CHAINS OF NEURONS IN STEADY-STATE ACTIVITY 9 



o-i = a(tr ~ h) , 



<r 2 = a(<T! — h) =a 2 <r — ah(a + 1), 



ui = a(a i _ 1 - h) =a*a - aMa*- 1 + a*" 2 + ••• + 1), 

 or 



(ah \ ah 

 «o+- )-- , a^l, (4) 



1 — a / 1 — a 



vi = o — ih , a = 1 . (5) 



Hence if a = 1 , the a; decrease in arithmetic progression until, for 

 some i ,<nfkh , after which all succeeding <r i+v are zero. Otherwise the 

 sequence 



<n + ah/(l — a) 



forms a geometric progression. The progression, and hence the se- 

 quence <Ji , increases when 



a > 1 , <t > ah/ (a — 1). 



When the second inequality is reversed but the first holds, the pro- 

 gression consists of negative terms which increase numerically until 

 some <n falls below threshold. If the first inequality is reversed, the 

 progression is decreasing. Finally, if the second inequality becomes 

 an equality, then every a* has the same value. 



So far we have been supposing that the only excitation intro- 

 duced from the outside — from receptors, that is, or from neurons 

 which are not members of the chain— was introduced at So alone. We 

 have further supposed all the neurons in the chain to be excitatory, 

 that is, asymptotically exciting while acting, since evidently, in such 

 a situation, no activity could occur beyond an inhibiting member of 

 the chain. We turn now to a somewhat more general situation in 

 which the chain may contain neurons which are asymptotically in- 

 hibiting while acting, and in which outside excitation may be present 

 at any or all the Si . Following a suggestion made by Pitts (1942b), 

 we now represent the function <j> as linear until the stimulus reaches 

 a certain maximal value, and constant at the upper limit thereafter. 

 This representation, though no doubt less accurate than the functions 

 (3) and (4) of chapter i, is at any rate a fair first approximation, 

 and is much more easily handled. Let St be the applied stimulus at 

 Si , and define the quantities 



£i = Si — hi , rji = £i + <Ti . (6) 



Then r\i represents the excess over the threshold of the total stimulus 



