44 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



ter i, replacing c by c' M _i and t by rT to calculate e»_i , and we use the 

 same equation to calculate e'„ by setting <j> = , replacing s by £„_i , 

 and £ by (1— r)T . When we do this we find by simple induction that 



e ' B = 0(l - e-^) (1 - e- naT )e- a ^ T /(l - e-" T ) , 



and as n becomes large the exponential containing it can be neglected. 

 The expression for j' n , similarly defined, is the same with b replacing 

 a. To obtain j(t) during the interval in question, we need only re- 

 place j by j' n and b by a in the same equation (8) . After taking the 

 difference e — j and performing elementary algebraical simplifica- 

 tions, we obtain finally the desired expression: 



— A p-bt 



a = <},e 



/ 1 - e- hil - r)T \ / 1 - e- a ^- r)r \ 



Now a reaches a maximum at t = t* given by 



t* = 



log 



a 



a(l- c*< 1 - r > f ) (1 - e- bT l 

 6(1 - e-^-'^Ml - <r aT ) 



(13) 



unless t* > rT , in which case the maximum value of a is a(rT). Set 

 a = h in equation (12) with t replaced by rT or t* according to the 

 case. Then for given r and T , that value of 5 which satisfies the 

 equation is the least stimulus that will produce a steady response 

 when repeated in this manner. 



For this r and S , let T* be the particular value of T employed. 

 Then f = 1/T* is a critical frequency separating response from no 

 response for the value of 5 in question. Then, if <j> = ft log S/hi and 

 H = h/0, for rT* < t* or f > r/t*, 



I 6-°' 

 V ~~1 



brT* _ g-br* 



-bT* 



irT* (>- aT * \ S 



log- = H, 



1 - e- aT * I K 



(14) 



and for rT* > t* or /* < r/t*, 



b 



1 -- e~ aT * 

 b 



-a(l-r)T* 



-. -a- 



1 - <r w * 



a 



9 -b(l-r)r* 



H 



L (a - b) log (S/h,) J 



a-b 



(15) 



For very large frequencies /*, we have from expression (14), for r 

 not too near zero or unity, 



a-b , S 



/* = r(l — r) log—. 



2H ha 



(16) 



This states that the frequency above which response fails to occur 

 is proportional to log S/Jh as well as to r(l-r), the function of r 



