26 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



For any point to the right of the first curve s x is decreasing, and for 

 any point above the second curve e 2 is decreasing. In case <£'(0) ^ 1 , 

 the two curves have always one and only one intersection ; at this 

 both e's are positive only if at least one £ > and the other is not too 

 small, and they define a stable equilibrium. In case ^'(0) > 1 there 

 will be one or three intersections. If there are three, one of these is 

 always the origin, and this is always a stable equilibrium; if there 

 is only one, this equilibrium is always stable and it may be the origin. 

 In the former case, the intersection farthest from the origin is also 

 stable. In particular, continuous activity following the withdrawal 

 of the outside stimuli can occur only in circuits for which ^'(0) > 1 , 

 and then only in case at least one of the initial e's has become suffici- 

 ently large. More detailed discussions of this type of circuit in which 

 an exponential form is assumed for the functions <j> , but these are 

 not assumed to be identical, have been given by Rashevsky (1938), 

 and Householder (1938b). Rashevsky (1938) has introduced these 

 circuits in his theory of conditioning [cf. chap. xi]. 



Circuits containing both excitatory and inhibitory neurons are 

 somewhat more interesting, because of the possibility of periodical 

 phenomena (Landahl and Householder, 1939; Sacher, 1942). The 

 scratch-reflex is one of numerous examples of a repetitive or fluctuat- 

 ing response to a stimulus. Consider the case of a single inhibitory 

 and a single excitatory neuron, both of which originate and termi- 

 nate at the same place, s . A self-stimulating neuron of mixed type 

 may be regarded formally as a special case. For a mixed neuron, it 

 is common to assume (Rashevsky, 1938) that the functions of <j> and \p 

 have a constant ratio for all values of their common argument, and we 

 shall make this assumption for simplicity. Then the equations may 

 be written 



de/dt = AE(£ + e - j) -as, 



dj/dt = BE(§ + s-j)-bj, (9) 



E — 04/ A = by/B . 



By dividing out a suitable factor from E and incorporating it into 

 A and B , we may suppose without making any restrictions that 



£7(0) =1. (10) 



Now if it should happen that a = b , we could subtract the sec- 

 ond of these equations (9) from the first, replace e — j everywhere 

 by <r , and have a single equation of the same form as equation (1). 

 Hence we suppose a ^ b . The pair of equations (9) in e and j can 

 be reduced to a single second-order equation in a as follows. Differen- 

 tiate equations (9) once each, and the equation 

 :i- ■ j ■ — j ■— a - (11) 



