
be pq*-1. So the average number of trials made will be

p + 2pq + 3pq2 + • • ■ + upq"-1 + • . . _ 1
p + pq + pq2 + . . . + pq"-1 -f . . . p

Temporary independence

22/8. The relation of variable to variable has been treated by
observing the behaviour of the whole system. But what of their
effects on one another ? Thus, if a variable changes in value, can
we distribute the cause of this change among the other variables ?
In general it is not possible to divide the effect into parts,
with so much caused by this variable and so much caused by that.
Only when there are special simplicities is such a division possible.
In general, the change of a variable results from the activity of
the whole system, and cannot be subdivided quantitatively.
Thus, if dx/dt = sin x + xey, and x = \ and y = 2, then in the
next 0-01 unit of time x will increase by 0-042, but this quantity
cannot be divided into two parts, one due to x and one to y.
Only when some special simplicity exists can the whole effect be
represented meaningfully as the sum of two effects, one from each.
Though not uncommon in theoretical physics, such simplicities are
rare in biological systems.

22/9. Given a state-determined system, its field, a line of
behaviour in it, and a particular portion P of the line; given also
that xp is a part-function, then the following are equivalent, in
that the truth (or falsity) of any one implies the truth (or falsity)
of all the others:

(1) xp is constant (inactive);

(2) dxp/dt = 0;

(S)fp{. . ., x0t . . .) =0;

(4) Xp = #jj independently of t;

(5) Fp(x°; t) == x°, with such values of t as do not take the line

out of P;

277

DESIGN FOR A BRAIN 22/10

all being understood to refer only to the region P. (The equiva-
lences follow readily from the properties of the equations of S. 19/9
and their integrals.)

22/10. Given a state-determined system and two transitions
from two initial states which differ only in their values of x°, (the
difference being A#9), the variable xk is independent of xt if xks
transition is identical in the two cases. Analytically, xk is inde-
pendent of Xj in the conditions given if

Fk{x°v . . . , 4 . . .; dt) = Fk{x\, . . . , x] + Aaf, . . .; dt) (1)

In other words, xk is independent of Xj if xks behaviour is invariant
when the initial state is changed by Ax9. (This ' change ' by

A#j must not be confused with the change dt.)

This narrow definition provides the basis for further develop-
ment. In practical application, the identity (1) may hold over all
values of Ax® (within some finite range, perhaps); and may also
hold for all initial states of xk (within some finite range, perhaps).
In such cases the test whether xk is independent of Xj is whether

faoFk(x<>; t) = 0.

The range over which the relation or equation holds must always
be specified (either explicitly or by implication).

Diagrams of effects

22/11. The diagram of immediate effects and the canonical
representation have a simple relation. Starting with the prag-
matic and empirical point of view of S. 2/7, we assume that the
observer gets his basic knowledge of the system by primary
operations. These operations will give him the functions Ft of
S. 19/7 and also (by S. 4/12) the diagram of immediate effects.
Now the test for whether to draw an arrow from x, to xk is essen-
tially the same as the test applied algebraically to see whether x°,
occurs effectively in Fk, and the outcomes must correspond. But
(by the Corollary, S. 19/9) whether Fk does or does not contain x°
effectively over a single step dt must correspond with whether fk
does or does not contain x, effectively. Thus, in the diagram of
immediate effects an arrow will run from Xj to xk if and only if,

278

22/14 THE EFFECTS OF CONSTANCY

in the canonical representation, Xj occurs effectively in /*. (The
range of /'s arguments is assumed to be specified.)

22/12. The diagram of ultimate effects can also be shown to have
the property that an arrow goes from Xj to x* if and only if, in
the equations of S. 19/7, x9 occurs effectively in Fk (over some
specified range).

(These matters were discussed more fully in the First Edition,
but need not be repeated at length.)

22/13. It is worth noticing that, given n arbitrary points, a
diagram of immediate effects can be drawn by the arbitrary
placing of any number of arrows. That of the ultimate effects
cannot, however, be so drawn ; for an arrow from p to q and one
from q to r imply an arrow from p to r. Thus, while diagrams
of immediate effects are, in general, unrestricted, those of ultimate
effects must be transitive.

22/14. The thesis of S. 12/10 can now be treated rigorously.
Figure 12/10/1 is given to be the diagram of immediate effects,
and the whole is assumed to be isolated and state-determined.
(For compactness below, the subscript A will be used to mean
' any variable in the ^4-set'; and similarly for B and C.) Then
the canonical representation of the whole must be of the form

xa =/a(xa, xb) 1

xb=/b(xa, xb, xc)\ . . (1)

XC =fc(XB, XC) J

with xc not in/,4, and xa not in/c. The two parts of the theorem
can now be proved.

(1) Suppose the 2?'s are null-functions (over some specified
range). They are therefore constant. Write their values col-
lectively as p. The topmost line of (1) then becomes

XA =fA(XA, ft)

which shows that the system composed of the variables xa is
state-determined (so long as (} is constant). Further, the integrals
Fa of these equations cannot contain x°, so the system A is inde-
pendent of the system C.

A similar proof will show that C is state-determined and

279

DESIGN FOR A BRAIN 22/14

independent of A. Thus a wall of constancies (the B's null-
functions) between the systems A and C is sufficient to leave them
each state-determined and independent of the other.

(2) Suppose the systems A and C are each found to be state-
determined and independent of one another; the whole is given
to be state-determined, and there is known to be no immediate
connexion from A to C or from C to A, but there is effective
connexion between A and B, and between C and B — what can
be deduced about the variables in B ?

The lack of connexion between A and C shows that the canonical
representation must have the form of (1) above. With

XA =fA(cCA, OCb)

the A's can be state-determined only if all the i5's are constant
(not effectively functions of the time). So the 2?'s must be null-
functions.

280

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282

Index

The reference is to the page. A number in bold-faced
type indicates a definition.

Accumulation, 142

Active, 170

Activity. 5

Adaptation, 58

Adaptation, serial, 200

Adaptation-time, 148

Adaptive behaviour, 58

Aim, 131

Algebra of machine, 242

All or none, 89

Alternating environments, 115

Ammonia, 77

Amoeboid process, 125

Amplification, 235

Ancillary regulations, 218, 229

Archimedes, 65

Automatic pilot, 108

Awareness, 11

Axioms, 8, 13

Bartlett, Sir F. C, 38
Behaviour, adaptive, 58
Behaviour, learned, 1
Behaviour, line of, 20
Behaviour, reflex, 1
Bladder, 92

Borrowed knowledge, 19
Bound, 48
Bourbaki, N., 241
Bovd, D. A., 119
Break, 92, 272
Burglar, 154

Cannon, W. B., 58, 64

Canonical representation, 244

Cardinal number, 32

Carey, E. J., 125

Change, 13

Characteristic equation, 255

Chewing, 7

Chimneys, 181

Circus, 3

Civilisation, 62

Clock, 14

Combination lock, 154, 205

Communication, 218

Complex system, 28, 33, 148, 192

Conditioned response, 2, 16, 113

Confluent, 185

Connectivity, 156
Consciousness, 11
Constancy, 164
Constraint, 139
Construction, random, 97
Control, 159
Control by error, 55
Co-ordinates, 41
Co-ordination, 57, 67, 103
Coughing, 2
Cowles, J. T., 201
Critical state, 91, 274
Critical surface, 274
Crystallisation, 153
Cullen, E., 74
Curare, 74
Cvcle, 49
Cycling, 11, 36

Dancoff, S. M., 130

Delay, 120

Demonstration, 12

Dependence, 159

Determinancy, 9, 95

Diabetes, 74

Diagram of immediate effects, 51, 122,

278
Diagram of ultimate effects, 163, 278
Dial reading, 14, 30
Dictionary, 236
Discontinuity, 120
Dispersion, 178
Disturbance, 130, 138
Dominance, 161
Ducklings, 39

Effectiveness, 162
Effects, 50
Elastic, 89, 272
End-plate, 125
Endrome, 125
Energy, 159
Engine driver, 6
Environment, 36
Environment, alternating, 115
Equations, simultaneous, 206
Equilibrial index, 175
Equilibrium, neutral, 44
Equilibrium, state of, 46, 254

283

INDEX

Equilibrium, unstable. 44
Equivalence relation, 242
Error-control, 55
Essential variables, 42
Even v. Odd, 232
Exponential function, 150
External composition, 242
Eye muscles, 104

Failure, 118

Falcon, 201

Feedback, 37, 228

Feedback, second-order, 83

Fencing, 67

Field, 23

Field, stable, 49

Fire, 12

Fraunhofer lines, 124

Full-function, 87

Function, full-, 87

Function, null-, 87

Function, part-, 87

Function, step-, 87

Fuse, 88

Gating mechanism, 144, 216
Gene-pattern, 8, 134
Gestalt, 156
Girden, E., 74
Goal-seeking, 54, 81, 131
Governor, 44, 48, 50
Grant, W. T., 68
Grindley, G. C, 112
Group, 244

Habituation, 185
Harrison, R. G., 125
Hertz, Heinrich, 52, 85
Hilgard, E. R., 113, 216
Holmes, S. J., 66
Homeostasis, 58
Homeostat, 100, 246
4 Homeostatic ', 100
Humphrey, G., 189
Hunter, 120
Hurvvitz, A., 256

/. to C, vii

Immediate effects, 51, 278
Impulsive stimulus, 76
Inactive, 170
Independence, 159, 278
Index of equilibria, 175
Initial state, 20
Instrumental learning, 113
Insulation, 165
Intelligence test, 121
Interference, 216
Introduction to Cybernetics, vii
Invariant, 242

Isolation, 35, 165
Iterated systems, 197

Jennings, H. S., 38, 189, 193
Joining, 76, 266

Kitten, 12, 62, 80
Kletsky, E. J., 100
Knowledge, borrowed, 19

Laplace, 28
Lashley, K. S., 182
Latent roots, 255
Learned behaviour, 2
Learning, 11, 234
Learning, instrumental, 113
Levi, G., 125
Line of behaviour, 20, 243
Linear system, 246
Localisation, 124, 181
Loeb, J., 126
Logic of mechanism, 241
Lorente de N6, R., 124

McCulloch, W. S., 125
McDougall, W., 65
Machine, 13

Machine with input, 25
Main variables, 93, 265
Mapping, 242, 263
Marina, A., 104
Marquis, D. G., 113, 216
Maze-running, 3
Memory, 10
Microscope, 3
Micturition, 92
Miniaturisation, 127
Monitoring, 220
Morgan, C. Lloyd, 39, 201
Motor end-plate, 125
Mowrer, O. H., 108
Miiller, G. E., 216
Multistage system, 209

Natural selection, 8, 134

Natural system, 25

Neuron, 4

Neutral equilibrium, 44

Nie, L. W., 119

Noise, 270

Normal stability, 253

Null-function, 87, 264

Number of parts, 148

Nyquist, H., 256

Objectivity, 9
Observation, 17
Odd v. Even, 232
Operation, primary, 18
Operational method, 9, 17

284

INDEX

Optimum, 224
Oscilloscope, 206
Outcome. 138

Pain, 43, 110

Pain insensitivity, 119

Paramecium, 189, 193

Parameter, 71, 262

Parker, G., 162, 197

Part and whole, 5, 78

Part-function. 87

Pauling, L., 127

Pavlov, I. P., 15, 65, 182

Pendulum, 15, 26, 54, 72

Phase-space, 22

Pigeon, 10

Pike, 111

Pilot, automatic, 108

Point, representative, 22

Poison, 119, 140

Poly stable system, 173

Pre-baiting, 140

Primary operation, 18, 243

Principle of interference, 216

Probability of stability, 258

Punishment, 110, 115

Quotient-law, 242

Radio, 52

Random construction, 97, 172

Random system, 150

Reaction, 53

Reciprocity of connexions, 124

Recurrent situation, 138

References, 281

Reflex behaviour, 1

Reflexologist, 156

Regular system, 243

Regulator, ancillary, 218

Relay, 90, 128

Re-organisation, 107

Representative point, 22

Requisite variety, 229

Response, conditioned, 2, 16, 113

Retro-active inhibition, 141, 214

Reward, 110

Robinson, E. S., 217

Rubin, H., 150

Runaway, 53

Running, 35

Sea- anemone, 162, 197
Second-order feedback, 83
Selection by equilibrium, 186, 231
Selection by veto, 79
Selection, natural, 8, 134
Self-correction, 55
Sense organs, 179
Serial adaptation, 200

Serving, 220

Servo-mechanism, 53

Shannon, C. E., 229, 270

Sherrington, C. S., 157

Shivering, 59

Signal, 6

Simultaneous equations, 206

Skaggs, E. B., 217

Skinner, B. F., 10

Smoking, 181

Snail, 189

Solving equations, 206

Sommerhoff, G., 69

Sparks, 174

Speidel, C. C, 125

Sperry, R. W., 104

Stability, 44, 253

Stable field, 49

Stable region, 48

Stable system, 49

Stalking, 120

Starling, E. H., 38, 64

State, 16, 243

State, critical, 91, 274

State-determined, 26

State of equilibrium, 46, 254

Steady state, 44

Step-function, 75, 87, 272

Step-mechanism, 91, 123

Stepping switch, 103

Stimulus, 75

Stimulus, impulsive, 76

Strategy, 28

Sub-goal, 82, 226

Subjective, 11

Surface, critical, 274

Survival, 8, 43, 65, 232

Swallows, 17

Switch, 88, 167

System, 16, 243

System, complex, 28, 33, 148, 192

System, linear, 246

System, multistable, 209

System, natural, 25

System, polystable, 173

System, random, 150

System, regular, 243

System, stable, 49

System, ultrastable, 80, 98, 276

Tabes dorsalis, 39
Teleology, 9, 54, 131
Temple, G., 28
Tennis, 220
Theory of sets, 241
Thermostat, 44
Threshold, 168
Thunderstorm, 17
Time, 16
Tissue culture, 125

285

INDEX

Tokens, 201
Topology, 241
Tract, anatomical, 218
Training, 4, 110
Trajectory length, 150
Transducer, 270
Transformation, 249
Transition, 18, 243
Trial and error, 82
Trial duration. 120
Type-problem, 12, 62, 80

Ultimate effects, 278
Ultrastable system, 80, 98, 27G
Uniselector, 103
Unstable equilibrium, 44

Unusual, 116
Urinary bladder, 92

Variable, 14, 30, 71, 243
Variable, essential, 42
Variable, main, 93
Veto, 79, 86
Vicious circle, 53
Vocal cords, 220

Watt's governor, 44, 48, 50
Whole system, 56, 156
Wireless, 52
Wolfe, J. B., 201

Young, J. Z., 124

286

mSm

BMffliiiirr^

111

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