42 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



If we let t* be large and if, for a fixed S , we restrict A S to the 

 least value it can have while U remains finite (i.e., the two roots t r ' 

 and tr + t of equation (3) coincide), we obtain, if we use equation 

 (6) of chapter i, 



AS/S = d = d , (5) 



da being a constant. That is, when a constant stimulus of intensity 

 S > h x has been applied for a long time, the smallest additional stim- 

 ulus A S necessary to produce the response must be a constant frac- 

 tion of the intensity of the original stimulus S . On the other hand, 

 US < hi we have 



6= (S + l)hJS- 1. (6) 



Thus d(S) decreases hyperbolically from oo to S as S varies from 

 zero to h x ; thereafter S is a constant. The quantity <3 of equations (5) 

 and (6) is essentially a Weber ratio, and its variation with 5 as de- 

 scribed in the above equations has the chief qualitative properties of 

 the experimental relation for most types of stimuli. This problem 

 will be discussed in more detail subsequently (chap. ix). 



Suppose that instead of replacing S by S + A S at time T, we 

 remove S for a time t' after which only A S is presented. This is the 

 experimental technique for studying the processes of adaptation and 

 recovery. Then for t > t* > t' , 



CT =z < £(S) [e-°<*-'*> - r*<*-** } + e- bt - e- at ] 



(7) 



At the time t*, we shall suppose a < h . Hence at some time t = t* + 

 t' + t" r , if A S is large enough, o- = h , and the response occurs. From 

 this relation, together with equation (7), we can determine the reac- 

 tion-time t" r (AS , S , t*, V) from the smaller root, and the duration 

 t" (AS , S , t\ t') of the response from the larger root, at any stage 

 of the process of recovery following preadaptation to the intensity S . 

 We can further determine the minimal AS required for stimulation 

 as well as the minimal interval of exposure at a given AS . 



' If t* > > 1/b , t' > > 1/a , t' > > t r " and t r " < < 1/b , we have 



cl>(AS) (1 - e-"*r") - ^(S) e-w - h = 0. (8) 



Thus 



t r " = - -log( 1 - *±-L—— , (9) 



so that the reaction time t r " increases with S but decreases with both 

 A S and t' . 



