28 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



- 16ji 2 v 2 = [a- b - (A + B)e i y - 4 AB e 2 . 



When the coefficient of E in equation (14) is negative, there is- 

 always a single non-positive root. If this is negative let <r denote it. 

 In the transformed equation (17) the coefficient of x is always posi- 

 tive, but the discussion is otherwise the same as before. 



We consider finally a circuit consisting of an inhibitory neuron 

 extending from s x to s 2 and an excitatory neuron from s 2 to s x . The 

 equations are 



de/dt = a [<j> (| 2 - j) - e] , 



dj/dt=bbp(i-i + e) -n. (20) 



There is always a single equilibrium obtained by equating to zero 

 the right members of these equations (Figure 4) . Let e, , j represent 



Figure 4 



the values at equilibrium, and let neither of them vanish. Then if 

 we set 



X = £ - £ , y = j - jo 



and expand, equations (20) have the form 



x' = — a(x + ay) +•••, 

 y' = b(fix-y) +..., 



(21) 



where —a and /? are the derivatives of 4> and of \p at /<> and at e , re- 

 spectively. The characteristic equation is 



A 2 + (a + b)X + ab (1 + a 0) — . 



(22) 



Since all parameters are positive, the real parts of the characteristic 

 roots are always negative and the equilibrium is stable. If, further 



