34 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



the y(t) in that expression being otherwise replaced by the nearest 

 limit. Let us introduce quantities a t (t), fti(t) denned as follows: 



0Li(t)=l when jf(f)=Vo 

 uj(O=0 when y(t)<yi, 



(10) 

 Pi(t)=l when y(0 =2/i", 



A(0 =0 when y(t) >yi". 



Then the contribution of the i-th chain to y(t + //;) may be written 



Si + A^imWfiiit) y(t) + tl-aiW] y{ + [1 -&(*)] 1/i" }. 



If we introduce the operator E defined by 



Ey(t) = y(t + l), 



let fi represent the largest of the ju t , and set 



P § = (i — fij , 

 then we have finally 



E*y{t)=Z + ^A i E» {*dt) pi(t)y(t) + [1 - a ,(t)] yi 



(11) 



+ ii-fawiyn. 



The functions a and /? are constant except when y crosses one of the 

 boundaries y' or y" associated with the corresponding chain. Hence 

 the difference-equation (11) can be solved on the assumption that 

 the a's and /3's are constant, and the solution is valid as long as it lies 

 on the particular interval associated with the assumed values of the 

 a's and /3's. 



If the numbers y,' and y" are arranged in order, they limit at 

 most 2n + 1 intervals (two of them infinite), and each interval is 

 associated uniquely with a particular set of values a, , fit . No other 

 set of the a ; , /?/ is possible. Associated with each of these sets a* , /J» , 

 is a unique value of y satisfying 



[l-2A»ai0i] y-^ + 24i [(1-002//+ (l-fii)yn (12) 



provided the coefficient of y is non-null. This defines a possible equi- 

 librium of the difference-equation. However, if this value of y does 

 not lie on the associated interval, then no equilibrium for the gen- 

 eral equation (11) exists on this interval. If, for the set a, , /3; , the 

 solution y of equation (12) does lie on the associated interval, the 

 solution y(v + t) of the difference-equation (11) corresponding to 

 the a ; , /?, equal to these constant values differs from this constant 

 value y by a sum of terms of the form p(v)A v , where p(v) is a poly- 

 nomial in v multiplied, possibly, by a sine or a cosine, and a is a real 



