32 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



after. Then, if y is now taken to represent the value of y during this 

 initial interval, we have 



y(t)=y (0^t<l), 



y(t + l)=Z + Ay(t) (t^O), (4) 



when y(t) lies between the limits y' and y", and when this is not the 

 case the nearest limit replaces y(t) in the latter equation. 



Now it is clear from the nature of the mechanism that the fol- 

 lowing- possibilities are exclusive and exhaustive: 



i) y(t) approaches asymptotically a value y x on the interval 

 from y to y" . 



ii) y(t) reaches and remains constant at a value in excess of 

 y" or else below y'. 



iii) y(t) ultimately alternates between two fixed values. 



If we set 



t = v + r (0^t<1), (5) 



where v is an integer, then when A ¥= 1 , the solution of the difference- 

 equation (4) has the form 



1- A v 



y(v + T ) Z + y A v = y„ + {y«-y„)A v (6) 



1- A 



where 



y„ = Z/(l-A), (7) 



until y falls outside the interval from y to y". Hence case (i) occurs 

 provided |A| < 1 and 



v''s§ v. =£ y". (8) 



If A > 1 , the interval between y(t) and y :X , wherever the latter may 

 be, continues to increase while y(t) lies on the interval, and after 

 having passed either limit — which will occur in a finite time — it re- 

 mains constant. If A < — 1 , the interval y — y^ increases numeri- 

 cally but with alternating sign until, after a finite time, one limit or 

 the other is passed. Thereafter, if equation (8) is satisfied without 

 the equality, (iii) occurs, while if equation (8) fails or if an equality 

 holds, then (ii) occurs. When A = — 1 , alternation between fixed 

 values starts immediately if relation (8) is satisfied without the equal- 

 ity, and otherwise (ii) occurs. Finally if A = 1 , it is evident from 

 the difference-equation (4) itself that the y(y) form an arithmetic 

 progression until one limit is passed, the later terms being identical. 

 Additional essential complications are involved in the discussion 

 of nets consisting of two or more circuits. However, certain simplifi- 

 cations can be performed at once. We wish to determine y at each 



