10 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



acting upon neuron A/* . In accordance with our description of the 

 0's, therefore, we have 



CTi+i = ^>i(^i) (7) 



where 



4>i (v> ) — when rji ^ , 



<f>i (rji) = a» rji when < rji < H % , (8) 



</>« (rji) = an Hi when rji ^ #; . 



The coefficient a* we shall call the activity-parameter of 2V< , and it 

 may be positive, for an excitatory, or negative, for an inhibitory neu- 

 ron, but is not zero. Our problem is the following: supposing Si , 

 5 2 , • • • , S n fixed, to express rj n as a function of rj — S — K , and, 

 more generally, to express rj i+v as a function of rji when S i+1 , ••• , S^ v 

 remain fixed. 



Since each rj i+1 varies linearly with rj x when the latter occupies a 

 certain restricted range, and rj in is otherwise constant, it is at once 

 apparent that the same is true of the variation of any iji+ v with rji . 

 Again, since any & may be so large that rji always exceeds Hi , or so 

 small that rji is always negative, it is evident that rju v may remain 

 constant for all values of rji . In such a case s i+v is said to be inac- 

 cessible to Si (Pitts, 1942a) ; otherwise it is accessible. More explic- 

 itly, if, as rji varies over all values from — oo to + oo while Si +1 , ••• , 

 Si+v remain fixed, the value of rj i+v remains constant, then s i+v is inac- 

 cessible to Si . Clearly s i+1 is always accessible to Si . If s i+v is inac- 

 cessible to Si , then s i+v is also inaccessible to any s , and further , 



i—v' 



any s is inaccessible to Si . Finally , it is clear that if Si +V is acces- 



i+v+v' 



sible to Si , then where rj^ v varies with rji , it decreases if there is an 

 odd number, increases if an even number of inhibitory neurons (with 

 negative a's) between Si and Si+ V . 



The above conditions for inaccessibility may be phrased thus: 



// any of the four following conditions holds: 



Mew Si+2 *s inaccessible to s% . Otherwise s i+2 is accessible to Si . 



These conditions are also sufficient for the inaccessibility of any 

 s i+2+ v to any s. , where v and v are non-negative integers, but the 



i-v' 



conditions are not necessary. 



The following conditions for accessibility are somewhat less ob- 

 vious. Let us say of a neuron AT* whose stimulus exceeds its threshold 



