THE GENERAL NEURAL NET 35 



root or the modulus of a complex root of the equation 



a* ~ 2 Ai ai fr xp* = . (13) 



As before, v is an integer for which 



t = V + T (O^T^l). 



Hence the equilibrium is unstable unless every root of equation (13) 

 has a modulus less than unity. 



In case for any of the intervals the coefficient of y in equation 



(12) vanishes, this equation has no solution unless the right member 

 also vanishes. But then equation (13) has a root unity and the corre- 

 sponding solution of (11) involves a simple polynomial of non-null 

 degree, so that no stable equilibrium occurs. Thus, in brief, in order 

 for any interval to possess a stable equilibrium, it is necessary and 

 sufficient that the solution y of equation (12) obtained from the 

 associated set a, , p t , shall lie on this interval, and that the equation 



(13) shall have every root of modulus less than unity. Fluctuating 

 equilibria of the sort met with in the simple circuit are here possible, 

 and also another sort arising from possible complex roots of the char- 

 acteristic equation (13) and leading to terms involving sines and 

 cosines. 



In the general case, let the synapses Si and the chains C K be sep- 

 arately enumerated, and let us define two sets of quantities P jK and 

 Qj K as follows : 



P jK = 1 if Sj is the origin of C K , 



Pjk = if Sj is not the origin of C K , 



Q iK = 1 if Si is the terminus of C K , 



Q iK = if s» is not the terminus of C K . 



All simplifications as described above having been made, we can de- 

 fine for each synapse s< a quantity 



Zi = Si + ^Q iK E Kt 



and for each chain C K the sets of quantities a K , p K with values or 1 

 according to the value of y , at the origin C K , relative to y K ' and y K ". 

 Then the difference equations satisfied by the y t have the form (Pitts, 

 1943a) 



E* yi = Zi + 2 Q^ A K E»«{(1- a K )y K ' 



(14) 



+ (1 -Pk)v" t- Ok & 2^*2//}. 



If there are n, chains originating at Si , the a's and p's associated with 

 these chains are able to take on at most 2/1* — 1 different sets of val- 

 ues, and there are therefore at most U (2nt — 1) sets of values for 



