Mculloch Pitts

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BULLETIN OF

MATHEMATICAL BIOPHYSICSVOLUME 5 1943

A L O G I C A L C A L C U L U S O F T H E

I D E A S I M M A N E N T I N N E R V O U S A C T I V I T Y

W A R R E N S . M C C U L L O CH A N D W A L T E R P I T T S

FROM THE UNIVERSITY OF ILLINOIS COLLEGE OF MEDICINI~

DEPARTMENT OF PSYCHIATRY AT THE ILLINOIS NEUROPSYCHIATRIC INSTITUTE

AND THE UNIVERSITY OF CHICAGO

B e c a u s e o f t h e a l l- o r -n o n e c h a r a c t e r o f n e r v o u s a c t i v i ty , n e u r a le v e n t s a n d t h e r e l a t i o n s a m o n g t h e m c a n b e t r e a t e d b y m e a n s o f p r o p o -s i t io n a l l o g ic . I t i s f o u n d t h a t t h e b e h a v i o r o f e v e r y n e t c a n b e d e s c r i b e di n th e s e t e r ms w i t h t h e a d d i ti o n o f mo r e c o mp l i c a t e d l o g i c a l m e a n s f o rn e t s c o n t a i n i n g c i r c l e s ; a n d t h a t f o r a n y l o g i c a l e x p r e s s i o n s a t i s f y i n gc e r t a i n c o n d i ti o n s o n e c a n fi n d a n e t b e h a v i n g i n t h e f a s h i o n i t d e s c r i b e s.I t i s s h ow n t h a t m a n y p a r t i c u l a r c h oi ce s a m o n g p o s s ib l e n eu r o p h y si o lo g i -c a l a s s u m p t i o n s a r e e q u i v al e n t i n t h e se n s e t h a t f o r e v e r y n e t b e h a v -i n g u n d e r o n e a s s u m p t i o n t h e r e e x i s ts a n o t h e r n e t w h i c h b e h a v e s u n -d e r th e o t h e r a n d g i v e s t h e s a m e r e s u l t s a l t h o u g h p e r h a p s n o t i n t h es a m e t i me . Va r i o u s a p p l i c a t i o n s o f t h e c a l c u l u s a r e d i s c us s e d .

I In t r o d u c t i o n

T h e o r e t i c a l n e u r o p h y s i o l o g y r e s t s o n c e r t a i n c a r d i n a l a s s u m p -

t i o n s . T h e n e r v o u s s y s t e m i s a n e t o f n e u r o n s e a c h h a v i n g a s o m a

a n d a n a x o n . T h e i r a d j u n c t i o n s o r s y n a p s e s a r e a l w a y s b e t w e e n th e

a x o n o f o n e n e u r o n a n d t h e s o m a o f a n o t h e r . A t a n y i n s t a n t a n e u r o n

h a s s o m e t h r e s h o l d w h i c h e x c i t a t i o n m u s t e x c e e d t o i n i t i a t e a n i m -

p u l se . T h i s e x c e p t f o r t h e f a c t a n d t h e t i m e o f i t s o c c u r r e n c e i s d e -

t e r m i n e d b y t h e n e u r o n n o t b y t h e e x c i t a t i o n . F r o m t h e p o i n t o f e x -

c i t a t i o n th e i m p u l s e i s p r o p a g a t e d t o a ll p a r t s o f t h e n e u r o n . T h e

v e l o c i t y a l o n g t h e a x o n v a r i e s d i r e c t l y w i t h i t s d i a m e t e r f r o m l e s s

t h a n o n e m e t e r p e r s e c o n d i n t h i n a x o n s w h i c h a r e u s u a l ly s h o r t t o

m o r e t h a n 1 5 0 m e t e r s p e r s e c o n d i n t h i c k a x o n s w h i c h a r e u s u a l ly

l o n g . T h e t i m e f o r a x o n a l c o n d u c t i o n i s c o n s e q u e n t l y o f l i t t le i m p o r -

t a n c e i n d e t e r m i n i n g t h e t i m e o f a r r i v a l o f i m p u l s e s a t p o i n t s u n -

e q u a l l y r e m o t e f r o m t h e s a m e s o u rc e . E x c i t a t i o n a c r o s s s y n a p s e s o c -

c u r s p r e d o m i n a n t l y f r o m a x o n a l t e r m i n a t i o n s t o s o m a t a . I t i s s t i l l a

m o o t p o i n t w h e t h e r t h i s d e p e n d s u p o n i r r e c i p r o c i t y o f i n d i v i d u a l s y n -

a p s e s o r m e r e l y u p o n p r e v a l e n t a n a t o m i c a l c o n f ig u r a t i o n s . T o s u p -

p o s e t h e l a t t e r r e q u i r e s n o h y p o t h e s i s a d h o c a n d e x p l a i n s k n o w n e x -

c e p t i o n s b u t a n y a s s u m p t i o n a s t o c a u s e is c o m p a t i b l e w i t h t h e c a l-

c u l u s t o c o m e . N o c a s e i s k n o w n i n w h i c h e x c i t a t i o n t h r o u g h a s i n g l e

s y n a p s e h a s e l ic i t e d a n e r v o u s i m p u l s e i n a n y n e u r o n w h e r e a s a n y

1 1 5



1 1 6 LOGIC L C LCULUS FOR NERVOUS CTIVITY

n e u r o n m a y b e ex c i te d b y i m p u l s e s a r r i v i n g a t a s uf fi c i en t n u m b e r o f

n e i g h b o r i n g s y n a p s e s w i t h i n t h e p e r i o d o f l a t e n t a d d i ti o n , w h i c h l a s ts

l es s t h a n o n e q u a r t e r o f a m i l l is e co n d . O b s e r v e d t e m p o r a l s u m m a t i o n

o f i m p u l s e s a t g r e a t e r i n t e r v a l s is im p o s s i b l e fo r s i n g l e n e u r o n s a n d

e m p i r i c a l ly d e p e n d s u p o n s t r u c t u r a l p r o p e r t i e s o f t h e n e t. B e t w e e n

t h e a r r i v a l o f i m p u l s e s u p o n a n e u r o n a n d i t s o w n p r o p a g a t e d i m -

p u l s e t h e r e i s a s y n a p t i c d e l a y o f m o r e t h a n h a l f a m i l l is e c o n d . D u r -

i n g t h e f i r s t p a r t o f t h e n e r v o u s i m p u l s e t h e n e u r o n i s a b s o l u t e l y re -

f r a c t o r y t o a n y s t i m u l a t i o n . T h e r e a f t e r i t s e x c i t a b i l i t y r e t u r n s r a p -

i dl y, in s o m e c a s e s r e a c h i n g a v a l u e a b o v e n o r m a l f r o m w h i c h i t s in k s

a g a i n t o a s u b n o r m a l v a l ue , w h e n c e i t r e t u r n s s l o w l y t o n o r m a l . F r e -

q u e n t a c t i v i t y a u g m e n t s t h i s s u b n o r m a l i t y . S u c h s p e c if ic i ty a s i s

p o s s e s s ed b y n e r v o u s i m p u l s e s d e p e n d s s o l el y u p o n t h e i r t i m e a n d

p l a c e a n d n o t o n a n y o t h e r s p e c i f i ci t y o f n e r v o u s e n e r g i e s . O f l at e

o n l y i n h i b i t i o n h a s b e e n s e r i o u s l y a d d u c e d t o c o n t r a v e n e t h i s t h e s i s .

I n h i b i t i o n i s t h e t e r m i n a t i o n o r p r e v e n t i o n o f t h e a c t i v i t y o f o n e

g r o u p o f n e u r o n s b y c o n c u r r e n t o r a n t e c e d e n t a c t i v i t y o f a s e co n d

g r o u p . U n t i l r e c e n t l y t h is c o u l d b e e x p l a in e d o n t h e s u p p o s i t i o n t h a t

p r e v i o u s a c t i v i t y o f n e u r o n s o f t h e s e c o n d g r o u p m i g h t s o r a i s e t h e

t h r e s h o l d s o f i n t e r n u n c i a l n e u r o n s t h a t t h e y c o u ld n o l o n g e r b e e x -

c i te d b y n e u r o n s o f th e f i r s t g r o u p , w h e r e a s t h e i m p u l s e s o f t h e f ir s tg r o u p m u s t s u m w i t h t h e i m p u l s e s o f t h e s e i n t e rn u n c i a l s t o ex c i te t h e

n o w in h i b i t ed n e u r o n s . T o d a y , s o m e i n h i b i t i o n s h a v e b e e n s h o w n t o

c o n s u m e le s s t h a n o n e m i l l is e c o n d . T h i s e x c l u d e s i n t e r n u n c i a l s a n d

r e q u i r e s s y n a p s e s t h r o u g h w h i c h im p u l s e s in h i b i t t h a t n e u r o n w h i c h

i s b e i n g s t i m u l a t e d b y i m p u l s e s t h r o u g h o t h e r s y n a p s e s . A s y e t e x -

p e r i m e n t h a s n o t s h o w n w h e t h e r t h e r e f r a c t o r i n e s s i s r e l a t i v e o r a b -

s o lu t e. W e w i l l a s s u m e t h e l a t t e r a n d d e m o n s t r a t e t h a t t h e d i f f er e n c e

i s i m m a t e r i a l to o u r a rg u m e n t . E i t h e r v a r i e t y o f r e f r a c t o r i n e s s c a n

b e a c c o u n t e d f o r in e i t h e r o f t w o w a y s . T h e i n h i b i t o r y s y n a p s e

m a y b e o f s u c h a k in d a s to p r o d u c e a s u b s t a n c e w h i c h r a Ss es t h et h r e s h o l d o f th e n e u r o n , o r i t m a y b e s o p l a c e d t h a t t h e l oc al d i s t u r b -

a n c e p r o d u c e d b y i ts e x c i t a ti o n o p p o s e s t h e a l t e r a t i o n i n d u c e d b y t h e

o t h e r w i s e e x c i t a t o r y s y n a p s e s . I n a s m u c h a s p o s it i o n i s a l r e a d y k n o w n

t o h a v e s u c h e f f e c ts in t h e c a s e o f e l e ct r ic a l s t i m u l a t i o n , t h e f i r s t h y -

p o t h e s i s ~s t o b e e x c l u d e d u n l e s s a n d u n t i l i t b e s u b s t a n t i a t e d , f o r t h e

s e c on d i n v o lv e s n o n e w h y p o t h e s i s . W e h a v e , th e n , t w o e x p l a n a t i o n s

o f i n h i b i t i o n b a s e d o n t h e s a m e g e n e r a l p r e m i s e s , d i f f e r i n g o n l y in

t h e a s s u m e d n e r v o u s n e t s a n d , c o n s eq u e n t l y , in th e t i m e r e q u i r e d f o r

i n h ib i ti o n . H e r e a f t e r w e s h al l r e f e r t o su c h n e r v o u s n e t s a s equiva

lent in the extended sense. S i n ce w e a r e c o n c e r n e d w i t h p r o p e r t i e so f n e t s w h i c h a r e i n v a r i a n t u n d e r e q u iv a le n c e, w e m a y m a k e t h e

p h y s i c a l a s s u m p t i o n s w h i c h a r e m o s t c o n v e n i e n t f o r t h e c al cu l us .





118 LOGIC L C LCULUS FOR NERVOUS CTIVITY

H . T h e T h e o r y : N e t s W i t h o u t C ir cl es

We shall make the following physical assumptions for our cal-

culus.

1. The activity of the neuron is an all-or-none process.

2. A certain fixed number of synapses mus t be excited within

the period of latent addition in order to excite a neuron at any time,

and this number is independent of previous activity and position on

the neuron.

3. The only significant delay within the nervous system is syn-

aptic delay.

4. The activity o f any inhibi tory synapse absolutely prevents

excitation of the neuron at that time.

5. The struc tur e of the net does not change with time.

To present the theory, the most appropr iate symbolism is that of

Lang uage II of R. Carnap (1938), augmented with various notations

drawn from B. Russell and A. N. Whitehead (1927), including the

P r i n c i p i a conventions for dots. Typographical necessity, however,

will compel us to use the upright E for the existential operator in-

stead of the inverted, and an arrow ('-~') for implication instead of

the horseshoe. We shall also use the Carnap syntactical notations, but

print them in boldface rather than German type; and we shall intro-

duce a functor S, whose value for a propert y P is the prop ert y which

holds of a number when P holds of its predecessor; it is defined by

S ( P ) ( t ) .= -. P ( K x ) . t ~ x')'; the brackets around its argument will

often be omitted, in which case this is understood to be the nearest

predicate-expression [Pr ] on the right. Moreover, we shall wr it e

S ~ P r for S (S (P r) ), etc.

The neurons of a given net ~ may be assigned designationsc1 , c~ , . . . , c~ . This done, we shall deno te the p rop ert y of a number,

that a neuron c~ fires at a time which is that number of synaptic de-

lays from the origin of time, by N with the numeral i as subscript,

so th at N~ (t) asse rts th at c~ fires at the t ime t. N~ is called the a c t i o n

of e~. We shall sometimes reg ard the subscripted numeral of N as

if it belonged to the object-language, and were in a place for a func-toral argument, so that it might be replaced by a number-variable

[z] and quantified; this enables us to abbreviate long but finite dis-

junctions and conjunctions by the use of an operator. We shall era-

ploy this locution quite generally for sequences of P r ; it may be se-cured formal ly by an obvious disjunctive definition. The predicates

'NI', 'N.~', .. ., comprise the syntact ical class 'N'.



W A R R E N S M C C U L L O C H A N D W A L T E R P I T T S 119

Let us define the p e r i p h e r a l a f y e r e n t s of ~ as the neurons of ~(

with no axons synapsin g upon them. Let N1, --. , Np denote the ac-

tion s of such neurons a nd N~+I, N~+~, .. ., N~ those of the r est. Th en a

s o l u t i o n o f $ ~ will be a class of senten ces of th e fo rm S~: Np+l (zl) .=.

Pr~ (N1, N~, .. . , Np, z~), where Pr~ contai ns no free vari able save zl

and no descriptive symbols save the N in the argument [Arg], and

possibly some constant sentences [ s a ] ; and such that each S~ is true

of ~ . Conversely, given a Pr l (l p~ , lp12, . . . , ~p~p, zl , s), cont aini ng

no free variable save those in its A r g , we shall say that it is r e a l i z a b l e

i n t h e n a r r o w s e n s e if the re exists a net ~ and a series of N~ in it

such that N l ( z ~ ) .----. P r I ( N ~ , N ~ , . . . , z l , sa ~ ) is true of it, where sa~

has the form N( 0) . We shall call i t r e a l i z a b l e i n t h e e x t e n d e d s e n s e ,

or simply r e a l i z a b l e , if for some n S ~ ( P r l ) ( p l , . , p p , z ~ , s ) is

realizable in the above sense, c~ is here the realiz ing neuron. We

shall say of two laws of nervous excitation which are such t hat every

S which is realizable in either sense upon one supposition is also re-

alizable, perhaps by a differen t net, upon the other, th at t hey are

equivalent assumptions, in that sense.

The following theorems about realizability all refer to the ex-

tended sense. In some cases, sharp er theo rems about nar ro w re~liz-

ability can be obtained; but in addition to greater complication instatement this were of little practical value, since our present neuroo

physiological knowledge determines the law of excitation only to ex-

tended equivalence, and the more precise theorems differ according

to which possible assum ptio n we make. Our less precise theorems,

however, are i nva ri ant und er equivalence, and are still sufficient fo r

all purposes in which the exact time for impulses to pass thro ugh t he

whole net is not crucial.

Our central problems may now be stated exactly: first, to find an

effective meth od of obtai ning a set of computable S co nsti tut ing a

solution of any given net; and second, to characterize the class ofrealizable S in an effective fashion. Mater iall y stated, the problems

are to calculate the behavior of any net, and to find a net which will

behave in a specified way, when such a net exists.

A net will be called c y c l i c if it contains a circle: i.e., if there ex-

ists a chain c~, c~§ ... of neuro ns on it, each member of the chain

synapsin g upon the next, with the same beginning and end. If a set

of its neur ons a~, c~ , .. ., cp is such th at its remov al from ~ leaves

it without circles, and no smaller class of neurons has this property,

the set is called a c y c l i c set, and its ca rdinal ity is the o r d e r of ~. In

an i mpo rt ant sense, as we shall see, the order of a net is an index ofthe complexity of its behavior. In partic ular, nets of zero order have

especially simple properties; we shall discuss them first.




L e t u s d e f i n e a t e m p o r a l p r o p o s i t i o n a l e x p r e s s i o n ( a T P E ) , d e s -

i g n a t i n g a t e m p o r a l p r o po s it io n a l f u n c t i o n ( T P F ) , b y t h e f o l lo w i n gr e c u r s i o n :

1. A ~p~ [z~] is a T P E , w h e r e p ~ i s a p r e d i c a t e - v a r i a b l e .

2 . I f S ~ a n d S~ a r e T P E c o n t a i n i n g t h e s a m e f r e e i n d iv i d u a l

v a r i a b l e , s o a r e SSI , S~vS~ , S I .S~ a n d S ~ . ~ S ~ .

3 . N o t h i n g e ls e i s a T P E .

T H E O R E M I

E v e r y n e t o f o r d e r 0 c ~ n b e s o l v e d i n t e r m s o f t e m p o r a l p r o p o s i -t iona l expres s ions .

L e t c~ b e a n y n e u r o n o f g~ w i t h a t h r e s h o l d t~ > 0 , a n d l e t c~ 1,

c ~ , . .. , c~, h a v e r e s p e c t i v e l y n ~ , n ~ : , . . . , n~p e x c i t a t o r y s y n a p s e s

u p o n i t. L e t c j~ , c i : , - - . , c~q h a v e i n h i b i t o r y s y n a p s e s u p o n i t. L e t ~

b e t h e s e t o f th e s u b c l a s s e s o f ( n ~ l, n ~ : , . ~ . , n~ ,) s u c h t h a t t h e s u m o f

t h e i r m e m b e r s e x c e e d s 6~. W e sh a l l t h e n b e a b l e t o w r i t e , i n a c c o r d -

a n c e w i t h t h e a s s u m p t i o n s m e n t i o n e d a b o v e ,

N~(z~) . - - . S t I~,~ ~ N j~ (z~ ) . ~ ,~ ~H N ~ ( z~ )} ( 1 )

w h e r e t h e 'F ~ ' a n d I I a r e s y n t a c t i c a l s y m b o l s f o r d i s j u n c t i o n s a n d

c o n j u n c t i o n s w h i c h a r e f in i te i n ea c h ca s e . S i n c e a n e x p r e s s i o n o f

t h i s f o r m c a n b e w r i t t e n f o r e a c h c~ w h i c h is n o t a p e r i p h e r a l a f f er -

e n t, w e ca n , b y s u b s t i t u t i n g t h e c o r r e s p o n d i n g e x p r e s s i o n i n ( 1 ) f o r

e a c h Ns, , o r N~ , w h o s e n e u r o n i s n o t a p e r i p h e r a l a f f e r e n t , a n d r e -

p e a t i n g t h e p r o c e s s o n t h e r e s u lt , u l t i m a t e l y c o m e t o a n e x p r e s s i o n

f o r N ~ i n t e r m s s o le ly o f p e r i p h e r a l l y a f f e r e n t N , s i n c e ~ i s w i t h o u t

c i rc l e s. M o r e o v e r , t h i s e x p r e s s i o n w i ll b e a T P E , s i n c e o b v i o u s l y ( 1 )

i s ; a n d J t f o l lo w s i m m e d i a t e l y f r o m t h e d e f i n it io n t h a t t h e r e s u l t o f

s u b s t i t u t i n g a T P E f o r a c o n s t i t u e n t p ( z ) i n a T P E i s a l s o o n e .

T H E O R E M I I .

E v e r y T P E i s r ea l i zab le by a ne t o f o rder z ero .

T h e f u n c t o r S o b v i o u s l y c o m m u t e s w i t h d i s j u n c t i o n , c o n j u n c t i o n ,

a n d n e g a t i o n . I t is o b v i o u s t h a t t h e r e s u l t o f s u b s t i t u t i n g a n y S ~ ,

r e a l iz a b l e i n th e n a r r o w s e n s e ( i . n .s .) , f o r th e p ( z ) i n a r e a l ~ i z a b l e e x -

p r e s s i o n 1 i s i t s e l f r e a l i z a b l e i .n .s . ; o n e c o n s t r u c t s t h e r e a l i z i n g n e t b yr e p l a c i n g t h e p e r i p h e r a l a f f e r e n t s i n t h e n e t f o r 1 b y t h e r e a l i z i n g n e u -

r o n s i n t h e n e t s f o r t h e S ~. T h e o n e n e u r o n n e t r e a l iz e s p l ( z l ) i .n .s . ,



WARREN S MCCULLOCH AND WALTER PITTS 121

and Figure 1-a shows a net that realizes S p l ( z l ) and hence S S 2 , i.n.s.,

if S: can be realized i.n.s. Now if S~ and $3 are realizab le then

SmS~ and S~S~ are realizable i.n.s., fo r suitable m and n. Hence so

are S~ S~ and S~ Sa. Now the nets of Figu res lb, c and d respectively

realize S (p~ ( z l ) v p~ ( z~ ) ) , S (p~(zD. p~ ( z~ ) ) , and S (p~ ( z~ ) . oo p~ ( z~ ) )

i.n.s. Hence S ~+'+~ (S , v S ~ ) , S ~§ (81 . 2 ) , and S ~§ (S1. o~ S2) are

real izab le i.n.s. Th er ef or e S, v $2 $1 . S~ S~ . oo $2 ar e real izab le if

$1 and S:~ are. By complete ind ucti on, all T P E are realizable. In this

way all nets may be regarded as built out of the fu ndam ent al elements

of Figures la, b, c, d, precisely as the temporal propositional expres-

sions are generated out of the operations of precession, disjunction,

conjunction, and conjained negation. In particular , corresponding to

any description of state, or distribution of the values t r u e and f a l s e

for the actions of all the neurons of a net save that which makes

the m all false, a single neuro n is constructi ble whose firing is a neces-

sary and sufficient condition for the validity of th at description. More-

over, there is always an indefinite nu mbe r of topologically diff erent

nets realizing any T P E .

THEOREM III

L e t t h e r e b e g i v e n a c o m p l e x s e n t e n c e S~ b u i lt u p i n a n y m a n n e r o u t

o f e l e m e n t a r y s e n t e n c e s o f th e f o r m p (z~ - zz) w h e r e zz is a n y n u -

m e r a l , b y a n y o f t h e p r o p o s i t i o n a l c o n n e c t io n s : n e g a t i o n , d i s ] u n c t io n ,

c o n ] u n c t io n , im p l i c a t i o n , a n d e q u i va l e n ce . T h e n $1 is a TPE wad

o n l y i f i t i s f a l s e w h e n i t s c o n s t i t u e n t p (zl - zz) a r e a l l a s s u m e d

f a l s e - - i. e . , r e p l a c e d b y f a l s e s e n t e n c e s - - o r t h a t t h e l a s t l in e i n i t s

t r u t h - t a b l e c o n t a i n s a n ' F ' , - - o r t h e r e i s n o t e r m i n i t s H i l b e r t d i s-

] u n c t iv e n o r m a l f o r m c o m p o s e d e x c l n s i v e ly o f n e g a t e d t e r m s .

These latter three conditions are of course equivalent (Hilbert

and Ack ermann, 1938). We see by induction tha t the first of the m is

necessary, since p ( z l - z z ) becomes false when it is replaced by afalse sentence, and $1 v S~, S~ . S~ and $1 . ~ $2 are all false if

both the ir co nstitu ents are. We see th at the l ast condition is sufficient

by remarking that a disjunction is a T P E when its constituents are,

and that any term

8 1 . 8 . ~ . . . . S ~ . ~ S , , ~ ~ . ~ 1 7 6 . . . ~ 8 .

can be writ ten as

( 8 ~ .8 . ~ . . . . S in ) . ~ (S~ , v S~ § . . . . v S , ) ,

which is clearly a T P E .The method of the last theorems does in fact provide a very con-

venient and workable procedure for constructing nervous nets to or-




d e r , f o r t h o s e c a s e s w h e r e t h e r e i s n o r e f e r e n c e t o e v e n t s i n d e f in i te l y

f a r i n t h e p a s t i n t h e s p e ci f ic a t i o n o f t h e c o n d it io n s . B y w a y o f e x -

a m p l e , w e m a y c o n s i d e r t h e c a s e o f h e a t p r o d u c e d b y a t r a n s i e n t

c o o l i n g .

I f a c ol d o b j e c t i s h e ld t o t h e s k i n f o r a m o m e n t a n d r e m o v e d ,

a s e n s a t i o n o f h e a t w i l l b e f e l t ; i f i t i s a p p l i e d f o r a l o n g e r t i m e , t h e

s e n s a t i o n w i l l b e o n l y of c o ld , w i t h n o p r e l i m i n a r y w a r m t h , h o w e v e r

t r a n s i e n t . I t i s k n o w n t h a t o n e c u t a n e o u s r e c e p t o r i s a f f ec t e d b y h e a t ,

a n d a n o t h e r b y c o ld . I f w e le t N 1 a n d N ~ b e t h e a c t i o n s o f t h e r e s p e c -

t i v e r e c e p t o r s a n d N ~ a n d N 4 o f n e u r o n s w h o s e a c t i v i t y i m p l i e s a s e n -

s a t i o n o f h e a t a n d c old , o u r r e q u i r e m e n t s m a y b e w r i t t e n a s

N ~ ( t ) : ~- : N l ( t - 1 ) . v . N ~ ( t - 3 ) . ~ N ~ ( t - 2 )

N 4 ( t ) . = - -. N ~ ( t - 2 ) . N ~ ( t - 1 )

w h e r e w e s u p p o s e f o r s i m p l i c i ty t h a t t h e r e q u i r e d p e r s i s t e n c e i n t h e

s e n s a t i o n o f c o ld is s a y t w o s y a a p t i c d e l ay s , c o m p a r e d w i t h o n e f o r

t h a t o f h e a t . T h e s e c o n d i t i o n s c l e a r l y f a l l u n d e r T h e o r e m I I I . A n e t

m a y c o n s e q u e n t l y b e c o n s t r u c t e d t o r e a li z e t h e m , b y t h e m e t h o d o f

T h e o r e m I I. W e b eg i n b y w r i t i n g t h e m i n a f a s h io n w h i c h e x h ib i t s

t h e m a s b u i lt o u t o f t h e i r c o n s t i t u e n t s b y t h e o p e r a t i o n s r e a l i z e d i n

F i g u r e s l a , b , c , d : i. e. , i n t h e f o r m~ t ) . - = . S ~ N 1 t ) v S E S N ~ t ) ) . ~ N ~ t ) ]}N 4 t ) . = - . S [ S Y ~ t ) ] . Y .~ t ) ) .

F i r s t w e c o n s t r u c t a n e t f o r th e f u n c t i o n e n c lo s e d i n th e g r e a t e s t

n u m b e r o f b r a c k e t s a n d p r o c e e d o u t w a r d ; i n t h i s c a s e w e r u n a n e t o f

t h e f o r m s h o w n i n F i g u r e l a f r o m cz t o s o m e n e u r o n c a , s a y , s o t h a t

Na t) . ~ . S N 2 ( t ) .

N e x t i n t r o d u c e t w o n e ts o f t h e f o r m s l c a n d l d , b o t h r u n n i n g f r o m

Ca a n d c : , a n d e n d i n g r e s p e c t i v e l y a t o4 a n d s a y c b . T h e n

N 4 ( t ) . ~ . S [ N ~ ( t ) . N~_ ( t ) ] . - ~ . S [ ( S N 2 ( t ) ) . N ~ ( t ) ] .

N b ( t ) . = --. S [ N r . ~ N ~ ( t ) ] . ~ . S [ ( S ~ [ 2 ( t ) ) . oo N ~ ( t ) ] .

F i n a l l y , r u n a n e t o f t h e f o r m l b f r o m Cl a n d c b t o c 8 , a n d d e r i v e

N~ t). ~-. S [ ~ t ) v Nb t)]9 ~ . S ( N ~ ( t ) v S [ ( S N : ( t ) ] . ~ N ~ ( t ) ) .

T h e s e e x p r e s s i o n s f o r N ~ ( t ) a n d N ~ ( t ) a r e t h e o n e s d e s i r e d ; a n d t h e

r e a l i z i n g n e t i n t o t o i s s h o w n i n F i g u r e l e .

T h i s i l l u si o n m a k e s v e r y c l e a r th e d e p e n d e n c e o f t h e c o r r e s p o n -d e n c e b e t w e e n p e r c e p t i o n a n d t h e e x t e r n a l w o r l d u p o n t h e s p ec if ic

s t r u c t u r a l p r o p e r t i e s o f t h e i n C e rv e n in g n e r v o u s n e t . T h e s a m e i l l u-



WA RREN S. MCCULLOCH AND WA LTER PITTS 123

s io n , o f c o u r s e , c o u ld a ls o h a v e b e e n p r o d u c e d u n d e r v a r i o u s o t h e r

a s s u m p t i o n s a b o u t t h e b e h a v i o r o f t h e c u t a n e o u s r e c e p to r s , w i t h c o r -

r e s p o n 4 i n g l y d i f f e r e n t n e t s .

W e sh a l l n o w c o n s i d e r s o m e t h e o r e m s o f e q u i v a l e n c e : i .e ., t h e o -

r e m s w h i c h d e m o n s t r a t e t h e e s s e n t i a l i d e n t i ty , s a v e f o r t im e , o f v a r i -

o u s a l t e r n a t i v e l a w s o f n e r v o u s e x c i t a t i o n . L e t u s f i rs t d i s c u s s t h e

c a s e o f relative inhibition B y t h i s w e m e a n t h e s u p p o s i ti o n t h a t t h e

f i ri n g o f a n i n h i b i t o r y s y n a p s e d o e s n o t a b s o l u t e l y p r e v e n t t h e f i r i n g

o f t h e n e u r o n , b u t m e r e l y r a i s e s i t s t h r e s h o ld , s o t h a t a g r e a t e r n u m -

b e r o f e x c i t a t o r y s y n a p s e s m u s t f ir e c o n c u r r e n t l y t o f ir e i t t h a n w o u l d

o t h e r w i s e b e n e e d ed . W e m a y s u p p o s e , lo s i n g n o g e n e r a l i t y , t h a t t h e

i n c r e a s e .in t h r e s h o l d i s u n i t y f o r t h e f i r i n g o f e ac h s u c h s y n a p s e ; w e

t h e n h a v e t h e t h e o r e m :

T H E O R E M IV .

Relative and absolute inhibition are equivalent in the extended

sense

W e m a y w r i t e o u t a l a w o f n e r v o u s e x c i t a t i o n a f t e r t h e f a s h i o n

o f ( 1 ) , b u t e m p l o y i n g t h e a s s u m p t i o n o f r e l a t iv e i n h i b i t i o n i n s t e a d ;

i n s p e c t i o n t h e n s h o w s t h a t t h i s e x p r e s s i o n i s a TPE A n e x a m p l e o ft h e r e p l a c e m e n t o f re l a t i v e i n h ib i t io n b y a b s o l u t e is g i v e n b y F i g u r e

l f . T h e r e v e r s e r e p l a c e m e n t i s e v e n e a s i e r ; w e g i v e t h e i n h i b i t o r y

a x o n s a f f e r e n t t o c~ a n y s u ff ic i en t ly la r g e n u m b e r o f i n h i b i t o r y s y n -

a p s e s a p i e c e .

S e c on d , w e c o n s i d e r t h e c a se o f e x t in c t io n . W e m a y w r i t e t h i s

i n t h e f o r m o f a v a r i a t i o n i n t h e t h r e s h o l d 0~ ; a f t e r t h e n e u r o n c~ h a s

f i re d ; to t h e n e a r e s t i n t e g e r - - a n d o n l y t o t h i s a p p r o x i m a t i o n i s t h e

v a r i a t i o n i n t h r e s h o l d s i g n if i ca n t in n a t u r a l f o r m s o f e x c i t a t i o n - - t h i s

m a y b e w r i t t e n a s a s e q u e n c e 0~ + b j f o r ] s y n a p t i c d e l a y s a f t e r f ir i n g ,

w h e r e b s ---- 0 f o r ~ l a r g e e n o u g h , s a y ] - - M o r g r e a t e r . W e m a y t h e ns t a t e

THEOREM V

Extinction is equivalent to absolute inhibition

F o r , a s s u m i n g r e l a t i v e i n h i b i ti o n t o h o l d f o r t h e m o m e n t , w e

n e e d m e r e l y r u n M c i rc u i t s T ~ , T , , . .. T . c o n t a i n i n g r e s p e c t i v e l y 1,

2 , - . . , M n e u r o n s , s u c h t h a t t h e f i r i n g o f e a c h l in k i n a n y i s s u f fi c ie n t

t o f ir e t h e n e x t , f r o m t h e n e u r o n c~ b a c k t o i t , w h e r e t h e e n d o f th e

c i r c u i t T j h a s j u s t b j i n h i b i t o r y s y n a p s e s u p o n c , . I t is e v i d e n t t h a tt h i s w i l l p r o d u c e th e d e s i r e d r e s u l t s . T h e r e v e r s e s u b s t i t u t i o n m a y b e

a c c o m p l i sh e d b y t h e d i a g r a m o f F i g u r e l g . F r o m t h e t r a n s i t i v i t y o f




r e p l a c e m e n t , w e i n f e r t h e th e o r e m . T o t h i s g r o u p o f t h e o r e m s a l so

b e l o n g s t h e w e l l - k n o w n

THEOREM V I

F a c i l i t a t i o n a n d t e m p o r a l s u m m a t i o n m a y b e r e pl a ce d b y s p a t i a l

s u m m a t i o n

T h i s i s o b v i o u s : o n e n e e d m e r e l y i n t r o d u c e a s u i t a b l e se q u e n c e

o f d e l a y i n g c h a in s , o f in c r e a s i n g n u m b e r s o f s y n a p s e s , b e t w e e n t h e

e x c i t i n g ce ll a n d t h e n e u r o n w h e r e o n t e m p o r a l s u m m a t i o n i s d e s i r e d

t o h o ld . T h e a s s u m p t i o n o f s p a t i a l s u m m a t i o n w i ll t h e n g i v e t h e r e -

q u i r e d r e s u l t s . S e e e .g . F, g u r e l h . T h i s p r o c e d u r e h a d a p p l i c a t i o n i n

s h o w i n g t h a t t h e o b s e r v e d t e m p o r a l s u m m a t i o n i n g r o s s n e t s d o e s n o t

i m p l y s u c h a m e c h a n i s m i n C he i n t e r a c t i o n o f in d i v i d u a l n e u r o n s .

T h e p h e n o m e n a o f l e a rn i n g , w h i c h a r e o f a c h a r a c t e r p e r s i s t -

i n g o v e r m o s t p h y s i o l o g ic a l c h a n g e s i n n e r v o u s a c t i v i ty , s e e m t o r e -

q u i r e t h e p o s s i b i l i ty o f p e r m a n e n t a l t e r a t io n s i n t h e s t r u c t u r e o f n e ts .

T h e s i m p l e s t s u c h a l t e r a t i o n i s t h e f o r m a t i o n o f n e w s y n a p s e s o r

e q u i v a l e n t l oc al d e p r e ss i o n s o f t h r e sh o l d . W e s u p p o s e t h a t s o m e a x -

o n a l t e r m i n a t i o n s c a n n o t a t f ir s t e x c i te t h e s u c c e e d i n g n e u r o n ; b u t i f

a t a n y t i m e t h e n e u r o n f ir es , a n d t h e a x o n a l t e r m i n a t i o n s a r e s i m u l -t a n e o u s l y e x c i te d , t h e y b e c o m e s y n a p s e s o f t h e o r d i n a r y k i n d , h e n ce -

f o r t h c a p a b l e o f e x c i t i n g t h e n e u r o n . T h e l o ss o f a n i n h i b i t o r y s y n -

a p s e g i v e s a n e n t i r e l y e q u i v a l e n t r e s u l t . W e s h a ll t h e n h a v e

THEOREM V I I .

A l t e r a b l e s y n a p s e s c a n b e r e p l a c e d b y c i r cl e s

T h i s is ac c o m p l i s h e d b y t h e m e t h o d o f F i g u r e l i . I t i s a l so t o b e

r e m a r k e d t h a t a n e u ro n w h i c h b e co m e s a n d r e m a i n s s p o n t a n e o u s ly

a c t i v e c a n l i k e w i s e b e r e p l a c e d b y a c ir c le , w h i c h i s s e t i n to a c t i v i t yb y a p e r i p h e r a l a f f e r e n t w h e n t h e a c t i v i t y c o m m e n c e s , a n d i n h i b i te d

b y o n e w h e n i t ce a s es .

I H T h e T h e o r y : N e t s w i t h C i rc le s

T h e t r e a t m e n t o f n e t s w h i c h d o n o t s a t i s f y o u r p r e v io u s a s s u m p -

t i o n o f f r e e d o m f r o m c ir c le s is v e r y m u c h m o r e d if fi cu lt t h a n t h a t c a se .

T h i s i s l a r g e l y a c o n s e q u e n c e o f t h e p o s s i b i l i t y t h a t a c t i v i t y m a y b e

s e t u p in a c i r c u i t a n d c o n t i n u e r e v e r b e r a t i n g a r o u n d i t f o r a n in -

d e f in i te p e r i o d o f t im e , s o t h a t t h e r e a l iz a b l e P r m a y i n v o l v e r e f e r -

e n c e t o p a s t e v e n t s o f a n i n d e f in i te d e g r e e o f re m o t e n e s s . C o n s i d e rs u c h a n e t ~ , s a y o f o r d e r p , a n d l e t c l , c ~, . . . , c~ b e a c y c li c s e t o f

n e u r o n s o f g ( . I t i s f i rs t o f a ll c le a r f r o m t h e d e f i n it io n t h a t e v e r y N ,



W A R R E N S M C C U L L O C H A N D W A L T E R P I T T S 125

of ~ can be expressed as a T P E , of N ~ , N 2 , . . . , N ~ and the absolute

affer ents; the solution of ~ involves then only the determination of

expressions for the cyclic set. This done, we shall derive a set of ex-

pressions [A] :

N ~ ( z~ ) . = .P r ~ [ S ~ , ~ N ~ ( z ~ ) , S ~ , - N ~ ( z l ) , . . . , S ~ , ~ N ~ ( z l ) ] , 2)

where Pr~ also involves periphera l afferents. Now if n is the least

common multiple of the n~j, we shall, by substituting their equiva-

lents according to 2) in 3) for the Nj , and repeating this process

often enough on the result, obtain S of the form

N~ za) . = . P r l [ S ~ N ~ ( z ~ ) , S ~ N ~ ( z l ) , . . . , S ~ N p ( z ~ ) ] . 3)

These expressions may be written in the Hilbert disjunctive normal

form as

N~ z~) . ~ . ~ S~ H S ~ N~ z~) H ~ S ~ Nj z~), for suitable ~, 4 )a e ~ i eK j ~~ a gK

where S~ is a T P E of the absolute afferents of g . There exist some

2p different sentences formed out of the p N~ by conjoining to the con-

junction of some set of them the conjunction of the negations of therest . Denumerati ng these by X~ zl) , X~ z~), --. , X z~), we ma y,

by use of the expresaions 4), a rrive at an equipollent set of equations

of the for m

~P

X ~ ( z l ) . =- . Y P r ~ j ( z l ) . S ~ X ~ ( z l ) . 5)i =

Now we import the subscripted numerals i , ] into the object-language:

i.e., define Prl and Pr: such that Prl ( z z l , z l ) . - - . X ~ ( z l ) and Pr~ ( z z l ,

z z ~ , z ~ ) . - - . P r ~ j ( z l ) are provable whenever zz~ and zz~ denote i and

j respectively.

Then we may rewrite 5) as

( z ~ ) z z p : P r l ( z ~ , z ~ )

9 = . ( E z ~ ) z z p . P r ~ ( z ~ , z ~ , z~ - z z ~ ) . P r l ( z ~ , z~ - z z ~ ) 6 )

where z z ~ denotes n and z z p denotes 2~. By repe ated s ubst itut ion we

arrive at an expression

( z ~ ) z z p : Pr~ z~, z z , zz ~ ) . = . ( E z ~ ) z z p ( E z ~ ) z z p . . . ( E z ~ ) z z j ~ .

P r ~ ( z ~ , z ~ , z z , ( z z~ ~ 1 ) ) . P r ~ ( z ~ , z 3 , z z ~ ( z z~ - 1 ) ) . . . . . ( 7 )

Pr~. z~_~, z~, 0). Pr l z~, 0), f or any numera l zz~ which denotes s.

This is easily shown by induction to be equipollent to




( z ~ ) z z p : . P r ~ ( z z , z z ~ z z = ) : = : ( E l ) ( z = ) z z 2 - - l f ( z 2 z z ~ )

z z p . f ( z z ~ z z= ) = z ~ . P r 2 ( f ( z z ~ ( z2 + 1) ) , ( 8 )

f ( z z ~ z 2) ) 9 P r ~ ( f ( O ) , O )

a n d s i n c e t h i s i s t h e c a s e f o r a l l z z 2 , i t i s a l s o t r u e t h a t

( z 4 ) ( z ~ ) z z ~ : P r ~ ( z~ , z ~ ) . ~ . ( E l ) ( z 2 ) (z 4 - - 1) . f ( z~)

<~ z z ~ . f ( z 4 ) - - z~ f ( z ~ ) - - z ~ . P r ~ [ f ( z ~ + 1 ) , f ( z~) , z2] . ( 9 )

P r ~ [ f ( r e s ( z 4 , z z ~ ) ) , r e s (z~ , zz ,~ ) ] ,

w h e r e z z , d e n o t e s n , r e s ( r , s ) i s t h e r e s i d u e o f r r o o d s a n d z z p d e n o t e s

2p. T h i s m a y b e w r i t t e n i n a l e ss e x a c t w a y a s

N ~ ( t ) . - - . ( E r ( x ) t - 1 . r _-< 2 . ~ ( t ) - - i .

P [ e p ( x + 1) , ~b (x ) . N~(o) ( 0 ) ] ,

w h e r e x a n d t a r e a l so a s s u m e d d i v is i b le b y n , a n d P r ~ d e n o t e s P .

F r o m t h e p r e c e d in g r e m a r k s w e sh a ll h a v e

T H EO R EM V I I I .

T h e e x p r e s s i o n ( 9 ) f o r n e u r o n s o f t h e c yc l ic se t o f a n e t ~ t o g e t h e r

w i t h c e r t a i n T P E e x p r e s s i n g t h e a c t w n s o f o t h e r n e u r o n s i n t e r m s

o f t h e m , c o n s t i t u t e a s o l u t i o n o f ~ ( .

C o n s i d e r n o w t h e q u e s t i o n o f th e r e a l iz a b i l i t y o f a se t o f S ~ , A

f i rs t n e c e s s a r y c o n d i t i o n , d e m o n s t r a b l e b y a n e a s y i n d u c ti o n , i s t h a t

9 i z 2 ) -= z 2 ) . 3 . -= / p l 1 1 0 )P2

s h o u l d b e t r u e , w i t h s i m i l a r s t a t e m e n t s f o r t h e o t h e r f r e e p i n S ~: i .e .,

n o n e r v o u s n e t c a n ta k e a c c o u n t o f f u t u r e p e r i p h e r a l a f f e r e n ts . A n y

S~ s a t i s f y i n g t h i s r e q u i r e m e n t c a n b e re p l a c e d b y a n e q u i p o l l en t S o f

t h e f o r m( g f ) ( z 2 ) z l ( z o )z z p : f e P r , ~

: f z l , z : , z 3 ) ~ 1 . - - . P z~ Z ~ ) 1 1 )

w h e r e z z p d e n o t e s p , b y d e f in i n g

P r ~ - - f [ ( z ~ ) ( z ~ ) z ~ ( z ~ ) z z p : . f ( z l , z ~ , z ~ ) - - 0 . v . f ( z ~ , z ~ , z ~ )

- - 1 : f z ~ , z 2 , z ~ ) - = 1 . = - . p ~ z D : - -> : S ~ ] .

C o n s i d e r n o w t h e s e s e r i e s o f c la s se s a s , f o r w h i c h

N ~ ( t ) : - - : ( E r ( x ) t ( m ) q : ~ea~ : N ~ ( x ) . - . r - - 1 .

[ i - - q + 1 , . . . , M ] ( 1 2 )



W A R R E N S . M C C U L LO C H A N D W A L T E R P I T T S 1 2 7

h o l d s f o r s o m e n e t . T h e s e w i ll b e c al le d p r e h e n s i b l e c l a s s e s . L e t u s

d e f i n e t h e B o o l e a n r i n g g e n e r a t e d b y a c l a s s o f c la s s e s ~ a s t h e a g g r e -

g a t e o f t h e c l a s se s w h i c h c a n b e f o r m e d f r o m m e m b e r s o f ~ b y r e -

p e a t e d a p p l i c a t i o n o f t h e l o g ic a l o p e r a t i o n s ; i .e ., w e p u t

( , , ) = p ' ~ [ ( ~ , ~ ) : ~ ~ K

- - ) a e ~ : a , ~ . - - ' . - a , a . , a v ~ ] .

W e s h a l l a l s o d e f in e

a n d

B

" a G O 9 - - . ~ ( , , ) - ~ ' p . . . . , , ,

~ ( , , ) = p ' ~ [ ((~ , ~ ) : a s k - - , a s , ~ . - ) .

R . ( D = ~ ( K ) - ~ ' p ' - " , , ,

- a , a . f l , a v f l , S a e ~ ]

a 9 , t ) - - - - - ~ [ m ) . + t + l , t , m ) = ~ m ) ] .T h e c la s s ~( ,(K ) i s f o r m e d f r o m K i n a n a l o g y w i t h ~ ( K ) , b u t b y r e -

p e a t e d a p p l i c a t i o n n o t o n l y o f t h e l o g ic a l o p e r a t i o n s b u t a ls o o f t h a t

w h i c h re p l a c e s a c l a s s o f p r o p e r t i e s P s a b y S ( P ) s S a . W e s h a l l

t h e n h a v e t h e

I m M M A

P r l (P l , P~ , "" , Pro, z l ) is a T P E i f a n d o n l y if

z ~ ) p ~ , . . . , p , ~ ) E pm + ~) : p ~ §( 1 3 )

Pm +l Z l ) ------- r l (p~ , P2 , , P,~ , z; )

i s t r u e ; a n d i t i s a T P E n o t i n v o l v i n g ' S ' i f a n d o n l y i f t h is h o l d s w h e n

' ~ e ' is re p l a c e d b y '~ ' , a n d w e th e n o b t a i n

T H E O R E M I X .

A s e r i e s o f c l a s s e s a l , a ~.,

a n d o n l y i f

( E ra ) ( E n ) ( p ) n ( i ) ( 9 ) : . ( x ) m ~ ( x ) z 0 v ~ ( x ) - - 1 : -+ : ( E f t )

( E y ) m . ~ ( y ) - - O . f l s ' t ~ [~ ( ( E i ) . ? ~ - a ~ ) ) . v . ( x ) m .

~ ( x ) - - O . f l s ~ [ r ( ( E i ) . r ~ - a i ) ] : ( t ) ( ~ ) : ~ e a ~ .

q ( ~ , n t + p ) . - ~ . ( E f ) . f e f l . ( w ) m ( x ) t - 1 .

~ ( n ( t + l ) + p , n x + p , w ) - - ~ f ( n t + p , n x + p , v ~ ) .

9 . as i s a s e r i e s o f p r e h e n s i b l e c l a s s es i f

( 1 4 )



128 LOGICAL CALCULUS FOR NERVOUS ACTIVITY

The proof here follows directly fro m the lemma. The condition is

necessary, since every net for which an expression of the form (4)can be wri tte n obviously verifies it, the ~ s being the charact erist ic

functions of the S~ and the p for each ~ being the class whose desig-

nation has the form H P r ~ H P r s , where P r ~ denotes a~ fo r all k. Con-~ea ie Sa

versely, we may write an expression of the form (4) for a net g(

fulfilling prehensible classes satisfying (14) by putting for the Pra

P r deno ting {he ~ s, and a P r , written in the analogue for classes of

the disjunctive normal form, and denoting the a corresponding to tha t

, conjo ined to it. Since eve ry S of the form (4) is clearly realizable,

we have the theorem.It is of some ,interest to consider the extent to which we can by

knowledge of the present determine the whole past of various specialnets: i.e., when we may construct a net the firing of the cyclic set of

whose neurons requires the peripheral afferents to have had a set of

past values specified by given functions ~ . In this case the classes

at of the last theorem reduced to unit classes; and the condition may

be transformed into

E m , n ) p ) n i , ~f) E l ) : . (x)m: ~(x) - - - -0.v.~(x)--1 :

r + p ) : -~ : w ) m x ) t - 1 . ~ n t + 1)+ p , n x + p , w ) ~ - ~ j n t + p , n x + p , w ) : .

u , v ) w ) m . g )~ n u + 1) + p, nu + p, w)

- - r l ) + p , n v + p , w ) .

On account of limitations of space, we have presented the above

argument very sketchily; we propose to expand it and certain of its

implications in a further publication.

The condition of the last theorem is fairly simple in principle,

though not in deta`il; its application to practical cases would, however,

require the exploration of some 22. classes of functions, namely themembers of ~({a~ , .. . , a~)). Since each of these is a possible fl ofTheorem IX, this res ult canno t be sharpened. But we may obtain a

sufficient condition for the realizability of an S which is very easily

applicable and probably covers most practical purposes. This is given

y

THEOREM X

Let us define a set K of S by the following recursion :

1. Any T P E and any T P E whose arguments have been replacedby members of K belong to K;

2 . I f P r l z l ) is a member of K, then z ~ ) z l . Prl(z~), E z ~ ) z l .



WARREN S MCCULLOCH AND WALTER PITTS 29

P r l z ~ ) , a n d C ~ . ( z l) . s b e l o n g to i t , w h e r e C ~ d e n o t e s t h e p r o p e r t y

o f b e i n g c o n g r u e n t t o m m o d u l o n , m < n .

3 . T h e s e t K h a s n o f u r t h e r m e m b e r s

T h e n e v e r y m e m b e r o f K i s r e a li z ab l e .

F o r , i f P r l ( z l) i s r e a l iz a b l e , n e r v o u s n e t s f o r w h i c h

N i z ~ ) . = -- . P r l z l ) . S N ~ z ~ )

N ~ z ~ ) . = . P r , z l ) v S N ~ z ~ )

a r e t h e e x p r e s s i o n s o f e q u a t i o n ( 4 ) , r e a li z e z ~ )z ~ . P r l z D a n d

E z ~ ) z l . P r ~ z D r e s p e c t i v e l y ; a n d a s i m p l e c i r c u i t , c ~ , c~ , . . . , e ~ ,o f n l i n k s , e a c h s u f f ic i e n t t o e x c i t e t h e n e x t , g i v e s a n e x p r e s s i o n

N m z ~ ) . - - . N I O ) . C ~

f o r t h e l a s t f o r m . B y i n d u c t i o n w e d e r i v e t h e th e o r e m .

O n e m o r e t h i n g i s t o b e r e m a r k e d i n c o n c l u s io n . I t i s e a s i l y

s h o w n : f ir s t, t h a t e v e r y n e t , i f f u r n i s h e d w i t h a t a p e , s c a n n e r s c o n -

n e c t e d to a f f e re n t s , a n d s u i t a b l e e f f e r e n ts t o p e r f o r m t h e n e c e s s a r y

m o t o r - o p e r a t i o n s , C all c o m p u t e o n l y su c h n u m b e r s a s c a n a T u r i n g

m a c h i n e ; s ec o n d , t h a t e a c h o f t h e l a t t e r n u m b e r s c a n b e c o m p u t e d b y

s u c h a n e t ; a n d t h a t n e t s w i t h c i r c le s c an b e c o m p u t e d b y s u c h a n e t ;a n d t h a t n e t s w i t h c i r c l es c a n c o m p u t e , w i t h o u t s c a n n e r s a n d a t a p e ,

s o m e o f t h e n u m b e r s t h e m a c h i n e c a n , b u t n o o th e r s , a n d n o t a l l o f

t h e m . T h i s is o f i n t e r e s t a s a f f o r d i n g a p s y c h o l o g i c a l j u s t i f i c a t i o n o f

t h e T u r i n g d e f in i ti o n o f c o m p u t a b i l i t y a n d i ts eq u i v a l e n ts , C h u r c h ' s

~ d e f i n ab i l i t y a n d K l e e n e ' s p r i m i t i v e r e c u r s iv e n e s s : I f a n y n u m b e r

c a n b e c o m p u t e d b y a n o r g a n i s m , i t is c o m p u t a b l e b y t h e s e d e f i ni ti o ns ,

a n d c o n v e r s e l y .

I V . C o n s e q u e n c e s

C a u s a l i t y , w h i c h r e q u i r e s d e s c r i p t i o n o f s t a t e s a n d a l a w o f n e c -

e s s a r y c o n n e c t i o n re l a t i n g t h e m , h a s a p p e a r e d i n s e v e r a l f o r m s i n

s e v e r a l s c ie n c e s , b u t n e v e r , e x c e p t i n s ta t i s t i c s , h a s i t b e e n a s i r r e -

c i p r o c a l a s i n t h i s t h e o r y . S p e c i f i c a ti o n f o r a n y o n e t im e o f a f f e r e n t

s t i m u l a t i o n a n d o f t h e a c t i v i t y o f al l c o n s t i t u e n t n e u r o n s , e a c h a n

a l l - o r - n o n e a f f a i r , d e t e r m i n e s th e s t a t e . S p e c i f i c a ti o n o f t h e n e r -

v o u s n e t p r o v i d e s t h e l a w o f n e c e s s a r y c o n n e c t i o n w h e r e b y o n e c an

c o m p u t e f r o m t h e d e s c r i p t io n o f a n y s t a t e t h a t o f t h e s u c c e e d i n g

s t a te , b u t t h e i n c lu s i o n o f d i s j u n c t i v e r e l a t i o n s p r e v e n t s c o m p l e t e d e -

t e r m i n a t i o n o f t h e o ne b e f o r e . M o r e o v e r , t h e r e g e n e r a t i v e a c t i v i t yo f C o n s t it u e n t c i r c le s r e n d e r s r e f e r e n c e i n d e f i n i t e a s t o t i m e p a s t .

T h u s o u r k n o w l e d g e o f th e w o r l d , i n c lu d i n g o u r s e lv e s , i s i n c o m p l e t e



13 L O G I CA L C A L C U L U S F O R N E R V O U S A C T I V IT Y

b

~

h

F I G ~ E

d



WARREN S. MCCULLOCH AND WALTER PITTS 3

a s to s p a c e a n d i n d e f in i t e a s t o t i m e . T h i s i g n o r a n c e , i m p l i c i t i n a l l

o u r b r a in s , i s t h e c o u n t e r p a r t o f t h e a b s t r a c t i o n w h i c h r e n d e r s o u r

k n o w l e d g e u s e fu l . T h e r o le o f b r a i n s i n d e t e r m i n i n g t h e e p is t e m i c

r e l a t io n s o f o u r t h e o r i e s t o o u r o b s e r v a t i o n s a n d o f t h e s e t o th e f a c t s

i s a ll to o c le a r, f o r i t is a p p a r e n t t h a t e v e r y i d e a a n d e v e r y s e n s a t i o n

is r ea l iz e d b y a c t i v i t y w i t h i n t h a t n e t, a n d b y n o su c h a c t i v i t y a r e t h e

a c t u a l a f f e r e n t s f u l ly d e t e r m i n e d .

T h e r e is n o t h e o r y w e m a y h o l d a n d n o o b s e r v a t i o n w e c a n m a k e

t h a t w i l l r e t a i n s o m u c h a s i ts o ld d e f e c t iv e r e f e r e n c e t o t h e f a c t s i f

t h e n e t b e a l t e re d . T i n i t u s , p a r a e s t h e a i a s , h a l l u c i n a t io n s , d e l u s io n s ,

c o n f u s i o n s an d d i s o r ie n t a t i o n s i n t e r v e n e . T h u s e m p i r y co n f i r m s t h a t

i f o u r n e t s a r e u n d e f i n e d , o u r f a c t s a r e u n d e f in e d , a n d t o t h e r e a l

w e c a n a t t r i b u t e n o t s o m u c h a s o n e q u a l i t y o r form. W i t h d e t e r m i -

n a t i o n o f t h e ne t , t h e u n k n o w a b l e o b j e c t o f k n o w l e d g e , th e t h i n g i n

i t s e l f , c e a s e s t o b e u n k n o w a b l e .

T o p s y c h o l o g y , h o w e v e r d ef in e d , sp e c i f ic a t io n o f t h e n e t w o u l d

c o n t r i b u t e a ll t h a t c o u l d b e a c h i e v e d i n t h a t f ie ld e v e n i f t h e a n a l y s i s

w e r e p u s h e d t o u l t i m a t e p s y c h i c u n i t s o r p s y c h o n s , f o r a p s y c h o n

c a n b e n o le s s t h a n t h e a c t i v i t y o f a si n g l e n e u r o n . S i n c e t h a t a c t i v i t y

i s i n h e r e n t l y p r o p o s i t i o n a l , a ll p s y c h i c e v e n t s h a v e a n i n t e n t io n a l , o r

s e m i o t i c , c h a r a c t e r . T h e a l l - o r - n o n e l a w o f t h e s e a c t i v i ti e s , a n dt h e c o n f o r m i t y o f t h e i r r e l a t i o n s t o t h o s e o f th e l og ic o f p r o p o s i t i o n s ,

i n s u r e t h a t t h e r e l a ti o n s o f p s y c h o n s a r e t h o s e o f t h e t w o - v a l u e d l o g ic

o f p r o p o s i t io n s . T h u s i n p s y c h o l o g y , i n t r o s p e c ti v e , b e h a v i o r i s t ic o r

p h y s i o lo g i c a l, t h e f u n d a m e n t a l r e l a ti o n s a r e t h o s e o f t w o - v a l u e d l og ic .

EXPRESSION FOR THE FIGURES

In the figure the neuron c~ is always marked with the numeral i upon the

body of the cell, and the corresponding action is denoted by N with i as sub-

script, as in the text.

F i gu r e l a N ~ ( t ) . ~ -- . N l ( t - - 1)

F i g u r e l b N z ( t ) . ~ . N l ( t - - 1 ) v N ~ ( t - - 1 )

Ffgure l c N ~ ( t ) . ~ . N ~ ( t - - 1 ) . N . e ( t - - 1 )

F i g u r e l d N 3( t ) . ~ . N ~ ( t - - 1 ) . o ~ N a ( t - - 1 )

Fig ure le N3(t) : ~ : N ~ ( t - - 1) . v. N ~ ( t - - 3). N2 t _ 2)

N 4 ( t ) . ~ . N 2 ( t - - 2) . N~ ( t - - 1 )

F i g u r e l f N 4 ( t ) : ~ : e ~ N ~ ( t - - 1 ) . N ~ ( t - - 1) vNa(t - - 1) .v .N ~( t -- 1).

N ~ ( t - - 1). N 3 ( t - - 1)

N 4 ( t ) : - ~ : c o N ~ ( t - - 2). N ~ ( t - - 2:) v N ~ ( t - - 2) . v 9 N ~ ( t - - 2).

N ~ ( t - - 2). N 3 ( t - - 2)

Figure lg N a ( t ) . - ~ . N ~ ( t - - 2). ~ N ~ ( t - - 3)F i g u r e l h N 2 ( t ) . - ~ . N ~ ( t - - 1) . N~ ( t - - 2 )

F igure l i N 3 ( t ) :~--: N ~ ( t - - 1) . v . N~ ( t - - 1 ) . ( E x ) t - - 1 . N ~ ( x ) . N , ( x )



132 L O G I C L C L C U L U S F O R N E R V O U S C T I V I T Y

Hence arise constructional solutions of holistic problems involving

the, differentiated continuum of sense awareness and the normative,

perfective and resolvent properties of perception and execution. From

the irreciprocity of causality it follows that even if the net be known,

though we may predict future from present activities, we can deduce

neither afferent from central, nor central from efferent, nor past from

present activities--conclusions which are reinforced by the contra-

dictory testimony of eye-witnesses, by the difficulty of diagnosing

differentially the organically diseased, the hyster ic and the malingerer,

and by comparing one's own memories or recollections with his con-

temporaneous records. Moreover, systems which so respond to the

difference between afferents to a regenerative net and certain activity

within that net, as to reduce the difference, exhibit purposive beha-

vior; and organisms are known to possess many such systems, sub-

serving homeostasis, appetition and attention. Thus both the formal

and the final aspects of that activity which we are wont to call ment l

are rigorously deduceable from p resent neurophysiology. The psy-

chiatrist may take comfort from the obvious conclusion concerning

causali ty--t hat, for prognosis, history is never necessary. He can

take little from the equally valid conclusion that his observables are

explicable only in terms of nervous activities which, until recently,have been beyond his ken. The crux of this ignorance is tha t infer-

ence from any sample of overt behavior to nervous nets is not unique,

whereas, of imaginable nets, only one in fact exists, and may, at any

moment, exhibit some unpredictab le activity. Certainly for the psy-

chiatr, ist it is more to the point that in such systems Mind no longer

goes more ghostly than a ghost. Instead, diseased mental ity can

be understood without loss of scope or rigor, in the scientific ~erms of

neurophysiology. For neurology, the theory sharpens the distinction

between nets necessary or merely sufficient for given activities, and

so clarifies the relations of disturbed structure to disturbed function.In its own domain the difference between equivalent nets and nets

equivalent in the narrow sense indicates the appropri'ate use and im-

portance of temporal studies of nervous activity: and to mathemati-

cal biophysics the theory contributes a tool for rigorous symbolic

treatment of known nets and an easy method of constructing hypo-

thetical nets of required properties.



WARREN S. MCCULLOCH AND WALTER PITTS 133

LITERATURE

Carnap R. 19~8. The Logical Syntax of Language New York: Harcour t Braceand Company.

Hilbert D. und Ackermann W. 1927. Grundi2ge tier The~retischen Logik Ber-lin: J. Springer.

Russell B. and Whitehead A. N. 1925 . Princip~ Ma~hematica Cambridge:Cambridge University Press.

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