18 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



Evidently if some but not all the Ni are responding, there must be 

 some ra ^ 1 such that N lf N 2 , ••• , N m are responding, N m+1 , • • • , N n 

 are not. Then for i = m, a m as given by equation (6) must exceed, 

 o- OT+ i fail to exceed h" , and if S represents the mean of all the Si this 

 leads to the relation 



nfiS + ah- (n-l)ph' + h" 



£>m - > " = Om+1* (10) 



a + p 

 In particular if 



Oi == O2 — — • • • — — o«i — — o , 

 &in+l Orft+2 * * * o n O , 



then these relations are equivalent to 



[a-/?(ra-l)] S' - P(n-m)S" > ah- (n-l)ph' + h" 



(11) 



^ [a - P(n-m-l)] S" - fimS' . 



In either case the m stimuli Si produce their response and prevent 

 the occurrence of the response to the other n—m stimuli (cf. Rashev- 

 sky, 1938, chap, xxii ; Landahl, 1938a) . 



Receptors in the skin and the retina are far too numerous to be 

 treated by the simple algebraic methods so far employed. Here we 

 must think in terms of statistical distributions. The receptors, or at 

 least the origins of the neurons to be discussed, may have a one-, a 

 two-, or a three-dimensional distribution. According to the case, let 

 the letter x stand for the coordinate, the coordinate-pair, or the co- 

 ordinate-triple of the origin of any neuron. Let x' represent the co- 

 ordinate, the coordinate-pair, or the coordinate-triple of any terminus 

 of one of these neurons. Running from the region x , dx (consisting 

 of points whose coordinates fall between the limits x and x + dx) to 

 the region x', dx' may be excitatory or inhibitory neurons, or both. 

 If we consider only the linear representation of the functions <j> and 

 tp , each neuron is characterized by the two parameters a and h . To 

 consider a somewhat more general type of structure than the one just 

 discussed for the discrete case, let 



N (x , x' , a , h) dx dx' do. dh 



represent the number of neurons originating within the region x , dx , 

 terminating in x' , dx' , and characterized by parameters on the ranges 

 a, da and h , dh . Then, S(x) being the stimulus-density at x , the 

 a-density which results from these neurons alone at x' is 



N(x , x' , a , h) [S(x) — h] dx da dh 



