PARALLEL, INTERCONNECTED NEURONS 19 



provided h < S (x) . Hence the total a-density, obtained by summing 

 over the entire region (x), over all values of a (positive and nega- 

 tive), and over all values of h < S(x) is 



a(x') =J (x J^J s o ^aN(x,x',a,h)[S(x) -K] dhdadx. (12) 



Corresponding expressions can be derived, of course, on the suppo- 

 sition that <f> and xp are non-linear of any prescribed form (Rashev- 

 sky, 1938, chap. xxii). 



Instead of writing the special form of expression (12) for the 

 strict analogue of the discrete case considered above, let us suppose 

 next that the inhibitory neuron Na , rather than passing from Si to s'j , 

 passes from s\ to s'j (Rashevsky, 1938, chap. xxii). The net a at any 

 s'j is then equal to the a produced by excitatory neurons terminating 

 here diminished by the amount of inhibition produced by the inhibi- 

 tory neurons originating at the other s' ( - , whereas it is this net a at 

 the s'i which acts as the stimulus for these inhibitory neurons. We 

 have therefore to solve an integral equation in order to determine the 

 net a. 



In the continuous case, let a(x) be the gross a-density produced 

 by the excitatory neurons terminating in the region x , dx . For the 

 inhibitory neurons, let — p = a represent the activity parameter. Let 

 N(x' y x , /} ,'h) dx' dx dp dh represent the number of inhibitory neu- 

 rons which run from the region x' , dx' to x , dx , and have activity 

 parameters and thresholds limited by the ranges p , dp and h , dh . 

 Then 



a(x) =o(x) — 



(13) 

 f x > lo Io iX,) P N ( X ' 'X'P'W&ix') ~ &] dhdfidx'. 



This is an integral equation, but one in which the unknown func- 

 tion enters as one of the limits of integration. If we interchange 

 orders of integration, we have as a form equivalent to equation (13) 



a(x) =7(x) — 



(14) 

 J"/ 00 / 0N(x' ,x,p,h)[<r(x') -h] dx'dfidh. 



<T(X')>h 



Then if, as a particular case, all inhibitory neurons have the same ft 

 and the same h , this becomes 



a(x) =a(x) -ft J N(X' ,x)[<t(x') - K\ dx' . (15) 



It is easy to solve this equation in certain special cases. Suppose 

 N(x' , x) is independent of x' and x . It is clear that the integral is 

 then independent of x , and we may write 



