36 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



them altogether. Each set of values is associated with a region in 

 y-spa.ce which may contain a single point whose coordinates yi rep- 

 resent a steady-state of the net. For this to be so (the equilibrium 

 being stable) two conditions must be fulfilled: The constant y- x de- 

 fined by equations (14) when the a K and /5 K are given these values and 

 the operator E is taken to be the unit operator must define a point 

 which lies in this region ; and a certain algebraic equation (the char- 

 acteristic equation of these difference-equations) must have only roots 

 whose moduli are less than one. In principle, therefore, the steady- 

 state activity of nets of any degree of complexity can be determined, 

 though admittedly the procedure could become exceedingly laborious. 

 Thus given three synapses, joined each to each by a total of six chains, 

 27 regions in ?/-space may exist and require separate consideration 

 as possible locations of equilibria. Moreover, persistent fluctuations 

 may arise, no steady-state being approached at any time. 



While the solution of the direct problem of describing the output 

 of any given net is complete, at least in principle, the general inverse 

 problem is still open. However, in the special case where the output 

 function is such that the a's and /5's remain constant, Pitts (1943a) 

 has shown how to construct a rosette to realize this function. 



This concludes our purely formal discussion of neural structures, 

 and we turn now to some special structures and their possible rela- 

 tion to concrete types of response. 



