BULLETIN OF I~IATHEMATICAL BIOPHYSICS

VOLUME 13, 1951

A NOTE ON FACILITATION AND THRESHOLD PHENOMENA

NORMAN ANDERSON AND ANATOL RAPOPORT THE UNIVERSITY OF CHICAGO AND COMMITTEE ON ~([ATHEMATICAL BIOLOGY,

THE UNIVERSITY OF CHICAGO

A n e u r o n sub jec ted to a Poisson shower o2 s t imul i r e sponds on ly i f h s t imul i imp inge upon i t w i t h i n the t ime i n t e rva l a . I t is shown t h a t t he d e r i v a t i v e of the i n p u t - o u t p u t cu rve c a n n o t exceed un i ty .

One of the authors (Rapoport, 1950) derived the input-output curve for a single neuron under the following conditions:

1. The neuron is subject to a Poisson shower of stimuli. 2. The neuron responds only to the hth stimulus of a group

which falls within the interval ~, provided none of the h stimuli im- pinges during the refractory period 5.

The expression for the input-output curve turned out to be

(1) x[1--e-~" Eh_~(X ,7) ]

Z S x [ 1 - - e - ~ " E h 2 ( x , 7 ) ] + 1 + ( h - - l ) [1--e-~'Eh_l(xa)] '

k where Ek(z) -- ~ zJ/]!.

j=O

The shape of the curve is sigmoid (cf. Rapoport, 1950, Fig. 3). A question naturally arises whether by a proper choice of the para- meters, a , 0, and h , the curve can be made to exhibit a "quasi- threshold" effect, i.e., a sudden rise from a very small value to prac- tically its asymptotic value. Such a "threshold" effect is certainly present if the stimuli are regularly spaced, as has been shown in the

�9 above-mentioned paper. It was conjectured, however, that for the case of a Poisson shower (randomized stimuli) no such threshhold effect could be obtained by any choice of parameters. This is in con- sequence of the fact that the derivative ~f/3x of the input-output

47

48 FACILITATION AND THRESHOLD PHENOMENA

curve (1) is bounded for all values of s , a , 6, and h . This proper ty will now be proved.

In expression (1) call the threshold hi instead of h and let h - - hi - - 2 . We also introduce the following notat ions

( x ~ ) ' ah ( x a) : 1 - - e - ~ Eh ( x a ) : e - ~ E ~ , ( 2 )

h§ t!

D = 6 x [ 1 - - e - ~ E h _ 2 ( x a ) ] + 1 + ( h - - l ) [1--e-~aE~_l(Xa)] ( 3 )

= 6 x a h + l + (h + l)ah+~.

With this notation,

X ah f - - (4)

6 x a ~ + l + ( h + l ) a ~ , l

Let pr imed quanti t ies represent derivat ives with respect to x . Then

f - - - - - ~ ( ah + x ah') D - - x ah D '

" - ( a h + X a ~ ' ) ( 6 X a h + l + (h + 1)ah§ - - X a h ( 6 a h + 6 X a h '

+ (h 4- 1)a'h.1) [ (5)

= - ~ 6xah ~ + a h + ( h + l ) ahah+~+6x ' a ~ a h ' + x a h '

+ ( h + l ) x a h ' a h . ~ - - 6 X a h 2 ( h + l ) x a ~ , a h . ~ - - 6 2 ,] - - ' X Clh g.h �9

Now

and

( x ~ ) ~ ~ ( x ~ ) ~ ( x ~ ) ~ a ~ = - - a e - ~ " E ~ + a e - ~ ' E - - - - a e -~" , (6)

k§ t!. ~ t! k!

(X q)k+l x a~' = (k + 1) e -~

( k + l ) ! = (k + 1) (ak§ a~,~). (7)

Thus we see tha t

NORMAN ANDERSON AND ANATOL RAPOPORT 49

I f I'---~21ah(l+(h+l)a~+l)- (h+ 1 ) ( h + 2 ) a ~ ( a ~ - - a h + 3 )

~ L

(s) 1

+ (h + 1) (ah+l-- ah+2) (1 § (h § 1)ah+~) [ m

- $

We have 0 < ak+l < ak -< 1 for all k . Also

0 < D - - 5 xah ~- 1 + (h + 1)ah+~ >- 1 Jr (h + 1)ah+l. (9)

Hence,

[1 + (h + 1)a~,~] [-~ + (h + 1) ( a ~ - - ah,,)] _< (1o)

[1-4- (h -}- 1)ah+~] ~

ah + (h + 1) (ah§ - - ah+2)

1 + (h § 1) ah+~

Then f' < 1, since

ah + (h + 1) (a~+1 -- ah+~) --< I. (II)

i + (h+l)ah+1

This last inequality is true because ah -< 1 and (h + 1) (ah+~ -- ah+~) --< (h ~- l)ah+1.

This work was aided in part by a grant from the Dr. Wallace C. and Clara A. Abbott Memorial Fund of The University of Chicago.

L I T E R A T U R E

Rapoport, A. 1950. "Contribution to the Prohabilist ie Theory of Neural Nets : II . Faci l i ta t ion and Threshold Phenomena." Bull. Math. Biophysics, 12, 187- 97.