THE DYNAMICS OF SIMPLE CIRCUITS 



25 



dj/dt = b [y(£- j) - ?] . 



(6) 



Then y> > only if | > , but in this case there is always a single, 

 stable, equilibrium. The result is that the applied 5 is decreased at 

 equilibrium by a certain amount j. . Also j (> increases as S increases, 

 although if \p has a finite asymptotic value, j cannot exceed this, 

 whatever the value of & . We may note, however, that the presence 

 of additional circuits of this kind, with higher thresholds, which add 

 their effects with increasing S , would provide an effective damping 

 mechanism over an arbitrarily large range. 



Consider next a two-neuron circuit, with one neuron passing 

 from s 2 to s 2 , and the other from s 2 to Sj . Suppose, first, that these 

 are both excitatory, and, for simplicity, that they are identical in 

 character. Let | t be the excess of Si over the threshold of the neuron 

 originating at s x , let £i represent the excitation produced here by the 

 other neuron, and let | 2 and e 2 represent the corresponding quantities 

 at s 2 . Then, still neglecting the conduction time, we have 



de l /dt = a [<j}(£ 2 + £2) ~ £1] » 

 de 2 /dt = a [<£(!i + £1) — £ 2 ] . 



(7) 



If it happens that £1 = £ 2 and that the initial values of the £'s 

 are equal, then it follows from symmetry that £1 = £ 2 for all t , and 

 the pair of equations (7) can be replaced by a single equation of the 

 form (2). In the general case for any £ t and £ 2 , an equilibrium is 



Figure 3 



determined by an intersection of the curves in the (e x , e 2 ) -plane de- 

 fined by the two equations [Figure 3] 



£i = <M£ 2 + £ 2>» ei=<j>l£i + £1). (8) 



