Full text of "Design for a brain"

See other formats

&m/£.

DESIGN FOR A BRAIN

W. ROSS ASHBY

M.A., M.D. (Cantab), D.P.M.

Director of Research
Barnwood House, Gloucester

REPRINTED
(with corrections)

NEW YORK
JOHN WILEY & SONS INC,

440 FOURTH AVENUE
1954

First Published . . . 1952

Reprinted (with corrections) . 1954

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

Summary and Preface*

The book is not a treatise on all cerebral mechanisms but an
attempt to solve 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
adaptively and the hypothesis that it is essentially mechanistic ;
it proceeds on the assumption that these two data are not irre-
concilable. 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. Many other workers have proposed
theories on the subject, but they 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.

Proceeding in this way I have deduced that any system which
shows adaptation must (1) contain many variables that behave
as step-functions, (2) contain many that behave as part-functions,
and (3) be assembled largely at random, so that its details are
determined not individually but statistically. The last require-
ment may seem surprising : man-made machines are usually
built to an exact specification, so we might expect a machine
assembled at random to be wholly chaotic. But it appears that
this is not so. Such a system has a fundamental tendency, shown
most clearly when its variables are numerous, to so arrange its
internal pattern of action that, in relation to its environment, it
becomes stable. If the system were inert this would mean little ;
but in a system as active and complex as the brain, it implies that
the system will be self-preserving through active and complex
behaviour.

The work may also be regarded as amplifying the view that
the nervous system is not only sensitive but ' delicate ' : that its
encounters with the environment mark it readily, extensively,
and permanently, with traces distributed according to the
4 accidents ' of the encounter. Such a distribution might be
expected to produce a merely chaotic alteration in the nervous
system's behaviour, but this is not so : as the encounters multiply
there is a fundamental tendency for the system's adaptation to
improve, for the traces tend to such a distribution as will make
its behaviour adaptive in the subsequent encounters.

* The summary is too brief to be accurate ; the full text should be con-
sulted for the necessary qualifications.

SUMMARY AND PREFACE

The work also in a sense develops a theory of the ' natural
selection ' of behaviour-patterns. Just as, in the species, the
truism that the dead cannot breed implies that there is a funda-
mental tendency for the successful to replace the unsuccessful,
so in the nervous system does the truism that the unstable tends
to destroy itself imply that there is a fundamental tendency for
the stable to replace the unstable. Just as the gene-pattern, in
its encounters with the environment, tends towards ever better
adaptation of the inherited form and function, so does a system
of step- and part-functions tend towards ever better adaptation
of learned behaviour.

These remarks give an impressionist picture of the work's
nature ; but a description in these terms is not well suited to
systematic exposition. The book therefore presents the evidence
in rather different order. The first five chapters are concerned
with foundations : with the accurate definition of concepts, with
basic methods, and especially with the establishing of exact
equivalences between the necessary physical, physiological, and
psychological concepts. After the development of more advanced
concepts in the next two chapters, the exposition arrives at its
point : the principle of ultrastability, which in Chapter 8 is defined
and described. The next two chapters apply it to the nervous
system and show how it explains the organism's basic power of
adaptation. The remainder of the book studies its developments :
Chapters 11 to 13 show the inadequacy of the principle in systems
that lack part-functions, Chapters 14 to 16 develop the properties
of systems that contain them, and Chapters 17 and 18 offer
evidence that the principle's power to develop adaptation is
unlimited.

The thesis is stated twice : at first in plain words and then in
mathematical form. Having experienced the confusion that
tends to arise whenever we try to relate cerebral mechanisms to
psychological phenomena, 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 meaning of terms, or adding assump-
tions, 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 everybody. As the basic thesis, however,
rested 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 Appen-
dix (Chapters 19-24) contains the definitive theory in mathema-

SUMMARY AND PREFACE

tical form. So far as is possible, the main account and the
Appendix have been written in parallel to facilitate cross-reference.

Since the reader will probably need cross-reference frequently,
the chapters have been subdivided 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 : Fig.
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.

Figs. 8/8/1 and 8/8/2 are reproduced by permission of the
Editor of Electronic Engineering.

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
helpful criticism.

W. ROSS ASHBY

Contents

CHAPTER PAGE

Summary and Preface v

1 The Problem 1

2 Dynamic Systems 13

3 The Animal as Machine 29

4 Stability 43

5 Adaptation as Stability 57

6 Parameters 72

7 Step-functions 80

8 The Ultrastable System 90

9 Ultras tability in the Living Organism 103

10 Step-functions in the Living Organism 125

11 Fully Connected Systems 130

12 Iterated Systems 139

13 Disturbed Systems and Habituation 144

14 Constancy and Independence 153

15 Dispersion 166

16 The Multistage System 171

17 Serial Adaptation 179

18 Interaction between Adaptations 190

APPENDIX

19 The Absolute System 203

20 Stability 216

21 Parameters 226

22 Step-functions 232

23 The Ultrastable System 237

24 Constancy and Independence 241

Index 255

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.
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 many other ways. At the same time the psychologists
have established beyond doubt that the living organism, whether
human or lower, can produce behaviour of the type called 4 pur-
poseful ' or ' intelligent ' or ' adaptive ' ; for though these words
are difficult to define with precision, no one doubts that they
refer to a real characteristic of behaviour. These two character-
istics of the brain's behaviour have proved difficult to reconcile,
and some workers have gone so far as to declare them incom-
patible.

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 principle hitherto
little used in machines. I hope to show that by the use of this
principle a machine's behaviour may be made as adaptive as we
please, and that the principle may he capable of explaining even
the adaptiveness 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 be not far from
its solution.

1/2 DESIGN FOR A BRAIN

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 perhaps an
over-simplification, but it will be sufficient for our purpose.

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

THE PROBLEM 1/5

reflex 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
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.

A kitten approaching a fire for the first time is unpredictable
in its first reactions. The kitten may walk almost into it, or
may spit at it, or may dab at it with a paw, or may try to sniff
at it, or may crouch and ' stalk ' it. The initial way of behaving
is not, therefore, determined by the animal's species.

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
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 i better ' will be discussed in

1/5 DESIGN FOR A BRAIN

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
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 kitten's behaviour in the presence of
a fire changes from being such as may cause injury by burning to
an accurately adjusted placing of the body so that the cat's body
is warmed by the fire neither too much nor too little. The circus
animals' behaviour changes from some random form to one deter-
mined by the trainer, who applied punishments and rewards.
The animals' later behaviour is such as has decreased the punish-
ments 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 movements, which were dis-
orderly and ineffective.

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 property ?

But before the solution is attempted we must first glance at the
peculiar difficulties which will be encountered.

THE PROBLEM 1/7

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. And
by their control of the muscles, the nerve cells can rouse to
activity engines of high mechanical 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
of 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. 20/12 ;
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. It was decided in S. 1/5 that after the learning process the
behaviour is usually better adapted than before. We ask, there-
fore, what property must be possessed by the neurons, or by the
parts of a mechanical ' brain ', so that the manifestation by

5 B

1/8 DESIGN FOR A BRAIN

the neuron of this property shall result in the whole animal's
behaviour being improved.

Even if we allow the neuron all the properties of a living
organism, it is still insufficiently provided. For the improvement
in the animal'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 reflexes,
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 ' lubri-
cant ' can have any direct relevance or meaning. Again, a rat
learns to run through a maze without mistakes ; 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
excitation 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.

1/8. In some cases there may be a simple mechanism which
uses the method 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
might be transmitted through the nervous system in the appro-
priate way.

The simplicity of the arrangement is due to the fact that we
are supposing that the two reactions are using two completely
separate and 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 ' incorrect ' neural activities are alike composed of excita-
tions, of inhibitions, and of other changes that are all physiological,

THE PROBLEM 1/8

so that the correctness is determined not by the process itself but
by the relations which it bears to the 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
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 4 correct ' or ' incorrect ', 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 cor-
rectness 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, it undergoes changes which
result in its behaviour becoming better adapted to the environ-
ment. The behaviour depends on the activities of the various
parts whose individual actions compound for better or worse into
the whole action. Why, in the living brain, do they always
compound for the better ?

If we wish to build an artificial brain the parts must be specified
in their nature and properties. But how can we specify the
4 correct ' properties for each part if the correctness depends not
on the behaviour of each part but on its relations to the other
parts ? Our problem is to get the parts properly co-ordinated.
The brain does this automatically. What sort of a machine can
be ^//-co-ordinating ?

This is our problem. It will be stated with more precision in
S. 1/12. But before this statement is reached, some minor topics
must be discussed.

1/9 DESIGN FOR A BRAIN

The genetic control of cerebral function

1/9. The various species of the animal kingdom differ widely
in their powers of learning : Man's intelligence, for instance, is
clearly a species-characteristic, for the higher apes, however well
trained, never show an intelligence equal to that of the average
human being. Clearly the power of learning is determined to
some extent by the inherited gene-pattern. In what way does
the gene-pattern exert its effect on the learning process ? In
particular, what part does it play in the adjustments of part to
part which the previous section showed to be fundamental ?
Does the gene-pattern determine these adjustments in detail ?

In Man, the genes number about 50,000 and the neurons number
about 10,000,000,000. The genes are therefore far too few to
specify every neuronic interconnection. (The possibility that a
gene may control several phenotypic features is to some extent
balanced by the fact that a single phenotypic feature may require
several genes for its determination.)

But the strongest evidence against the suggestion that the
genes exert, in the higher animals, a detailed control over the
adjustments of part to part is provided by the evidence of S. 1/4.
A dog, for instance, can be made to respond to the sound of a
bell either with or without salivation, regardless of its particular
gene-pattern. It is impossible, therefore, to relate the control of
salivation to the particular genes possessed by the dog. This
example, and all the other facts of which it is typical, show that
the effect of the gene-pattern on the details of the learning process
cannot be direct.

The effect, then, must be indirect : the genes fix permanently
certain function-rules, but do not interfere with the function-rules
in their detailed application to particular situations. Three
examples of this type of control will be given in order to illustrate
its nature.

In the game of chess, the laws (the function rules) are few and
have been fixed for a century ; but their effects are as numerous
as the number of positions to which they can be applied. The
result is that games of chess can differ from one another though
controlled by constant laws.

A second example is given by the process of evolution through
natural selection. Here again the function-rule (the principle of

THE PROBLEM 1/10

the survival of the fittest) is fixed, yet its influence has an infinite
variety when applied to an infinite variety of particular organisms
in particular environments.

A final example is given in the body by the process of inflam-
mation. The function-rules which govern the process are gene-
tically determined and are constant in one species. Yet these
rules, when applied to an infinite variety of individual injuries,
provide an infinite variety in the details of the process at particular
points and times.

Our aim is now clear : we must find the function-rules. They
must be few in number, much fewer than 50,000, and we must
show that these few function-rules, when applied to an almost
infinite number of circumstances and to 10,000,000,000 neurons,
are capable of directing adequately the events in all these circum-
stances. The function-rules must be fixed, their applications
flexible.

(The gene-pattern is discussed again in S. 9/9.)

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.

The nervous system, and living matter in general, will be
assumed to be identical with all other matter. So no use of any
1 vital ' property or tendency will be made, and no deus ex machina
will be invoked. No psychological 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 demonstration. 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.

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

1/11 DESIGN FOR A BRAIN

would, of course, involve a circular argument ; for our purpose
is to explain the origin of behaviour which appears to be teleo-
logically directed.

It will be further assumed that the nervous system, living
matter, and the matter of the environment are all strictly deter-
minate : that if on two occasions they are brought to the same
state, the same behaviour will follow. Since at the atomic level
of size the assumption is known to be false, the assumption implies
that the functional units of the nervous system must be sufficiently
large to be immune to this source of variation. For this there is
some evidence, since recordings of nervous activity, even of single
impulses, show no evidence of appreciable thermal noise. But
we need not prejudge the question. The work to be described
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.

Consciousness

1/11. The previous section has 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 aspects of the mind-body relation,
and with an aspect — 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 move-
ment hundreds of times, are quite unconscious of making it.
The direct intervention of consciousness is evidently not necessary
for adaptive learning.

THE PROBLEM 1/12

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.

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 demonstrate
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

1/12. 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. 8/1).

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. Later, however, when
adult, its reactions are different. It approaches the fire and seats
itself at that place where the heat is moderate. If the fire burns
low, it moves nearer. If a hot coal falls out, it jumps away.
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.

With this as specific example, we may state the problem

1/12 DESIGN FOR A BRAIN

generally. We commence with the concepts that the organism
is mechanistic in action, that it is composed of parts, and that
the behaviour of the whole is the outcome of the compounded
actions of the parts. Organisms change their behaviour by
learning, and 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.

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 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. When constructed, it must satisfy these
axiomatic demands: — (1) Its procedure for obtaining informa-
tion must be wholly objective. (2) It must obtain its information
solely from the ' machine ', no other source being permitted.
(3) It must be applicable, at least in principle, to all material
4 machines ', whether animate or inanimate. (4) It must be
precisely defined.

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 : electronic circuits, rats in
mazes, solutions of reacting chemical substances, automatic
pilots, or heart-lung preparations. The method proposed here

2/3 DESIGN FOR A BRAIN

must have the peculiarity that it is applicable to all ; it must,
so to speak, specialise in generality.

Variable and system

2/3. The first step is to record the behaviours of the machine's
individual parts. To do this we identify any number of suitable
variables. A variable is a measurable quantity which at every
instant has a definite numerical value. A ' grandfather ' 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 4 variable ' I shall use the criterion whether
it can be represented by a pointer on a dial. I shall, in fact,
assume that such representation is always used : that the
experimenter is observing not the parts of the real ' machine '
directly but the dials on which the variables are displayed, as
an engineer watches a control panel.

Only in this way can we be sure of what sources of information
are used by the experimenter. 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 actually see them 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 useful, 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 sufficiently reliable : the unexpected
sometimes happens ; and the only way to be certain of the rela-
tion between two parts in a new ' machine ' is to test the rela-
tionship directly.

All the quantities used in physics, chemistry, biology, physio-
logy, and objective psychology, are variables in the defined sense.
Thus, the position of a limb can be specified numerically by co-
ordinates of position, and movement of the limb can move a pointer

DYNAMIC SYSTEMS 2/5

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 quan-
tity 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 S. 3/4.

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. ; c 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. A system is any arbitrarily selected set of variables. It is
a list nominated by the experimenter, and is quite different from
the real ' machine '.

At this stage no naturalness of association is implied, and the
selection is arbitrary. (' Naturalness ' is discussed in S. 2/14.)

The variable ' time ' will always be used, so the dials will
always include a clock. But the status of ' time ' in the method
is unique, so it is better segregated. I therefore add the qualifi-
cation that ' time ' is not to be included among the variables of
a system.

The Method

2/5. It will be appreciated that every real i machine ' embodies
no less than an infinite number of variables, most 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

2/6

DESIGN FOR A BRAIN

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 his system. Thus, an experimenter once
drew up Table 2/5/1. He thereby defined a three- variable

Time
(mins.)

Distance of
secondary
coil (cm.)

Part of skin
stimulated

Secretion of

saliva during

30 sees.

(drops)

• • •

. . .

• • •

Table 2/5/1

system, ready for testing. This experiment being finished, he later
drew up other tables which included new variables or omitted
old. By definition these new combinations were new systems.

2/6. The variables being decided on, the recording apparatus
is now assumed to be attached to the ' machine ' and the experi-
menter ready to observe the dials. We must next specify what
power the experimenter has over the experimental situation.

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.

DYNAMIC SYSTEMS 2/7

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 we cannot create a thunder-
storm, we can observe how swallows react to one simply by
waiting till one occurs 4 spontaneously '.

2/7. The 4 machine ' will be studied by applying the primary
operation, defined thus : The variables are brought to a selected
state (S. 2/9) by the experimenter's power of control (S. 2/6) ;
the experimenter decides which variables are to be released and
which are to be controlled ; at a given moment the selected
variables are released, so that their behaviour is controlled pri-
marily by the ' machine ', while the others are forced by the
experimenter to follow their prescribed courses (which often
includes their being held constant) ; the behaviours of the vari-
ables are then recorded. This operation is always used in the
practical investigation of dynamic systems. Here are some
examples.

In chemical dynamics the variables are often the concen-
trations 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 posi-
tions 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-
tures of some places are held constant, the variations of the
other temperatures is 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

2/8 DESIGN FOR A BRAIN

blood. 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 4 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 sur-
gically '. The second variable is permanently under the experi-
menter'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.

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 :
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 prescribed courses 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 4 machine ' may be made to yield more and more
complex information about its inner organisation.

2/8. All our concepts will eventually be defined in terms of
this method. For example, ' environment ' is so defined in S. 3/8,
' adaptation ' in S. 5/8, and ' stimulus ' in S. 6/6. If any have
been omitted it is by oversight ; for I hold that this procedure
is sufficient for their objective definition.

The Field of a System

2/9. 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.

DYNAMIC SYSTEMS 2/10

Two states are equal if and only if the corresponding pairs of
numerical values are all equal.

2/10. 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. One
primary operation yields one line of behaviour.

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

Time — *-

Figure 2/10/1 : 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 conditioned stimulus : the sound
of a buzzer. Line D records by a vertical stroke (F) the application of
the electric shock. (After Liddell et al.)

The graphical method is exemplified by Figure 2/10/1. 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.

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

2/11

DESIGN FOR A BRAIN

acquaintance with elementary mathematics would suggest. With
biological material it is rare.

Another form is the tabular, of which an example is Table 2/10/1.
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)

■a

7-35
156-7

7-26
154-6

7-28
1541

7-29
151-5

110-3

116-7

118-3

118-5

22-2

15-3

150

14-6

Table 2/10/1 : Blood changes after a dose of ammonium chloride, a;
= serum pH ; x = serum total base ; y = serum chloride ; z = serum
bicarbonate ; (the last three in m. eq. per 1.).

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.

2/11. 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.

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 values
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 2/11/1. The representative
point of a state is the 'point whose co-ordinates are respectively
equal to the values of the variables. By S. 2/4 ' time ' is not to
be one of the axes.

2/12

Figure 2/11/1.

2/12. Suppose next that a system of two variables gave the
line of behaviour shown in Table 2/12/1. The successive states
will be graphed, by the method, at positions B, C, and D (Figure
2/11/1). 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

0
1
2
3

7
5

10
9

7
4

Table 2/12/1.

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

21 c

2/13

DESIGN FOR A BRAIN

correspondence is maintained exactly no matter how numerous
the variables.

2/13. 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 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 usually want to

15-

c
_o
O

10 15

Weight of dog (kg.

Figure 2/13/1 : Arrow-heads show the direction of movement of the
representative point ; cross-lines show the positions of the representa-
tive point at weekly intervals.

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 four primary operations, gave the field shown in
Figure 2/13/1. Other examples occur frequently later.

It will be noticed that a field is defined, in accordance with
S. 2/8, by reference exclusively to the observed values of the

DYNAMIC SYSTEMS 2/14

variables and to the results of primary operations on them. It
is therefore a wholly objective property of the system.

The concept of ' field ' will be used extensively for two reasons.
It defines the characteristic behaviour of the system, replacing the
vague concept of what a system ' does ' or how it 4 behaves '
(often describable only in words) by the precise construct of a

4 field '. From this precision comes the possibility of comparing
field with field, and therefore of comparing behaviour with
behaviour. 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 ' typical way of behaving \

The Natural System

2/14. In S. 2/4 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. 2/3, 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.

If, on repeatedly applying a primary operation to a system, it
is found that all the lines of behaviour which follow an initial state

5 are equal, and if a similar equality occurs after every other initial
state S', S", . . . , then the system is regular.

Whether a system is regular or not may be decided by first
constructing and then examining its field. For if the system is
regular, from each initial state will go only one line of behaviour,

2/15 DESIGN FOR A BRAIN

the subsequent trials merely confirming the first. The concept
of ' regularity ' thus conforms to the demand of S. 2/8 ; for it
is definable in terms of the field and is therefore wholly objective.

The field of a regular system does not change with time.

If, on testing, a system is found to be not regular, the experi-
menter is faced with the common problem of what to do with a
system that will not give reproducible results. Somehow he
must get regularity. The practical details vary from case to
case, but in principle the necessity is always the same : he must
try a new system. This means that new variables must be
added to the previous set, or, more rarely, some irrelevant variable
omitted.

From now on we shall be concerned mostly with regular sys-
tems. We assume that preliminary investigations have been
completed and that we have found a system, based on the real
* machine ', that (1) includes the variables in which we are specially
interested, and (2) includes sufficient other variables to render
the whole system regular.

2/15. For some purposes regularity of the system may be
sufficient, but more often a further demand is made before the
system is acceptable to the experimenter : it must be ' absolute \*
It will be convenient if I first define the concept, leaving the
discussion of its importance to the next section.

If, on repeatedly applying primary operations to a system, it is
found that all the lines of behaviour which follow a state S are equal,
no matter how the system arrived at S, and if a similar equality
occurs after every other state S', S", . . . , then the system is
absolute.

Consider, for instance, the two- variable system that gave the
two lines of behaviour shown in Table 2/15/1.

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, 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, y = 1*8 and not x = 0-2,
y == 2*1. As the two lines of behaviour that follow the state
x = 0, y — 2-0 are not equal, the system is not absolute.

A well-known example of an absolute system is given by the

* (O.E.D.) Absolute : existent without relation to any other thing ; self-
sufficing ; disengaged from all interrupting causes.

DYNAMIC SYSTEMS

2/15

simple pendulum swinging in a vertical plane. It is known that
the two variables — (x) angle of deviation of the string from

Time (seconds)

Line

Variable

0-2

0-3

0
20

0-2
2-1

0-4
2-3

0-6
2-6

- 0-2
2-4

- 01
2-2

0
20

01
1-8

Table 2/15/1.

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.)

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.

An absolute system is thus ' state-determined ', and this is its
most important property : the occurrence of a state is sufficient
to determine the line of behaviour that ensues. The property

2/16 DESIGN FOR A BRAIN

is both necessary and sufficient ; so all state-determined systems
are absolute. We shall use this fact repeatedly.

The field of an absolute system is characteristic : from every
point there goes only one line of behaviour whether the point is
initial on the line or not. The field of the two-variable system
just mentioned is sketched in Figure 2/15/1 ; through every point
passes only one line.

These relations may be made clearer if this field is contrasted
with one that is regular but not absolute. Figure 2/15/2 shows

such a field (the system is described
in S. 19/15). 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
absolute ; for to say that the repre-
sentative point is leaving C is insuf-
ficient to define its future line of
behaviour, which may go to A' or B' .
Figure 2/15/2 : The field Even if the lines from A and B always
Figure ijfivi.8110™ in 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 absolute, the lines CA', CB\ and CD would
coincide.

A system's absoluteness is determined by its field ; the property
is therefore wholly objective.

An absolute system's field does not change with time.

2/16. We can now return to the question of what we mean when
we say that a system's variables have a c 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.
The third property makes clear that we cannot expect a proposed
definition to be established by a few lines of verbal argument :

DYNAMIC SYSTEMS 2/17

it must be treated as a working hypothesis and used ; only experi-
ence can show whether it is faulty or sound.

From here on I shall treat a ' natural ' system as equivalent to
an i absolute ' system. Various reasons might be given to make
the equivalence plausible, but they would prove nothing and I
shall omit them. Much stronger is the evidence in the Appendix.
There it will be found that the equivalence brings clarity where
there might be confusion ; and it enables proof to be given to
propositions which, though clear to physical intuition, cannot
be proved without it. The equivalence, in short, is indispens-
able.

Why the concept is so important can be indicated briefly.
When working with determinate systems the experimenter always
assumes that, if he is interested in certain variables, he can find
a set of variables that (1) includes those variables, and (2) has
the property that if all is known about the set at one instant the
behaviour of all the variables will be predictable. The assump-
tion is implicit in almost all science, but, being fundamental, it
is seldom mentioned explicitly. Temple, though, refers to ' . . .
the fundamental assumption of macrophysics that a complete
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.

2/17. To conclude, here is an example to illustrate this chapter's
method.

Suppose someone constructed two simple pendulums, hung them
so that they swung independently, and from this l machine '
brought to an observation panel the following six variables :

(v) the angular deviation of the first pendulum

(w) „ „ „ „ „ second

(x) the angular momentum of the first pendulum

{y) 9, » » ,, a second „

(z) the brightness of their illumination

(t) the time.
The experimenter, knowing nothing of the real 4 machine ', or of

2/17 DESIGN FOR A BRAIN

the relations between the five variables, sits at the panel and
applies the defined method.

He starts by selecting a system at random, constructs its field
by S. 2/13, and deduces by S. 2/15 whether it is absolute. He
then tries another system. It is clear that he will eventually be
able to state, without using borrowed knowledge, that just three
systems are absolute : (v, w, x, y), {v, x\ and (w, y). He will add
that z is unpredictable. He has in fact identified the natural
relations existing in the ' machine '. He will also, at the end of
his investigation, be able to write down the differential equations
governing the systems (S. 19/20). Later, by using the method
of S. 14/6, he will be able to deduce that the four-variable system
really consists of two independent parts.

References

Eddington, A. S. The nature of the physical world. Cambridge, 1929 ;

The philosophy of physical science. Cambridge, 1939.
Liddell, H. S., Anderson, O. D., Kotyuka, E., and Hartman, F. A.

Effect of extract of adrenal cortex on experimental neurosis in sheep.

Archives of Neurology and Psychiatry, 34, 973 ; 1935.
Temple, G. General principles of quantum theory. London. Second edition,

1942.

CHAPTER 3

The Animal as Machine

3/1. 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 numerical 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, the essential event 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 DESIGN FOR A BRAIN

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 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 4 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 4 high ', ' medium ', and i 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 4 unlit '. More
complex examples occur frequently in psychological experiments.
Table 2/5/1, for instance, contains a variable ' part of skin stimu-

THE ANIMAL AS MACHINE 3/3

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 4 stimulus ' which takes such values as ' bubbling water ',
4 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 l 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 4 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 sys-
tem ; 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. 21/1) 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 4 situation of the experiment '
might be allotted the arbitrary value of 4 1 ' if the experiment
occurs in the animal house, and 4 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
a single variable provided the arrangement of the experiment is

3/4 DESIGN FOR A BRAIN

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 so much with qualities as 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. 1/11) 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
demonstrate that the method is exact and that it can be extended
to any extent without loss of precision.

THE ANIMAL AS MACHINE 3/5

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 :

(a?) 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

Figure 3/5/1 : Effect of haemor-
rhage on the rate of blood-flow
through : x, the inferior vena cava ;
y, the muscles of a leg ; and z, 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
[pc) •>■> >» >> leit ,, ,, ,, ,,

{y) 99 „ » right „ „ „ right tibia

\~) 5) 5? ?> leit ,, ,, ,, leit ,,

In w and x the angle is counted positively when the knee comes

3/6

DESIGN FOR A BRAIN

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

0-7

0-8

-10

-20

-35

J2
C8

-35

-10

-20

-35

f-t

170

180

1G0

120

100

170

120

100

170

180

160

120

Table 3/5/1.

The organism as system

3/6. In a physiological experiment the nervous system is usually
considered to be absolute. That it can be made absolute is
assumed by every physiologist before the work starts, for he
assumes that it is subject to the fundamental assumption 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 its absoluteness (S. 2/15). So unless there are
special reasons to the contrary, the nervous system in a physio-
logical experiment has the properties of an absolute system.

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 in-
tended to ensure that this was so. So unless there are special
reasons to the contrary, the animal in an experiment with con-
ditioned reflexes has the properties of an absolute system.

THE ANIMAL AS MACHINE 3/10

The environment

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 represent-
able by dials, is explorable (by the experimenter) by primary
operations, and is intrinsically determinate.

Organism and environment

3/9. The theme of the chapter can now be stated : the free-
living organism and its environment, taken together, form an
absolute 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. The
two parts act and re-act on one another (S. 3/11), and are there-
fore properly regarded as two parts of one system. And since
we have assumed that the conjoint system is state-determined,
we may treat the whole as absolute.

3/10. As example, that the organism and its environment form
a single absolute 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 canjbe 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.)

3/11 DESIGN FOR A BRAIN

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
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 absolute. 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
denned by S. 3/8 to be the 4 environment ' of the rider. (Whether
the fourth variable is allotted to ' rider ' or to i environment ' is
optional (S. 3/12) ). To make the system absolute, 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 eye 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 bird and the
air. The whole may justifiably be assumed absolute.

3/11. The organism affects the environment, and the environ-
ment affects the organism : such a system is said to have 4 feed-
back ' (S. 4/12).

The examples of the previous section provide illustration.
The rider's arm moves the handlebars, causing changes in the

THE ANIMAL AS MACHINE 3/11

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. 1/12 — the
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 : —

' 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 achieve-
ment of the task '.

(Bartlett.)

4 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

37 D

3/11 DESIGN FOR A BRAIN

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-
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.

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

4 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.
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.'

THE ANIMAL AS MACHINE 3/12

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. 4/14) 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
introduces properties which can be explained only by reference
to the properties of 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 4 organism ' and
4 environment ' becomes partly conceptual, and to that extent
arbitrary. Anatomically and physically, of course, there is a
unique and obvious distinction between the two parts of the sys-
tem ; but if we view the system functionally, ignoring purely
anatomical facts as irrelevant, the division of the system into
4 organism ' and ' 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 mechan-
ism 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 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 chew without
biting his own tongue ; functionally both bread and tongue are
part of the environment of the cerebral cortex. But the environ-
ments with which the cortex has to deal are sometimes even deeper

3/13 DESIGN FOR A BRAIN

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 (S. 17/4 and Chapter 18) are not unreasonable.
There it is intended to treat one group of neurons in the cerebral
cortex 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 w the system ' means
not the nervous system but the whole complex of the organism
and its environment. Thus, if it should be shown that i 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 i 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 when a dog puts its foot on a sharp and
immovable stone, the latter does not seem particularly dynamic.
Yet the dog can cause great changes in this environment — by
moving its foot away. Again, 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.

Static systems (like the sharp stone) can always be treated as if
dynamic (though not conversely), for we have only to use the
device of S. 2/3 and treat the static variable as one which is
undergoing change of zero degree. The dynamic view is therefore

THE ANIMAL AS MACHINE 3/14

the more general. For this reason the environment will always
be treated as wholly dynamic.

Essential variables

3/14. The biologist must view the brain, not as being the seat of
the ' mind ', nor as something that ' thinks ', but, like every other
organ in the body, as a specialised means to survival. We shall
use the concept of ' survival ' repeatedly ; but before we can use
it, we must, by S. 2/8, transform it to our standard form. What
does it mean 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
— they fetch quite different prices in the market. The distinc-
tion must be capable of objective definition.

It is suggested that the definition may be obtained in the fol-
lowing way. That an animal should remain ' alive ' certain
variables must remain within certain ' physiological ' limits.
What these variables are, and what the limits, are fixed when
we have named the species we are working with. In practice
one does not experiment on animals in general, one experiments
on one of a particular species. In each species the many physio-
logical 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 re-
lated 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 method of

3/14 DESIGN FOR A BRAIN

S. 2/8 ? 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
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. This dis-
tinction may not be quite clear, for an animal's variables cannot
be divided sharply into ' essential ' and ' not essential ' ; but
exactness is not necessary here. All that is required 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.

References

Bartlett, F. C. The measurement of human skill. British Medical Journal,

1, 835 ; 14 June 1947.
Jennings, H. S. Behavior of the lower organisms. New York, 1906.
Morgan, C. Lloyd. Habit and instinct. London, 1896.
Rein, H. Die physiologischen Grundlagen des Kreislaufkollapses. Archiv

fur klinische Chirurgie, 189, 302 ; 1937.
Starling, E. H. Principles of human physiology. London, 6th edition, 1933.

CHAPTER 4

Stability

4/1. The words ' 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 connections 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 bath causes a change which in its turn causes the heating to
become more intense or more effective ; and vice versa. The

4/2 DESIGN FOR A BRAIN

result is that if any transient disturbance cools or overheats the
bath, the thermostat brings the 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
states 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 concept of stability must therefore be defined in
terms of the basic primary operations (S. 2/3).

4/3. Consider next 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 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 4 distance of the ball laterally ' is absolute and
has a definite, permanent field, which is sketched in the Figure.

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

A [

Figure 4/3/1.

STABILITY

4/5

4/4. This preliminary remark begins to justify the emphasis
placed on absoluteness. Since stability is a feature of a field,
and since only regular systems have unchanging fields (S. 19/16)
it follows that to discuss stability in a system we must suppose
that the system is regular : we cannot test the stability of a
thermostat if some arbitrary interference continually upsets it.
But regularity in the system is not sufficient. If a field had
lines criss-crossing like those of Figure 2/15/2 we could not make
any simple statement about them. Only when the lines have a
smooth flow like those of Figures 4/5/1, 4/5/2 or 4/10/1 can a
simple statement be made about them. And this property
implies (S. 19/12) that the system must be absolute.

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 an absolute system which
has two variables :

(cc) the angle 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

O O o a 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.

Y v

Figure 4/5/2.

4/5 DESIGN FOR A BRAIN

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 (x) and parallel (y) to the lower
edge. The field will resemble that
sketched in Figure 4/5/2. Displace-
ment from the origin 0 to A is followed
by a return of the representative point
to 0, and this return corresponds to the
stability. Displacement from 0 to B is
followed by a departure from the region
under consideration, and this departure
corresponds 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 quan-
tities because they are rigidly
connected). If, now, a disturb-
ance 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
of behaviour must resemble that

Figure 4/5/3 : One line of behav-
iour in the field of the Watt's
governor. For clarity, the resting
state of the system has been used
as origin. The system has been
displaced to A and then released,

STABILITY 4/8

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 dif-
ferent 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,
to a regular small oscillation. We shall be little interested in the
details of what happens at the exact centre.

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 limit 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. 4/12) 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 definition which will
be used. Given an absolute system and a region within its field,
a line of behaviour from a point within the region is stable if it
never leaves the region. Within one absolute system a change of
the region or of the line of behaviour may change the result of
the criterion.

Thus, in Figure 4/3/1 the stability around A can be decided
thus : make a mark on each side oi" A so as to define the region ;
then as the line of behaviour from any point within this region
never leaves it, the line of behaviour is stable. On the other
hand, no region can be found around B which gives a stable line
of behaviour. Again, consider Figure 4/5/2 : a boundary line
is first drawn to enclose A, 0 and B, in order to define which
part of the field is being discussed. The line of behaviour from

4/9 DESIGN FOR A BRAIN

A is then found to be stable, and the line from B unstable. This
example makes it obvious that the concept of ' stability ' belongs
primarily to a line of behaviour, not to a whole field. In particular
it should be noted that in all cases the definition gives a unique
answer once the line, the region, and the initial state are given.

The examples above have been selected to test the definition
severely. Sometimes the fields are simpler. In the field of the
cube, for instance, it is possible to draw many boundaries, each oval
in shape, such that all lines within the boundary are stable. The
field of the Watt's governor is also of this type. It will be noticed
that before we can discuss stability in a particular case we must
always define which region of the phase-space we are referring to.

A field within a given region is ' stable ' if every line of behaviour
in the region is stable. A system is ' stable ' if its field is stable.

4/9. A resting state is one from which an absolute system does
not move when released. Such states occur in Figure 4/3/1 at
A and B, and in Figure 4/5/1 at the origin.

Although the variables do not change value when at a resting
state this invariance does not imply that the ' machine ' itself is
inactive. Thus, a steady Watt's governor implies that the engine
is working at a non-zero rate. And a living muscle, even if
unchanging in tension, is continually active in metabolism.
4 Resting ' applies to the variables, not necessarily to the ' machine '

that yields the variables.

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
behaviour 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.

STABILITY 4/12

4/11. This definition of stability conforms to the requirement
of S. 2/8 ; for the observed behaviour of the system determines
the field, and the field determines the stability.

Feedback

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
4 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.)

Lest the diagram should seem based on some metaphysical
knowledge of causes and effects, its derivation from the actual
machine, using only primary operations, will be described.

Suppose the relation between ' speed of engine ' and 4 distance
between weights ' is first investigated. The experimenter would
fix the variable ' velocity of flow of steam '. 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. He
need know nothing of the nature of the ultimate physical linkages,
but he would observe the fact. Then, still keeping i 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.

4/12 DESIGN FOR A BRAIN

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 4 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, or is held, at Ax ; and also its behaviour
when A starts, or is held, at A2. If these behaviours of B are the
same, then there is no immediate effect from A to B. But if the
B's behaviours are unequal, and regularly depend on what value
A starts from, or is held at, then there is an immediate effect,
which we symbolise by A — > B.

By interchanging A and B in the process we can test for
B — > A . And by using other pairs in turn we can determine
all the immediate effects. The process is clearly defined, and
consists purely of primary operations. It therefore uses no
borrowed knowledge. We shall frequently use this diagram of
immediate effects.

If A has an immediate effect on B, and B has an immediate
effect on A, the relation will be represented by A ^± B. If A
affects B, and B also affects C, but A does not affect C directly, the
relation will be shown by A — > B — > C. If there is a sequence of
arrows joined head to tail and we are not interested in the inter-
mediate steps, the sequence may often be contracted without
ambiguity to A — > C. The diagram will be used only for illustration
and not for rigorous proofs, so further precision is not required. (It
should be carefully distinguished from the diagram of ' ultimate '
effects, but this is not required yet and will be described in S. 14/6.
At the moment we regard the concept of one variable ' having an
effect ' on another as well understood. But the concept will
be examined more closely, and given more precision, in S. 14/3.)

A gas thermostat also shows a functional circuit or feedback ;

STABILITY 4/12

for if it is controlled by a capsule which by its swelling moves a
lever which controls the flow of gas to the heating flame, the
diagram of immediate effects would be :

Temperature

Diameter

of capsule

> '

Size of
gas flame

Position
of lever

Velocity

Position

of ga

s 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-potential

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-
potential 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 a circuit
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

4/13 DESIGN FOR A BRAIN

of searchlight, anti-aircraft guns, rockets, and torpedoes, and
facilitated by the great advances that had occurred in electronics.
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. 4/14.

The nature, degree, and polarity of the feedback has a decisive
effect on the stability or instability of the system. In the Watt's
governor or in the thermostat, for instance, the connection 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 rela-
tion 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

4^=^3 3^ 4

Figure 4/12/1.

and larger deviation from the resting state. The phenomenon
is identical with that referred to as a ' vicious circle '.

The examples shown have only a simple circuit. But more
complex systems may have many interlacing circuits. If, for
instance, as in S. 8/8, four variables all act on each other, the
diagram of immediate effects would be that shown in Figure
4/12/1 (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.

4/13. It will be noticed that stability, as denned, in no way
implies fixity or rigidity. It is true that stable systems may have
a resting state at which they will show no change ; but the lack

STABILITY

4/14

of change is deceptive if it suggests rigidity : they have only to be
disturbed to show that they are capable of extensive and active
movements. They are restricted only in that they do not show
the unlimited divergencies of instability.

Goal-seeking

4/14. Every stable system has the property that if displaced
from a resting state and released, the subsequent movement is
so matched to the initial displacement that the system is brought
back to the resting state. 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

— Time

Figure 4/14/1 : Tracing of the temperature (solid line), of a thermostatically
controlled bath, and of the control setting (broken line).

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 4/14/1 shows how, as the control setting of a thermostat
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-

53 E

4/15 DESIGN FOR A BRAIN

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.

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 resting state (which, in these examples,
can be moved by an outside operation). The thermostat is
affected by the difference between the actual and the set tem-
peratures. The searchlight is affected by the difference between the
two directions. So it will be seen that 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.

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 ' or ' intelligent ' 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 mini-
mise 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 feed-
back may be both wholly automatic and yet actively and complexly
goal-seeking. There is no incompatibility.

4/15. 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 con-
sideration of the third diagram of S. 4/12 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.

STABILITY 4/16

(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 S. 21/5-7.)

(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.

(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.

4/16. 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
(as contrasted with instability) 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

4/16 DESIGN FOR A BRAIN

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. 20/12 : 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.

Reference
Wiener, Norbert. Cybernetics. New York, 1948.

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 be given in terms that
conform to the demand of S. 2/8.

5/2. The suggestion that an animal's behaviour is ' adaptive '
if the animal ' 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 4 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
4 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 ' adapted ' to its end.

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

variables within physiological limits.
57

5/4 DESIGN FOR A BRAIN

(3) Almost all the behaviour of an animal's vegetative system
is due to such mechanisms.

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 sur-
rounding 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

ADAPTATION AS STABILITY 5/4

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.

The water content of the blood is disturbed by the intake of
water at drinking and eating, by the output during excretion
and secretion, and by sweating. When the water content is
lowered, sweating, salivation, and the excretion of urine are all
diminished ; thirst is increased, leading to an increased intake,
and the tissues of the body pass some of their water into the
blood-stream. When the water content is excessive, all these
activities are reversed. By these means the body tends to
maintain the water-content of the blood within limits.

The pressure of the blood in the aorta may be disturbed by
haemorrhage or by exertion. When the pressure falls, centres
in the brain and spinal cord make the heart beat faster, increasing
the quantity of blood forced into the aorta ; they make the small
arteries contract, impeding the flow of blood out of it. If the
pressure is too high, these actions are reversed. By these and
other mechanisms the blood pressure in the aorta is maintained
within limits.

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.

5/5 DESIGN FOR A BRAIN

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 circulat-
ing blood ' and ' oxygen supplied to the tissues ' within normal
limits.

Every fast-moving animal is liable to injury by collision with
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, depend-
ing 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 nar-
rowing is the objective form 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

ADAPTATION AS STABILITY 5/6

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 activi-
ties ; the principles are uniform throughout.

5/6. Just the same criteria for ' adaptation ' may be used in
judging the behaviour of the free-living animal in its learned
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 temperature 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 hap-
pened to the kitten's brain, we can at least say that whereas

5/6 DESIGN FOR A BRAIN

at first the kitten's behaviour was not homeostatic 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 surround-
ings 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
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 hap-
pened 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,

ADAPTATION AS STABILITY 5/7

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
4 stress on bone ' within narrower limits. Good headlights keep
the luminosity of the road within limits narrower than would
occur in their absence.

The thesis that ' 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.
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 ' 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
1 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 8.

5/8 DESIGN FOR A BRAIN

5/8. We can now recognise that 4 adaptive ' behaviour is equi-
valent 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) :

1 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.)

1 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 con-
stancy.'

(Cannon.)

4 Every material system can exist as an entity only so long
as its internal forces, attraction, cohesion, etc., balance the
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 4 stability ' explicitly, but
when describing the type of behaviour which he considered to
be most characteristic of the living organism, he wrote :

c Take a billiard 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

ADAPTATION AS STABILITY 5/11

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. The forces of the environment, and even the drift of time,
tend to displace the essential variables by amounts to which we
can assign no limit. For survival, the' essential variables must
be kept within their physiological limits. In other words, the
values of the essential variables must stay within some definite
region in the system's phase-space. It follows therefore that
unless the environment is wholly inactive, stability is necessary
for survival.

5/10. If an animal's behaviour always maintains its essential
variables within their physiological limits, then the animal can
die only of old age. Disease might disturb the essential variables,
but the processes of repair and immunity would tend to restore
them. But it is equally clear that the environment sometimes
causes disturbances for which the body's stabilising powers are
inadequate ; infections may prove too virulent, cold too extreme,
a famine too severe, or the attack of an enemy too swift.

The possession of a mechanism which stabilises the essential
variables is therefore of advantage : against moderate disturb-
ances it may be life-saving even if it eventually fails at some
severe disturbance. It promotes, but does not guarantee, survival.

5/11. 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 ?

5/11 DESIGN FOR A BRAIN

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
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 5/11/1.

Behaviour -

r Non-adaptive

Sustentative

f Self-
maintaining,

Adaptive-

Race-
maintaining

Useless tropistic reaction.
Misdirected instinct.
Abnormal sex behaviour.
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.
Preparation of food.

Against enemies — fight,

flight.
Against inanimate forces.
Reactions to heat, gra vity,

chemicals.
^Against inanimate objects.

Ameliorative Rest, sleep, play, basking.

(with these we are not concerned,

Protective

S. 1/3).
Table 5/11/1 : 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 w adaptive-
ness ' of behaviour is properly measured by its tendency to
promote the organism's survival.

ADAPTATION AS STABILITY 5/13

The stable organism

5/12. 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
formulation transfers the centre of interest to the resting
state, to the fact that the essential variables of the adapted
organism change less than they would if they were unadapted.
Which is important : constancy or change ?

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 impor-
tant only in so far as it contributes to this end.

5/13. So far the discussion has traced the relation between the
concepts of ' adaptation ' and of 4 stability '. It will now be
proposed that ' motor co-ordination ' also has an essential con-
nection with stability.

c 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 move-
ment follows accurately its
appropriate path. Con-
trasted to it are the concepts
of clumsiness, tremor, ataxia,
athetosis. It is suggested Figure 5/13/1.

that the presence or absence

of co-ordination may be decided, in accordance with our methods,
by observing whether the movement does, or does not, deviate
outside given limits.

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 5/13/1, which shows the line
traced by the point of an expert fencer's foil during a lunge.

5/14

DESIGN FOR A BRAIN

Any inco-ordination would be shown by a divergence from the

intended line.

A second example is given by the record of Figure 5/13/2.
The subject, a patient with a tumour in
the left cerebellum, was asked 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 sug-
gested 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-ordination

of motor activity.

Figure 5/13/2 : 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.)

5/14. So far we have noticed in stable systems only their pro-
perty of keeping variables within limits. But such systems 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 ' intelligence ' not
shared by non-living systems. In these two instances the assump-
tion 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
resting state (e.g. Figure 4/5/3). When this occurs, variables
may be observed to move away from their values in the resting
state, only to return to them later. Thus, suppose in Figure
5/14/1 that the field is stable and that at the resting 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-

ADAPTATION AS STABILITY 5/15

sentative point moves towards B, y hardly alters ; but x, which

started at X', moves to X 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 its resting state 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 „

\ Figure 5/14/1.

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 a rudimentary skill in getting back to its resting
state in spite of obstacles.

5/15. 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 5/15/1 ; the variables have been divided by the
dotted line into ' animal ' on the right and 4 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
arrived at the ' adapted ' condition (S. 5/7). If disturbed, its
changes will show co-ordination of part with part (S. 5/14), and

69 f

5/16

DESIGN FOR A BRAIN

this co-ordination will hold over the whole system (S. 4/15). 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.

In the higher organisms, and especially in man, the power to
react correctly to something not immediately visible or tangible

-< — | ,

-< — | .

: ►-•

■ >- .

Figure 5/15/1.

has been called i imagination ', or i 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
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.

5/16. At this stage it is convenient to 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 certain 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 two chapters.

ADAPTATION AS STABILITY 5/16

References

Ashby, W. Ross. Adaptiveness and equilibrium. Journal of Mental Science,

86, 478 ; 1940.
Idem. The behavioral properties of systems in equilibrium. American

Journal of Psychology, 59, 682 ; 1946.
Cannon, W. B. The wisdom of the body. London, 1932.
Conference on Teleological Mechanisms. Annals of the New York

Academy of Science, 50, 187 ; 1948.
Grant, W. T. Graphic methods in the neurological examination : wavy

tracings to record motor control. Bulletin of the Los Angeles Neurological

Society, 12, 104 ; 1947.
Holmes, S. J. A tentative classification of the forms of animal behavior.

Journal of Comparative Psychology, 2, 173 ; 1922.
McDougall, W. Psychology. New York, 1912.
Pavlov, I. P. Conditioned reflexes. Oxford, 1927.
Rosenbluetii, A., Wiener, N., and Bigelow, J. Behavior, purpose and

teleology. Philosophy of Science, 10, 18 ; 1943.
Sommerhoff, G. Analytical biology. Oxford, 1950.

CHAPTER 6

Parameters

6/1. So far, we have discussed the changes shown by the vari-
ables of an absolute 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 be-
haviour ; 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 absolute system is formed by selecting some variables out
of the totality of possible variables. ' Forming a system '
means dividing all possible variables 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 will be
described as 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 :
their change of 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 ' viscosity of the air ' in relation to
the same system. And finally, for completeness, may be men-
tioned the infinite number of parameters that are without detect-
able 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

PARAMETERS

6/3

may be ignored ; but the relationship of an effective parameter
to a system must be clearly understood.

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

constant at zero),

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

constant),

(4) the position (co-ordinates) 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 an absolute system of a change of parameter-
value will now be shown. Table 6/3/1 shows the results of four

Length
(cm.)

Line

Vari-
able

Time

005

010

015

0-20

0-25 | 0-30

0
147

7
142

14
129

20
108

28
48

29
12

14
129

108

28
48

29
12

29
- 24

-58

0
147

7
144

14
135

21
121

26
101

34
51

21
121

26
101

34
51

36
23

36
- 6

35
- 36

Table 6/3/1.
73

6/4 DESIGN FOR A BRAIN

primary operations applied to the two-variable system mentioned

above, x is the angular deviation 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 absolute.
The line of behaviour is shown
solid in Figure 6/3/1. In these

Iqq. 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

Figure 6/3/1. system is aSain absolute- 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 parameter bears to the two variables

is therefore as follows :

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

(2) After the parameter changes from one constant value to
another, the system of x and y becomes again absolute, 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.

The converse proposition is also true. Suppose we form a
system's field and find it to be absolute. If our control of its
surroundings has not been complete, and we test it later and find
it to be again absolute but to have a changed field, then we may
deduce, by S. 22/5, that some parameter must, in the interval,
have changed from one constant value to another constant value.

6/4. The importance of distinguishing between change of a
variable and change of a parameter, that is, between change of

PARAMETERS

6/4

300-

200

state and change of field, can hardly be over-estimated. In
order to make the distinction clear I will give some examples.

In a working clock, the single variable defined by the reading
of the minute-hand on the face is absolute 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 absolute 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 c blood -glucose ' is not
absolute, 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 vari-
able, however, such as ' rate
of change of blood-glucose ',
which may be positive or
negative, we obtain a two-
variable system which is
sufficiently absolute for
illustration. The field of
this two-variable system
will resemble 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, Fig. 6/4/2), it is seen to be not the same
as that of the normal subject. The change of value of the
parameter 4 degree of diabetes present ' has thus changed the
field.

Girden and Culler developed a conditioned reflex in a dog which
was under the influence of curare (a paralysing drug). When
later the animal was not under its influence, the conditioned reflex
could not be elicited. But when the dog was again put under its
influence, the conditioned reflex returned. We need not enquire
closely into the absoluteness of the system, but we note that two

2 3

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

6/5

DESIGN FOR A BRAIN

characteristic lines of behaviour (two responses to the stimulus)
existed, and that one line of behaviour was shown when the

3\£

St- 200-

blood
100 ml

6
o

° -100' ^^.y

a. '00 200 100 200 300

BLOOD GLUCOSE (mg. per 100 ml.)

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.

parameter ' concentration of curare in the tissues ' had a high
value, and the other when the parameter had a low value.

6/5. The physicist, studying systems whose variables are all
clearly marked and controllable, seldom confuses change of state
with change of field. The psychologist, however, studies systems
whose variables, even in the simplest systems, are so numerous
that he cannot, in practice, make an exact list of them : his
grasp of the situation must be intuitive rather than explicit.
In his practical work he seldom fails to distinguish between the
variables he is observing and the parameters he is controlling ;
it is chiefly in his theoretical work, especially when he discusses
cerebral mechanisms, that he is apt to allow the distinction to
become blurred. To preserve the distinction between variable
and parameter we must discuss, not the real ' machine ', with
its infinite richness of variables, but a defined system. The
advantage to be gained will become clearer as we proceed.

Stimuli

6/6. 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.

PARAMETERS 6/6

In some cases the animal, at some resting state, is subjected to
a sudden change in the value of the stimulator, and the second
value is sustained throughout the observation. Thus, the pupil-
lary 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.

Sometimes a parameter is changed sharply and is immediately
returned to its initial value, as when the experimenter applies a
single electric shock, a tap on a tendon, or a flash of light. 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 resting state are restored, and the
representative point returns to the resting state. Such a stimulus
reveals a line of behaviour leading to the resting state.

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 4 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

6/7 DESIGN FOR A BRAIN

parameter and thus avoid using unnecessarily large numbers
of parameters.

A more extensive generalisation is provided if we replace
4 change of parameter ' by ' change of initial state '. It will be
shown (S. 7/7 and 21/4) that if a variable, or parameter, stays
constant over some period it may, within the period, be regarded
indifferently as inside or outside the system — as variable or para-
meter. If, therefore, a contact switch, once set, stays as the
experimenter leaves it, we may, if we please, regard it as part of
the system. Then what was a comparison between two lines of
behaviour from two fields (of a set of variables a, b, c, say) under
the change of a parameter p from p' to p", becomes a comparison
between two lines of behaviour of the four-variable system from
the initial states a, b, c, p' and a, b, c, p" . 4 Applying a stimulus '
is now equivalent to ' releasing from a different initial state ' ;
and this will be used as its most general representation.

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 concentra-
tions reach the resting state. If the mixture was originally derived
from pure ammonia, the single variable 4 percentage dissociated '
forms a one-variable absolute system. Among its parameters
are temperature and pressure. As is well known, changes in these
parameters affect the position of the resting state.

Such a system is simple and responds to the changes of the
parameters with only a simple shift of resting state. 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,
resting states may be moved, single resting states may become
multiple, resting states may become cycles ; and so on. Figure
21/5/1 provides an illustration.

Here we need only the relationship, which is reciprocal : in

PARAMETERS 6/7

an absolute 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.

Reference

Girden, E., and Culler, E. Conditioned responses in curarized striate
muscle in dogs. Journal of Comparative Psychology, 23, 2G1 ; 1937.

CHAPTER 7

Step-Functions

7/1. 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 ' simple harmonic oscillator '
the meaning of the phrase is well understood. Here we shall be
more interested in the extent to which a variable displays con-
stancy. Four types may be distinguished, and are illustrated in

-^v

TIME— ►-

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

Fig. 7/1/1. (A) The full-function has no finite interval of con-
stancy ; 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. 14/12. (C) The step-function
has finite intervals of constancy separated by instantaneous jumps.

STEP-FUNCTIONS 7/2

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 Fig. 2/10/1 will be found to be part-,
full-, step-, and null-, functions respectively.

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.

7/2. 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.

(3) The viscosity of water, measured as the temperature

passes 0° C, changes similarly.

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

mately proportional to its length. The constant of
proportionality 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.

(5) If a trajectory is drawn through the air, a few feet above the

ground and parallel to it, the resistance it encounters as it
meets various objects varies in step-function form.
81

7/3 DESIGN FOR A BRAIN

(6) A stone, falling through the air into a pond and to the

bottom, would meet resistances varying similarly.

(7) The temperature of a match when it is struck changes in

step-function form.

(8) If strong acid is added in a steady stream to an un-

buffered alkaline solution, the pH changes in approxi-
mately step-function form.

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

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

(10) As the pH is changed, the amount of adsorbed substance

often changes in approximately step-function form.

(11) By quantum principles, many atomic and molecular

variables change in step-function form.

(12) The blood flow through the ductus arteriosus, when ob-

served over an interval including the animal's birth,
changes in step-function form.

(13) The sex-hormone content of the blood changes in step-

function form as an animal passes puberty.

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

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

7/3. Few variables other than the atomic can change instan-
taneously ; a more minute examination shows that the change
is really continuous : the fusing of an electric wire, the closing of a
switch, and the snapping of a piece of elastic. But if the event
occurs in a system whose changes are appreciable only over some
longer time, it may be treated without serious error as if it oc-
curred instantaneously. Thus, if x — tanh t, it will give a graph
like A in Figure 7/3/1 if viewed over the interval from t = — 2
to t = +2. But if viewed over the interval from t — — 40 to
t = -|- 40, it would give a graph like B, and would approximate
to the step-function form.

In any experiment, some ' order ' of the time-scale is always
assumed, for the investigation never records both the very quick
and the very slow. Thus to study a bee's honey-gathering flights,
the observer records its movements. But he ignores the movement
caused by each stroke of the wing : such movements are ignored
as being too rapid. Equally, over an hour's experiment he ignores

STEP-FUNCTIONS

7/4

the fact that the bee at the end of the hour is a little older than it

was at the beginning : this change is ignored as being too slow.

Such changes are eliminated by being treated as if they had

their limiting values. If a single rapid change occurs, it is

Time — *•

Figure 7/3/1 : The same change viewed : (A) over one interval
of time, (B) overman interval twenty times as long.

treated as instantaneous. If a rapid oscillation occurs, the
variable is given its average value. If the change is very slow,
the variable is assumed to be constant. In this way the concept
of ' step-function ' may legitimately be applied to real changes
which are known to be not quite of this form.

7/4. 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 .
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

7/5' DESIGN FOR A BRAIN

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 step-function form, ap-
proximating 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. 10/5.

Critical states

7/5. In any absolute 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 an absolute system with a step-function at a particular value,
all the states with the step-function 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-function'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 ' constant of proportionality ' of an elastic
strand is the length at which it breaks.

An example from physiology is provided by the urinary bladder
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
a certain value, the centre becomes inactive and the bladder refills.
A graph of the two variables would resemble Figure 7/5/1 . The
two- variable system is absolute, for it has the field of Figure 7/5/2.
The variable y is approximately a step-function. When it is at 0,
its critical state is x = X2, y = 0, for the occurrence of this state

7/6

TIME-*-

Figure 7/5/1 : 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.

'1 ~2

Figure 7/5/2 : Field of the changes shown in Figure 7/5/1,

determines a jump from 0 to F. When it is at Y, its critical
state is x = Xv y = Y, for the occurrence of this state determines
a jump from Y to 0.

7/6. A common, though despised, property of every machine is
that it may 4 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
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-

85 G

7/7 DESIGN FOR A BRAIN

spread tendency for systems to show changes of step-function
form if their variables are driven far from some usual value.
Later (S. 10/2) it will be suggested that the nervous system is not
exceptional in this respect.

Systems containing full- and null-functions
7/7. We shall now consider the properties shown by absolute
systems that contain step-functions. But the discussion will be
clearer and simpler if we first examine some simpler systems.

Suppose we have an absolute system composed wholly of full-
functions and we ignore one of the variables. Every experimenter
knows only too well what happens : the behaviour of the system
becomes unpredictable. Every experimenter has spent time
trying to make unpredictable experiments predictable ; he does
it by identifying the unknown variable. The unknown variable
may be scientifically trivial, like a loose screw, or important, like
a co-enzyme in a metabolic system ; but in either case, he cannot
establish a definite form of behaviour until he has identified and
either controlled or observed the unknown variable. To ignore a
/wZZ-function in an absolute system is to render the remainder non-
absolute, so that no characteristic form of behaviour can be
established.

On the other hand, an absolute system which includes null-
functions may have the null-functions removed from it, or other
null-functions added to it, and the new system will still be absolute.
(The alteration is done, of course, not by interfering physically
with the 4 machine ', but by changing the list of variables.) Thus, if
the two-variable system of the pendulum (S. 6/3) is absolute, and
if the length of the pendulum stays constant once it is adjusted,
then the system composed of the three variables :

(1) length of pendulum

(2) angular deviation

(3) angular velocity)

is also absolute. A formal proof is given in S. 21/4, but it follows
readily from the definitions. (The reader should first verify that
every null-function is itself an absolute system.) Conversely, if
three variables A, B, N, are found to form an absolute system,
and N is a null-function, then the system composed of A and B
is absolute.

Unlike the full-function, then, the null-function may be

STET-F UNCTIONS

7/8

omitted from a system, for its omission leaves the remainder still
producing predictable behaviour.

Systems containing step-functions

7/8. Suppose that we have a system with three variables,
A, B, S ; that it has been tested and found absolute ; that A
and B are full-functions ; and that S is a step-function. (Vari-
ables A and B, as in S. 21/3, will be referred to as main variables.)
The phase-space of this system will resemble that of Figure 7/8/1
(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
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

Figure 7/8/1 : Field of an absolute 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.

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 c 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 7/8/2,
and sometimes a field like II, the one or the other appearing ac-
cording to the value that S happens to have at the time.

7/9 DESIGN FOR A BRAIN

The behaviour of the system A B, 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 A B is caused by the inner mechanisms of the
4 machine ' itself.

The property may now be stated in general terms. Suppose,
in an absolute system, that some of the variables are step-functions,
and that these are ignored while the remainder (the main variables)
are observed on many occasions by having their field constructed.

Figure 7/8/2 : The two fields of the system composed of A and B.
P is in the same position in each field.

Then so long as no step-function changes value during the con-
struction, the main variables will be found to form an absolute
system, and to have a definite field. But on different occasions
different fields may be found. The number of different fields shown
by the main variables is equal to the number of combinations of
values provided by the step-functions.

1J9. 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 ?

STEP-FUNCTIONS 7/9

The presence of step-functions in an absolute 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 spontaneous
change are speaking of the main variables only. For the whole
system, which includes the step-functions, is absolute, has one field
only, and is completely state-determined (like Figure 7/8/1). But
the system of main variables may show as many different forms of
behaviour (like Figure 7/8/2, I and II) as the step-functions
possess combinations of values. And if the step-functions 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.

The argument may seem plausible, but it is stronger than that.
It may be proved (S. 22/5) that if a 4 machine ', known to be
completely isolated and therefore absolute, produces several
characteristic forms of behaviour, i.e. possesses several fields, then
there must be, interacting with the observed variables and included
within the c machine ', some step-functions.

CHAPTER 8

The Ultrastable System

8/1. Our problem, stated briefly at the end of Chapter 5, can
now be stated finally. The type-problem was the kitten whose
behaviour towards a fire was at first chaotic and unadapted,
but whose behaviour later became effective and adapted. We
have recognised (S. 5/8) that the property of being ' adapted '
is equivalent to that of having the variables, both of the animal
and of the environment, so co-ordinated in their actions on one
another that the whole system is stable. We now know, from
S. 6/3 and 7/8, that an observed system can change from one
form of behaviour to another only if parameters have changed
value. Since we assumed originally that no deus ex machina
may act on it, the changes in the system must be due to step-
functions acting within the whole absolute system. Our problem
therefore takes the final form : Step-functions by their changes in
value are to change the behaviour of the system ; what can ensure
that the step functions shall change appropriately ? The answer is
provided by a principle, relating step-functions and fields, which
will now be described.

8/2. In S. 7/8 it was shown that when a step-function changes
value, the field of the main variables is changed. The process
was illustrated in Figures 7/8/1 and 7/8/2. This is the action
of step-function on field.

8/3. There is also a reciprocal action. Fields differ in the rela-
tion of their lines of behaviour to the critical states. Thus, if
a representative point is started at random in the region to the
left of the critical states in Figure 8/3/1, the proportion which
will encounter critical states is, in I — 1, in II — 0, and in III —
about a half. So, given a distribution of critical states and a
distribution of initial states, a change of field will, in general,

THE ULTRASTABLE SYSTEM

8/5

[II

Figure 8/3/1 : Three fields. The critical states are dotted.

change the proportion of representative points encountering
critical states.

The ultrastable system

8/4. The two factors of the two preceding sections will now be
found to generate a process, for each in turn evokes the other's
action. The process is most clearly shown in what I shall call
an ultrastable system : one that is absolute and contains step-
functions in a sufficiently large number for us to be able to ignore
the finiteness of the number. Consider the field of its main
variables after the representative point has been released from
some state. If the field leads the point to a critical state, a
step-function will change value and the field will be changed.
If the new field again leads the point to a critical state, again
a step-function will change and again the field will be changed ;
and so on. The two factors, then, generate a process.

8/5. Clearly, for the process to come to an end it is necessary
and sufficient that the new field should be of a form that does
not lead the representative point to a critical state. (Such a
field will be called terminal.) But the process may also be de-
scribed in rather different words : if we watch the main variables
only, we shall see field after field being rejected until one is
retained : the process is selective towards fields.

As this selectivity is of the highest importance for the solution
of our problem, the principle of ultrastability will be stated
formally : an ultrastable system acts selectively towards the fields
of the main variables, rejecting those that lead the representative
point to a critical state but retaining those that do not.

This principle is the tool we have been seeking ; the previous

8/6 DESIGN FOR A BRAIN

chapters have been working towards it : the later chapters will
develop it.

8/6. In the previous sections, the critical states of the step-
functions were unrestricted in position ; but such freedom does
not correspond with what is found in biological systems (S. 9/8),
so we will examine the behaviour of an ultrastable system whose
critical states are so sited that they surround a definite region
in the main-variables' phase-space. (At first we shall assume
that the main variables are all full-functions, though the defini-
tion makes no such restriction. Later (S. 11/8) we shall examine
other possibilities.)

8/7. The simplest way to demonstrate the properties of this
system is by an example. Suppose there are only two main
variables, A and B, and the critical states of all the step-functions

Figure 8/7/1

Changes of field in an ultrastable system. The critical
states are dotted.

are distributed as the dots in Figure 8/7/1. Suppose the first
field is that of Figure 8/7/1 (I), and that the system is started
with the representative point at X. The line of behaviour from

THE ULTRASTABLE SYSTEM 8/8

X is not stable in the region, and the representative point follows
the line to the boundary. Here (F) it meets a critical state
and a step-function changes value ; a new field, perhaps like II,
arises. The representative 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 stable resting state,
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, the line of behaviour from
Z is stable with regard to the region. So the representative
point moves to the resting state 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 critical states surround a region, the ultra-
stable system is selective for fields that are stable within the region.
(This statement is not rigorously true, for a little ingenuity
can devise fields of bizarre type which are not stable but which
are, under the present conditions, terminal. A fully rigorous
statement would be too clumsy for use in the next few chapters ;
but the difficulty is only temporary, for S. 13/4 introduces some
practical factors which will make the statement practically true.)

The Homeostat

8/8. So far the discussion of step-functions and of ultrastability
has been purely logical. In order to provide an objective and
independent test of the reasoning, a machine has been built
according to the definition of the ultrastable system. This
section will describe the machine and will show how its behaviour
compares with the prediction of the previous section.

The homeostat (Figure 8/8/1) consists of four units, each of
which carries on top a pivoted magnet (Figure 8/8/2, M in
Figure 8/8/3). 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

8/8

DESIGN FOR A BRAIN

•:*iia

JbSfe*

Figure 8/8/1 : The homeostat. Each unit carries on top a magnet and
coil such as that shown in Figure 8/8/2. 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 8/8/3.

Figure 8/8/2 : 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 8/8/3.

THE ULTRASTABLE SYSTEM

8/8

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 // is at
180 V. ; so £ 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

s©tS

K<*&

u _^

&^s,

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

in the text.)

in one direction or the other. So after E is adjusted, the output
is approximately proportional to il/'s deviation from its central
position.

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

8/8 DESIGN FOR A BRAIN

(J£), 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 absolute. To this system the commutators 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
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 c runaway ' occurs and the magnets diverge from
the central positions with increasing velocity.

So far, the system of four variables has been shown to be
dynamic, to have Figure 4/12/1 (A) as its diagram of immediate
effects, and to be absolute. 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 com-
ponents 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. In addition, the coil G of each uniselector is
energised when, and only when, the magnet M diverges far from
the central position ; for only at extreme divergence does the
output-current reach a value sufficient to energise the relay F
which closes the coil-circuit. A separate device, not shown,
interrupts the coil-circuit regularly, making the uniselector move
from position to position as long as F is energised.

The system is now ultrastable ; its correspondence with the

THE ULTRASTABLE SYSTEM 8/8

definition will be shown in each of the three requirements.
Firstly, the whole system, now of eight variables (four of the
magnet- deviations and four of the uniselector-positions), is abso-
lute, because the values of the eight variables are sufficient to
determine its behaviour. Secondly, the variables may be divided
into main variables (the four magnet-deviations), and step-func-
tions (the variables controlled by the uniselector-positions).
Thirdly, as the uniselectors provide an almost endless supply of
step-function values (though not all different) we do not have to
consider the possibility that the supply of step-function changes
will come to an end. In addition, the critical states (those
magnet-deviations at which the relay closes) are all sited at about
a 45° deviation ; so in the phase-space of the main variables they
form a 4 cube ' around the origin.

It should be noticed that if only one, two, or three of the
units are used, the resulting system is still ultrastable. It will
have one, two, or three main variables respectively, but the critical
states will be unaltered in position.

Time

Figure 8/8/4 : Behaviour of one unit fed back into itself through a uniselector.
The upper line records the position of the magnet, whose side-to-side
movements are recorded as up and down. The lower line (U) shows
a cross-stroke whenever the uniselector moves to a new position. The
first movement at each D was forced by the operator, who pushed the
magnet to one side to make it demonstrate the response.

Its ultrastability can now be demonstrated. First, for sim-
plicity, is shown a single unit arranged to feed back into itself
through a single uniselector coil such as A, D being shorted out.
In such a case the occurrence of the first negative setting on
the uniselector will give stability. Figure 8/8/4 shows a typical
tracing. At first the step-functions gave a stable field to the
single main variable, and the downward part of Dl9 caused by
the operator deflecting the magnet, is promptly corrected by the
system, the magnet returning to its central position. At Rv

8/8 DESIGN FOR A BRAIN

the operator reversed the polarity of the output-input junction,
making the system unstable (S. 20/7). As a result, a runaway
developed, and the magnet passed the critical state (shown by
the dotted line). As a result the uniselector changed value. As
it happened, the first new value provided a field which was
stable, so the magnet returned to its central position. At D2,
a displacement showed that the system was now stable (though
the return after Rx demonstrated it too).

At R2 the polarity of the join was reversed again. The value
on the uniselector was now no longer suitable, the field was
unstable, and a runaway occurred. This time three uniselector
positions provided three fields which were all unstable : all were

Time

Figure 8/8/5 : Two units (1 and 2) interacting. (Details as in Fig. 8/8/4.)

rejected. But the fourth was stable, the magnet returned to
the centre, no further uniselector changes occurred, and the
single main variable had a stable field. At D3 its stability was
again demonstrated.

Figure 8/8/5 shows another experiment, this time with two
units 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-function values combined to give
stability, shown by the responses to Dv 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 D1 in 2 caused an upstroke in 1, it

THE ULTRASTABLE SYSTEM 8/9

caused a down stroke in 1 after Rv showing that the action 2 — ► 1
had been reversed by Jhe 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
comparison of the effect of Z)3 on 1 with that of D2 shows that
compensation has occurred again.

The homeostat can thus demonstrate the elementary facts of
ultrastability.

8/9. In what way does an ultrastable system differ from an
ordinary stable system ?

In one sense the two systems are similar. Each is assumed
absolute, and if therefore we form the field of all its variables,
each will have one permanent field. Given a region, every line
of behaviour is permanently stable or unstable (see Figure 7/8/1).
Viewed in this way, the two systems show no essential difference.
But if we compare the variables of the stable system with only
the main variables of the ultrastable, then an obvious difference
appears : the field of the stable system is single and permanent,
but in the ultrastable system the phase-space of the main vari-
ables shows a succession of transient fields concluded by a terminal
field which is always stable, The distinction in actual behaviour
can best be shown by an example. The automatic pilot is a
device which, amongst other actions, keeps the aeroplane hori-
zontal. 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 devia-
tion made the step-functions start changing. On the occurrence

8/10 DESIGN FOR A BRAIN

of the first suitable new value, the homeostat would act to stabilise
instead of to overthrow ; it would return the plane to the hori-
zontal ; and it would then be ordinarily self-correcting for dis-
turbances.

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.

Another difference can be seen by considering the number of
factors which need adjustment or specification in order to achieve
stability. Less adjustment is needed if the system is ultrastable.
Thus an automatic pilot must be joined to the ailerons with care,
but an ultrastable pilot could safely be joined to the ailerons at
random. Again, a linear system of n variables, to be made stable,
needs the simultaneous adjustment of at least n parameters
(S. 20/11, Ex. 3). If n is, say, a thousand, then at least a thou-
sand parameters must be correctly adjusted if stability is to be
achieved. But an ultrastable system with a thousand main
variables needs, to achieve stability, the specification of about
six factors ; for this is approximately the number of independent
items in the specification of the system (S. 9/9). A large system,
then, can be made stable with much less detailed specification
if it is made ultrastable.

8/10. 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 4 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 an absolute 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.
Now the ultrastable system proceeds to a terminal field which

100

THE ULTRASTABLE SYSTEM

8/10

is stable in conjunction with all the system's parameter-values
(and it is clear by the principle of ultrastability 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-function values which is so related to the
particular set of parameter-values that, in conjunction with them,
the system is stable. If the parameters have unusual values,

Time

Figure 8/10/1 : Three units interacting. At J, units 1 and 2 were con-
strained to move together. New step-function 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.

the step-functions will also finish with values that are compen-
satingly unusual. To the casual observer this adjustment of the
step-function 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

101 H

8/10 DESIGN FOR A BRAIN

made by joining the front two magnets by a light glass fibre
so that they had to move together. Figure 8/10/1 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-function 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-function
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.

References

Ashby, W. Ross. Design for a brain. Electronic Engineering, 20, 379 ; 1948.
Idem. The cerebral mechanisms of intelligent behaviour, in Perspectives in

Neuropsychiatry, edited D. Richter. London, 1950.
Idem. Can a mechanical chess-player outplay its designer? British Journal

for the Philosophy of Science, 3, 44 ; 1952.
Fisher, R. A., and Yates, F. Statistical tables. Edinburgh, 1943.

102

CHAPTER 9

Ultrastability in the
Living Organism

9/1. The principle of ultrastability has so far been treated as
a principle in its own right, true or false without reference to
possible applications. This separation has prevented the possi-
bility of a circular argument ; but the time for its application
has now come. I propose, therefore, the thesis that the living
organism uses the principle of ultrastability as an automatic
means of ensuring the adaptiveness of its learned behaviour.
At first I shall cite only facts in its favour, leaving all major
criticisms to Chapter 11. We shall have, of course, to assume
that the animal, and particularly the nervous system, contains
the necessary variables behaving as step-functions : whether this
assumption is reasonable will be discussed in the next chapter.

Examples of adaptive, learned behaviour are so multitudinous
that it will be quite impossible for me to discuss, or even to
mention, the majority of them. I can only select a few as
typical and leave the reader to make the necessary modifications
in other cases.

The best introduction is not an example of learned behaviour,
but Jennings' classic description of the reactions of Stentor, a
single-celled pond animalcule. I shall quote him at length :

4 Let us now examine the behaviour [of Stentor] under
conditions which are harmless when acting for a short time,
but which, when continued, do^ interfere with the normal
functions. Such conditions may be produced by bringing a
large quantity of fine particles, such as India ink or carmine,
by means of a capillary pipette, into the water currents
which are carried to the disc of Stentor.

1 Under these conditions the normal movements are at
first not changed. The particles of carmine are taken into
the pouch and into the mouth, whence they pass into the
internal protoplasm. If the cloud of particles is very dense,

103

9/1 DESIGN FOR A BRAIN

or if it is accompanied by a slight chemical stimulus, as is
usually the case with carmine grains, this behaviour lasts
but a short time ; then a definite reaction supervenes. The
animal bends to one side ... It thus as a rule avoids the
cloud of particles, unless the latter is very large. This
simple method of reaction turns out to be more effective
in getting rid of stimuli of all sorts than might be expected.
If the first reaction is not successful, it is usually repeated
one or more times . . .

4 If the repeated turning toward one side does not relieve
the animal, so that the particles of carmine continue to come
in a dense cloud, another reaction is tried. The ciliary
movement is suddenly reversed in direction, so that the
particles against the disc and in the pouch are thrown off.
The water current is driven away from the disc instead of
toward it. This lasts but an instant, then the current is
continued in the usual way. If the particles continue to
come, the reversal is repeated two or three times in rapid
succession. If this fails to relieve the organism, the next
reaction — contraction — usually supervenes.

4 Sometimes the reversal of the current takes place before
the turning away described first ; but usually the two
reactions are tried in the order we have given.

4 If the Stentor does not get rid of the stimulation in either
of the ways just described, it contracts into its tube. In
this way it of course escapes the stimulation completely,
but at the expense of suspending its activity and losing all
opportunity to obtain food. The animal usually remains
in the tube about half a minute, then extends. When its
body has reached about two-thirds its original length, the
ciliary disc begins to unfold and the cilia to act, causing
currents of water to reach the disc, as before.

4 We have now reached a specially interesting point in
the experiment. Suppose that the water currents again
bring the carmine grains. The stimulus and all the external
conditions are the same as they were at the beginning.
Will the Stentor behave as it did at the beginning ? Will
it at first not react, then bend to one side, then reverse the
current, then contract, passing anew through the whole
series of reactions ? Or shall we find that it has become
changed by the experiences it has passed through, so that
it will now contract again into its tube as soon as stimulated ?

4 We find the latter to be the case. As soon as the car-
mine again reaches its disc, it at once contracts again. This
may be repeated many times, as often as the particles come
to the disc, for ten or fifteen minutes. Now the animal
after each contraction stays a little longer in the tube than
it did at first. Finally it ceases to extend, but contracts

104

ULTRASTABILITY IN THE LIVING ORGANISM 9/1

repeatedly and violently while still enclosed in its tube. In
this way the attachment of its foot to the object on which
it is situated is broken and the animal is free. Now it
leaves its tube and swims away. In leaving the tube it may
swim forward out of the anterior end of the tube ; but if
this brings it into the region of the cloud of carmine, it
often forces its way backwards through the substance of
the tube, and thus gains the outside. Here it swims away,
to form a new tube elsewhere.

1 . . . the changes in behaviour may be summed up as
follows :

(1) No reaction at first ; the organism continues its normal

activities for a time.

(2) Then a slight reaction by turning into a new position.

(3) ... a momentary reversal of the ciliary current . . .

(4) . . . the animal breaks off its normal activity com-

pletely by contracting strongly . . .

(5) ... it abandons its tube . . . '

The behaviour of Stentor bears a close resemblance to the
behaviour of an ultrastable system. The physical correspon-
dences necessary would be as follows : — Stentor and its environ-
ment constitute an absolute system by S. 3/9 ; for Jennings,
having set the carmine flowing, interferes no further. They
consequently correspond to the whole ultrastable system, which
is also absolute by the definition of S. 8/4. The observable
(here : visible) variables of Stentor and its environment corre-
spond to the main variables of the ultrastable system. In Stentor
are assumed to be variables which behave like, and correspond
to, the step -functions of the ultrastable system. The critical
states of the organism's step-functions surround the region of
the normal values of the organism's essential variables so that
its step-functions change value if the essential variables diverge
widely from their usual, normal values. These critical states
must be nearer to the normal value than the extreme limits of
the essential variables, for these critical states must be reached
before the essential variables reach the extreme limits compatible
with life.

Now compare the behaviour of the ultrastable system, de-
scribed in S. 8/7, with the behaviour of organisms like Stentor,
epitomised by Jennings in these words :

1 Anything injurious to the organism causes changes in
its behaviour. These changes subject the organism to new

105

9/2 DESIGN FOR A BRAIN

conditions. As long as the injurious condition continues,
the changes of behaviour continue. The first change of
behaviour may not be regulatory [what I call ' adaptive '],
nor the second, nor the third, nor the tenth. But if the
changes continue, subjecting the organism successively to
all possible different conditions, a condition will finally be
reached that relieves the organism from the injurious action,
provided such a condition exists. Thereupon the changes
in behaviour cease and the organism remains in the favourable
situation.'

The resemblance between my statement and his is obvious.
Jennings grasped the fundamental fact that aimless change can
lead to adaptation provided that some active process rejects the
bad and retains the good. He did not, however, give any physical
(i.e. non-vital) reason why this selection should occur. He records
only that it does occur, and that its occurrence is sufficient to
account for adaptation at the primitive level.

The first example therefore suggests that, provided we are
willing to assume that Stentor contains step-functions which (a)
affect Stentor* s behaviour, and (b) have critical states that are
encountered before the essential variables reach their extreme
limits, Stentor may well achieve its final adaptation by using
the automatic process of ultrastability.

9/2. The next example includes more complicating factors but
the main features are clear. 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 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-
sidering the known actions of part on part in the real system
we can construct the diagram of immediate effects. Thus, the

106

ULTRASTABILITY IN THE LIVING ORGANISM

9/2

excitations in the motor cortex certainly control the rat's bodily
movements, 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

Events in

sensory cortex

motor cortex

Excitation
in skin

Position of
limbs

>
4

Voltage

Position

on |

shrill

of pedal

a positive physical action all the time ; but here, in accordance
with S. 2/3, we regard them as permanently linked though some-
times 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 absolute. To apply our thesis, we
assume that the cerebral part, represented by the boxes around
arrow 6, contains step -functions whose critical states will be
transgressed if stimuli of more than physiological intensity are
sent to the brain.

We now regard the system as straightforwardly ultrastable,
and predict what its behaviour must be. It is started, by hypo-
thesis, 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 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-functions to change value, thus causing different patterns
of body movement to occur. The step-functions act directly
only at stage 6, but changes there will (S. 14/11) affect the field

107

9/3 DESIGN FOR A BRAIN

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 connec-
tion, 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. 5/15.

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

9/3. The two examples have taken a known fact of animal
behaviour and shown its resemblance to the behaviour of the
ultrastable system. Equally, the behaviour of the homeostat,
a system known to be ultrastable, shows some resemblance to
that of a rudimentary nervous system. The tracings of Figures
8/8/4 and 8/8/5 show its elementary power of adaptation. In
Figure 8/8/5 the reversal at Rx might be regarded as the action
of an experimenter who changed the conditions so that the i aim '
(stability and homeostasis) could be achieved only if the ' organ-
ism ' (Unit 1) reversed its action. Such a reversal might be
forced on a rat who, having learned a maze whose right fork
led to food, was transferred to a maze where food was to be
found only down the left fork. The homeostat, as Figure 8/8/5
shows, develops a reversed action in Unit 1, and this reversal
may be compared with the reversal which is usually found to
occur in the rat's behaviour.

108

ULTRASTABILITY IN THE LIVING ORGANISM 9/3

A more elaborate reaction by the homeostat is shown in
Figure 9/3/1.

Time

Figure 9/3/1 : Three units interacting. At R the effect
of 2 on 3 was reversed in polarity.

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 St : 2 followed 1 downward (similar movement), and 3
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
downward movement of 1, at S2, demonstrated the regained
stability.

109

9/4 DESIGN FOR A BRAIN

The tracing, however, deserves closer study. The action 2 — ► 3
was reversed at R, and the responses of 2 and 3 at S2 demon-
strate 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 altera-
tion made in the environment by the experimenter, and finally
a reorganisation within the nervous system, compensating for
the experimental alteration. The homeostat can thus show, in
elementary form, this power of self-reorganisation.

The necessity of ultrastability

9/4. In the previous sections a few simple examples have sug-
gested that the adaptation of the living organism may be due
to ultrastability. But the argument has not excluded the possi-
bility that other theories might fit the facts equally well. I shall
now give, therefore, evidence to show that ultrastability is not
merely plausible but necessary : the organism must be ultra-
stable.

First the primary assumptions : they are such as few scientists
would doubt. It is assumed that the organism and its environ-
ment form an absolute system, and that the organism sometimes
changes from one regular way of behaving to another. The
crucial question is whether we can prove that the organism's
mechanism must contain step-functions. In S. 22/5 is given
such a proof, stated in mathematical form ; but its theme is
simple and can be stated in plain words.

Suppose a ' machine ' or experiment behaves regularly in one
way, and then suddenly changes to behaving in another way,
again regularly. Suppose, for instance, a pharmacologist, test-
ing the effect of a new drug on the frog's heart, finds at every
test all through one day that it causes the pulse-rate to lessen.
Next morning, taking records of the effect, he finds at every

110

ULTRASTABILITY IN THE LIVING ORGANISM 9/4

attempt that it causes the pulse-rate to increase. He will almost
certainly ask himself 4 What has changed ? '

Such facts provide valid evidence that some variable has
changed value. I need not elaborate the logic for no experi-
menter would question it. What has been sometimes overlooked
though, is that we are also entitled to draw the deduction that
the variable, being as it is an effective factor towards the system,
must, throughout the previous day, have remained constant ;
for otherwise the reactions observed during the day could not
have been regular. For the same reason, it must also have been
constant throughout the next morning. And further, the two
constant values cannot have been equal, for then the hearts'
behaviours would not have been changed. Assembling these
inferences, we deduce that the variable must have behaved as
a step-function. Exactly the same argument, applied to the
changes of behaviour shown by Jennings' Stentor, leads to the
deduction that within the organism there must have been vari-
ables behaving as step-functions.

Is there any escape from this conclusion ? It rests primarily
on the simple thesis that a determinate system does not, if started
from identical states, do one thing on one day and something
else on another day. There seems to be no escape if we assume
that the systems we are discussing are determinate. Suppose,
then, that we abandon the assumption of determinism and allow
indeterminism of atomic type to affect heart, Stentor, or brain
to an observable extent. This would allow us to explain the
' causeless ' overnight change ; but then we would be unable to
explain the regularity throughout the previous day and the next
morning. It seems there is no escape that way. Again, we
could, with a little ingenuity, construct a hypothesis that the
pharmacologist's experiment was affected by a small group of
variables, whose joint action produced the observed result but
not one of which was a step-function ; and it might be claimed
that the theorem had been shown -false. But this is really no
exception, for we are not concerned with what variables ' are '
but with how they behave, and in particular with how they
behave towards the system in question. If a group of variables
behaves towards the system as a step-function, then it is a step-
function ; for the ' step-function ' is defined primarily as a form
of behaviour, not as a thing.

Ill

9/5 DESIGN FOR A BRAIN

Once it is agreed that a system, such as that of Mowrer's rat,
contains step-functions, then all it needs is that they should
not be few for the system to be admitted as ultrastable.

After this, we can examine the qualifications that were added
when considering Stentor as an ultrastable system. Are they,
too, necessary ? Not with the assumptions made so far in this
section, but they become so if we add the postulates that the
system ' adapts ' in the sense of S. 5/8, and that it does so by
' trial and error'. In order to be definite about what 'trial and
error ' implies, here is the concept defined explicitly :

(1) The organism makes trials only when ' dissatisfied ' or

4 irritated ' in some way.

(2) Each trial persists for a finite time.

(3) While the irritation continues, the succession of trials

continues.

(4) The succeeding trial is not specially related to the preced-

ing, nor better than it, but only different.

(5) The process stops at the first trial that relieves the irritation.
The argument goes thus. As each step-function forms part

of an absolute system, its change must depend on its own and
on the other variables' values ; there must, therefore, be certain
states — the critical — at which it changes value. When, in the
process of adaptation by trial and error, the step-function changes
value, its critical states must have been encountered ; and since,
by (1) above, the step-functions change value only when the
organism is ' dissatisfied ' or ' irritated ', the critical states must
be so related to the essential variables that only when the organism
is driven from its normal physiological state does its representa-
tive point encounter the critical states. This knowledge is suffi-
cient to place the critical states in the functional sense : they
must have values intermediate between those of the normal state
and those of the essential variables' limits. The qualifications
introduced in S. 9/1 are thus necessary.

Training

9/5. 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. ' Punish-

112

ULTRASTABILITY IN THE LIVING ORGANISM 9/5

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. 7/6 and
10/2). The concept of c reward ' is more complex. It usually
involves the supplying of some substance (e.g. food) or condition
(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
complications. We shall assume that the training is by pain,
i.e. by some change which threatens to drive the essential vari-
ables outside their normal limits ; and we shall assume that
training by reward is not essentially dissimilar.

It will now be shown that the process of ' training ' necessarily
implies the existence of feedback. But first the functional rela-
tionship of the experimenter to the experiment must be made
clear.

The experimenter 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 experi-
ment 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 pre-
scribed 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

113

9/5 DESIGN FOR A BRAIN

at the minnows he struck the glass. The following immediate
effects can be clearly distinguished :

Activities in

motor cortex

muscles

Activities in

Pressure on

sensory

cortex

nose

The arrow 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 led 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

Activities in

Amount of

sensory

cortex

carrot p

resented

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

114

ULTRASTABILITY IN THE LIVING ORGANISM 9/5

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 Grindley's
experiment, the value of the variable 4 food ' depended on the
animal's response : if the head turned to the left, ' food ' was
' no carrot ', while if the head turned to the right, ' food ' was
4 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, there-
fore, allows no effect from the variable 4 animal's behaviour ' to
4 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 4 conditioning ' and
4 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.
Culler et al. performed an experiment in which feedback, at first
absent, was added at an intermediate stage : as a result, the
dog's behaviour changed. They gave the dog a shock to the
leg and sounded a tone. The reaction to the shock was one of
generalised struggling movements of the body and retraction of
the leg. After a few sessions the tone produced generalised
struggling and retraction of the leg. So far there had been no
feedback ; but now the conditions were changed : the shock was
given at the tone only if the foot was not raised. As a result
the dog's behaviour changed : the response rapidly narrowed to
a simple and precise flexion of the leg.

It will be seen, therefore, that the 4 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-functions
are, of course, assumed to be confined to the animal ; both
because the human trainer may be replaced in some experiments

115

9/5 DESIGN FOR A BRAIN

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
stimulation 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.)

The process can be shown on the homeostat. Figure 9/5/1
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
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.
At Sv 1 was moved and 2 moved similarly. This is the * for-
bidden ' response ; so at D1} 3 was forced by the trainer to an
extreme position. Step-functions changed value. At S2, the
homeostat was tested again : again it produced the forbidden
response ; so at Z)2, 3 was again forced to an extreme position.
At £3, the homeostat was tested again : it moved in the desired
way, so no further deviation was forced on 3. And at *S4 and
$5 the homeostat continued to show the desired reaction.

116

ULTRASTABILITY IN THE LIVING ORGANISM 9/6

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

Another property of the whole system should be noticed.
When the movement-combination 4 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
any inanimate system which behaved in this way we would

Figure 9/5/1 : 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.

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 i punished ' the 4 animal ' is equiva-
lent to saying in our terms that the system has a set of step-
function 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.

9/6. 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 thirty 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

117 I

9/6 DESIGN FOR A BRAIN

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,
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.

In S. 3/12 it was decided that the anatomical criterion for
dividing the system into ' animal ' and ' 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.

I have already shown, in S. 8/10 and in Figure 8/10/1, that
the ultrastable system arrives at a stability in which the values
of the step-functions are related to those of the parameters of
the system, i.e. to the surrounding fixed conditions, and that
the relation will be achieved whether the parameters have values
which are * normal ' or are experimentally altered from those
values. If these conclusions are applied to the experiments of
Marina and Sperry, the facts receive an explanation, at least in
outline. To apply the principle of ultrastability we must add
an assumption that ' 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 in those cerebral mechanisms that
determine the actions. (The plausibility of this assumption will

118

ULTRASTABILITY IN THE LIVING ORGANISM 9/7

be discussed in S. 9/8.) Ultrastability will then automatically
lead to the emergence of behaviour which produces binocular
vision or normal progression. For this to be produced, the step-
function values must make appropriate allowance for the par-
ticular characteristics of the environment, whether ' crossed ' or
4 uncrossed '. S. 8/10 and Figure 9/3/1 showed that an ultra-
stable system will make such allowance. The adaptation shown
by Marina's monkey is therefore homologous with that shown
by Mowrer's rat, for the same principle is responsible for both.

9/7. ' Learning ' and ' memory ' are vast subjects, and any
theory of their mechanisms cannot be accepted until it has been
tested against all the facts. It is not my intention to propose
any such theory, since this work confines itself to the problem
of adaptation. Nevertheless I must indicate briefly the relation
of this work to the two concepts.

4 Learning ' and ' memory ' have been given almost as many
definitions as there are authors to write of them. The concepts
involve a number of aspects whose interrelations are by no means
clear ; but the theme is that a past experience has caused some
change in the organism's behaviour, so that this behaviour is
different from what it would have been if the experience had not
occurred. But such a change of behaviour is also shown by a
motor-car after an accident ; so most psychologists have insisted
that the two concepts should be restricted to those cases in
which the later behaviour is better adapted than the earlier.

The ultrastable system shows in its behaviour something of
these elementary features of ' learning '. In Figure 9/3/1, for
instance, the pattern of behaviour produced at S2 is different
from the pattern at Sv The change has occurred after the
1 experience ' of the instability at R. And the new field pro-
duced by the step-function change is better adapted than the
previous field, for an unstable field has been replaced by a stable.

An elementary feature of ' memory ' is also shown ; for further
responses, S3, S^ etc. would repeat S%s pattern of behaviour,
and thereby might be said to show a ' memory ' of the reversal
at R ; for the later pattern is adapted to the reversal at R, and
not adapted to the original setting.

The ultrastable system, then, shows rudimentary ' learning '
and ' memory '. The subject is resumed in S. 11/3.

119

9/8

DESIGN FOR A BRAIN

The control of aim

9/8. 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.
In this section will be described some methods by which the
goal may be varied.

If the critical states' distribution in the main-variables' phase-
space is altered by any means whatever, the ultrastable system

Figure 9/8/1.

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. 8/7). Thus if (Figure
9/8/1) for some reason the critical states moved to surround B
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 9/8/2. 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 A's effectors and receptors. If
A should have a marked effect on the ultra-
stable 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 a, 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

120

Figure 9/8/2.

ULTRASTABILITY IN THE LIVING ORGANISM 9/8

A go outside the range 4 to 6, it will make U go outside the
range 0 to 10, and this will destroy the field. So U becomes
w 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 9/8/1, 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.

121

9/9 DESIGN FOR A BRAIN

These considerations may clarify the relations between the
change of concentration of a sex-hormone in the blood of a
mammal and its consequent sexual goal-seeking behaviour. A
simple alternation between ' present ' and 4 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.)

Ultrastability and the gene-pattern

9/9. In S. 1/9 it was pointed out that although the power of
adaptation shown by a species ultimately depends on its genetic
endowment, yet the number of genes is, in the higher animals,
quite insufficient to specify every detail of the final neuronic
organisation. It was suggested that in the higher animals, the
genes must establish function-rules which will look after the
details automatically.

As the minimal function-rules have now been provided (S. 8/7)
it is of interest to examine the specification of the ultrastable
system to see how many items will have to be specified geneti-
cally if the ovum is to grow into an ultrastable organism. The
items are as follows :

(1) The animal and its environment must form an absolute

system (S. 3/9) ;

(2) The system must be actively dynamic ;

(3) Essential variables must be defined for the species (S. 3/14) ;

(4) Step-functions are to be provided (S. 8/4) ;

(5) Their critical states are all to be similar (S. 8/6) ;

(6) The critical states are to be related in value to the limiting

values of the essential variables (S. 9/1).
122

ULTRA STABILITY IN THE LIVING ORGANISM 9/10

From these basic rules, an ultrastable system of any size can
be generated by mere repetition of parts. Thus each critical
state is to have a value related to the limits of the essential vari-
ables ; but this requirement applies to all other critical states
by mere repetition. The repetition needs fewer genes than would
be necessary for independent specification.

It is not possible to give an exact estimate of the number of
genes necessary to determine the development of an ultrastable
system. But the number of items listed above is only six ; and
though the number of genes required is probably a larger number,
it may well be less than the number known to be available. It
seems, therefore, that the requirement of S. 1/9 has been met
satisfactorily.

9/10. If the higher animals are made ultrastable by their genetic
inheritance, the gene-pattern must have been shaped by natural
selection. Could an ultrastable system be developed by natural
selection ?

Suppose the original organism had no step-functions ; such an
organism would have a permanent, invariable set of reactions.
If a mutation should lead to the formation of a single step-func-
tion 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-function affected in any
way the reaction between the organism and the environment,
then such a step-function might increase the organism's chance
of survival. A single mutation causing a single step-function
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 ultrastable
can therefore be made by a long series of small changes, each
of Avhich improves the chance of survival. The change is thus
possible under the action of natural selection.

References

Culler, E., Finch, G., Girden, E., and Brogden, W. Measurements of
acuity by the conditioned-response technique. Journal of General
Psychology, 12, 223 ; 1935.

123

9/10 DESIGN FOR A BRAIN

Grindley, G. C. The formation of a simple habit in guinea-pigs. British

Journal of Psychology, 23, 127 ; 1932-3.
Hilgard, E. R., and Marquis, D. G. Conditioning and learning. New

York, 1940.
Marina, A. Die Relationen des Palaeencephalons (Edinger) sind nicht fix.

Neurologisches Centralblatt, 34, 338 ; 1915.
Mowrer, O. H. An experimental analogue of ' regression ' with incidental

observations on ' reaction-formation '. Journal of Abnormal and Social

Psychology, 35, 56 ; 1940.
Sperry, R. W. Effect of crossing nerves to antagonistic limb muscles in

the monkey. Archives of Neurology and Psychiatry, 58, 452 ; 1947.

124

CHAPTER 10

Step-Functions in the
Living Organism

10/1. In S. 9/4 the existence of step-functions in the living
organism was deduced from the observed facts. But so far
nothing has been said, other than S. 7/6, about their physio-
logical nature. What evidence is there of a more practical nature
to support this deduction and to provide further details ?

Direct evidence of the existence of step-functions in the living
organism is almost entirely lacking. What evidence exists will
be reviewed in this chapter. But the lack of evidence does not,
of course, prove that such variables do not occur, for no one, so
far as I am aware, has made a systematic search for them.
Several reasons have contributed to this neglect. Their signi-
ficance has not been appreciated, so if they have been mentioned
in the literature they were probably mentioned only casually;
and since they show a behaviour bordering on total immobility,
they would usually have been regarded as uninteresting, and
may not have been recorded even when observed. It is to be
hoped that the recognition of the fundamental part which they
play in the processes of adaptation, of integration, and of co-
ordination, may lead to a fuller knowledge of their actual nature.
' The anatomical localisation ', said Claude Bernard, c is often
revealed first through the analysis of the physiological process.'
Here I can do no more than to indicate some possibilities.

10/2. 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 behave as step-functions.

125

10/3 DESIGN FOR A BRAIN

If the cell is sufficiently sensitive to be affected by changes of
atomic size, then such changes would usually 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 here (S. 1/10).

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 w delicate '
can mean little other than 4 easily broken ' ; and it was observed
in S. 7/6 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-functions 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.'

10/3. 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 fx in size, which is actively
amoeboid, continually throwing out processes as though explor-
ing the medium around. Levi studied tissue-cultures by micro-
dissection, so that individual cells could be stimulated. He found

126

STEP-FUNCTIONS IN THE LIVING ORGANISM 10/5

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. Reasons have been given in S. 9/1
and 9/4 suggesting that adaptation by step-functions is as old as
protoplasm itself. So the hypothesis that neurons are amoeboid
assumes only that they have never lost their original property.
It seems possible, therefore, that step-functions might be provided
in this way.

10/4. A variable which can take only the two values ' all ' or
4 nothing ' obviously provides a step-function. It may not always
conform to the definition of a step-function, for its change is not
always sustained ; but such variables may well provide changes

w -Time-*- - Time —

Figure 10/4/1.

which appear elsewhere in step-function form. Such would hap-
pen, for instance, if the change of the variable X (Figure 10/4/1 A)
resulted in some accumulative change Y, which would vary as in
B. Variables like X could therefore readily yield step-functions.

10/5. Step-functions could also be provided by groups of neurons
acting as a whole.

Lorente de No has provided abundant histological evidence that

127

10/6

DESIGN FOR A BRAIN

neurons form not only chains but circuits. Figure 10/5/1 is taken
from one of his papers. Such circuits are so common that he has
enunciated a c 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 '.
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 :

Figure 10/5/1 : Neurons and their connections in the trigeminal
reflex arc. (Semi-diagrammatic ; from Lorente de N6.)

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 starting 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. The reader will notice that the ' endrome '
exemplifies the principle of S. 7/4.

10/6. The definition of the ultrastable system might suggest
that an almost infinite number of step-functions is necessary if
the system is not to keep repeating itself ; and the reader may
wonder whether the nervous system can supply so large a number.
In fact the number required is not large. The reason can be
shown most simply by a numerical illustration.

If a step-function can take two values it can provide two fields
for the main variables (Figure 7/8/1). If another step-function
with two values is added, the total combinations of value are four,
and each combination will, in general, produce its own field
(S. 21/1). So if there are n step-functions, each capable of taking

128

STEP-FUNCTIONS IN THE LIVING ORGANISM 10/6

two values, the total number of fields available will be 2n. This
number would have to be lessened in practical cases for practical
reasons, but even if it is only approximate, it still illustrates the
main fact : the number of fields is moderate when n is moderate,
but rapidly becomes exceedingly large when n increases. Ten
step-functions, for instance, will provide over a thousand fields,
while twenty step-functions will provide over a million. The
number of fields soon becomes astronomic.

The following imaginary example emphasizes the relation be-
tween the number of fields and the number of step-functions
necessary to provide them. If a man used fields at the rate of
ten a second day and night during his whole life of seventy years,
and if no field was ever repeated, how many two-valued step-
functions would be necessary to provide them ? Would the
reader like to guess ? The answer is that thirty-five would be
ample ! Quantitatively, of course, the calculation is useless ; but
it shows clearly that the number of step-functions can be far less
than the number of fields provided. So if the human nervous
system produces a very large number of fields, we need not deduce
that it must have a very large number of step-functions.

References

Carey, E. J., Massopust, L. C, Zeit, W., Haushalter, E., and Schmitz, J.
Studies of ameboid motion and secretion of motor end plates : V, Experi-
mental pathologic effects of traumatic shock on motor end plates in
skeletal muscle. Journal of Neuropathology and experimental Neurology,
4, 134 ; 1945.

Harrison, R. G. Observations on the living developing nerve fiber. Pro-
ceedings of the Society for Experimental Biology and Medicine, 4, 140 ;
1906-7.

Levi, G. Ricerche sperimentali sovra elementi nervosi sviluppati ' in vitro \
Archiv fiir experimented Zellforschung, 2, 244 ; 1925-6.

Lorente de No, R. Vestibulo-ocular reflex arc. Archives of Neurology and
Psychiatry, 30, 245 ; 1933.

McCulloch, W. S. A heterarchy of values determined by the topology of
nervous nets. Bulletin of mathematical Biophysics, 7, 89 ; 1945.

Speidel, C. C. Studies of living nerves ; activities of ameboid growth
cones, sheath cells, and myelin segments as revealed by prolonged observa-
tion of individual nerve fibers in frog tadpoles. American Journal of
Anatomy, 52, 1 ; 1933.

Idem. Studies of living nerves ; VI, Effects of metrazol on tissues of frog
tadpoles with special reference to the injury and recovery of individual
nerve fibers. Proceedings of the American Philosophical Society, 83,
349 : 1940.

129

CHAPTER 11

Fully Connected Systems

11/1. In the preceding chapters all major criticisms were post-
poned : the time has now come to admit that the simple ultra-
stable system, as represented by, say, the homeostat, is by no
means infallible in its attempts at adaptation.

But before we conclude that its failures condemn it, we must
be clear about our aim. The designer of a new giant calculating
machine and we in this book might both be described as trying
to design a ' mechanical brain '. But the aims of the two
designers are very different. The designer of the calculator wants
something that will carry out a task of specified type, and he
usually wants it to do the work better than the living brain can
do it. Whether the machine uses methods anything like those
used by the living brain is to him a side-issue. My aim, on the
other hand, is simply to copy the living brain. In particular,
if the living brain fails in certain characteristic ways, then I want
my artificial brain to fail too ; for such failure would be valid
evidence that the model was a true copy. With this in mind,
it will be found that some of the ultras table system's failures
in adaptation occur in situations that are well known to be just
those in which living organisms also are apt to fail.

(1) If an ultrastable system's critical surfaces are not disposed
in proper relation to the limits of the essential variables (S. 9/1),
the system may seek an inappropriate goal or may fail to take
corrective action when the essential variables are dangerously
near their limits.

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

130

FULLY CONNECTED SYSTEMS 11/1

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
recently 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.

(2) Even if the ultrastable system is suitably arranged — if the
critical states are encountered before the essential variables reach
their extreme 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
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 ' 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

131

11/1 DESIGN FOR A BRAIN

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.

(3) 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
Fig. 8/7/1 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 i success ' to declare itself ;
and too prolonged a testing of a wrong trial may allow serious
damage to occur. Up to now I have said nothing of this necessity
for delay between one trial and the next, but there is no doubt
that it is an essential part of the ultrastable system's method
of adaptation. Thus the homeostat needed a device, not shown
in Fig. 8/8/3, for allowing the uniselectors to move only at about
every 2-3 seconds.

In animals, little is known scientifically about the optima for
such delays. But there can be little doubt that on many occa-
sions 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 same difficulty, then, seems to confront both ultrastable
system and living organism.

(4) If we grade an ultrastable system's environments according
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 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 '

132

FULLY CONNECTED SYSTEMS 11/4

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 ultras table system and the living organism are likely
to fail together.

It seems, then, that the ultrastable system's modes of failure
support, rather than discredit, its claim to resemble the living
brain.

11/2. Now we can turn to those features in which the simple ultra -
stable system, as represented by the homeostat, differs markedly
from the brain of the living organism. One obvious difference
is shown by the record of Figure 8/8/4, in which the homeostat
made four attempts at finding a terminal field. After its first
three trials its success was zero ; then, after its next trial, its
success was complete. The homeostat can show no gradation in
success, though this is almost universally observable in the living
organism : day by day a puppy becomes steadier on its legs ;
year by year a child improves its education.

11/3. A second difference is seen in their powers of conservation.
If the homeostat adapts to an environment A and then to an
environment B, and is then returned to A again, it has no adapta-
tion immediately ready, for its old adaptation was destroyed in
the readjustments to B ; it does not even start with a tendency
to adapt more quickly than before : its second adaptation to
A takes place as though its first adaptation had never occurred.
This, of course, is not the case in living organisms, except perhaps
in the extremely primitive : a child, by learning what two times
three is, does not thereby destroy its acquired knowledge of what
is two times two.

11/4. Although the homeostat, in adapting to B, usually de-
stroys its adaptation to A, this is not the case necessarily, and
we should notice a property, inherent in the ultrastable system,
that might enable it to adapt to more than one environment.
It will be described partly for its intrinsic interest, as it will be
referred to later, and partly to show that it is insufficient to
remove the main difficulty.

133 K

11/4 DESIGN FOR A BRAIN

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
11/4/1 illustrates the process. At R19 R2, jR3, and R± the hand-

Jj J, Rj *3 R4

lA A.

* H-

X h 3l-

Time

Figure 11/4/1 : 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.

controlled commutator H was reversed. At first the change of
value caused a change of field, shown at A. But the second
uniselector position happened to provide a field which gave
stability with both values of H. So afterwards, the changes of
H no longer caused step-function changes. The responses to the
displacements Z), forced by the operator, show that the system
is stable for both values of //. 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-function
values which give stability for both values of an alternating
parameter.

11/5. Such a process would occur in a biological system if an
animal had to adapt by one internal arrangement to two environ-

1.34

FULLY CONNECTED SYSTEMS 11/6

ments which affected the animal alternately. Such alternations
do occur. A cat, for instance, must learn to catch mice, which
tend to run towards corners, and birds, which tend to fly upwards ;
and the diving birds alternate between aerial and submarine
environments. Were the bird's nervous system like the homeo-
stat, step-function changes would occur until there arose a set of
values giving behaviour suitable to both environments, and this
set would then be terminal. That such a set is not impossible is
shown by the snake's mode of progression, which is suitable in
both undergrowth and water.

11/6. But it is easily seen that the process cannot answer the
problems of this chapter. First, the process shows, contrary
to requirement, no gradation : when there occurs a set of step-
function values terminal for both environments, the animal be-
comes adapted ; prior to that it was unadapted. The second
reason is that any extensive adaptation in this way is very
improbable.

This brings us to the most serious of the difficulties. A suc-
cessful trial, or a terminal field, is useful for adaptation only if it
occurs within some reasonable time : success at the millionth trial
is equivalent to failure. Consequently, the principle of ultra -
stability, while it guarantees that a field of a certain type will
be retained, guarantees much less than it seems to. If the delay
in reaching success were slight, a general increase in the system's
velocity of action might give sufficient compensation ; but in
fact the delay is likely to exceed the utmost possible compensa-
tion. For definiteness, take a numerical example. Suppose that
in some ultrastable system each field has a one in ten chance
of being stable with any given environment, and that the chances
are independent. Then the chance of a field being stable to two
environments will be one in a hundred, and to N environments
will be one in 10 N. The time that a system takes on the average
to find a stable field is proportional to the reciprocal of the prob-
ability (S. 23/2). Suppose that when N = 1 the average time
t taken to find a terminal field is 1 second, then

t = ^.10Y seconds.

Try the effect of different values of N. Three environments will
require about a minute and a half. This might be tolerable.

135

11/7 DESIGN FOR A BRAIN

But if N is twenty the time becomes 3,200,000,000 centuries, which
for our purpose, is equivalent to ' never '. Other examples,
though quantitatively different, would lead to the same general
conclusion : when the number of environments is more than a few,
the time taken by this method to find a field stable to all exceeds
the allowable. Evidently our brains do not use this method :
success by it is too improbable.

11/7. In the previous section we regarded the animal as having
to adapt to a variety of environments, but we can also regard
them as constituting a single 4 total ' environment. This makes
the number of variables in the system increase. What will be
the effect of this increase on the time taken to find a terminal
field ? For instance, could the homeostat adapt if it consisted
of a hundred units instead of four ? The question cannot be
ignored, for the human brain contains about 10,000,000,000
nerve-cells, and to this we must add the number of variables in its
environment. What is the chance that a field should be terminal
when it occurs in a system with this number of variables ?

If the system worked as a magnified homeostat then, although
exact calculation is impossible, the evidence, reviewed in S. 20/12,
is sufficient to show that, for practical purposes, there is no chance
at all. If we were like homeostats, waiting till one field gave us,
at a stroke, all our adult adaptation, we would wait for ever.
But the infant does not wait for ever ; on the contrary, the
probability that he will develop a full adult adaptation within
twenty years is near to unity. Some extra factor must therefore
be added if the large ultrastable system is to get adapted within
a reasonable time.

11/8. It may seem that we have now proved that the whole
solution must be wrong. But if we re-trace the argument, we find
that to some extent the difficulty has been unnecessarily magnified.
From S. 8/6 onwards we assumed for convenience of discussion
that every main variable was in full dynamic interaction with
every other main variable, so that every change in every variable
at once affected every other variable. This gives a system that is
extremely active and that unquestionably acts as a whole, not as
a collection of small parts acting independently. As an intro-
duction it has distinct advantages, but it raises its own difficulties.

136

FULLY CONNECTED SYSTEMS 11/8

Our present difficulties are, in fact, largely due to this assumption.
By modifying it we shall not only lessen the difficulties but we
shall obtain a model more like the real brain.

The views held about the amount of internal connection 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 reflex might leave a pre-
viously established reflex 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 related to what was happening at 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 '. . . .
4 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 connected-
ness '.

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 demands 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 mount

137

11/8 DESIGN FOR A BRAIN

the pavement. 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.

We now, therefore, no longer maintain the restriction of S. 8/6 :
from now on the main variables may be of any type : full-, part-,
step-, or null-functions. This freedom makes possible new types
of ultrastable system, systems still ultrastable and still selective
for stable fields, but no longer necessarily fully inter-connected
internally. In particular, if many of the main variables are part-
functions, the system is able to avoid the earlier-mentioned diffi-
culty in getting adapted ; it does this by developing partial,
fluctuating, and temporary independencies within the whole
without losing its essential wholeness. The study of such systems
will occupy the remainder of the book.

References

Boyd, D. A., and Nie, L. W. Congenital universal indifference to pain.

Archives of Neurology and Psychiatry, 61, 402 ; 1949.
Sherrington, C. S. The integrative action of the nervous system. New

Haven, 1906.

138

CHAPTER 12

Iterated Systems

12/1. Whereas in the previous chapters we studied a system
whose main variables were all in intimate connection with one
another, so that a disturbance applied to any one immediately
disturbed all the others, we shall now study, for contrast, a system
composed of the same number of main variables but divided into
many parts. Each part is assumed to be wholly separated from
the other parts, and to contain only a few main variables. The
diagram of immediate effects might appear as in Figure 12/1/1
which shows, at A, what we have considered in Chapters 8-11,
and at B what we shall be considering in this chapter. (For
simplicity, the diagram shows lines instead of arrows.)

Figure 12/1/1.

As before, it is assumed that each of the five systems in B
consists partly of variables belonging to the animal and partly of
variables belonging to the environment. The relation between
animal and environment is shown more clearly in Figure 12/1/2.

Environment.

f\\

//
7

Animal.
Figure 12/1/2 : Diagrammatic representation of an animal of eight main
variables interacting with its environment as five independent systems.

139

12/2 DESIGN FOR A BRAIN

Such an arrangement would be shown by any organism that
reacted to its environment by several independent reactions. In
such an arrangement each system, still assumed to be ultrastable,
can change its own step-functions and find its own terminal field
without effect on what is happening in the others. We shall say
that the whole consists of iterated ultrastable systems.

Since each system is ultrastable it can adapt and learn inde-
pendently of the others. That such independent, localised learn-
ing can occur within one animal was shown by Parker in the
following experiment :

4 If a sea-anemone is fed from one side of its mouth, it will
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.'

12/2. If we start a set of iterated ultrastable systems, and
observe the set's behaviour, noting particularly at each moment
how many of the systems have arrived at a terminal field, we
shall find that the set, regarded as a whole, shows the following
properties.

The proportion which is adapted is no longer restricted to the
two values ' all ' or ' none '. In fact, if the systems are many,
the degrees of adaptation which the whole can show will be as
many. A whole which consists of iterated systems will therefore
show in its adaptation a gradation which was seen (S. 11/2) to
be lacking in the fully-connected ultrastable system.

A second property is that when one system has arrived at a
terminal field, the changes of the other systems will not cause the
loss of the first field. In other words, while the later adaptations
are being found, the earlier are conserved. A whole which con-
sists of iterated systems will therefore show some conservation of
adaptations.

A third property is that, as time passes, the number of systems

140

ITERATED SYSTEMS 12/3

which are adapted will increase, or may stay constant, but cannot
decrease (in the conditions assumed here : more complex conditions
are discussed later). If the number of stable systems is regarded
as measuring, in a sense, the degree of adaptation achieved by the
whole, then, in a whole which consists of iterated systems, the
degree of adaptation tends always to increase. The whole will
therefore show a progression in adaptation.

12/3. Let us now compare the two types of system, (a) the
fully connected, and (b) the iterated, in the times they take,
on the average, to reach terminal fields, other things, including
the number of main variables, being equal. (The calculation
can only be approximate but the general conclusion is unam-
biguous.)

We start with a system of N main variables and want to find,
approximately, how long the system will take on the average to
reach the condition where all N main variables belong to systems
with stable fields , Three arrangements will be examined ; they are
extreme in type, but they illustrate the possibilities. (1) All the
N main variables belong to one system, so that to stabilise all
N a field must stabilise all simultaneously. (2) Each main vari-
able is in a system which includes it alone, and where the systems
are related in such a way that only after the first is stabilised can
the second start to get stabilised, and so on in succession. (3)
Each system, also containing only one main variable, proceeds
independently to find its own stability.

In order to calculate how long the three types will take, suppose
for simplicity that each main variable has a constant and inde-
pendent probability p of becoming stable in each second.

The type in which stability can occur only when all the N
events are favourable simultaneously will have to wait on the

average for a time given by 1\ = —. The type in which sta-
bility can occur only by the variables achieving stability in succes-
sion will have to wait on the average for a time given by 1\ = N/p.
And the type in which the variables proceed independently to
stability will have to wait on the average for a time which is
difficult to specify but which will be of the order of T3 = 1/p.
These three estimates of the time taken are of interest, not for
their quantitative exactness, but for the fact that they tend to

141

12/4 DESIGN FOR A BRAIN

have widely different values. Some numerical values will be

calculated in order to demonstrate the differences. The values

have not been specially selected, and if the reader will substitute

some values of his own he will probably find that his values lead

to essentially the same conclusions as are reached here.

Suppose that the chance of any one variable becoming stable

in a given second is a half. If we are testing a system with a

thousand variables, then N = 1000

and J\ = 21000 sees.

rr 100°

1 2 = — £- sees.

T3 = about \ sec.
When these are converted to more ordinary numbers, we find that
the three quantities differ widely. Tz is about a half-second, T2
is about 8 minutes, and 1\ is about 3 X 10291 centuries. The
last number, if written in full, would consist of a 3 followed by
about five lines of zeros. 1\ and T2 are moderate, but T1 is so
vast as to be outside even astronomical duration.

This example is typical. What it means in general is that
when N is large, it is not possible to get stability if all N must
find some favourable feature simultaneously. The calculation
confirms the statement of S. 11/7 that it is not reasonable to
assume that 1010 neurons have formed a stable field by waiting
for the fortuitous occurrence of one field which stabilises all.

12/4. The argument may also be viewed from a different angle.
When the system of a thousand variables could achieve stability
only by the occurrence of a field which was favourable to all at
once, it had to wait, on the average, through 3 X 10291 centuries.
But if its conditions were changed so that the variables could
become stable in succession or independently, then the time taken
dropped to a few minutes or less. In other words, what was, for
all practical purposes, an impossibility under the first condition
became, under the second and third conditions, a ready possibility.
It is difficult to find a real example which shows in one system
the three ways of progression to stability, 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

142

ITERATED SYSTEMS 12/5

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,
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 impos-
sible. But if we allow success to be achieved by first finding a
1 0 ', then finding a 4 1 ', and so on until a l 9 ' is seen, then the
number of cars which must pass will be about fifty, and this
number makes 4 success ' easily achievable.

12/5. 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, then crystallisation could never occur :
it would be too improbable. But in fact crystallisation can occur
by succession, 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 crystallisation can pro-
ceed by stages, and the time taken resembles T2 rather than 2\.
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.

Reference

Parker, G. The evolution of man. New Haven, 1922.

143

CHAPTER 13

Disturbed Systems and Habituation

13/1. We have seen that ultrastable systems are subject to two
conflicting requirements : complexity and speed. The system
with abundant internal connections, though able to represent
a complex and well-integrated organism and environment,
requires, at least in the form so far studied, almost unlimited time
for its adaptation. On the other hand, the same number of main
variables, divided into many independent parts, achieves adapta-
tion quickly, but cannot represent a complex biological system.
There are, however, intermediate forms that can combine, to
some extent, the advantages of these two extremes. Since the
properties of the intermediate forms are somewhat subtle we
shall have to proceed by small steps. As a first step I shall
examine in this chapter the properties of ultrastable systems that
are no longer completely isolated, as has been assumed so far, but
are subject to some slight disturbance from the outside.

13/2. Before entering the subject, I must make clear a point of
method that will be used frequently. In Chapter 8, the discussion
of the ultrastable system necessarily paid so much attention to the
process by which the terminal field was reached that some loss of
proportion occurred; for the focusing of attention suggested
that the system spent most of its time reaching a terminal field,
whereas in the living organism this process may occupy only a few
moments — a time unimportant in comparison with the remainder
of the organism's life during which the terminal field will act
repeatedly to keep the essential variables within limits.

From this point of view the terminal field is more important
than the preceding fields simply because it is permanent while
the others are transient. As we increase the time over which the
system is observed, so do the transient fields become negligible.
The same principle is used in the Darwinian theory of natural

144

DISTURBED SYSTEMS AND HABITUATION 13/4

selection where, although it is recognised that mutations and re-
combinations of defective viability can occur, yet as the processes
of evolution are viewed over an increasing range of time, so
do these defectively adapted individuals sink into insignificance.
Statistical mechanics, too, uses the same principle, for it excludes
an event by proving that its occurrence is not impossible but
infrequent. Sometimes we shall not be able to distinguish even
the transient from the permanent, but only the lesser from the
greater persistence. Nevertheless, the distinction may be im-
portant, especially if the small difference acts repeatedly and
cumulatively ; for what is feeble on a single action may be over-
whelming on incessant repetition.

It will be suggested later (S. 16/6) that the animal's behaviour
depends not on one system but on many, so that what counts is
not the peculiarity of one particular field but the average proper-
ties of many. In the discussion we shall therefore notice the
average properties and the tendencies rather than the individual
peculiarities of the various fields.

Effects of small random disturbances

13/3. A disturbance may affect variables or parameters. If it
affects the variables, the system will undergo a sudden change of
state ; in the phase-space the representative point would be
displaced suddenly from one line of behaviour to another. If
it affects a parameter, there will occur a sudden change of field :
the representative point will be affected only mediately.

13/4. We shall now examine the effect on an ultrastable system
of small, occasional, and random disturbances applied to the
variables. I assume at first that the displacements are distributed
in all directions in the phase-space.

A displacement may make the representative point meet a
critical state it would not otherwise have met ; then the dis-
placement destroys the field. The three fields of Figure 13/4/1
show some of the consequences. In fields A and C the undis-
turbed representative points will go to, and remain at, the resting
states. 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 Cs field may survive

145

13/4 DESIGN FOR A BRAIN

a displacement that destroyed A's. Similarly a displacement
applied to the representative point on the resting cycle in B is
more likely to change the field than if applied to C. A field like
C, therefore, with its resting states compact and near the centre
of the region, tends to have a higher immunity to displacement
than fields whose resting states or cycles go near the edge of the

Figure 13/4/1 : 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.)

region. (A quantitative discussion of the tendency is given in
S. 23/4.)

If the disturbances fall on a large number of iterated ultrastable
systems, the probabilities become actual frequencies. We can
then predict that if iterated ultrastable systems are subjected to
repeated small occasional and random disturbances, the average
terminal field will tend to the form C.

13/5. How would this tendency show itself in the behaviour of
the living organism ?

In S. 8/7 we noticed that a field may be terminal and yet
show all sorts of bizarre features : cycles, resting states near the
edge of the region, stable and unstable lines mixed, multiple
resting states, multiple resting cycles, and so on. These possi-
bilities 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-

146

DISTURBED SYSTEMS AND HABITUATION 13/7

ment by fields of ' normal ' stability (S. 20/2) that look like C
of Figure 13/4/1 and act simply to keep the representative point
well away from the critical states.

Effects of repeated stimuli

13/6. So far we have studied only the effects of irregular dis-
turbances : what of the regular ? By the argument of S. 6/6, all
such can be considered as * stimuli ' and are of two types : a
sudden change of parameter-value, and a sudden jump of the
representative point. The two types will be considered separately.

The effect on an ultrastable system of an alternation of a
parameter between two values has already been described in S.
11/4, where Figure 11/4/1 showed how the ultrastable system is
automatically selective for any set of step-function values which
gives stability with both the parameter- values.

The facts can also be seen from another point of view. If we
start alternating the parameter and observe the response of the step-
functions we shall find that at first they change, and that after a
time they stop changing. The responses, in other words, diminish.

13/7. Next consider the effect of repetitions of the other type of
stimulus — the displacement of
the representative point. Its
effect can readily be found by

asking what sort of field can be • '

terminal. Suppose, for instance, '

that the displacement was a A ' Q

movement to the left through »

the distance shown by the arrow
below Figure 1 3/7/1 . It is easy
to see that a field, to be terminal
in spite of this displacement,
must have its resting state
within B. If the constant dis-
placement is applied from time *"* •
to time to an ultrastable system FlGURf 13/7/1 : Region of an ultra-
, r. i t , . stable system. The representative
whose fields have resting states point must stay in B if the field is
distributed over both A and B, to bf imm"ne to a displacement
., • i /> i i • , . equal and parallel to that shown
then terminal fields with resting by the arrow.

147

13/8 DESIGN FOR A BRAIN

state in A will be destroyed ; but the first with resting state in B
will be retained. The displacement will then stop causing step-
function changes. So if we regard the application of the constant
displacement as • stimulus ', and the step-function and main-
variable changes as ' response ', then we shall find that the
response to the stimulus tends to diminish.

13/8. This particular process cannot be shown on the homeostat,
for its resting state is always at the centre, but it will demon-
strate a related fact. If two fields (Figure 13/8/1) each have a

Figure 13/8/1.

resting state at the centre and the line of one (A) from a constant
displacement returns by a long loop meeting critical states while
the return path of the other (B) is more direct, then the applica-
tion of the displacement will destroy A but not B. In other
words, a set of step-function values which gives a large ampli-
tude of main-variable movement after a constant displacement
is more likely to be replaced than a set which gives only a small
amplitude.

The process is shown in Figure 13/8/2. Two units were joined
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 tended to diminish.

The same process in a more complex form is shown in Figure

148

DISTURBED SYSTEMS AND HABITUATION

13/9

13/8/3. Two units are interacting : 1 ^± 2. Both effects go
through the uniselectors, so the whole is ultrastable. At each D,
the operator displaced l's magnet through a constant distance.
On the first c stimulation ', 2's response brought the system
to its critical states, so the ultrastability found a new terminal

Time

Figure 13/8/2 : 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 states (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.

-\rv

Time

Figure 13/8/3 : 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.

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.

13/9. For completeness we will now consider the effects of these
disturbances in combination. The combination of repeated
constant displacements with small random disturbances yields
little of interest. But the combination of an alternation of para-
meter-value with small random disturbances is worth notice.

149 l

13/9

DESIGN FOR A BRAIN

From S. 11/4 it is known that the alternation of a parameter
p between two values, p' and p", will result in the emergence of
two stable fields. We might get a pair like A and B of Figure
13/9/1. As the parameter p alternates between p' and p", so

Figure 13/9/1 : Two possible terminal fields : A, when p has the value p'
and B, when it has the value p" . (Critical states shown as dots.)

Figure 13/9/2 : When the parameter is constant at p', the representative
point will follow the path from )3 to a ; when at p" the point follows
the path from a to j8.

will the field of the system's main variables alternate between
A and B. If p alternates slowly in comparison with the move-
ment of the representative point, the point will follow the circuit
of Figure 13/9/2, going from a to /? when p is changed from
p' to p'\ and returning to a when p is returned to p' .

Suppose now that small random disturbances are applied to

150

DISTURBED SYSTEMS AND HABITUATION 13/10

two such systems (C and D) with circuits such as arc shown in
Figure 13/9/3. We can predict (by S. 13/4) that a system of
type D, with a short and central circuit, will have a higher
immunity to random disturbance than a system of type C.
Maximal immunity will be shown by systems in which a and ft
coalesce at the centre of the region.

Figure 13/9/3.

If there are many systems like C and D, the probabilities become
actual frequencies. As the less resistant fields are destroyed while
the more resistant remain, the average movement of the repre-
sentative point, as the parameter is alternated between p' and
p", will change from a large circuit like C towards a small central
circuit like D. So both the number of step-function changes and
the range of movement evoked by the stimulus p will diminish.

Habituation

13/10. Some uniformity is now discernible in the responses of an
ultrastable system to repeated stimuli. There is a tendency for
the response, whether measured by the number of step-functions
changing or by the range of movement of the main variables,
to diminish. The diminution is not due to any triviality of
definition or to any peculiarity of the homeostat : it follows from
the basic fact, inseparable from any delicate or ultrastable system,
that large responses tend, if there is feedback, to destroy the

151

13/11 DESIGN FOR A BRAIN

conditions that made them large, while small responses do not
destroy the conditions that made them small.

13/11. 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 :

4 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 J% 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.

The nature of habituation has been obscure, and no explanation
has yet received general approval. The results of this chapter
suggest that it is simply a consequence of the organism's ultra-
stability, a by-product of its method of adaptation.

Reference
Humphrey, G. The nature of learning. London, 1933.

152

CHAPTER 14

Constancy and Independence

14/1. 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 absolute, must be 4 properly
isolated ' ; some parameters in S. 6/2 were described as 4 ineffec-
tive ' ; and iterated ultrastable systems were defined as ' wholly
independent ' of each other. So far a simple understanding 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 l independence ' of two dynamic systems
might at first seem simple : is not a lack of material connection
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-
nection 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 con-
troversy. The criterion of connection or separation is thus useless.

14/2. This attempted criterion obtained its data by a direct
examination of the real ' machine '. The examination not only
failed in its object but violated the rule of S. 2/8. What we
need is a test that uses only information obtained by primary
operations.

It is convenient to approach the subject by first clarifying
what we mean by one system c controlling ' or ' affecting ' another.

153

14/2

DESIGN FOR A BRAIN

Our understanding has been greatly increased by the development
during recent years of the science of 4 cybernetics '. The word —
from KvfiepvrjTf]£, a steersman — was coined by Professor Wiener to
describe the science which, though really dating back to Watt
and his governor for steam-engines, has developed partly as a
result of the extraordinary properties of the thermionic vacuum
tube, and partly as a result of the urgent demands during the
last war for complex calculating and controlling machinery such
as predictors for gun- and bomb-sights, automatically controlled
searchlights, automatically controlled anti-aircraft guns, and
electronic computors.

When a radar-installation passes information about the posi-
tion of an aeroplane to a predictor, and the predictor emits
instructions which determine, either manually or automatically,
the laying of an anti-aircraft gun, we can write simply enough : —

Radar
set

Predictor

> Gun

but what do the arrows mean ? what is transmitted from box
to box ? Energy ? No, says cybernetics — information.

If we turn to simple machines for guidance, we will probably
be misled. 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

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 can allow air to escape and

154

CONSTANCY AND INDEPENDENCE

14/3

can cause the pressure in the cylinder to fall. Now suppose a
stranger comes along ; he knows nothing of the internal mech-
anism, 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 Z?, 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.

14/3. The factual content of the concept of one variable c con-
trolling ' another is now clear. A ' controls ' B if B's behaviour
depends on A, while A's does not depend on B. But first we
need a definition of ' independence '. Given a system that includes
two variables A and B, and two lines of behaviour ivhose initial
states differ only in the values of B, A is independent of B if A's
behaviours on the tzvo lines are identical. The definition can be
illustrated on the data in Table 14/3/1. On the two lines of

Time

Variable

Line

Table 14/3/1 : Two lines of behaviour of a three- variable absolute system.

behaviour the initial states are equal except for the values of B.
The subsequent behaviours of A on the two lines are identical.
So A is independent of B. (Independence within the range
covered by the table in no way restricts what may happen
outside it.) By the definition, C is not independent of B.

By ' dependent ' will be meant simply 4 not independent '.

The definition is given primarily by reference to two lines of
behaviour, for only in this form is the result of the criterion

155

14/4 DESIGN FOR A BRAIN

always unambiguous. Other criteria might be confused by some
of the fields that ingenuity can construct. But often a simple
uniformity holds. Two variables may be independent over all
such pairs of initial states ; and sometimes all variables of one
set may be independent of all variables of another set : a system
R is independent of a system S if every variable in It is independent
of every variable in S, all possible pairs being considered. Some
region of the field is understood to be given before the test is
applied.

14/4. 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 but with all other constituents equal ; he seeds
them with equal numbers of organisms ; and he observes how
the increasing numbers of organisms compare in the two tubes
from hour to hour. Thus he is observing the numbers of organisms
after two initial states that differed only in the concentrations
of chemical.

To test whether an absolute system is dependent on a para-
meter, i.e. to test whether the parameter is ' effective ', we observe
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 one sets the regulator at some
value, checks that the temperature is at its usual value, and
records the subsequent behaviour of the temperature ; then one
returns the temperature to its previous value, changes the posi-
tion of the regulator, and observes again. A change of behaviour
implies an effective regulator. (Here we have used the fact that
by S. 21/4 we can take a null-function into the system without
altering its absoluteness, for the change is only formal.)

Finally, an example from animal behaviour. Parker tested
the sea-anemone to see whether the behaviour of a tentacle was
independent of its connection with the body.

* 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.'

156

CONSTANCY AND INDEPENDENCE 14/6

(He has established that the behaviour is regular, and that the
system of tentacle-position and food-position is approximately
absolute. He has described the line of behaviour following the
initial state : tentacle extended, food on tentacle.)

4 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 null-function ' con-
nection 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 :

4 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
cases of providing a clear and precise foundation. By its use
the independencies within a system can be proved by primary
operations only.

14/5. In an absolute system it is not generally possible to
assign the dependencies and independencies arbitrarily. For if
x is dependent on y, and y is dependent on z, then x must neces-
sarily be dependent on z. This is evident, for when s's initial
state is changed, i/'s behaviour is changed ; and these changed
values of y, acting in an absolute system, will cause x,s behaviour
to change. So the observer will find that a change in z9s initial
state is followed by a change in as's behaviour. (A formal proof
is given in S. 24/11.)

14/6. We can now see that the method for testing an imme-
diate effect, described in S. 4/12, is simply a test for independence
applied when all the variables but two are held constant. The
relation can be illustrated by an example. Suppose three real
machines are linked so that their diagram of immediate effects is

% — > y — > x.
157

14/6

DESIGN FOR A BRAIN

The system's responses to tests for independence will show that
y is independent of x, and that z is independent of both. The
same set of independencies would be found if we tested the three
machines when their linkages were

The distinction appears when we test the immediate effect be-
tween z and x. For if in both cases we fix y, we shall find in
the first that x is independent of 2, but in the second that x is
not independent of z.

Given a system's diagram of immediate effects, its diagram
of ultimate effects is formed by adding to every pair of arrows
joined tail to head a third arrow going from tail to head, like
2 — > x above, and by repeating this process until no further
additions are possible. Thus, the diagram of immediate effects
I in Figure 14/6/1 would yield the diagram of ultimate effects II.

>■ 2

Figure 14/6/1.

The diagram of ultimate effects shows directly and completely
the independencies in the system. Thus, from II of the figure
we see that variable 1 is independent of 2, 3, and 4, and that
the latter three are dependent on all the others.

14/7. If, in a system, some of the variables are independent
of the remainder, while the remainder are not independent of
the first set, then the first set dominates the remainder. Thus,
in Figure 14/6/1, variable 1 dominates 2, 3, and 4. And in
the diagram of S. 6/6 the animal dominates the recorders.

The effects of constancy

14/8. So far the independencies have been assumed permanent :
we now study the conditions under which they can alter.
Suppose an absolute system of eight variables has the diagram

158

CONSTANCY AND INDEPENDENCE

14/9

of immediate effects shown in Figure 14/8/1. What properties
must the three variables B have if the systems A and C are to
become independent and absolute ? 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 tem-

Figure 14/8/1.

perature ; 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 ' removed '. In fact the
answer is capable of proof (S. 24/15) : that A and C should he
independent and absolute it is necessary and sufficient tlmt the
variables B should be null-functions. In other words, A and C
must be separated by a wall of constancies.

14/9. Here are some illustrations to show that the theorem
accords with common experience.

(a) If A (of Figure 14/8/1) 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 absolute, 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 con-

159

14/10

DESIGN FOR A BRAIN

tinuity is not necessary : a segment may be poisoned, or anaes-
thetised, 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.

14/10. The example of Figure 14/8/1 showed one way in which
the constancy of a set of variables 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 are held constant. The
rule is proved in S. 24/14 : — 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, complete the diagram of ultimate effects, using

1-^2

\/\

4-*

R C D F

1^=§=±=2 \-±-*z2 1r< 2 1 t 2

[IXIIXH/I

^3 4± — ^3 4-« 3 4-*

Figure 14/10/1 : 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.

the rule of S. 14/6. The resulting diagram will be that of the
ultimate effects, and therefore of the independencies, when V is
constant. (It will be noticed that the effect of making V constant
cannot be deduced from the diagram of ultimate effects alone.)
Thus, if the system of Figure 14/10/1 has the diagram of imme-
diate 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

160

CONSTANCY AND INDEPENDENCE 14/11

independence show a remarkable variety. Thus, in C, 1 domi-
nates 3 ; but in D, 3 dominates 1. As the variables become
more numerous so does the variety increase rapidly.

The multiplicity of inter-connections possible in a telephone
exchange is due primarily to the widespread use of temporary
constancies. The example serves to remind us that 8 switching '
is merely one of the changes producible by a re-distribution of

+*•

**•

Figure 14/10/2.

constancies. For suppose a system has the diagram of imme-
diate effects shown in Figure 14/10/2. 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
4 switched ' to the BE branch, B must be freed and A must
become constant. Any system with a ' switching ' process must
use, therefore, an alterable distribution of constancies. Con-
versely, a system whose variables can be sometimes fluctuating and
sometimes constant is adequately equipped for switching.

14/11. At this point it is convenient to consider what degree
of independence is shown in a system if some part is not directly
affected by some other part. To take an extreme case, to what
extent are two parts joined functionally if they have only a

W x \

A B

Figure 14/11/1.

single variable in common — the parts A and B in Figure 14/11/1,
for instance, which share only the variable x ? It is shown in
S. 24/17 that if x is a full-function capable of unrestricted varia-
tion, then the two parts A and B are as effectively joined as if

161

14/12 DESIGN FOR A BRAIN

they had many more direct effects bridging the gap. Construc-
tion of the diagram of ultimate effects provides a simple proof.
The explanation is that each system affects, and is affected by,
not only aj's value but also a?'s first, second, and higher derivatives
with respect to time. These act to provide a richness of func-
tional connection that is not evident at first glance.

Part- functions

14/12. In S. 14/8 we saw that if a whole system is to be divided
into independent parts some intervening variables must become
constant. It follows that if the independence is to be tem-
porary, being sometimes present and sometimes absent, the inter-
vening variables must be sometimes constant and sometimes
varying : they must, in short, be part-functions. This class of
variable will therefore now be considered.

A part-function was defined in S. 7/1 as a variable which,
over some interval of observation, was constant over some finite
intervals and fluctuated over some finite intervals. It is not
implied that the constant values are all equal. The definition
refers solely to the variable's observed behaviour, making no
reference to any cause for such behaviour ; though there will
usually be some definite physical reason to account for this way
of behaving. A part-function will be said to be ' active ', or
4 inactive ', at a given moment according to whether it is, or
is not, varying. As the amount of time spent active tends
to 'all', or 'none', so does the part-function tend to full-, or
step-, function form. The part-function thus fills the gap be-
tween the two types, and may be expected to have intermediate
properties.

14/13. Here are some examples. Like the step-function, it is
met with much more commonly in the real world than in books.
— The pressure on the brake-pedal during a car journey. The
current flowing through a telephone during the day. The posi-
tion co-ordinates of an animal, such as a frog or grasshopper,
that moves intermittently. The pressure on the sole of the
foot during walking. The activity of pain receptors, if they are
activated only intermittently. The rate of secretion of saliva in
an experiment on the conditioned reflex. The rate at which

162

CONSTANCY AND INDEPENDENCE 14/15

water is being swallowed (ml. /sec.) by a land animal observed
over several days. The sexual activities of a stag during the
twelve months. The activity in the mechanisms responsible for
reflexes which act only intermittently : vomiting, sneezing,
shivering.

14/14. The property of ' threshold ' leads often to behaviour of
part-function form. For if x dominates y, and if, when x is less
than some value, y remains constant, while if, when x is greater
than the value, y fluctuates as some function of x, then, if x is
a full-function and fluctuates across the threshold, y will behave
as a part-function. In the nervous system, and in living matter
generally, threshold properties are widespread ; part-functions
may therefore be expected to be equally widespread.

Systems containing part-functions

14/15. Having earlier examined the properties of systems con-
taining null-functions (S. 7/7), and step-functions (S. 7/8 et seq.),
we will now examine the properties of systems containing part-
functions. It is convenient to suppose at first that the system
is composed of them exclusively.

Even when not at a resting state, some of such a system's
variables may be constant. If the system is composed of part-
functions which are active for most of the time, the system will
show little difference in behaviour from one composed wholly of
full-functions. But if the part-functions are active only at in-
frequent intervals then, as the system traverses some line of
behaviour, inspection will show that only some of the variables
are changing, the remainder being constant. Further, if observed
on two lines of behaviour, the set of variables which were active on
the first line will in general be not the same as the set active
on the second. That this may be so can be seen by considering
its field.

The field of an absolute system which contains part-functions
has the peculiarity that the lines of behaviour often run in a
sub-space. Thus, over an interval when all the variables but
one are inactive, the line will run in a straight line parallel to
the axis of the active variable. If all but two are inactive, the
line will run in a plane parallel to that which contains the axes

163

14/15

DESIGN FOR A BRAIN

of the two active variables ; and so on. If all the variables
are inactive, the line becomes a point. Thus a three-variable

system might give the line of behaviour
shown in Figure 14/15/1.

An absolute system composed of
part-functions has also the property
that if a variable changes from inactive
to active, then amongst the variables
which affect that variable directly
there must, at that moment, have
been at least one which was active.
One might say, more vividly but
less accurately, that activity in one
variable can be obtained only from
activity in others. A proof is given
in S. 24/16, but the reason is not
difficult to see. Suppose for simplicity that a variable A is
directly affected only by B and C, so that the diagram of
immediate effects is

B C

Figure 14/15/1. In the dif-
ferent stages the active
variables are : A, y ; B, y
and z; C, z; D, x\ E, y;
F, x and 2.

Suppose that over a finite interval of time all three have been
constant, and that the whole is absolute. If B and C remain
at these constant values, and if A is started at the same value
as before, then by the absoluteness A's behaviour must be the
same as before, i.e. A must stay constant. The property has
nothing to do with energy or its conservation ; nor does it attempt
to dogmatise about what real 4 machines ' can or cannot do ; it
simply says that if B and C remain constant and A changes from
inactive to active, then the system cannot be absolute — in other
words, it is not completely isolated.

The sparks which wander in charred paper give a vivid picture
of this property : they can spread, one can become multiple, or
several can converge ; but no spark can arise in an unburning
region.

14/16. Part-functions were introduced primarily in the hope
that they would provide a system more readily stabilised than
one of full-functions. It can now be shown that this is so.

164

CONSTANCY AND INDEPENDENCE 14/16

First, what do we mean by ' difficulty of stabilisation ' ?
Consider an engineer designing, on the bench, an electronic
system. He has before him an apparatus which he wants to
be stable at some particular state. The apparatus contains a
number of adjustable constants, parameters, and he has to find
a combination of values that will give him what he wants. The
4 difficulty ' of stabilisation may be defined as, and measured
by, the proportion of all possible parameter-values that fail to
give the required stability. The definition has the advantage
that it is directly applicable to the homeostat and any similar
mechanism that has to search through combinations of values.

With this definition it can be shown that if a system of N
part-functions has on the average k of its variables active, then
its difficulty of stabilisation is the same, other things being equal,
as that of a system of k full-functions.

The proof is given in S. 24/18, but the theorem is clearly
plausible. When a system of part-functions is in a region of
the phase-space where k variables are active and where all the
other variables are constant, the k variables form a system which
is absolute and which is not essentially different from any other
absolute system of k variables. The fact that we have been
thinking of it differently does not affect the intrinsic nature of
the situation. Equally, whenever we have postulated an abso-
lute system, we have assumed that its surrounding variables are
constant, at least for the duration of the experiment or observa-
tion. Yet these surrounding variables are usually not constant
for ever. So our ' absolute system ' was quite commonly only
a portion of a larger system of part-functions. There is there-
fore no intrinsic difference between an absolute system of k
full-functions and a subsystem of k active variables within a
larger system of part-functions. That being so, there is no reason
to expect any difference in their difficulties of stabilisation.

The theorem is of great importance to us, for it means that
the time taken to stabilise a system of N part-functions will,
very roughly, be more like T2 of S. 12/3 than Tx ; so the change
to part-functions may change the stabilisation from ' impossible '
to ' possible '. The subject will be developed in S. 17/3.

165

CHAPTER 15

Dispersion

15/1. Systems of part-functions have the fundamental property
that each line of behaviour may leave some of the variables
inactive. Dispersion occurs when the set of variables made
active by one line of behaviour differs from the set made active
by another. We will begin to consider the physiological applica-
tions of this fact.

First consider a system of full-functions. Suppose we record
a few of its variables' behaviours while it traverses first one line
of behaviour and then another. The records would show the
variables always fluctuating, and the two records would differ
only in their patterns of fluctuation. Now suppose we have a
system of part-functions. Again we record some of the variables'
behaviours. It may happen that from one initial state the line
of behaviour leaves all the recorded variables inactive, while the
line from another shows some activity. Since, by S. 6/6, the
change of initial state corresponds to ' applying a stimulus ', a
by-standing physiologist would describe the affair as a simple
case of a mechanism ' responding ' to a stimulus. Since living
organisms' responses to stimuli have been sometimes offered as
proof that the organism has some power not possessed by
mechanisms, we must examine these reactions more closely.

In S. 6/6 we saw that the most general representation of a
1 stimulus ' was a change from one initial state to another. Now
in general, even though the system is absolute, the course of
the line of behaviour from one initial state puts no restriction
on the course from another initial state. From this lack of
restriction follow several consequences.

15/2. The first consequence is that, in a system known only
to be complex, however small the difference between the initial
states — however slight or simple the stimulus — we can put no
limit to the greatness of the difference between the subsequent

166

DISPERSION 15/3

lines of behaviour. Thus Pavlov conditioned a dog so that it
gave no salivary response when subjected to the compound
stimulus of :

the experimental room, the harness, the feeding apparatus,
the sound of a metronome beating at 104 per minute, and
the sound of a No. 16 organ pipe,
but gave a positive salivary response when subjected to :

the experimental room, the harness, the feeding apparatus,
the sound of a metronome beating at 104 per minute, and
the sound of a No. 15 organ pipe.
Such a ' discrimination ' has been considered by some to be beyond
the powers of mechanism, but this is not so : all that is neces-
sary is that the system should be complex and should contain
part-functions.

15/3. The same point of view helps to make clear the physio-
logical concept of ' adding ' stimuli. In the simple case it is
easy enough to see what is meant by the ' addition ' of two
stimuli. If a dog has developed one response to a flashing light
and another to a ticking metronome, it is easy to apply simul-
taneously the flashes and the ticks and to regard this stimulation
as the c sum ' of the two stimuli. But the application of such
' sums ' was found in many cases to lead to no simple addition
of responses : a dog could easily be conditioned to salivate to
flashes and to ticks and yet to give no salivation when both were
applied simultaneously. Some physiologists have been surprised
that this could happen. Let us view the events in phase-space.
Suppose our system has variables a, b, c, . . . and that the
basal, c control ', behaviour follows the initial state aQ9 b0, c0, . . .
Suppose the effect of stimulus A corresponds to the line of
behaviour from the initial state al9 b0, c0, . . . , and that of
stimulus B to the line from a0, bl9 c09 . . . Then the behaviour
after the initial state al9 blt c0, . . . would correspond to the
response to the simultaneous presentation of A and B. If we
know the behaviours after A and after B separately, what can
we predict of the behaviour after their presentation simultane-
ously ? The possibility is illustrated in Figure 15/3/1, which
shows at once that the lines of behaviour from II and J in no
way restrict that from K, which represents, in this scheme, the
4 sum ' response.

167

15/3 DESIGN FOR A BRAIN

It will be seen, therefore, that in a complex system, every
group of stimuli will have a holistic quality, in that the response
to the whole group will not be predictable from the responses
to the separate stimuli, or even to sub-groups. The dog that

salivated to each of two stimuli but not to the two together is
therefore behaving in no way surprisingly, and such behaviour
is no evidence of any ' supra-mechanistic ' power. In complex
systems such non-additive compoundings are to be expected.

15/4. Another variation in stimulus-giving occurs when a
pattern is varied in some mode of presentation without the
pattern itself being changed, as when an equilateral triangle is
shown both erect and inverted. The same argument as before
prevents us from expecting any necessary relation between the
two evoked responses.

In some cases the two evoked responses are found to be the
same, and to be characteristic of the particular pattern even
though its presentation may have been much changed : an
object may be recognised though its image falls on a part of
the retina never before stimulated by it. This power of Gestalt-
recognition was also sometimes thought to demand ' supra-
mechanistic ' powers. But in 1947 Pitts and McCulloch showed
that any mechanism can show such recognition provided it can
form an invariant over the group of equivalent patterns. As
the formation of such invariants demands nothing that cannot
be supplied by ordinary mechanism, the subject need not be
discussed further here.

To sum up, these examples have shown that no matter how
small the difference between stimuli, or initial states, we can,
in general, if the system is complex, put no limits to the differ-
ence that may occur between the subsequent lines of behaviour.
From this we may deduce that if the system is one with many

168

DISPERSION 15/5

part-functions, we can put no limit to the difference there may
be between the two sets of variables made active in the two
responses ; or in other words, there is, in general, no limit to
the degree of dispersion that may occur other than that imposed
by the finiteness of the mechanism.

15/5. 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 facts are well known, so I can
be brief.

The fact that the sense-organs are not identical enforces an
initial dispersion. Thus if a beam of radiation of wave-length
0-5 jli 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^, 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.

The sense of taste depends on four histologically-distinguishable
types of receptors each sensitive to only one of the four qualities
of salt, sweet, sour, and bitter. If change from one solution to
another changes the excitation from one type of receptor to
another, then dispersion has occurred.

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 4 pain ' type of receptor ; i.e. dispersion
occurs.

In the cochlea, sounds differing in pitch vibrate different parts
of the basilar membrane. As each part has its own sensitive
cells and its own nerve-fibres, a change in pitch will shift the
excitation from one set of fibres to another.

The three semicircular canals are arranged in planes mutually
at right-angles, and each has its own sensitive cells and nerve-
fibres. A change in the plane of rotation of the head will there-
fore shift the excitation from one set of fibres to another.

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

169

15/6 DESIGN FOR A BRAIN

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.

15/6. 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 cerebral processes, especially those of cellular
magnitude, frequently show threshold, the fact that this property
generates part-functions (S. 14/14), and the fact that part-
functions cause dispersion (S. 14/15) have already been treated.
The deduction that dispersion must occur within the nervous
system can hardly be avoided.

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 strong
reasons for believing that dispersion plays an important part in
the nervous system. What that part is will be discussed in the
next three chapters.

Reference

Pitts, W., and McCulloch, W. S. How we know universals : the pereeption
of auditory and visual forms. Bulletin of mathematical Biophysics,
9, 127 ; 1947.

170

CHAPTER 16

The Multistable System

16/1. The systems discussed in the previous chapter contained
no step-functions, and the effect of ultrastability on their pro-
perties was not considered. In this chapter, ultrastability will
be re-introduced, so we shall now consider what properties will
be found in systems which show both ultrastability and dispersion.

To study the interactions of these two properties we might
start by examining the properties of an ultrastable system whose
main variables are all part-functions. But it has been found
simpler to start by considering a system defined thus : a multi-
stable system consists of many ultrastable systems joined main
variable to main variable, all the main variables being p art-functions .

The restriction to part-functions is really slight, for the part-
function ranges all the way from the full- to the step-function.
It will further be noticed that, as the ultrastable, or ' sub- ',
systems are joined main variable to main variable only, each
step-function will now be restricted in two ways. The critical
states which determine whether a particular step-function shall
change value depend only on those main variables that belong
to the same subsystem. And when a step -function has changed
value, the immediate effect is confined to that subsystem to
which it belongs. In the definition of the ultrastable system
(S. 8/6) no such limitation was imposed.

This type of system has been defined, not because it is the
only possible type, but because the exactness of its definition
makes possible an exact discussion. When we have established
its properties, we will proceed on the assumption that other
systems, far too varied for individual study, will, if they approxi-
mate to the multistable system in construction, approximate
to it in behaviour.

171

16/2 DESIGN FOR A BRAIN

16/2. The multistable system is itself ultrastable. The pro-
position may be established by considering the class of ' all
ultrastable systems '. Such a class will include every system
not incompatible with the definition of S. 8/6. It will, for
instance, contain systems whose main variables are all full-func-
tions, systems some of whose main variables are part-functions,
and systems whose main variables are all part-functions (S. 11/8).
Further, the class will include both those whose step-functions
are wide in their immediate effects and those whose step-functions
act directly on only a few main variables. The class will there-
fore include those systems defined as ' multistable '.

From this fact it follows that all the properties possessed gener-
ally by the ultrastable system will be possessed by the multi-
stable. In particular, the multistable system will reject all
unstable fields of its main variables but will retain the first
occurring stable field. In other words, the multistable system
will ' adapt ' just as will any other ultrastable system.

On the other hand, the faults discussed in Chapter 11 were
due to the fact that the systems considered before that chapter
had main variables which were all full-functions. Now that the
main variables have become all part-functions we shall find, in
this and the next two chapters, that the faults have been reduced
or eliminated.

16/3. In a multistable system, if no step-function changes in
value, the main variables, being all part-functions, will form a
system identical with that discussed in S. 14/15. In particular,
it will show dispersion : two lines of behaviour will make active
two sets of variables ; the two sets will usually not be identical,
and may perhaps have no common member.

16/4. It is now possible to deduce the conditions that must
hold if a system, multistable or not, is to be able to acquire a
second adaptation without losing a first.

We may view the process in two ways, which are really equi-
valent. First, I will suppose that we have an ultrastable system
which can be connected to either of two environments (as Units
3 and 4 of the homeostat, representing the adapting system,
might be joined to either Unit 1 or Unit 2, representing the
two environments). Suppose that the system has been joined

172

THE MULTISTABLE SYSTEM 16/5

to environment a, has adapted to it, and has thus reached a
terminal field. To record this ' first adaptation ', we disturb a
slightly in various ways and record the system's responses. Give
the variables activated in these responses the generic label A.
Next, remove a, join on environment /?, and allow ultrastability
to establish a ' second adaptation '. Give the generic label S
to all step-functions that were changed by this process. Finally,
remove fi, restore a, and again test the system's responses to
small disturbances applied to a ; compare these responses with
those first recorded to see whether the first adaptation has been
retained or lost. For the responses to he unchanged — -for the first
adaptation to be retained — it is necessary and sufficient that during
the responses there should be a wall of null-functions between the
variables A and the step-functions S. The condition is necessary,
for if an S is not so separated from an A, then at least one A's
behaviour will be changed. It is also sufficient, for if the wall
of constancies is present, then by S. 14/8 the A's are independent
of the S's, and the *S"s changes will not affect the A's responses.

(The other way of viewing the process is to allow a parameter
P to affect the ultrastable system, the two environments being
represented by two values P' and P". The ' disturbance from a '
becomes a transient variation in the value of P. The reader
can verify that this view leads to the same conclusion.)

The necessary wall of constancies can be obtained in more
than one way. Thus, if the system really consisted of two
permanently unconnected parts, one of which was joined to a
and the other to {3, then the addition of a second adaptation
would be possible ; so the present discussion includes the case
of the iterated ultrastable systems. More interesting now is
the possibility that the constancies have been provided by part-
functions, for this enables the connections to be temporary and
conditional. The multistable system is certainly not incapable
of so acquiring a second adaptation. The facts that set A will
often be only a fraction of the whole, that part-functions are
ubiquitous, and that all step-functions are only local in their
effects makes the separation of A and S readily possible.

16/5. As a further step towards understanding the multistable
system, suppose that we are observing two of the subsystems,
that their main variables are directly linked so that changes of

173

16/5 DESIGN FOR A BRAIN

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. 14/8).

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
field. Its behaviour will not be essentially different from that
recorded in Figure 8/8/5. If, however, we regard the same
series of events as occurring, not within one ultrastable whole,
but as interactions between two subsystems, then we shall observe
behaviours homologous with those observed when interaction
occurs between ' animal ' and l 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 variables of the double system within the
region of its critical states.

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 connection
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-functions, together with
those of C, give a stable field to the main variables of B and C,
then that set of J5's step-function values will persist indefinitely ;
for when B rejoins A the original stable field will be re-formed.
But if Z?'s set with C's does not give stability, then it will be
changed to another set. It follows that B's step-functions 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. 11/4 should be noted.)

174

THE MULTISTABLE SYSTEM

16/5

The process can be illustrated on the homeostat. Tliree 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 8/8/2) of 2 and 3 so that they could
move only in the direction recorded as downwards in Figure
16/5/1, while 1 could move either upwards or downwards.
If 1 was above the central line (shown broken), 1 and 2 inter-
acted, and 3 was independent ; but if 1 was below the central
line, then 1 and 3 interacted, and 2 was independent. 1 was

u jVl^m* n

—\j — \r

Time

Figure 16/5/1 : 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.

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
connections (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 connections which made both systems stable. The values
of the step-functions are now permanent ; 1 can interact repeatedly
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-functions

175

16/6 DESIGN FOR A BRAIN

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-functions 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
join. Always we can predict that their step-functions 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-function values
which provide stability ; for the fundamental interaction between
step-function and stability, the principle of ultrastability described
in S. 8/5, still rules the process.

16/6. 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 during the whole process. What modifications must
be made when we allow for the fact that in the multistable system
the number and distribution of subsystems active at each moment
fluctuates ?

It is readily seen that the principle of ultrastability holds equally
whether dispersion is absent or present ; for the proof of Chapter 8
was independent of special assumptions about the type of vari-
able. The chief effect of dispersion is to destroy the individuality
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 is restored. Thus, suppose that a multistable
system's field of all its main variables is stable, and that its repre-
sentative point is at a resting state R. If the representative point
is displaced to a point P, or to Q, the lines from these points 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

176

THE MULTISTABLE SYSTEM 16/7

and apparent confusion. Travel along the other line, from Q 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 subsystems when a multi stable system adapts. What
will happen is that instability, and consequent step-function
change, will cause combination after combination of subsystems
to become active. So long as instability persists, so long will
new combinations arise. But when a stable field arises not
causing step-functions to change, it will, as usual, be retained.
If now the multistable system's adaptation be tested by dis-
placements of its representative point, 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 refer 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.

16/7. In S. 12/4 it was shown that the division of a system
into parts reduced markedly the time necessary for adaptation.
The multistable system, being able to adapt by parts (S. 16/5),
can adapt by this quicker method. But no reason has yet been
given why this quicker method should be taken if offered. There
is, however, a well-known principle which ensures this.

When changes can occur by two processes which differ in their
speeds of achievement, the faster process, by depriving the
slower of material, will convert more material than the slow ;
and if we imagine the material marked in some way according
to its mode of change, then the major part of the material will
bear the mark of the faster process. If the difference between
the speeds is great, then for practical purposes the slow process
may not be in evidence at all. The important fact here is that
we can predict a priori that if the change be examined, it will

177

16/7 DESIGN FOR A BRAIN

be found to occur by the fast process ; and we can make this
prediction without any reference to the particular physical or
chemical details of the particular change.

The principle is well known in chemical dynamics. Thus there
is a reaction whose initial and final states are described by the
equation

6FeCl2 + KCIO3 + 6HC1 = 6FeCl3 + KC1 + 3H20.
There are at least two processes leading from the initial to the
final states : one corresponding to the reaction (of the thirteenth
order) as written above, and one composed of a series of reactions
of low order of which the slowest is the reaction (of the third
order)

2FeCl2 + Cla = 2FeCl3.
The first is slow, for it has to wait for an appropriate collision
of thirteen molecules, while the second is fast. We can predict
that the fast will be preferred ; and direct testing has shown that
the reaction occurs by the second, and not the first, process.

From this we may draw several deductions. First, the multi-
stable system will similarly tend to adapt by its fast rather than
by its slow process. Secondly, since the fast process, by S. 12/4,
is that of adaptation by a series of small independent parts, any
multistable system will behave as if it 4 preferred ' to adapt by
many small independent adaptations rather than by a few com-
plex adaptations : it ' prefers ' to adapt piecemeal if this is
possible. Finally, by using the fast process, the time it takes
in getting adapted will tend to the moderate T2 (of S. 12/3)
rather than to the immoderate Tv It is therefore at least partly
free from the fault of excessive slowness described in S. 11/7.

178

CHAPTER 17

Serial Adaptation

17/1. We have now reached a stage where we must distinguish
more clearly between the organism and its environment, for the
concept of the l multistable ' system clearly refers primarily to
the nervous system. From now on we shall develop the theme
that the nervous system is approximately multistable, and that
it is joined to, or interacts with, an environment. But before
discussing the events in the nervous system we must be clear on
what we mean by an ' environment '. So far we have left the
meaning very open : now we want to know what we mean exactly.
The question occurs in its most urgent form to the designer of a
4 mechanical brain ', for if he has designed this successfully he
still has to decide with what it shall interact : having made a
model of the brain, he must confront it with a model of the
environment. What model could represent the environment
adequately in principle ?

It seems clear that we can, in general, put no limit to what
may confront the organism. The last century's discoveries have
warned us that the universe may be inexhaustible in surprises, so
we should not attempt to define the environment by some formula
such as ' that which obeys the law of conservation of energy ',
for the formula may be obsolete before it is in print. In general,
therefore, the nature of the environment must be left entirely
open.

On the other hand, we may obtain a partial definition of some
practical use by noticing that the living organism on this earth
adapts not to the whole universe but to some part of it. It is
often not unlike the homeostat, adapting to a unit or two within
its immediate cognisance and ignoring the remainder of the world
around it. Yet, given a particular organism, especially if human,
we cannot with certainty point to a single variable in the universe
and say ' this variable will never affect this organism '. This

179

17/1 DESIGN FOR A BRAIN

possibility makes the homeostat unrepresentative ; for a man
does not, like a prince in a fairy tale, pass instantaneously from one
world to another, but has rather a series of environments that are
interrelated, neither wholly separate nor wholly continuous. We
are, in fact, led again to consider the properties of a system whose
connections are fluctuating and conditional — the type encoun-
tered before in S. 11/8, and therefore treatable by the same
method. I suggest, therefore, that many of the environments
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 environments.
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
depths of the pond, will vary as part-functions. A discontinuity
like a surface will generate part-functions in a variety of ways.

1 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.

4 ... it approaches a region where the sun has been . . .
heating the water.'

Temperature of the water will behave sometimes as part-, sometimes
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 of these substances is constant at
zero.

4 Other Paramecia . . . often strike together ' (collide).

180

SERIAL ADAPTATION 17/1

The pressure on the Paramecium's anterior end varies as a part-
function.

4 The animal may strike against stones.'
Similar part-functions.

4 Our animal comes against a decayed, softened, leaf.'

More part-functions.

4 . . . till it comes to a region containing more carbon dioxide
than usual.'

Concentration of carbon dioxide, being generally uniform with
local increases, will vary as a part-function.

4 Finally it comes to the source of the carbon dioxide — a large
mass of bacteria, embedded in zoogloea.'

Another part-function due to contact.

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 will therefore tend, as in S. 14/15, to be
fluctuating and conditional rather than invariant. As an animal
interacts with its environment, the observer will see that the
activity is limited now to this set, now to that. If one set per-
sists active for a long time and the rest remains inactive and in-
conspicuous, the observer may, if he pleases, call the first set 4 the '
environment. And if later the activity changes to another set he
may, if he pleases, call it a 4 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 com-
posed of subsystems which sometimes have individuality and inde-
pendence but which from time to time show linkage. The alter-
nation 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 pavement
and pedestrian ; he has to learn to control the accelerator and
its relation to engine-speed, learning neither to race the engine nor

181 N

17/2 DESIGN FOR A BRAIN

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 steering did not exist. But on an ordinary
journey the relations vary. For much of the time the three
systems

driver + steering wheel -f . . .
driver -f- accelerator 4- . . .
driver -f gear lever -f . . .

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
round a corner should be preceded 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.

17/2. Before supposing that the nervous system, in its con-
struction and function, resembles the multistable, we may ask
to what extent the supposition is necessary. S. 9/4 showed the
necessity for ultrastability ; is the hypothesis of multistability
equally necessary ?

Our basic facts and assumptions are now as follows :

(1) the nervous system adapts by the process of ultrastability

(S. 9/4),

(2) it can retain one adaptation during the acquisition of

another (S. 11/3),

(3) this independence is not achieved by a division of the

nervous system into permanently separate parts (S. 11/8),

(4) no special mechanism is to be postulated for special en-

vironmental conditions (S. 1/9) : if possible, the variables
are to be statistically homogeneous.
Given these, what can be deduced ?

In the system, label the main variables M and the step-functions
S. Call those variables immediately affected by the first environ-

182

SERIAL ADAPTATION 17/3

ment, M1 ; those immediately affected by the second, M2 ; those
step-functions which changed during the second adaptation, S2 ;
and those main variables that S2 directly affects, M3. It is not
assumed that the M-classes are exclusive.

After the step-functions S2 have changed value, the behaviours
of the variables Mx are unchanged, by postulate 2 ; so Mx is in-
dependent of *S2 (S. 14/3). But S2 affects M3 ; so Mx must be
independent of M3 (S. 14/5). There must therefore be a wall of
constancies between them (S. 14/8), which must be only temporary,
by postulate 3. We can deduce therefore that some of the main
variables must be part-functions.

Since M1 is independent of S2, it follows that the step-functions
S2 can have no immediate effect on the main variables Mv In
other words, some of the step-functions' immediate effects are
restricted to a few of the main variables.

If we now use the fourth postulate, that these particular main
variables and step-functions are typical, it follows that part-
functions must be common, and step-functions must usually be
restricted in the variables they immediately affect. We conclude,
therefore, that if the nervous system is to show the listed proper-
ties, the main features of the multistable system are necessary.

17/3. We can now start to examine the thesis that the nervous
system is approximately multistable. We assume it to be joined
to an environment that contains many part-functions, and we
ask to what extent the thesis can explain not only elementary
adaptation of the type considered earlier but also the more complex
adaptations of the higher animals, found earlier to be beyond the
power of a simple system like the homeostat.

We may conveniently divide the discussion into stages accord-
ing to the complexity of the environment. First there is the
environment that, though perhaps extensive, is really simple, for
it consists of many parts that are independent, so that they can
be adapted to separately. Such an environment was sketched
in Figure 12/1/2. It will be considered in this section. Then
there is the environment that has some connection between its
parts but where the adaptation can proceed from one part to
another, perhaps in some order. It will be considered in the
remainder of this chapter. Then there is the environment that
is richly interconnected but in which there is still some transient

183

17/3 DESIGN FOR A BRAIN

subdivision into parts, where there are many subsystems, some
simple, some complex, acting sometimes independently and some-
times in conjunction, where an adaptation produced for one part
of the environment may conflict with an adaptation produced for
another part, and where the adaptations themselves have to be
woven into more complex patterns if they are to match the com-
plex demands of the environment. It will be considered in
Chapter 18. Beyond this, for completeness, are the environments
of extreme complexity ; but they hardly need discussion, for at
the limit they go beyond any possibility of being adapted to —
at least, in the present state of our knowledge.

The environment of the first type, that composed of indepen-
dent parts, would, if joined to a multistable system, form an
ultrastable whole (S. 16/2). Adaptation will, therefore, tend to
occur. But as the whole is also multistable the process will show
modifications. Dispersion will occur, so that at each moment
only some of the whole system's variables will be active. This
allows the possibility that though the whole may contain a great
number of variables yet little subsystems may occur containing
only a few. A subsystem may become stable before all the rest
are stable. By the usual rule such stable subsystems will tend
to be self-preserving. There is therefore the possibility that the
multistable system will adapt piecemeal, its final adaptation
resembling that of a collection of iterated ultrastable systems,
like that of Figure 12/1/2. The present system will, however,
differ in that the constancies that divide subsystem from sub-
system are not unalterable but conditional.

Such a multistable system, having arranged itself as a set
of iterated systems, will show the features previously noticed
(S. 12/2) : its adaptation will be graduated ; it can conserve its
old adaptations while developing new ; and, most important, the
time taken before all its variables become stabilised will
be reduced from the impossibly long to the reasonably short
(S. 12/4).

This is what may happen ; but will it actually occur ? The
tendency to adaptation may be persistent, but why should the
process take the favourable course ? First we notice that as
adaptation in some form or other is inevitable the only question
is what form it will take. For simplicity, consider an eight-
variable environment that can be stabilised either in two inde-

184

SERIAL ADAPTATION 17/4

pendent parts of four variables each or in one of eight. During
the random changes of trial and error, a field stabilising one of the
sets of four will occur many times more frequently than will a
field stabilising all eight (S. 20/12). Such a four, once stabilised,
will retain its field leaving only the other four to find a stable field.
Consequently, before the process starts we can predict that the
eight- variable system is much more likely to arrive at stability by
a sequence of four and four than by a simultaneous eight. The
fast process is the more probable (S. 16/7).

We can predict, therefore, that in general if a multistable
system adapts to an environment composed of P independent
parts it will tend to develop P independent subsystems, each
reacting to one part. The nervous system, if multistable, will
thus tend to adapt to a fragmented environment by a fragmented
set of reactions, each complete in itself and having no relation to
the other reactions. It will do this, not because this way is the
best but because it must. But even though unavoidable, the
method is by no means unsuitable. It has the great advantage
of speed — it reduces to a minimum the dangerous period of
error-making — and there is no point in the nervous system's
attempting to integrate the reactions when no integration is
required.

17/4. The second degree of complexity occurs when the environ-
ment is neither divided into independent parts nor united
into a whole, but is divided into parts that can be adapted to
individually provided that they are taken in a suitable order and
that the earlier adaptations are used to promote adaptation later.
Such environments are of common occurrence. A puppy can
learn how to catch rabbits 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
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.

185

17/5 DESIGN FOR A BRAIN

As a clear illustration of such a process I quote from Lloyd
Morgan on the training of a falcon

' 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 4 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.

4 . . . 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.

17/5. To what process in the multistable system does serial
adaptation correspond ? It is sullicient if we examine the
relation of a second adaptation to a first, for a series consists only
of this primary relation repeated.

We assume then that the multistable system has learned one
reaction and that it is now faced with an environment that can
be adapted to only by the system developing some new reaction
that uses the old. It is convenient, for simplicity, to assume
here that the first reaction is no longer able to be disrupted by
subsequent events. The assumption demands little, for in the
next chapter we shall examine the contrary assumption ; and
there is, in fact, some evidence to suggest that, in the mammalian

186

SERIAL ADAPTATION 17/5

brain, step-functions that were once labile may become fixed.
Duncan, for instance, let rats run through a maze, and at various
times after the run gave them a convulsion by giving an electric
shock to the brain. He found that if the shock was given within
about half an hour of the run, all memory of the maze seemed to
be lost ; but if the shock was given later, the memory was retained.
In his words : ' It is suggested that newly learned material under-
goes a period of consolidation or perseveration. Early in this
period a cerebral electroshock may practically wipe out the effect
of learning. The material becomes more resistant to such disrup-
tion ; at the end of an hour no retroactive effect was found.'
Such a consolidation could easily occur in the animal brain :
many proteins, for instance, if kept in unusual ionic conditions
undergo irreversible changes. But with the details we are hardly
concerned : we simply assume the possibility.

If, then, the first learned reaction is unbreakable, the whole
system becomes simple, at least in principle, for as it is an ultra-
stable system adapting to a system not subject to step-function
change (i.e. to the complex of environment and first reaction-
system acting together), the situation is homologous with that
already treated in Chapter 9 — the adaptation and ' training ' of
an ultrastable system by an environment. It is therefore not
playing with words, but expressing a fundamental parallelism to
say that, in serial learning, the first reaction-system and the
environment together i train ' the second. They train it by not
allowing the second to follow lines of behaviour incompatible
with their own requirements.

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 step-function values
which give a field such that the system composed 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 immediate effects will therefore
resemble Figure 17/5/1. This system will be referred to as part A,
the c avoiding ' system.

187

17/5

DESIGN FOR A BRAIN

As the animal must now get its own food, the brain must
develop a set of step-function values 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

BRAIN -«-

EYES

SKIN

MUSCLE S

-^OBJECTS

Figure 17/5/1 : Diagram of immediate effects of the ' avoiding '
system. Each word represents many variables.

normal limits (S. 5/6). (This system will be referred to as part B,
the 4 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 may allow
the two reactions to use common variables. When the systems
interact, the diagram of immediate effects will resemble Figure
17/5/2.

BLOOD
GLUCOSE"

-^~ BRAIN -<-

EYES

FOOD
SUPPLY

SKIN

-«-

MUSC LES

^-OBJECTS

Figure 17/5/2.

As the * avoiding ' system A is not subject 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.
B will therefore change its step-function 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-functions to change. We know from S. 8/10 that, whatever
the peculiarities of A, B's terminal field will be adapted to them.

188

SERIAL ADAPTATION 17/5

It should be noticed that the seven sets of variables (Figure
17/5/2) 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.

If, next, the second reaction becomes unbreakable, by mere
repetition third and subsequent reactions can similarly be added.

The multistable system will thus show the phenomenon of serial
adaptation, not only in its seriality but in the proper adaptation
of each later acquisition to the earlier.

References

Cowles, J. T. Food-tokens as incentives for learning by chimpanzees.
Comparative Psychology Monographs, 14, No. 71 ; 1937-8.

Culler, E. A. Recent advances in some concepts of conditioning. Psycho-
logical Review, 45, 134 ; 1938.

Duncan, C. P. The retroactive effect of electroshock on learning. Journal
of comparative and physiological Psychology, 42, 32 ; 1949.

Wolfe, J. B. Effectiveness of token-rewards for chimpanzees. Compara-
tive Psychology Monographs, 12, No. 60 ; 1935-6.

189

CHAPTER 18

Interaction between Adaptations

18/1. At this stage it is convenient to consider in more detail
the question of ' localisation ' in a multistable system : how will
the pattern of activity be distributed within it ? In treating the
brain as a multistable system we followed the incoming sensory
stimuli through the sense-organs to the sensory cortex (S. 15/5
and 15/6) ; we have now to consider what happens in the areas of
4 association ', not of course in detail but sufficiently to develop a
clear picture of what we would expect to see there.

Some functions in the cortex are, of course, unquestionably
localised : the reception of retinal stimuli at the area striata for
instance. With such I shall not be concerned. I shall consider
only the localisation of learned reactions, especially of those to
situations, such as puzzle-boxes containing food, for which the
organism has no detailed inborn preparation. In such a case the
simplest hypothesis, the one to be tried first, is that the dispersion
occurs at random. By this I mean that at each elementary point,
at each synapse perhaps, the functional details are determined by
factors of only local significance and action : whether two pieds
terminaux make contact or three, whether the nucleus happens to
be on this side of the cell or that, whether five dendrons converge
or seven (I use these examples only as illustrations of my mean-
ing). Such details have been determined by primary genetic
factors modified by merely local incidents in embryological
development and perhaps by local incidents in past learning.

But though the local details were once decided by some trifling
local event, I assume that they persist with some tenacity; for
learned behaviour can, in the absence of disruptive factors, per-
sist for many years. I will quote a single example. By differ-
ential 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

190

INTERACTION BETWEEN ADAPTATIONS 18/1

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.

4 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.'

I assume, therefore, that the system's behaviour is locally
regular, in the sense of S. 2/14.

How will responses be localised in such a system ? Under-
standing is easier if we first consider the distribution over a town
of the chimneys that ' smoke ' when the wind blows from a given
direction. The smoking or not of a particular chimney will be
locally determinate ; for a wind of a particular force and direc-
tion, striking the chimney's surroundings from a particular angle,
will regularly produce the same eddies, which will regularly deter-
mine the smoking or not of the chimney. But geographically the
smoking chimneys are distributed more or less at random ; for if
we mark a plan of the town with a black dot for every chimney that
smokes in a west wind, and a red dot for every one that smokes
in a north wind, and then examine the j^lan, we shall find the
black and red dots intermingled and scattered irregularly. The
phenomenon of ' smoking '. is thus localised in detail yet dis-
tributed geographically at random.

Such is the 4 localisation ' shown by the multistable system.
We are thus led to expect that the cerebral cortex will show a
' localisation ' of the following type. The events in the environ-
ment will provide a continuous stream of information which will
pour through the sense organs into the nervous system. The set
of variables activated at one moment will usually differ from
the set activated at a later moment ; for in this system there
is nothing to direct all the activities of one reaction into one set
of variables and all those of another reaction into another set.
On the contrary, the activity will spread and wander with as

191

18/1 DESIGN FOR A BRAIN

little orderliness as the drops of rain that run, joining and separat-
ing, down a window-pane. But though the wanderings seem
disorderly, the whole is regular ; 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 reflexes in dogs, removed

various parts of the cerebral cortex, and observed the effects on

the conditioned reflexes. 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 reflex 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 it was

found that the removal of cerebral cortex from other parts of the

brain gave vague results. Removal of almost any part caused

some disturbance, no matter from where it was removed or what

type of reflex or habit was being tested ; and no part could be

found whose removal would destroy the reflex 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 multistable system the whole

reaction will be based on step-functions and 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 scattered elements will tend to have more

immunity to localised injury than one whose elements are few and

compact.

192

INTERACTION BETWEEN ADAPTATIONS 18/3

18/2. 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 physiologically 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
untidiness 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 hardly 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 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-functions 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 that there are good reasons for
believing that its tendency will actually be towards ever-increasing
adaptation.

18/3. Before considering these reasons we should notice that
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

193

18/4 DESIGN FOR A BRAIN

occur if the nervous system were multistable. Pavlov, for in-
stance, records that ' . . . the addition of new positive, and
especially of new negative, reflexes exercises, in the great majority
of cases, an immediate, though temporary, influence upon the
older reflexes '. And in experimental psychology ' 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. Miiller 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 ' 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 this the nervous system resembles the multistable ; but the
resemblance is even closer. In a multistable system, the more the
stimuli used in new learning resemble those used in previous learn-
ing, 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-functions. In psycho-
logical experiments it has repeatedly been found that the more the
new learning resembled the old the more marked was the inter-
ference. Thus 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.

18/4. One factor tending always to lessen the amount of inter-
action between subsystems is c habituation ', already shown in
Chapter 13 to be an inevitable accompaniment of an ultrastable
system's activities. There it was shown that an ultrastable

194

INTERACTION BETWEEN ADAPTATIONS 18/5

system, coupled to a source of disturbance, tends to change its
step-functions to such values as will render it independent of the
source. Such a change must also occur if one part of a multis table
system is repeatedly disturbed by another part ; for the reacting
system possesses the essential properties, and the origin of the
disturbance is irrelevant. A subsystem is safe from such dis-
turbance when and only when its variables are independent of all
the other variables in the system. There is no necessity for me to
repeat the evidence here, for it is identical with that of Chapter 13.
It can therefore be predicted that as the various subsystems of a
multistable system act on one another, the tendency will be, as
time goes on, for the various subsystems to upset each other less
and less.

If the nervous system is multistable it would show the same
tendency. It would thus show habituation twice : once in its
interactions with its environment and again between its various
component subsystems. Such ' intracerebral ' habituation will
tend to lessen the disturbing actions of part on part, and it will
therefore contribute to lessening the chaos described in S. 18/2.
But such a process will not always lead to complete adaptation ;
for its tendency, being always to remove interaction, is to divide
the whole into many independent parts. With some simple
environments such subdivision may be sufficient, as was noticed
in S. 17/3 ; but it contributes nothing towards the co-ordination
of reactions when a complex environment can be controlled only
by an intricate co-ordination in the nervous system.

18/5. In the turbulence of many subsystems interacting, the
principle of ultrastability still holds and still acts persistently in
the direction of tending to improve the organism's adaptation
to its environment. It will still act selectively towards the
useful interactions. Suppose first that two subsystems interact
in such a way that, though individually adaptive, their com-
pounded reactions are non-adaptive/ A kitten, for instance, has
already learned that when it is cold it should go right up to the
warmth of its mother, and that when it is hungry it should go
right up to the redness of a piece of meat. If later, when it is
both cold and hungry, it sees a fire, it would probably tend, in the
absence of other factors, to go right up to it. But the very fact
that the interaction leads to non-adaptive behaviour provides the

195

18/6 DESIGN FOR A BRAIN

cause for its own correction : step-functions change value, and that
particular form of interaction is destroyed. Then the step-func-
tions' new values provide new forms of interaction, which are again
tested against the environment. The process can stop when and
only when the step-functions have values that, acting with the
environment, give behaviour that keeps the essential variables
within normal limits. Interactions are thus as subject to the
requirements of ultrastability as are the other characteristics of
behaviour.

Ultrastability thus works in all ways towards adaptation.
The only question that remains is whether it is sufficiently effective.

18/6. Is the principle of ultrastability really sufficient to over-
come the tendency to chaos ? Is it really sufficient to co-ordinate
the activities of, say, 1010 neurons when they interact with an
extremely complicated environment ? Let me admit at once that
the problem will require a great deal of further study before a final
answer can be given. The mathematical study of such systems
has yet hardly begun, so no rigorous proof can be given. The
available physiological evidence is slight, and the physiologist
who tries to get direct evidence will encounter formidable diffi-
culties. Nevertheless, we are not wholly without evidence on the
subject.

Consider first the spinal reflexes. If we examine a mammal's
reflexes, examining them in relation to its daily life, we shall
usually find, not only that each individual reflex is adapted to the
environment but that the various reflexes are so co-ordinated in
their interactions that they work together harmoniously. Nor
is this surprising, for species whose reflexes are badly co-ordinated
have an obviously diminished chance of survival. The principle
of natural selection has thus been sufficient to produce not only
well-constructed reflexes but co-ordination between them.

A second example is given by the many complex biochemical
processes that must be co-ordinated successfully if an organism is
to live. Not only must a complicated system like the Krebs'
cycle, involving a dozen or more reactions, be properly co-ordinated
within itself, but it must be properly co-ordinated into all the other
cycles and processes with which it may interact. Biochemists
have already demonstrated something of the complexity of these
systems and the future will undoubtedly reveal more. Yet in

196

INTERACTION BETWEEN ADAPTATIONS 18/6

the normal organism natural selection has been sufficient to
co-ordinate them all.

If now from the principle of natural selection we remove all
reference to its cytological details, there remains a process strik-
ingly similar in the abstract to that impelled by the principle of
ultrastability. Thus natural selection co-ordinates the reflexes
by repeated application of the two operations :

(1) test the organism against the environment ; if harmful

interactions occur, remove that organism ;

(2) replace it by new organisms, differing randomly from the

old.
And it is known that in general these rules are sufficient, given
time, to achieve the co-ordination. Similarly, the principle of
ultrastability leads to the repeated application of the two
operations :

(1) test the organism against the environment ; if a harmful

interaction occurs, change the values of the step-functions
responsible for it ;

(2) let the new values provide new forms of behaviour, differing

randomly from the old.
The analogy between genes and step-functions is most interesting
and could be developed further ; but it must not distract us now
The point at issue is : if natural selection's method of action is
sufficient to account for the co-ordination between spinal reflexes,
may not ultrastability's method of action also be sufficient to
account for the co-ordination between cerebral responses, consider-
ing that the two processes are abstractly almost identical ?

It may be objected that the spinal reflexes do not have to fear
disorganisation by new learning ; but the objection will not stand.
We are comparing the ontogenetic progress of the cerebral re-
sponses with the phylogenetic progress of the spinal reflexes. As
the species evolves, its environment changes and new reflexes have
to be developed to suit the new conditions. Each new reflex,
though suitable in itself, may cause difficulties if it compounds
badly with the pre-existing reflexes. Thus, a bird that developed
a new reflex for pecking at a new type of white, round, edible
fungus might be in danger of using the same reflex on its eggs.
Evolution has thus often had to face the difficulty that a harmoni-
ous set of reflexes will be disorganised if an extra reflex is added.
Again, consider the biochemical systems. As most, if not all,

197 o

18/6 DESIGN FOR A BRAIN

genes have biochemical effects, the acquisition of many a new
favourable mutation has meant that a harmonious set of bio-
chemical reactions has had to be reorganised to allow the incor-
poration of the new reaction.

Evolution has thus had to cope, phylogenetically, with all the
difficulties of integration that beset the individual ontogenetically.
The tendency to ' chaos ', described in S. 18/2, thus occurs in the
species as well as in the individual. In the species, the co-ordinat-
ing power of natural selection has shown itself stronger than the
tendency to chaos.

Natural selection is effective in proportion to the number of
times that the selection occurs : in a single generation it is negli-
gible, over the ages irresistible. And if the unrepeated action of
ultrastability seems feeble, might it not become equally irresistible
if the nervous system was subjected to its action on an equally
great number of occasions ?

How often does it act in the life of, say, the average human
being ? I suggest that in those reactions where interaction is
important and extensive, the total duration of the learning process
is often of ' geological ' duration when compared with the duration
of a single incident, in that the total number of incidents contri-
buting to the final co-ordination is very large. I will give a single
example. Consider the adult's ability to make a prescribed move-
ment without hitting a given object — to put the cap on a fountain-
pen, say, without damaging the nib. This skill demands co-
ordinated activity, but the co-ordination has not been developed
by a single experience. Here are some of the incidents that will
probably have contributed to this particular skill :

Putting the finger into the mouth (without hitting lips or teeth) ;
putting the finger into the handle of a cup (without striking the
handle) ; dipping pen into inkwell (without striking the rim) ;
putting button into button-hole ; passing a shoe-lace through its
hole ; inserting a collar-stud into the neck-band ; putting pen-nib
into pen ; making a knot by passing an end through a loop ;
replacing the cork in a bottle ; putting a key into a key-hole ;
threading a needle ; placing a gramophone record on the turn-
table's central pin ; putting the finger into a ring ; inserting a
funnel into a flask ; putting a cigarette into a holder ; putting
a cuff-link into a cuff ; inserting a pipe-cleaner ; putting a screw
into a nut ; and so on.

198

INTERACTION BETWEEN ADAPTATIONS 18/6

Not only could the list be extended almost indefinitely, but
each item is itself representative of a great number of incidents,
carried out on a variety of occasions in a variety of ways. The
total number of incidents contributing to the adult's skill may thus
be very large.

So by the time a human being has developed an adult's skill
and knowledge, he has been subjected to the action of ultrastability
repetitively to a degree which may be comparable with that to
which an established species has been subjected to natural selec-
tion. If this is so, it is not impossible that ultrastability can
account fully for the development of adaptive behaviour, even
when the adaptation is as complex as that of Man.

References

Ashby, W. Ross. Statistical machinery. Thales, 7, 1 ; 1951.

Lashley, K. S. Nervous mechanisms in learning. The foundations of
experimental psychology, edited C. Murchison. Worcester, 1929.

Muller, G. E., and Pilzecker, A. Experimentelle Beitrage zur Lehre
vom Gedachtniss. Zeitschrift fur Psychologie und Physiologie der Sinnes-
organe, Erganzungsband No. 1 ; 1900.

Robinson, E. S. Some factors determining the degree of retroactive inhibi-
tion. Psychological Monographs, 28, No. 128 ; 1920.

Skaggs, E. B. Further studies in retroactive inhibition. Psychological
Monographs, 34, No. 161 ; 1925.

Skinner, B. F. Are theories of learning necessary ? Psychological Review,
57, 193 : 1950.

199

APPENDIX

CHAPTER 19

The Absolute System

(Some of the definitions already given are re-
peated here for convenience)

19/1. A system of n variables will usually be represented by
xv . . . , xn, or sometimes more briefly by x. n will be assumed
finite ; a system with an infinite number of variables (e.g. that
of S. 19/23), where xi is a continuous function of i, will be
replaced by a system in which i is discontinuous and n finite,
and which differs from the original system by some negligible
amount.

19/2. Each variable x% is a function of the time t ; it will
sometimes be written as xi(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.

19/3. The state of a system at a time t is the set of numerical
values of x^t), . . . , xn(t). Two states are ' equal ' if n equalities
exist between the corresponding pairs.

19/4. A line of behaviour is specified by a succession of states
and the time-intervals between them. Two lines of behaviour
which differ only in the absolute times of their initial states are
equal.

19/5. A geometrical co-ordinate space with n axes xv . . . , xn,
and a dynamic system with variables xv . . . , xn provide
a one-one correspondence between each point of the space
(within some region) and each state of the system. The region
is the system's ' phase-space '.

19/6. A primary operation discovers the system's behaviour by

203

19/7 DESIGN FOR A BRAIN

finding how it behaves after being released from an initial state
#J, . . . , x„. It generates one line of behaviour.

The field of a system is its phase-space filled with such lines
of behaviour.

19/7. If, on repeatedly applying primary operations to a
system, it is found that all the lines of behaviour which follow
an initial state S are equal, and if a similar equality occurs after
every other initial state S', S", . . . then the system is regular.
Such a system can be represented by equations of form

xx = Fx(x\9 ...,«£; 01

xn — t n\x^ . . . , xn ; t)J

Obviously, if the initial state is at t = 0, we must have

Fi(x°v . . . , x°n ; 0) = a£ N,l n).

The equations are the written form of the lines of behaviour ;
and the forms F{ define the field. They are obtained directly
from the results of the primary operations.

19/8. If, on repeatedly applying primary operations to a system,
it is found that all lines of behaviour which follow a state S
are equal, no matter how the system arrived at S, and if a
similar equality occurs after every other state S\ S", . . . then
the system is absolute.

19/9. A system is ' state-determined ' if the occurrence of a
particular state is sufficient to determine the line of behaviour
which follows. Reference to the preceding section shows that
absolute systems are state-determined, and vice versa.

The equations of an absolute system form a group
19/10. Theorem. That the equations

Xi = Fi{x\t . . . , a£ ; t) (i = l n)

should be those of an absolute system, it is necessary that, re-
garded as a substitution converting #J, . . ., #° to xv . . ., xn,

204

THE ABSOLUTE SYSTEM

19/10

they should form a finite continuous (Lie) group of order one
with t as parameter.

(1) The system is assumed absolute. 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

x»—

Figure 19/10/1.

state let time t" elapse so that x' changes to x". As the system
is absolute, the same line of behaviour will be followed if the
system starts at x° and goes on for time f -f- t". So

x- = Fi{xi . . . , a£ ; t") = Fi(x°v . . . ,' x\ ; t' + t")

(i = 1, . .

but

Fi{x°l9

,o .

(* = i,

., n)

giving

FilF^x"; n • .» Fn(af>; t'); t"}

= Fi{x°v . . . , xQn ; t' + n (i = 1, . . . , n)

for all values of x°, f and t" over some given region. The equation
is known to be one way of defining a one-parameter finite con-
tinuous group.

(2) The group property is not, however, sufficient to ensure
absoluteness. Thus consider x = (1 -f- t)x° ; the times do not
combine by addition, which has just been shown to be necessary.

Example : The system with lines of behaviour given by

Xl = x\ + x\t -f Z21

Xo — Xo -f- "I J

is absolute, but the system with lines given by

xx = x\ + x\t + V

is not.

'2 — 2 ~" •

205

19/11 DESIGN FOR A BRAIN

The canonical equations of an absolute system

19/11. Theorem : That a system xv . . . , xn should be absolute
it is necessary and sufficient that the #'s, as functions of t, should
satisfy differential equations
dx1

i — fl\xl> - • • t xn)

~rr — JnK^n • • • » xn)

(i)

where the /'s are single-valued, but not necessarily continuous,
functions of their arguments ; in other words, the fluxions of
the set xv . . . , xn can be specified as functions of that set and
of no other functions of the time, explicit or implicit.

(The equations will be written sometimes as shown, sometimes
as dxi/dt =fi(xv . . . , #») [i == 1, . . . , n) . (2)

and sometimes abbreviated to x = f(x), where each letter repre-
sents the whole set, when the context indicates the meaning
sufficiently.)

(1) Start the absolute system at x\, . . . , a% at time t = 0
and let it change to xv . . . , xn at time t, and then on to
xx + dasl9 . . . , xn + dxn at time t + dt. Also start it at
SBl9 . . . , xn at time t = 0 and let time dt elapse. By the group
property (S. 19/10) the final states must be the same. Using
the same notation as S. 19/10, and starting from a£, Xi changes
to Fi(x° ; t + dt) and starting at x% it gets to Fi(x ; dt).
Therefore

Fi(x° ; t + dt) = Fi{x ; dt) (i = 1, . . . , n).

Expand by Taylor's theorem and write ^rFi(a ; b) as F'i{a ; b).

Then

Fi{x° ; t) + dt.Fl(x° ; t) = Fi{x ; 0) + dt.F'lx ; 0)

{% = 1, . . . , n)

But both Fi(x° ; /) and Fi(x ; 0) equal xi.

Therefore F^x0 ; t) = F^x ; 0) {i = 1, . . . , n) . (3)

But Xi = Fi(x° ; t) \i = 1 n)

206

THE ABSOLUTE SYSTEM 19/11

so, by (3), °ft = F,(x ; 0) (* = 1, . . . , n)

which proves the theorem, since F\(x ; 0) contains t only in
xv . . . , xn and not in any other form, either explicit or
implicit.

Example 1 .* The absolute system of S. 19/10, treated in this
way, yields the differential equations

dxx

~dt

dx2

~dt

The second system may not be treated in this way as it is not
absolute and the group property does not hold.
Corollary :

— #2

Ji\Xi, . . . , Xn) =

■Fi(xv . . . , xn ; t)\ (i = 1, . . . , n)

(2) Given the differential equations, they may be written
dxi =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
dxi will occur in each variable xi during the next time-interval
dt. By integration this defines the line of behaviour from that
state. The system is therefore absolute.
Example 2 : By integrating

dx1

~dt

dx2

dt
the group equations of the example of S. 19/10 are regained.

Example 3 : The equations of the homeostat may be obtained
thus : — If xi is the angle of deviation of the ith. magnet from
its central position, the forces acting on xi are the momentum,
proportional to xi, the friction, also proportional to xi, and the
four currents in the coil, proportional to xv x2, x3 and #4. If
linearity is assumed, and if all four units are constructionally
identical, we have

jt(mxi) = — kxi + l(p — q)(ailx1 + . . . + a^x^

(i = 1, 2, 3, 4)
207

19/11 DESIGN FOR A BRAIN

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 = * t j = _ then the equations can be written

m m

dxi _ .
~dt~Xi

dx~
— = h(ailx1 + . . . + ai4oj4) — jxi

(i = 1, 2, 3, 4)

which shows the 8 -variable system to be absolute.
They may also be written

dxi _ .

dt ~ m\ k {ChlXl + * * * + a^l*>

Let m — > 0. dxi/dt becomes very large, but not dxi/dt.
So xi tends rapidly towards

k ^Xl + • • • + ^4^4)

while the CD's, changing slowly, cannot alter rapidly the value
towards which xi is tending. In the limit,

^ = £i = fcZ-%^ + . . . + aiixt) (i = 1, 2, 3, 4)

Change the time-scale by r = , 1 ;

— = ailx1 + . . . + ai^ (i = 1, 2, 3, 4)

showing the system xv . . . , x± to be absolute and linear. The
a's are now the values set by the hand-controls of Figure 8/8/3.

19/12. That a system should be absolute, it is necessary and
sufficient that at no point of the field should a line of behaviour

208

THE ABSOLUTE SYSTEM 19/15

bifurcate. The statement can be verified from the definition or
from the theorem of S. 19/11. The statement does not prevent
lines of behaviour from running together.

19/13. The theorems of the previous four sections show that
the following properties, collected for convenience, in a system
xlt . . . , xn, are all equivalent in that the possession of any one
of them implies the others :

(1) From any point in the field departs only one line of

behaviour (S. 19/8) ;

(2) the system is state-determined (S. 19/9) ;

(3) the system has lines of behaviour whose equations specify

a finite continuous group of order one ;

(4) the system has lines of behaviour specified by differential

equations of form

-^ =fi(xv • • • > ®n) (i = 1, • • • , n)

where the right-hand side contains no functions of t
except those whose fluxions are given on the left.

19/14. From the experimental point of view the simplest test
for absoluteness is to see whether the lines of behaviour are
state-determined. An example has been given in S. 2/15. It
will be noticed that experimentally one cannot prove a system
to be absolute — one can only say that the evidence does not
disprove the possibility. On the other hand, one value may be
sufficient to prove that the system is not absolute.

19/15. A simple example of a system which is regular but not
absolute 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 19/15/1). Looking down on
it from above, we can mark across it a rectangular 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,

209

19/16 DESIGN FOR A BRAIN

would find the 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 toD! He would say
that he could make nothing of the system ; for although each

Figure 19/15/1.

line of behaviour is accurately reproducible, the different lines
of behaviour have no relation to one another.

This lack of relation means that they do not form a ' group '.
But whether the experimenter agrees with this or not, he will,
in practice, reject this 2- variable system and will not rest till
he has discovered, either for himself or by following Newton,
a system that is state-determined. In my theory I insist on the
systems being absolute because I agree with the experimenter
who, in his practical work, is similarly insistent.

19/16. That the field of a system should not vary with time,
it is necessary and sufficient that the system be regular. The
proof is obvious.

19/17. One reason why a system's absoluteness is important is
because the system is thereby shown to be adequately isolated
from other unknown and irregularly varying parameters. This
demonstration is obviously fundamental in the experimental
study of a dynamic system, for the proof of isolation comes,

210

THE ABSOLUTE SYSTEM 19/20

not from an examination of the material substance of the system
(S. 14/1), which may be misleading and in any case presupposes
that we know beforehand what makes for isolation and what
does not, but from a direct test on the behaviour itself.

Closely related to this in a fundamental way is the fact that
Shannon's concept of a ' noiseless transducer ' is identical in defini-
tion with my definition of an absolute system. Thus he defines
such a transducer as one that, having states a and an input
xy will, if in state an and given input xn> change to a new state
a„+i that is a function only of xn and an :

a«+l = g(Vn, an)
Though expressed in a superficially different form, this equation
is identical with my ' canonical ' equation, for it says simply
that if the parameters x and the state of the system are given,
then the system's next step is determined. Thus the com-
munication engineer, if he were to observe the physicist and the
psychologist 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 far-reaching consequences
and the possibilities of rigorous deduction.

19/18. A second feature which makes absoluteness important
is that its presence establishes, by appeal only to the behaviour,
that the system of variables is complete, i.e. that it includes all
the variables necessary for the specification of the system.

19/19. When we assemble a machine, we usually know the
canonical equations directly. If, for instance, certain masses,
springs, magnets, be put together in a certain way the mathe-
matical physicist knows how to write down the differential
equations specifying the subsequent behaviour.

His equations are not always in our canonical form, but they
can always be converted to this form provided that the system
is isolated, i.e. not subjected to arbitrary interference, and is
determinate.

19/20. In general there are two methods for studying a dynamic
system. One method is to know the properties of the parts
and the pattern of assembly. With this knowledge the canonical

211

19/21 DESIGN FOR A BRAIN

equations can be written down, and their integration predicts the
behaviour of the whole system. The other method is to study
the behaviour of the whole system empirically. From this
knowledge the group equations are obtained : differentiation of
the functions then gives the canonical equations and thus the
relations between the parts.

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 manipulations
will be shown in the next few sections.

19/21. 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 yv . . . , ym equal in number to the
old and related in some way

y% = </>i(xl9 ...,#„) (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. It is easily shown that such
a change of co-ordinates does not change the absoluteness.

19/22. In the ' homeostat ' example of S. 19/11 a derivative
was treated as an independent variable. I have found this
treatment 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 an absolute system
we can write them as

& —M®v . . . , as*) = 0 (» = i, . . . , n)

treating them as n equations in 2n algebraically independent
variables xv . . . , xn, xv . . . , xn. Now differentiate all the
equations q times, getting (q + l)n 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 ?i, we can eliminate the other qn variables,
using up qn equations. If the variables selected were zl9 . . . , 0»
we now have n equations, in 2n variables, of type

&i(zv . . . , z„, zv . . . , in) = 0 (f = 1, . . . , n)

212

THE ABSOLUTE SYSTEM 19/24

These have only to be solved for zv . . . , zn in terms of
zl9 . . . , zn and the equations are in canonical form. So the
new system is also absolute.

This transformation implies that in an absolute system we can
avoid direct reference to some of the variables provided we use
derivatives of the remaining variables to replace them.

Example : xl = xx — x

Xn i)X 1 ~\~ Xaj

can be changed to omit direct reference to x2 by using xx as a
new independent variable. It is easily converted to

dxx

~dt ~~
dx.

which is in canonical form in the variables xx and xv

19/23. Systems which are isolated but in which effects are
transmitted from one variable to another with some finite delay
may be rendered absolute by adding derivatives as variables.
Thus, if the effect of xx takes 2 units of time to reach a?2, while
x2's effect takes 1 unit of time to reach xv and if we write x(t)
to show the functional dependence,

then ^=/i(«i(ft «#-»»

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 an absolute system which resembles it as closely
as we please.

19/24. If a variable depends on some accumulative effect so

that, say, xt =f\\ <j>{cc2)dt>, then if we put <f>{x2)dt = y, we get

213 P

19/25 DESIGN FOR A BRAIN

the equivalent form

^?- etc

dt — ' ' etC*

which is in canonical form.

19/25. If a variable depends on velocity effects so that, for
instance

dx1 _ fdx2 \

dt -Jl\df *v xv

-jj£ = J2\XV X2)

dx
then if we substitute for -=-* in/i(. . .) we get the canonical form

—jr = fxifiPufitl* xv X2)

2 S" I \

~fa —J2\XV X2)

19/26. 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.

19/27. Explicit solutions of the canonical equations

dxi/dt =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
x\ as a power series in t, is given by

an = <*xx\ (t = 1 n) . . (1)

where X is the operator

/i«. ■ • • . Ogjg + • • • +/"('* • • • > xl)hn ■ (2)

and e'x = l+tX+£x*+~X» + . . . . (3)

214

THE ABSOLUTE SYSTEM 19/28

It has the important property that any function @(xlf . . . , xn)

can be shown as a function of t, if the aj's start from x®, . . . , x„,

by 0(xv . . . , xn) = <^<Z>(^, . . . t 3,0) _ (4)
(2) If the functions jfi are linear so that

-5— = ttj^j -f- di2X2 l • • • 1" a\n&n ~T "1

dxn

dt '

— ClniX^ -j- ^712^2 1 • • ■ "T" "nn^n 1 ^n

(5)

then if the fr's are zero (as can 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
[ciij]. 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.

19/28. Any comparison of an absolute system with the other
types of system treated in mechanics and in thermodynamics
must be made with caution. Thus, it should be noticed that the
concept of the absolute system makes no reference to energy or
its conservation, treating it as irrelevant. It will also be noticed
that the absolute system, whatever the ' machine ' providing it,
is essentially irreversible. This can be established either by
examining the group equations of S. 19/10, the canonical equa-
tions of S. 19/11, or, in a particular case, by examining the field
of the common pendulum in Figure 2/15/1.

Reference

Shannon, C. E. A mathematical theory of communication. Bell System
technical Journal, 27, 379-423, 623-56 ; 1948.

215

CHAPTER 20

Stability

20/1. ' Stability ' is defined primarily as a relation between a
line of behaviour and a region in phase-space because only in
this way can we get a test that is unambiguous in all possible
cases. Given an absolute system and a region within its field,
a line of behaviour from a point within the region is stable if it
never leaves the region.

20/2. 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.

20/3. A resting state can be defined in several ways. In the
field it is a terminating point of a line of behaviour. In the
group equations of S. 19/10 the resting state Xv . . . , Xn is
given by the equations

Xi = Lim Fi(x° ; t) [i = 1, . . . , n) . (1)

t >-00

if the n limits exist. In the canonical equations the values satisfy

fi(Xl3 . . ., Xn)=0 (t = l n) . (2)

A resting state is an invariant of the group, for a change of t
does not alter its value.

dxj

be symbolised by J, is not identically zero, then there will be
isolated resting states. If J = 0, but not all its first minors are
zero, then the equations define a curve, every point of which
is a resting state. If J = 0 and all first minors but not all second
minors are zero, then a two-way surface exists composed of
resting states ; and so on.

If the Jacobian of the /'s, i.e. the determinant

which will

20/4. Theorem : If the /'s are continuous and differentiable,
an absolute system tends to the linear form (S. 19/27) in the
neighbourhood of a resting state.

216

STABILITY

20/5

Let the system, specified by

dxi/dt =fi(xv • • • » xn) (t = 1, . . . , n)
have a resting state Xj, . • . , Xn, so that

fi(Xv . . . , Xn) = 0 (t = 1 n)

Put Xi = Xi -f- & (i = 1, . . . , m) so that xi is measured as a

deviation ft from its resting value. Then

(Xi + ft) =fi(X1 + & xn + ft.)

(i = 1, . . . , n)

Expanding the right-hand side by Taylor's theorem, noting that
dXi/dt = 0 and that/i(Z) = 0, we find, if the £'s are infinitesimal,
that

d£i _ dfi dft

si T" • • • ~r 37?n

(* = L

., n)

* aii * ' ' " ' ' din-

The partial derivatives, taken at the point Xv . . . , Xn, are
numerical constants. So the system is linear.

20/5. 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 de-
scribed are widely applicable.

Let the linear system be
dxi
dt

&%-iX-,

0>i2p2 T" • • • i Min^n

(i = i,

n) (1)

or, in the concise matrix notation (S. 19/27)

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 resting state. The determinant

'ii

■X a

22 ■

-;. .

when expanded gives a polynomial

'\n
' 2 n

(Inn A.

in A of degree n which, when

equated to 0, gives the characteristic equation of the matrix A

217

20/6

DESIGN FOR A BRAIN

20/6. Each coefficient rm is the sum of all i-vowed principal
(co-axial) minors of A, multiplied by (— 1)*. Thus,

ml = — («11 + «22 + • • • + ann) \ Wl„ = (— l)n | A |.

Example : The linear system

dxjdt = — 5x± + 4a?2 — 6^3!

7a; 1

6x2 + 8x3
4#3,

dxjdt =

dx3/dt = — 2xx + 4^2

has the characteristic equation

A3 + 15A2 + 21 + 8 = 0

20/7. Of this equation the roots Xlt . . . , AB are the latent

roots of ^4. The integral of the canonical equations gives each
X{ as a linear function of the exponentials eV, . . . , eV. For
the sum to be convergent, no real part of Als . . . , An must be
positive, and this criterion provides a test for the stability of
the system.

Example : The equation A3 + 15A2 -f 2 A + 8 = 0 has roots
— 14-902 and — 0-049 ± 0-729 V^^T, so the system of the
previous section is stable.

20/8. 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.

mx 1

m1 1 0

mx 1

m3 m2

in 3 vi 2 m1

mz m 2

m5 7?i4 mz

m5 m±

m7 m6

(where, i

f q > n,

= 0), are all

sitive.

Example : The system with characteristic equation
P + 15A2 + 2A + 8 = 0
yields the series

+ 15,

8
0

0
15

These have the values + 15, + 22, and + 176.
is stable, agreeing with the previous test.

218

So the system

STABILITY

20/11

20/9. If the coefficients in the characteristic equation are not
all positive the system is unstable. But the converse is not
true. Thus the linear system whose matrix is

i V6 °

— V6 i °

0 0 —3^

has the characteristic equation A3+A2 + A + 21=0; but the
latent roots are + 1 ± V— 6 and — 3 ; so the system is unstable.

20/10. Another test, related to Nyquist's, states that a linear
system is stable if, and only if, the polynomial

ln + mj"-1 + m2Xn~2 +--•+«■
changes in amplitude by nn when A, a complex variable
(A = a + hi where i = V— 1), goes from - t oo to + t co along
the fr-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 renders the system
non-absolute and is therefore based on an approach different from
ours.

20/11. Some further examples will illustrate various facts
relating to stability.

Example 1 : If a matrix [a] of order n x n has latent roots
Al9 . . . , An, then the matrix, written in partitioned form,

0 ! /

of order 2n x 2/?, where / is the unit matrix, has latent roots

± VI7, . . . , ± a/A„. It follows that the system

d2x-

— ! = di1x1 + ai2x2 + . . . + ainXn (« = 1, . . . , «)

of common physical occurrence, must be unstable.

Example 2 : The diagonal terms an represent the intrinsic
stabilities of the variables ; for if all variables other than xi are
held constant, the linear system's i-th. equation becomes
dxi/dt — auxi + c,
219

20/11 DESIGN FOR A BRAIN

where c is a constant, showing that under these conditions Xi
will converge to — c/au if an be negative, and will diverge without
limit if an be positive.

If the diagonal terms an are much larger in absolute magnitude
than the others, the roots tend to the values of an. It follows
that if the diagonal terms take extreme values they determine
the stability.

Example 3 : If the terms aij 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 4 : The matrix of the homeostat equations of S, 19/11
is

a^h a12h a13h alih
a9,h a99h a9Ji a9Ji

24'

-J

h a»Ji a^Ji anji

33'

-J

_ailh ai9h ai3h a^Ji

-1

If j = o, the system must be unstable (by Example 1 above).
If the matrix has latent roots fiv . . . , /u8, and if Al5 . . . , A4
are the latent roots of the matrix [a%jh]9 and if j ^ 0, then the
A's and ^'s are related by Xp = jbt2q -f- jjuq. As ; — > oo the 8-variable
and the 4-variable systems are stable or unstable together.

Example 5 : In a stable system, fixing a variable may make
the system of the remainder unstable. For instance, the system
with matrix

6 5 - 10"

- 4 - 3 - 1
4 2 - 6 .

is stable. But if the third variable is fixed, the system of the
first two variables has matrix

L-4 -3J

and is unstable.

Example 6 : Making one variable more stable intrinsically

220

STABILITY

20/12

(Example 2 of this section) may make the whole unstable. For
instance, the system with matrix

is stable. But if alx becomes more negative, the system becomes
unstable when axl becomes more negative than — 4 J.
Example 7 : In the n x n matrix

c i d

in partitioned form, [a] is of order k X k. If the k diagonal
elements an become much larger in absolute value than the rest,
the latent roots of the matrix tend to the k values an and the
n — k latent roots of [d]. Thus the matrix, corresponding to [d],

1
.1
has latent roots -- 1-5 i l-658«, and the matrix

— 1 2 0"
100-1 2

— 3 1—3

— 1 1 2.

has latent roots -- 101-39, — 98-62, and + 1-506 ± 1-720*.

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 8 : The holistic nature of stability is well shown by
the system with matrix

— 3 — 2 2'
-6-5 6

— 5 2 — 4_
in which each variable individually, and every pair, is stable ;
yet the whole is unstable.

100

— 2

The probability of stability

20/12. The probability that a system should be stable can be
made precise by the point of view of S. 14/16. We consider

221

20/12 DESIGN FOR A BRAIN

an ensemble of absolute systems

da%/dt =fi(xlt . . . , xn ; olv . .-.) (i = 1, . . . , n)

with parameters oy, such that each combination of a-values gives
an absolute system. We nominate a point Q in phase-space, and
then define the ' probability of stability at Q ' as the proportion
of a-combinations (drawn as samples from known distributions)
that give both (1) a resting state at Q, and (2) stable equilibrium
at that point. The system's general ' probability of stability ' is
the probability at Q averaged over all Q-points. As the proba-
bility will usually be zero if Q is a point, we can consider instead
the infinitesimal probability dp given when the point is increased
to an infinitesimal volume dV.

The question is fundamental to our point of view ; for, having
decided that stability is necessary for homeostasis, we want to
get a system of 1010 nerve-cells and a complex environment
stable by some method that does not demand the improbable.
The question cannot be treated adequately without some quan-
titative study. Unfortunately, the quantitative study involves
mathematical difficulties of a high order. Non-linear systems
cannot be treated generally but only individually. Here I shall
deal only with the linear case. It is not implied that the nervous
system is linear in its performance or that the answers found
have any quantitative application to it. The position is simply
that, knowing nothing of what to expect, we must collect what
information we can so that we shall have at least some fixed
points around which the argument can turn.

The applicability of the concept of linearity is considerably
widened by the theorem of S. 20/4.

The problem may be stated as follows : 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 non-positive 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 -f- a. Nevertheless,
some answer is desirable, so the ' rectangular ' distribution (integers
evenly distributed between — 9 and + 9) was tested empirically.
Matrices were formed from Fisher and Yates' Table of Random

222

STABILITY

20/12

Numbers, and each matrix was then tested for stability by Hurwitz'
rule (S. 20/8 and S. 20/9). 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 as it is unstable by S. 20/9. The testing becomes
very time-consuming when the matrices exceed 3x3, for the time
taken increases approximately as /i5. The results are summarised
in Table 20/12/1.

Order of
matrix

Number
tested

Number
found
stable

Per cent
stable

2x2
3x3

4x4

320
100
100

77
12

24
12

Table 20/12/1.

The main feature is the rapidity with which the probability
tends to zero. The figures given arc compatible (x2 = 4-53,
P = 0-10) with the hypothesis that the probability for a matrix
of order n x n is l/2n. 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 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

223

20/12

DESIGN FOR A BRAIN

2 the probability is 3/4 (instead of 1/4). Some empirical tests
gave the results of Table 20/12/2.

Order of
matrix

Number
tested

Number
found
stable

Per cent
stable

2x2
3X3

120

100

87
55

72
55

Table 20/12/2.

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, and the percentage of stable combinations found when
the number of units was two; the percentage was then found
for the same general conditions except that three units interacted ;

ioo

2 3

number of variables
Figure 20/12/1.

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 encour-
aged, 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 units was increased. The results
are given in Figure 20/12/1.

224

STABILITY 20/12

These results prove little ; but they suggest that the proba-
bility of stability is small in large systems assembled at random.
It is suggested, therefore, that large systems should be assumed
unstable unless evidence to the contrary can be produced.

References

Ashby, W. Ross. The effect of controls on stability. Nature, 155, 242 ;

1945.
Idem. Interrelations between stabilities of parts within a whole dynamic

system. Journal of comparative a?id jihysiological Psychology, 40, 1 ;

1947.
Idem. The stability of a randomly assembled nerve-network. Electro-
encephalography and clinical neurophysiology, 2, 471 ; 1950.
Frazer, R. A., and Duncan, W. J. On the criteria for the stability of small

motions. Proceedings of the Royal Society, A, 124, 642 ; 1929.
Hurwitz, A. t)ber die Bedingungen, unter welchen eine Gleichung nur

Wurzeln mit negativen reellen Teilen besitzt. Mathematische Annalen,

46, 273 ; 1895.
Nyquist, H. Regeneration theory. Bell System technical Journal, 11, 126 ;

1932.

225

CHAPTER 21

Parameters

21/1. With canonical equations

— * = fi(x1, . . . , xn) {i = 1, . . . , ri),

the form of the field is determined by the functional forms ft
regarded as functions of xv . . . , xn. If parameters av a2, . . .
are taken into consideration, the system will be specified by
equations

-j— = Ji\&ii . . . , xn ; fit 1? &2, • • •/ [t = x9 , m , 9 71).

If the parameters are constant, the #'s continue to form an absolute
system. If the a's can take m combinations of values, then the
oj's form m different absolute systems, and will show m different
fields. If a parameter can change continuously (in value, not in
time), no limit can be put to the number of different fields which
can arise.

If a parameter affects only certain variables directly, it will
appear only in the corresponding /'s. Thus, if it affects only
xx directly, so that the diagram of immediate effects is

(X * X -i ^ Xn)

then a will appear only in fx :

dxj&t =f1(x1, x2; a)
dxjdt =f2{xlf x2).

But it will in general appear in all the F's of the integrals (S. 19/10).
The subject is developed further in Chapter 24.

Change of parameters can represent every alteration which can
be made on an absolute system, and therefore on any physical
or biological ' machine '. It includes every possibility of experi-
mental interference. Thus if a set of variables that are joined
to form the system x = f(x) are changed in their relations so
that they form the system x = <j>(x), then the change can equally

226

PARAMETERS 21/2

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) =yj(a:; 1)
(j)(x) = ip(x ; 2)
then the two representations are identical.

As example of its method, the action of S. 8/10, 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 xlt x2 and x3 were used, and that the
magnets of 1 and 2 were joined. Before joining, the equations
were (S. 19/11)

dxjdt = a11x1 + a12x2 + a13x3^\

dx2/dt = a21X± -f- «22^2 + a22X3 f

dxjdt = a31x± + a32x2 + a33x3)

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 = (a±1 + a12 + a21 + a22)xx + {a13 + a^)^
dxjdt = (a31 + 032)^ + a32x3

It is easy to verify that if the full equations, including the parameter

bt were :

dxjdt = {alx + b(a12 + a21 + a22)}x1 + (1 - b)a12x2

+ (« is + ^23)^3
dxjdt = a21xx + a22x2 + o23x3

dxjdt = (a31 + ^32)^! + (1 — b)a32x2 + a33x3_

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.)

21/2. A variable x^ 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 x%

(2) In the canonical equations, fk(xlt . . . , xn) is identically

zero.

227

21/3 DESIGN FOR A BRAIN

(3) In the group equations, Fk(x®, . . . , a?J ; t) = x^.
(Some region of the phase-space is assumed given.)
Since we usually consider absolute systems, we shall usually
require the parameters to be held constant. Since null-functions
also remain constant, the properties of the two will often be
similar. (A fundamental distinction by definition is that para-
meters are outside, while null-functions may be inside, the given
system.)

21/3. In an absolute system, the variables other than the step-
and null-functions will be referred to as main variables.

21/4. Theorem : In an absolute system, the system of the main-
variables forms an absolute subsystem provided no step-function
changes from its initial value.

Suppose xl9 . . . , Xjt are null- and step-functions and the main-
variables are Xk+u . . . , xn. The canonical equations of the
whole system are

dxjdt = 0

dxjc/dt — 0
dxk+i/dt = fk+i(xv . . . , xk, xk+i, . . . , xn)

dXn/dt =fn(xv . . . , X*, Xk+1, . . . , Xn)

The first k equations can be integrated at once to give xx = x\,
. . ., Xk = xQk. Substituting these in the remaining equations
we get :

dxk+i/dt = fk+i{x\, . . ., x% xk+if . . ., xnT\

ClXn/dt — Jn\pC\i • • • 5 #jfc> Xk+1, • • •» xn) J

The terms x^, . . ., x^ are now constants, not effectively functions
of t at all. The equations are in canonical form, so the system is
absolute over any interval not containing a change in a?J, . . . , x^.
Usually the selection of variables to form an absolute system
is rigorously 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 ' machine ' itself. The theorem,
however, shows that without affecting the absoluteness we may take

228

PARAMETERS 21/6

null-functions into the system or remove them from it as we
please.

It also follows that the statements : ' parameter a was held con-
stant at a0 \ and c the system was re-defined to include a, which,
as a null-function, remained at its initial value of a0 ' are merely
two ways of describing the same facts.

21/5. The fact that the field is changed by a change of parameter
implies that the stabilities of the lines of behaviour are changed.
For instance, consider the system

dx/dt = — x -f ay, dy/dt = x — y -f 1
where x and y have been used for simplicity instead of x± and x2.
When a — 0, 1, and 2 respectively, the system has the three
fields shown in Figure 21/5/1.

Figure 21/5/1 : Three fields of x and y when a has the values (left to
right) 0, 1, and 2.

When a = 0 there is a stable resting state at a? = 0, y = 1 ;

when a = 1 there is no resting state ;

when a = 2 there is an unstable resting state at x = — 2,

y = -l.
The system has as many fields as there are values to a.

21/6. 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 group equations.

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
' joining ' two systems. A better method is to equate para-
meters in one system to variables in the other. When this is

229 q

21/7 DESIGN FOR A BRAIN

done, the second dominates the first. If parameters in each are
equated to variables in the other, then a two-way interaction
occurs. For instance, suppose we start with the 2-variable
system

dx/dt = fJx, y; a)\ , .. „ . . . _ . _

, ,, _ // \ fand the 1 -variable system dz/dt = 6(z; b)
ay /at — j2(x, y) j

then the diagram of immediate effects is

a— > x+±y b—> z

If we put a = z, the new system has the equations

dx/dt =f1{x, y; z)\

dy/dt =f2{x, y) >

dz/dt = cf>(z ; b) J
and the diagram of immediate effects becomes

b — > z — ► x ^=t y.
If a further join is made by putting b = y, the equations become

dx/dt —fiix, y; z)
dy/dt =f2(x, y)
dz/dt = <j>(z ; y)
and the diagram of immediate effects becomes

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.

21/7. The stabilities of separate systems do not define the
stability of the system formed by joining them together.

In the general case, when the/'s are unrestricted, this propo-
sition is not easily given a meaning. But in the linear case (to

230

PARAMETERS 21/7

which all continuous systems approximate, S. 20/4) the meaning is
clear. Several examples will be given.

Example 1 : Two systems may be stable if joined one way, and
unstable if joined another. Consider the 1 -variable systems
dx/dt = x + 2px -f Vz and dy/dt = — 2r — 3y. If they are
joined by putting r = x, px = y, the system becomes

dx/dt = x -f 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 : Several systems, all stable, may be unstable when
joined. Join the three systems

dx/dt = — x — 2q — 2r
dy/dt = — 2p — y -f- r
dz/dt = p -f a — 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: Systems, each unstable, may be joined to form a
stable whole. Join the 2-variable system

dx/dt = Sx — Sy — Sp

dy/dt = 3x — 9y — 8p^
which is unstable, to dz/dt = 21 g -j- 3r -J- 3^, which is also
unstable, by putting q — x, r = y, p = z. The whole is stable.
Example 4 : If a system
dxi/dt =fi{xv . . . , xn ; al9 . . .) (i = 1, . . . , n)

is joined to another system, of ?/'s, by equating various a's and i/'s,
then the resting states that were once given by certain com-
binations of x and a will still occur, so far as the ^-system is
concerned, when the ?/'s take the values the a's had before. The
zeros of the/'s are thus invariant for the operations of joining and
separating.

231

CHAPTER 22

Step-Functions

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.
The term ' step-function ' will also be used, for convenience, to
refer to any physical part whose behaviour is typically of this
form.

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 centimetre 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

/ dx\
{mdt)

where g is the acceleration due to gravity. This equation is not
in canonical form, but may be made so by writing x = xXi
dx/dt = x2, when it becomes

(2)

232

STEP-FUNCTIONS

22/2

If the elastic breaks, k becomes 0, and the equations become

dx1

dt
dx2
~dt

(3)

Assume that the elastic breaks if it is pulled longer than X.

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 x2, 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 {xx 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 : scl9 x2, and k. This system is absolute, and has
one field, shown in Figure 22/2/2.

In this form, the step-function must be brought into the
canonical equations. A possible form is :

dk (K K
dt = q[-2 +2

where K is the initial value of the variable k, and q is large and
positive. As q— ■> oo, the behaviour of k tends to the step-
function form.

Another method is to use Dirac's ^-function, defined by S(u) = 0
if u ?±0, while if u = 0, d(u) tends to infinity in such a way that

rco

I d(u)du = 1.

J —00

233

+ - tanh {q(X - x,)} - k

(4)

22/2

DESIGN FOR A BRAIN

Then if du/dt = <5{</>(w, v, . . .)}, du/dt will be usually zero ; but
if the changes of w, v , . . . take </> through zero, then d(u) becomes
momentarily infinite and n will change by a finite jump. These

Figure 22/2/2 : Field of the 3-variable system.

representations are of little practical use, but they are important
theoretically in showing that a step-function can be represented
in the canonical equations.

22/3. In an absolute 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
and to the right of the line xx = X are critical states for the step-
f unction 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

<f>(kt X{, . . . , Xn) = 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 x1 = 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

231

STEP-FUNCTIONS 22/5

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
two, three or four units are in use, the critical surfaces will form a
square, cube, or tesseract respectively in the phase-space around
the origin. The critical states will fill the space outside this sur-
face. 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
sometimes to break and sometimes to replace the elastic, and if
we were to test the behaviour of the system xv x2 over a prolonged
time including many such actions, we would find that the system
was often absolute with a field like A of Figure 22/2/1, and often
absolute 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 intermittent change by some other,
unobserved system, a system might be found to have r fields.

22/5. The argument can, however, be reversed : if we find that
a subsystem has r fields we can deduce, subject to certain restric-
tions, that the other variables must include step-functions.

Theorem : If, within an absolute system xv . . . , xn, xp, . . . , xs,
the subsystem xl9 . . . , xn is absolute within each of r fields
(which persist for a finite time and interchange instantaneously)
and is not independent of xPi . . . , xs ; then one or more of
Xp, . . . , xs must be step-functions.

Consider the whole system first while one field persists. Take
a generic initial state x\, . . . , x% x^, . . . , x°s and allow time tx
to elapse ; suppose the representative point moves to a?i, . . . , xn,

x'v xs, where each x' is not necessarily different from x°.

Let further time t2 elapse, the point moving on to x'u • • • > #n>
x'p, . . . , x". Now consider the line of behaviour that follows
the initial state x[, . . . , xn, x°p, xq, . . . , x's, differing from the

235

22/5 DESIGN FOR A BRAIN

second point only in the value of xp : as the subsystem is absolute,
an interval t2 will bring its variables again to x\, • • • , ®»» i-e. these
variables' behaviours are the same on the two lines. Now xp
either is, or is not, equal to x°r If unequal, then by definition
(S. 14/3) x19 . . . , xn is independent of xv. So the behaviour
of x19 . . . , xn over t2 will show either that x'v = x® (i.e. that

xp did not change over t±) or that xx xn is independent of

xp. Similar tests with the other variables of the set xv, . . . , xs
will enable them to be divided into two classes : (1) those that
remained constant over tv and (2) those of which the subsystem
Xl9 . . . , Xn is independent. By hypothesis, class (2) may not
include all of xp, . . . , xg ; so class (1) is not void.

When a field of xv . . . , xn changes, some parameter to this
system must have changed value. As xlt . . . , xn, xp, . . . , x8
is isolated, the ' parameter ' can be none other than one or more of
xPi . . . , xs. As the field has changed, the parameter cannot be
in class (2). At the change of field, therefore, at least one of
those in class (1) changed value. So class (1), and therefore the
set xp, . . . , Xs, contains at least one step-function.

Reference

Ashby, W. Ross. Principles of the self-organising dynamic system. Journal
of general Psychology, 37, 125 ; 1947.

236

CHAPTER 23

The Ultrastable System

23/1. The definition and description already given in S. 8/6 and
7 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 Xi
and of step-functions at, so that the whole is absolute :

-£ =fi(x; a) (i = 1, . . . , n)

d^ = gi(x; a) (i = l, 2, . . .)

The functions gi must be given some form like that of S. 22/2.
The system is started with the representative point within the
critical surface cf)(x) = 0, contact with which makes the step-
functions change value. When they change, the new values are
to be random samples from some distribution, assumed given.

Thus in the homeostat, the equations of the main variables are
(S. 19/11) :

dx'

-± = ailx1 + ai2 x2 + ai3 x3 + ati x± (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 Xj
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

237

23/2 DESIGN FOR A BRAIN

computed to any degree of accuracy by a numerical method. The
proof given in Chapter 8, though verbal, is adequate to establish
the elementary properties of the system. A rigorous statement
and proof would add little of real value.

23/2. 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,

i/P.

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 w-th field will
be pqu~A. So the average number of trials made will be
p -f 2pq + 3pq2 + . . . + upqu~x + . . . _ 1
V + V9. + Vf + • • • + Vt~x + • • • ~ V

23/3. In an ultrastable system, a field may be terminal and yet
show little resemblance to the ' normal ' equilibrium which is
necessary if the system is to show, after each of a variety of dis-
placements, a return to the resting state. A field, for instance,
might have a resting state at which only a single line of behaviour
terminated : if the representative point were on that line the field
would be terminal ; but hardly any displacement would be
followed by a return to the resting state.

It can, however, be shown that if a proportion of the fields
evoked by the step-function changes are of this or similar type,
then the terminal fields will contain them in smaller proportion.
For, given a field and a closed critical surface, let k± be the pro-
portion of lines of behaviour crossing the boundary which are
stable. Thus in Figure 8/7/1, in I kx = 0, in II ht = 0, in III
/c1 = i approximately, and in IV k± = 1. To count the lines,
the boundary surface could be divided into portions of equal
area, small enough so that stable and unstable lines do not pass
through the same area. Then if we assume that in any field the
representative point is equally likely to start at any of the small
areas, a field's chance of being terminal is proportional to kv

238

THE ULTRASTABLE SYSTEM

23/4

It follows that if the changes of step-functions evoke fields whose
values of k1 are distributed so that the probability of a field having
a A^-value between kx and kt + dkx is ^(k-^dk^ then in the terminal
fields the probablity is

k1y)(k1)dk1

Figure 23/3/1 shows a possible distribution of values of k1 in

Figure 23/3/1 : Solid line : a distribution y^i) 5 broken line :
the corresponding distribution k^kj.

the original fields (solid line), and how k± would then be dis-
tributed in the terminal fields (broken line). The shift towards
the higher values of kx is clear.

Fields with a low value of kv unsatisfactory for adaptation,
tend therefore not to be terminal.

23/4. It was noticed in S. 13/4 that fields like A and B of Figure
13/4/1, though terminal, are defective in their persistence after
small random disturbances. This idea may be given more
precision.

Assume that the small random disturbances cause displace-
ments which have some definite probability distribution, Gaussian
say, so that if applied to the representative point when it is at
some definite position in the field, there is a definite probability
k2 that a random displacement will not carry the point beyond
the critical surface. Assume the representative point is always
at the resting state or resting cycle. Then any terminal field has
a unique value for k2. If the field contains a single resting state,
k2 for that field is the probability, when the representative point

239

23/4 DESIGN FOR A BRAIN

is at the resting state, that the application of a single random dis-
turbance will not take the representative point beyond the critical
surface. If the field has a resting cycle, k2 is the average of the
values when the representative point is on the many portions of
the cycle, the value for each portion being weighted according
to the time spent by the representative point in that portion. For
more complex fields, k2 could be defined, but a more detailed
study is not necessary here.

Suppose that the ultrastable system, when the step-functions
undergo random changes, yields terminal fields whose values of k2
are distributed so that the proportion falling between k2 and
k2 + dk2 is cf>(k2)dk2. If to such fields, with k2 lying between such
limits, we apply one random disturbance, a proportion k2 will
not be changed ; but the proportion 1 — k2 will be changed, and
will be replaced by new terminal fields ; their values of k2 will be
distributed again as <j)(k2)dk2, and this distribution will be added
to that of the unchanged fields. In this way it is easy to show
that the final distribution X{k2) equals

where A is a constant.

Examination of the form of the distribution k{k2) shows that
it is cf>{k2) heavily weighted in favour of the values of k2 near 1.
Such fields can only be those with the resting state or cycle near
the centre of the region. So the result confirms the common-
sense argument of S. 13/4. It will be noticed that the deduction
is independent of the particular form of the distribution of
disturbances.

References

Asiiby, W. Ross. The physical origin of adaptation by trial and error.

Journal of general Psychology, 32, 13 ; 1945.
Idem. The nervous system as physical machine : with special reference to

the origin of adaptive behaviour. Mind, 56, 1 ; 1947.

240

CHAPTER 24

Constancy and Independence

24/1. 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.

24/2. But a relationship which can be treated in detail is that of
' independence '. By the principle of S. 2/8 it must be defined in
terms of observable behaviour.

Given an absolute system and two lines of behaviour from two
initial states which differ only in their values of x® (the difference
being A#°), the variable xk is independent of Xj if xk's behaviour is
identical on the two lines. Analytically, xk is independent of Xj
in the conditions given if

Fk(4, . . . , x% . . . ; t) = Fk(xl . . . , ^ + AflJ, . . . ; t) (1)
as a function of t. In other words, xk is independent of Xj
if XfcS behaviour is invariant whemthe initial state is changed
by A4

This narrow definition provides the basis for further develop-
ment. In practical application, the identity (1) may hold over all
values of Aa?° (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

241

24/3 DESIGN FOR A BRAIN

r-Q Ffc(aj?, . . . , a?2 » /) = 0. (These relations and notations are

collected in S. 24/19 for convenience in reference.)
Example : In the system of S. 19/10

X-^ I= X^ -j- XyZ ~i~ t

x2 = x% + 2t
x2 is independent of xv but xx is not independent of x2.

24/3. We shall be interested chiefly in the independencies intro-
duced when particular variables become constant : when they are
part-functions, for instance. Such constancies are most naturally
expressed in the canonical equations, for here are specified the
properties of the parts before assembly (S. 19/19). We there-
fore need a method of deducing the independence from the
canonical equations, preferably without an explicit integration.
Such a method is developed below in S. 24/3 to 10. (The method
recently developed by Riguet, however, promises to be much
better.)

Given an absolute system

-^ =fi(x1, . . . , xn) (t = 1, . . - , n) . (1)

it is required to find whether or not x^ is independent of Xj, some
region of values being assumed. The region must not include
changes of values of step-functions or of activations of part-
functions ; for the derivatives required below may not exist,
and the independencies may change.

If the functions fi are expandable by Taylor's series around the
point X®, . . . , x„, we may write their integrals symbolically
(S. 19/27) as

Fi(4, . . . , xl ; t) = <**x\ {i = 1, . . . , n) . (2)

where X is the operator

(The zero superscripts will now be dropped as unnecessary.)

Expanding the exponential, and operating on (2) with =— ,
the test whether xk is independent of Xj becomes whether

sf—0 <*- 1. *...•> • • (3)

242

CONSTANCY AND INDEPENDENCE 24/5

By expanding ^— A':

£- X'+'a:* = £& £- X*xt + X ~ X»xk . (4)

OXj '—' OXj OXp OXj

Applying the test (3), if the test for /j, = m gives

5- XmXk = 0

OXj

then for /li = m -j- 1, by using (4) we need only see whether

?%k**- ■ ■ ■ ^

24/4. We now add the hypothesis that the system is linear (S.
19/27). The restriction is unimportant as no arguments are used
elsewhere which depend on linearity or on non-linearity. Further,
in the region near a resting state all systems tend to the linear
form (S. 20/4), and this region has our main interest.
Starting with ju = 1 the tests 24/3 (5) become

OXj

sfi a/, _

dxp dxj

~* — ' dxp dxa dxj
etc.

(i)

These tests now use only the /'s, as required. They are both
necessary and sufficient. They have been shown necessary ; and
by merely retracing the argument they are found to be sufficient.
Only the first n — 1 tests of (1) above are required, for products
which contain more than n — 1 factors must include products
already given, in the first n — 1 tests, as zero.

The tests are, however, clumsy. The simplicity and directness
can be improved by using the facts that we need distinguish only
between zero and non-zero quantities, and that the sums of (1)
above resemble the elements of matrix products. Sections 24/5-10
develop this possibility.

24/5. An R0- matrix has elements which can take only two
values : R (non-zero) and 0 (zero). The elements therefore

243

24/6 DESIGN FOR A BRAIN

combine by the rules

R + R = R, 0+0=0, # + 0=0 + ]? = #,
R x R = R, 0x0=0, Rx0=0xR=0.

A sum of such elements can therefore be zero in general only if

each element is zero.

24/6. In an 7?0-matrix of order n x n, the zeros are patterned
if, given any zero not in the principal diagonal, we can separate
the numbers 1, 2, . . . , n into two sets a and /? (neither being
void) so that the minor left after suppressing columns a and rows /?
is composed wholly of zeros which include the given zero. For
example, the 720-matrix

0 0

R R

0 R

0 0

R
R
R

has its zeros patterned. Selecting, for instance,
the third row, we can make a = 1, 3, 4 and /? = 2.
the minor

the zero in
This leaves

0 .
0 .

where dots indicate eliminated elements ; the remaining elements
are all zero, and they include the selected zero. The other zeros
in the original matrix can all be treated similarly.

24/7. Some necessary theorems will now be stated. Their proofs
are simple and need not be given here.
A matrix A is idempotent if A2 = A.
Theorem : If an 7?0-matrix has no zeros in the principal diagonal
a necessary and sufficient condition that the zeros be patterned
is that the matrix be idempotent.

24/8. Theorem : If A is an i?0-matrix of order n X w, and I
is the matrix with 72's in the principal diagonal and zeros elsewhere,
then the matrix

I + A + A2 + . . . + An~x
is idempotent.

244

CONSTANCY AND INDEPENDENCE 24/10

24/9. From the /'s of the canonical equations (24/3(1) ) form
the differential matrix [f] by inserting, in the (kj)-th position
(at the intersection of the A:-th row and the ;'-th column) an 0 or
R according as dfk/dxj is, or is not, zero (in the region of phase-
space considered). Then the square, cube, etc., of [/] will contain
in the (A^')-th position an element which is zero or non-zero as the
second, third, etc., tests of 24/4(1) are or are not zero. If now
these powers are summed, to S :

« = m + [/? + • • • + 1/]-1, . . (i)

a zero element in S at the (Jcj)-th position means that all the
terms of the series were zero, and therefore that Xk is independent

Of Xj.

The same independence will make zero the element at the {kj)-t\\
position in the matrix whose (^)-th element is zero or non-zero
as dFn/dJl is or is not zero. This integral matrix, [F], must
therefore satisfy

[*]=« (2)

24/10. The restriction is now added that the behaviour of each
variable xt is to depend on its own starting-point. (Physical
systems not conforming to this restriction are, so far as I am
aware, rare and peculiar.) The principal diagonal of [F] will
then be found to have all its elements non-zero. In such a case,
[F] is not altered if we add to it the matrix / of S. 24/8, and we may
sum up as follows :

If a dynamic system is specified by

— l =fi{x1, . . . , xn) (i = 1, . . . , n)

and if [/] is an jRO-matrix where each (ftj)-th element is 0 or R

as ~- is or is not zero respectively (in some region within which

CXj

the nullity does not change), and if [F] is an .RO-matrix where
each (kj)-th element is 0 or J? as Xk is or is not independent of Xj
respectively in the same region, and if each x's behaviour depends
on its own starting point, then

m = [/] + [/]« + • • ■ + [nn-1 ■ ■ a)

This equation gives the independencies when the differential
matrix is given; for xk is or is not independent of x5 as the

245 r

24/11 DESIGN FOR A BRAIN

element in the k-th row and the j-th column of the integral
matrix is or is not zero respectively.

The advantage of equation (1) is that the differential matrix
is often formed with ease (for only zero or non-zero values are
required), and often the first multiplication shows that [/]2 = [/].
When this is so, the integral matrix is at once proved to be equal
to [/], and all the independencies are obtained at once. A
further advantage is that the theory of partitioned matrices can
often be used, with considerable economy of time. The next
few sections provide some examples.

24/11. In an absolute system the independencies cannot be
assigned arbitrarily.

By the theorem of S. 24/8, the integral matrix, being the sum
of powers, is idempotent ; and therefore, by S. 24/7, has its zeros
patterned. The independencies of an absolute system must
always be subject to this restriction.

What is really the same line of reasoning may be shown in an
alternative form. The group property requires (S. 19/10) that

Fk{Fx(x« ; /), F2(x» ;*),...;«'} = *W4 4 - • - I * + O;
so if X/c is independent of Xj then x° will not appear effectively on
the right-hand side, and it must therefore not appear effectively
on the left. So if, say, Fm( . . . ; t) contains x°p then x ^ must not
occur in Fk ; so xk must be independent of xm as well.

24/12. If the variables of an absolute system are divisible into
two groups A and B, such that all the variables of A are inde-
pendent of B, but not all those of B are independent of 4> then
the subsystem A dominates the subsystem B.

Theorem : The subsystem A is itself absolute.

Write down the group equations of the A's :

FA{F^; t), F2(x°; /),...; t'} = FA{x°v «& . . .; t + t'}
where the subscript a refers to all the members of A in succession.
Each FA is independent of a?jj, so, omitting the unnecessary
symbols from each side both from the F's and from the x°'s, we get

FA{FA{xl; t), . . .; t'} = FA{x°A; t + t'}
where the change of subscript means that only the members of
A are now included. Inspection shows that these are the equa-

246

CONSTANCY AND INDEPENDENCE 24/14

tions of a finite continuous group in the variables A. So the
^4's form an absolute system.

The fact of dominance may be shown in the integral matrix by
finding that the deletion of columns A and rows B leaves only
zeros ; but the deletion of columns B and rows A leaves some
non-zero elements. (If the second operation also leaves only
zeros, then the system really consists of two completely inde-
pendent subsystems ; the whole system is 4 reducible '.)

24/13. If A, B, and C are systems such that they together form
one absolute system, and if A dominates B, and B dominates C,
then A dominates C.

On the information given, [F], in partitioned form, can be
filled in but for two elements, shown as dots :

It must be idempotent (S. 24/11). Trying the four possible
combinations of R and 0 for the two undefined elements, we find
that there must be 0 at the top right corner, and R at the bottom
left. A therefore dominates C.

The theorem illustrates again the importance of the concept of
' absoluteness ' ; for without this assumption the theorem,
obvious physically, cannot be proved (for lack of the group
property).

24/14. An account of the primary effects of part-functions on
the independencies within an absolute system can now be given.
The definition of a part-function xp implies that over finite
regions of values of xl9 . . . , xn fp[ocl9 . . . , xn) becomes zero.
Within such a region, i.e. while not activated, the canonical
equations include dxp/dt = 0, which can be integrated at once to
xp = a£ ; so Fp{x° ; t) = x°p ; and xp and Fp are both constant.
dFp/dxj is therefore zero for all values of j other than p. The
effect of a part-function xp being inactive is therefore to make
the whole of the p-th row of the differential and integral matrices
zero (except for the element in the main diagonal, which remains
an R).

247

24/14

DESIGN FOR A BRAIN

It will be recognised that [/] and [F], the differential and
integral matrices, are the matrix equivalents of the diagrams of

■^2
A

—►3

Figure 24/14/1.

immediate and ultimate effects respectively. Thus the diagram
of immediate effects A in Figure 24/14/1 yields the diagram of
ultimate effects B. For the system, [/] is

assuming (S. 24/10) that the terms in the main diagonal are all R.
Then

R 0

R R

0 R

R R

[f? =

and the sum /+[/]+ l/T + L/T gives

0 "

This, by S. 24/10, is the integral matrix. If it is compared directly
with B of the Figure, the agreement will be found complete.
Thus, it may be verified in both that xx dominates the system of

it/05 ^3 anu & a.

The rule of S. 14/10 for the formation of the diagram of ulti-
mate effects when a variable is an inactive part-function can now
be proved. For the effect on the differential matrix of a part-
function xi being inactive is to make all the elements in the i-th.
row zero, except the element in the main diagonal. Exactly
the same change is caused in the differential matrix if we remove
those arrows whose heads are at a&i. After these two changes

248

CONSTANCY AND INDEPENDENCE 24/15

the correspondence continues as before. Thus, A of Figure

14/10/1 has

[/] = 0 R 0 R and I**]

And a?3 is not independent of xv But if cc2 becomes inactive,

[/] =

[F] =

R R 0 R

0 R 0 0

0 R R 0

R 0 R R_

and x3 is now independent of a^.

The other diagrams, B, D and E, may be verified similarly.

24/15. We can now investigate the problem of S. 14/8 : the
separation of parts in a dynamic whole.

Theorem : If the variables of an absolute system are divisible
into three sets, A, B, and C such that no fA contains any of the
set a&c, and no fc contains any of xA, i.e. so that the diagram of
immediate effects is A ^ B ~^_ C, and if variables Xb remain
constant, then A is independent of C, and vice versa.

If all variables of set B are constant, the differential matrix,
in partitioned form, will be

It is idempotent, so this matrix is also the integral matrix. As
the elements at the top right and bottom left corners are zero,
A and C are independent of each other.

On the other hand, without further restrictions the constancy
is not necessary. Thus, suppose that A and C are independent
and that the differential matrix is

R R (T
P R Q
.0 R R^

where P and Q are to be determined. For the integral matrix
to have zeros in the top right and bottom left corners, it is easily

249

24/15 DESIGN FOR A BRAIN

found that P and Q must both be zero. So dfB/dxA and dfB/dxc
must be zero over the region. This can be achieved in several
ways without fn being zero, i.e. without Xb being constant. Two
examples will be given.

(1) If fn is a constant, then Xb will increase uniformly, i.e. will
not be constant, but Xa and xc will still be independent. Without
a fourth variable, the linear change is the most which xB can make
if the system is to remain absolute.

(2) If fn is a function of other variables not yet mentioned, y
is not restricted to a constant rate of change. Thus if there is a
variable u which dominates y we could have a system

dx
~dt

x + y

du _
dt~3

= sin u

dt
dz

dt=y+Zj
which is clearly absolute. Its solution is :

x = (xQ + y° +

sin u° + — cos u0)el — y°

cos IT
3

sin (wc

U = u° + 3t,

y = yQ -J- - COS U

cos {u° + 3/),

30 + gjj cos (u<> + 30,

(2° + y° + — sin u° + — cos u°)el

cos UK

sin (u° 4- St) + — cos (u° + 3f).

10 v ^ ; ^ 30 v ^ y

Not even the rate of change of y is constant, yet x and z are
independent.

Physically the conclusions are reasonable. The various con-
ditions which make x and z independent all have the effect of
lessening or abolishing x's and s's effect on y. The abolition can
be done either by making y constant, or by driving y exclusively
by some other variable (u). A well-known example of the latter
method is the ' jamming ' of a broadcast by the addition of some

250

CONSTANCY AND INDEPENDENCE 24/17

powerful fluctuating signal from another station. It may effec-
tively render the listener independent of the broadcaster.

If to the original conditions we add the restriction that the
system A is to become absolute on being made independent of C,
then constancy of the variables Xb becomes necessary. For the
possibilities examined in paragraphs (1) and (2) leave system A
subject to parameters xr which were assumed to be effective and
which are now changing. In such conditions A cannot be
absolute (S. 21/1) : constancy of the variables xn is therefore
necessary.

24/16. The statement of S. 14/15, that in an absolute system an
inactive variable cannot become active unless some variable
directly affecting it is active, will now be proved.

Theorem : If a variable xa is related to a set xb so that/0(. . .)
contains only xa and Xb, and if xa and Xb have all been constant
over a finite time, and if xa becomes active while the set xb stays
inactive, then the system cannot be absolute.

We are given that dxa/dt =fa(xa, xB). As xa remained con-
stant (at Xa, say) while the set Xb were constant (at Xb), it follows
that fa(Xa, Xb) = 0. But if xa starts to change value, dxa/dt is
no longer zero, nor is/a ; sofa(Xa, Xb) is a double- valued function
of its arguments, the system is not state-determined, and it is
therefore not absolute.

24/17. In the ' hour-glass ' system of S. 14/11, every variable
may be shown to be dependent on every other variable. As in
Figure 14/11/1, let systems A and B each act on, and be acted
on, by a variable x. The differential matrix, in partitioned form,
is

Its square contains only R's. So none of the A's are independent
of the Z?'s, and conversely.

The proof is confirmed by the theorem of S. 19/22, which shows
that, as far as system B is concerned, the values of the A's can
be replaced by the derivatives of x. The behaviours of all A's

251

24/18

DESIGN FOR A BRAIN

variables are therefore represented in a?'s behaviour by aj's deriva-
tives, and Z?'s variables are thus not independent of A's.

24/18. In S. 14/16 we wanted to compare two probabilities,
each that a system would be stable, one composed of part-
functions and the other of full-functions, other things being equal.
The method of S. 20/12 will define the individual probabilities.
The question of what we mean by ' other things ' may be treated
by postulating that, regarded as two random processes, (a) the
one system's full-functions and (b) the active sections of the other
system's part-functions are to have the same statistical properties
(when averaged over all lines of behaviour.) This postulate is
stated purely in terms of the systems' observable behaviour, so
that it would be easy, in a given case, to test whether the postu-
late was satisfied.

Now consider a system of n variables, part-functions that on
the average are active over a fraction p of the time. The average
number of variables active at one time will be pn = k, say.
Suppose that, at a point Q, the average number of variables are
active. For convenience, re-label the variables to list the active
first. Add parameters olv . . . to generate the distribution. At
Q we have

dxjdt = f1{x1, . . . , xn\ <*!, . . .y

dxk/dt =fk(xv . . . , xn; ocv . . .)

dxk+i/dt = 0

dxn/dt = 0

The differential matrix at this point will be formally of order
n X n, but the rows from k + 1 to n will be all zero. If now we
test the probability of stability at this Q we find that in fact it
depends on the probability that the a-combination has given (a)
fx= . . . =fk = 0, and (b) that the matrix

dfk
_dx1

dxk.

252

CONSTANCY AND INDEPENDENCE 24/19

passes Hurwitz' test. Whatever the probability may be, it is
clearly equal to the probability of stability given by a system of
k similar full-functions.

Conventions and symbols

24/19. The relations between the various entities defined in this
chapter are here summarised for convenience of reference.

(1) (In these four statements take the upper relation in the
braces in all, or the lower in all).

(a) xA^ not independent of Xj

(b) Fk{. . . , x] + A4 . . . ; ojljlW • • ' X°P • • ' ' ') ;

(d) The integral matrix has i -^ j> at the {kj)-th position
(2) The (hi)-th position is in the h-th row and ^-th column :

en) (12) (is) . . r

(21) (22) (23) . . .

(c)—0Fk{x°v . . ., x°n; 0| +

(3) The differential matrix has <^ j> at (pq) as dfp/dxq{ i ^0

„ integral „ „ ,, „ ,, „ dFp/dxq ,, „

(4) The following correspond :

(a) In the diagram of immediate effects : xr — ► xs ;

(b) In the canonical equations : f8(. . . , av, . . .) ;

(5) If sets A and B include all the variables, and if deletion
from the integral matrix of :

columns A and rows B leaves all zeros, and
columns B and rows A leaves not all zeros,
then A dominates B.

253

24/19

DESIGN FOR

BRAIN

(6)

If xp is a

part-function

and

inactive :

(a)

dxp/dt

= 0;

(b)

Xp = X

o .

Fp(x<> ;

t) - < ;

(d) a^/3ajj - 0 (all g + p) ;

(e) all elements (except (pp) ) of the p-th row of the

differential and integral matrices are zero.

Reference
Riguet, J. Sur les rapports entre les concepts de machine de multipole et de
structure algebrique. Comptes rendus des seances de V Academic des
Sciences, 237, 425 ; 1953.

254

INDEX

(The number refers to the page. A bold-faced number indicates a definition.)

Absolute system, 24, 45, 73, 105,

209, Chapter 19
Accumulation, 213
Activity, 67, 162, 164
Adaptation, 64, Chapter 5

accumulation of, 172

loss of, 133

needs independence, 137

of iterated systems, 141

of multistable system, 172

serial, Chapter 17

system to system, 174

time taken, 135

two meanings, 63
Adaptive behaviour, 64

classified by Holmes, 66
Addition of stimuli, 167
Aileron, 99
Aim : see Goal
Algebra, of machine, 254
All or nothing, 127
Ammonia, 78
Ammonium chloride, 20
Amoeba, 152

Amoeboid movement, 126
Animal-centred co-ordinates, 40
Archimedes, 65
Area striata, 170
Artificial limb, 39
Ashby, W. Ross, 71, 102, 199, 225,

231, 236, 240
Assembly, and canonical equations,

211
Association areas, 190
Assumptions made, 9
Automatic pilot, 99
Awareness, 10

Bacteria, 156
Bartlett, F. S., 37, 42
Behaviour, 14

classified, 66

reflex and learned, 2

representation of, 23
Bernard, Claude, 125
Bicycle, 35
Bigelow, J., 71
Binocular vision, 118
Biochemical co-ordination, 196
Body-temperature, 58
Borrowed knowledge, 14
Boyd, D. A., 131, 138

Break, 85, 126, 233
Burn, 131

Calculating machine, 130
Cannon, W. B., 57, 64, 71
Canonical equations, 206, 211
Cardinal number, 31
Carey, E. J., 127, 129
Carmine, 103
Causation, 49

partition of, 241
Cell, step-functions in, 125
Characteristic equation, 217
Chemical dynamics, 17, 143, 178
Chess, 8, 102

Chewing, co-ordination of, 7
Chimney, 191
Chimpanzee, 186
Civilisation, 62
Clock, as time-indicator, 15

variables of, 14
Cochlea, 169

Collision, 60, 131, 180, 187
Complexity, 121
Compound microscope, 3
Concepts, restrictions on, 9
Conditioned reflex, 2, 8, 16, 19, 34, 75,

115, 137, 167, 192, 194
Conduction of heat, 17
Congruence, 205

Connection between systems, 153
Consciousness, 10

Conservation, of adaptation, 133, 140,
184

of zeros, 231
Constancy, 67, 72, Chapter 14

classification of, 80
Constant, as null-function, 203

of proportionality, 81, 84, 232
Constraint, adaptation to, 102
Control, 16, 155

"by error, 54
Convulsion, 187
Co-ordinates, animal-centred, 40

change of, 212
Co-ordination, and stability, 55

motor, 67

of reflexes, 196
Cortex, localisation in, 191
Cough reflex, 2
Cowles, J. T., 186, 189
Critical state, 84

255

INDEX

Critical state and goal, 120

distribution of, 92

necessity of, 112

of endrome, 128
Critical surface, 234

and essential variables, 130
Crystallisation, 143
Cube, equilibrium of, 45
Culler, E., 75, 79, 115, 123, 186, 189
Curare, 75
Cybernetics, 154
Cycle, stable, 48
Cycling, 10, 35

DAMS, 171

Delay, between trials, 132

in canonical equations, 213
Delicacy, of neuron, 126
Demonstrability, 9, 11
Dependence, 155

Derivatives as variables, 162, 212
Determinateness, 10, 111
Diabetes, 75

Diagram of immediate effects, 50, 157
Diagram of ultimate effects, 158, 160
Dial-readings, 14, 29
Differential equations, 206
Differential matrix, 245
Digits, terminal, 142
Dirac's 8-function, 233
Discontinuity, in environment, 131
Discrimination, 167
Dispersion, 166, Chapter 15

and ultrastability, 176

in multistable system, 172

in sense organs, 169

of new learning, 194
Displacement, 145
Disturbance, 239, Chapter 13
Dominance, 158, 185, 246
Ducklings, 38
Duncan, C. P., 187, 189
Dynamic systems, Chapters 2 and 19

Eddington, A. S., 15, 28
Effect, 49, 241
Effector, 120
Elastic, 81, 232
Endrome, 128

Energy, 5, 43, 154, 164, 215
Engine driver, 6
Environment, 35

and homeostasis, 60

discontinuous, 131

functional criterion, 118

iterated, 139

nature of, 179

number of variables, 136

scale of difficulty, 132

types of, 183

Epistemology, 211
Equation, canonical, 206

characteristic, 217
Equilibrium, 43
Error, control by, 54

correction of, 54
Essential variable, 41

and adaptation, 64

and critical surfaces, 122, 130

and normal equilibrium, 146

and Stentor, 105
Evolution, 8, 196
Experience, 119
Experiment, structure of, 31
Experimenter, during training, 113
Eyeball, 117

Falcon, training of, 186
Fatigue, and habituation, 152
Feedback, 51

and stability, 52

demonstration of, 49

in homeostat, 95

in neuronic circuits, 128

in physiology, 37

in training, 114

organism-environment, 36
Feeding, 113
Fencer, 67
Fibrils, neuro-, 126
Field, 22

destroyed, 145

multiple, 88, 235

of absolute system, 26

of regular system, 210

parameter-change, 74

part-functions, 163

stability of, 84
Finite continuous group, 205
Fire, 3

Fisher, R. A., 96, 102, 222
Fit, 187

Fixing a variable, 55, 220
Fracture, 131
Frazer, R. A., 225
Freezing of spinal cord, 160
Full-function, 80

if ignored, 86
Function-rules, 8, 122
Fuse, as step-function, 81

critical state, 84

Gene-pattern, 8, 122, 197

Gestalt-recognition, 168

Gestalt school, 137

Girden, E., 75, 79

Glucose in blood, and diabetes, 75

homeostasis of, 58
Goal-seeking, 53

control of aim, 120

256

INDEX

Goal-seeking, inappropriate, 130
Governor, see Watt's governor
Gradation, in homeostat, 133
in iterated systems, 140
in multistable system, 184
Grant, W. T., G8, 71
Grindley, G. C, 114, 124
Group, equations of, 204
of equivalent patterns, 108

Habituation, 151

intracerebral, 195
Haemorrhage, 33
Harmonic oscillator, 80
Harrison, R. G., 120, 129
Hawking, 180

Hilgard, E. R., 115, 124, 194
Holmes, S. J., 00, 71
Homeostasis (Cannon), 57
Homeostat,

adapting, 109

construction, 93

delay between trials, 132

difficulty of stabilisation, 105

equations of, 207, 220

habituation, 148

interaction, 175

modes of failure, 130

trained, 110

two environments, 134
Hormone, 122
Hour-glass system, 101, 251
House, for homeostasis, 02
Humphrey, G., 152
Hunger, 58
Hunting, 132
Hurwitz, A., 225
Hurwitz' test, 218, 223

Idempotency, 244
Immediate effect, 50
Immunity to displacement, 140
Inactivity, 162

Independence, 155, Chapters 14
24

not arbitrary, 157

ultimate effects, 158
Individuality of subsystems, 170
Inflammation, 9
Information, 154, 211
Initial state, 19

control over, 10, 78
Insensitivity, 131
Instability, see Stability
Instrumental lean
Insulator, 153, 15*
Integral matrix, 245
Integration, 212, 214
Intelligence test, 132
Interaction, 174, Chapter 18

and

Interference, Principle of, 194

Intrinsic stability, 219

Invariant, 108, 231

Isolation, 153, 159, 210

Iterated systems, 140, 184, Chapter 12

Jacobian, 210

Jamming, 250

Jennings, H. S., 37, 42, 103, 152, 180

Joining systems, 55, 229

Kitten, 3, 11, 37, 01, 90, 195
Krebs' cycle, 190

Lashley, K. S., 192, 193, 199

Latent roots, 218

Law of Reciprocity of Connections,

128
Learning

and adaptation, 03

and consciousness, 10

and ultrastable system, 119

effect of new, 193

habituation as, 152

irreversible, 187

localisation of, 140, 190

serial adaptation, 180
Lens, 170

Lethal environment, 131
Levi, G., 120, 129
Liddell, H. S., 19, 28
Lie group, 205
Life, 29
Line of behaviour, 19

equality of two, 19

in absolute system, 20, 208

recording, 19

of part-functions, 104

stability of, 47
Linear system, 215
Localisation of learning, 190
Locomotion, 40
Loeb, Jacques, 120
Lorente de X6, R., 127, 129

Machine, 13

algebra of, 254

number of variables, 15
Main variables, 87, 228
Marina, A., 117, 124
Mathematics, learning, 185
Matrix,

differential, 245

integral, 245

notation, 215

R0— , 243
Marquis, D. G., 115, 124, 194
Maze, 3

McCulloch, W. S., 128, 129, 108, 170
McDougall, W., 04, 71

257

INDEX

Mechanical brain, 130, 179

Memory, 119

permanence of, 190

Method, 15

Metrazol, 127

Microscope, compound, 3

Micturition, 84

Minnows, 113

Morgan, C. Lloyd, 38, 42, 180

Motor-car, and homeostasis, 62

Motor co-ordination, see Co-ordina-
tion

Motor end-plate, 127

Mowrer, O. H., 106, 124

Miiller, G. E., 194, 199

Multistable system, 171, Chapters
16-18
necessity of, 182

Natural selection, 123, 144, 197

Natural system, 23

Nervous system, assumptions, 9, 34,
190

Neuron, 5

number of, in Man, 8

Neuronic circuit, 128

Neutral equilibrium, 43

Noise, 211

Normal equilibrium, 146, 216, 238

Null-function, 81, 227
and absolute system, 86
separates systems, 159, 173

Number, 29

Nyquist, H., 225

Nyquist's test, 219

Objectivity, 9
Observation, 15
Obstacle, 64, 69
Oesophageal fistula, 113
Order of time-scale, 82
Organisation, 7, 70, 110
Organism and environment, 35
Ovum, 122

Pain, insensitivity to, 131
Paramecium,

habituates, 152

environment of, 180
Parameter, 72, Chapters 6 and 21

alternation of, 134, 147

and disturbance, 145

and goal, 121

compound, 32

effective, 73, 153, 156

generating step-function, 83

reaction to, 101

stabilisation by, 165
Parker, G., 140, 143, 156
Part-function, 80

Part-function and dispersion, 166

and independence, 247

and stabilisation, 165

examples, 162

in environment, 180

in multistable system, 171
Parts, relation to whole, 5, 211
Pattern of reaction, 33
Pattern-recognition, 168
Patterned zeros, 244
Pavlov, I. P., 16, 64, 71, 115, 167,

192, 194
Pendulum, 15

as absolute system, 25

field of, 25

is goal-seeking, 53

parameters to, 72, 86
Perception, 11
Phase-space, 20, 203

step-function in, 87

part-function in, 164
Pike, 113, 131
Pitts, W., 168, 170
Poison, 130, 132
Predictor, 154
Primary operation, 17, 203

and field, 23

and independence, 153, 157

and line of behaviour, 19

and stability, 49
Principle of Interference, 194

of Ultrastability, 91
Problem, stated, 11, 70, 90
Processes, fast and slow, 177
Progression, in adaptation, 141
Protoplasm, 126
Protozoa, habituation in, 152
Psychological concepts, 9
Punishment, 112

Random numbers, 96, 222
Reaction, in radio, 51
Receptor, control of aim, 120

dispersion, 169
Reducible system, 247
Reflexes, 2

co-ordination of, 196

independence of, 137
Region of stability, 47
Regular system, 23, 204, 209
Rein, H., 33, 42
Representative point, 20
Response, to stimulus, 166

to repeated stimuli, 147
Resting state, 48, 216
Reverberating circuits, 128
Reversal, reaction to, 98
Reward, 112
Riguet, J., 242, 254
Rigidity, 52

258

INDEX

JBO-matrix, 243
Robinson, E. S., 194, 199
Rosenblueth, A., 71
Runaway, 52

Sea- anemone, 140, 156
Selection of fields, 91, 106
Self-correction, 54
Self-destruction, 5
Sense-organs, dispersion in, 169
Separation, 153, 160
Serum, 20

Servo-mechanism, 51
Sex hormone, 122
Shannon, C. E., 211, 215
Sherrington, C. S., 137, 138
Shivering, 58

Simple harmonic oscillator, 80
Skaggs, E. B., 194, 199
Skin, 169

Skinner, B. F., 190, 199
Snail, 152
Snake, 135
Sommerhoff, A., 71
Species, 8
Spectrum, 30
Speidel, C. C, 127, 129
Sperry, R. W., 117, 124
Spider monkey, 118
Spinal cord, transection, 159
Spinal reflexes, co-ordination between,
196

interaction between, 137
Spontaneity, 88
Stability, 47, Chapters 4 and 20

and parameter change, 78

examples, 55

is holistic, 54

nature of, 76

of homeostat, 96, 220

probability of, 56, 100, 136, 165, 221
Stalking, 132

Starling, E. H., 37, 42, 64
State, 18, 203

equality of two, 19
State-determined system, 25
Statistical machinery, 145
Steady state, 43
Stentor, 103
Step-function, 80, Chapters 7 and 22

and natural selection, 123

effect of omission, 88

in Stentor, 105

in the organism, Chapter 10

nature of, 111

necessity of, 110

number necessary, 128
Stepping switch, 96
Stimuli,

addition of, 167

Stimuli, and part-functions, 166

numericising, 30, 76

repeated, 147

response to, 57
Strychnine, 130
Subjective phenomena, 10, 32
Subspaces, 87, 163
Subsystems, 171, 181
Surface, critical, 234
Surgical alterations, 117
Survival, 42, 65, 197
Swallow, 17

Switch, as step-function, 81
Switching, and part-function, 161
Symbols, collected, 253
System, 15

absolute, 24, Chapter 19

containing null-functions, 86, 228

— part-functions, 163

— step-functions, 86
hour-glass, 161, 251
independence of, 156
infinite, 203
linear, 215
regular, 23, 204
symbols for, 203
stability of, 48
with feedback, 39

Tabes dorsalis, 38
Tadpoles, 127
Taste, 169
Teleology, 9, 71
Telephone exchange, 161
Temperature, homeostasis of, 58
Temple, G., 27, 28
Temporal cortex, 170
Terminal field, 91

normality, 146
Thermostat, 43, 51, 53, 54, 67, 156
Thirst, 59

Threshold, 163, 170
Thunderstorm, 17
Time,

as variable, 15

for adaptation, 135, 141, 177, 184
Tissue culture, 126
Tokens, as reward, 186
Tongue, 7, 39
Training, 112, 187
Transducer, 121

noiseless, 211
Transformations, 168
Transient, 144
Trap, 132
Tremor, 67

Trial and error, 112, 132, 174
Trials, number of, 238
Type-problem, 11
Typewriter, 154

259

INDEX

Ultrastability

and dispersion, 171, 17G

and habituation, 152

and interaction, 195
Ultrastable system, 91, Chapters 8
and 23

specified by genes, 122

formed by natural selection, 123
Uniselector, 96
Universals, 170
Unstable equilibrium, 43
Urinary bladder, 84

Variable, 14

and disturbance, 145
independence, 15G
replacement by derivative, 212
restricted meaning, 72

Velocity, 214
Vicious circle, 52
Visual cortex, 170
Vital properties, 9

Walking, 38

Water, homeostasis of, 59

Watt's governor, 43, 46, 49, 154

Weather, 32

Wholeness

of nervous system, 137

of environment, 184
Wiener, N., 56, 71, 154
Wolfe, J. B., 186, 189

Yates, F., 96, 102

Zeros, patterned, 244

Internet Archive Audio

Featured

Top

Images

Featured

Top

Software

Featured

Top

Books

Featured

Top

Video

Featured

Top

Mobile Apps

Browser Extensions

Archive-It Subscription

Save Page Now

Full text of "Design for a brain"

See other formats