24 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



£<0 



4>'io) >1 

 Figure 2 



e is eliminated from equation (5) and 



*'U + e)=l , 



and the resulting equation solved for £ = £<, , then £ < , and for 

 £o < I < we have the case for which equation (5) has two distinct 

 positive roots. 



Let £ represent the greatest of the roots (possibly zero). Then 

 Co represents a stable equilibrium of equation (2). Since we can have 

 £ > even for £ < (if also I > £ ) it is possible for the activity to 

 persist even after the withdrawal of the stimulus when £ = — h , 

 provided —h > £ , and provided the initial value £ = £i at the time 

 of withdrawal of the stimulus exceeds the smaller, unstable, positive 

 equilibrium obtained from equation (5) when £ — — h . 



But whatever the value of </>'(0), if b x exceeds the threshold h 

 at the time the stimulus is withdrawn, some activity will continue 

 for a time, if not permanently. In order to account for learning in 

 terms of activated circuits, the continuation must be permanent or 

 nearly so (cf. chap. xi). Very likely a number of circuits would be 

 involved in any act of learning, in which case forgetting could be 

 accounted for as a result of the gradual damping out of one after 

 another because of extraneous inhibition. In order to determine the 

 period of the continuation where it is not permanent, it is necessary 

 to know something about how the applied stimulus S disappears. If 

 S suddenly drops to zero, then the time required for the activity to 

 die out is given by equation (3) with £ = — h and the upper limit £ 

 of the integration equal to +h . But if S is itself an £ from another 

 neuron, a new set of equations must be written down and solved. 



If a circuit is formed by a single inhibitory neuron, the behavior 

 is described by the equation 



