50 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



u 



LU 



T 

 O 



x 



>- 

 u 



z 



<JJ 



3 



a 



time r 



1.0 



IN SECONDS 



1.5 



Figure 1. — Comparison of theory with experiment: adaptation to muscular 

 stretch. Curve theoretical, predicted by equation (1) ; points, experimental 

 (Matthews, 1931). Abscissa, duration of stretch; ordinate, frequency of response. 



obtain the same result regardless of the point of view adopted. 



The structure consisting of two afferents and one efferent where 

 the afferent N x is now simple excitatory leads to other interesting 

 results when the stimulating conditions are altered. Let Si > h^. and 

 S 2 > h 2 be constant stimuli applied at times t ± = and t 2 to the neu- 

 rons N x and N 2 respectively. Let h be, as before, the threshold of the 

 efferent neuron. Also let e x , e 2 and j 2 be zero initially. Then the effer- 

 ent neuron is excited at the time t' when a = e x + e 2 — j 2 = h , or 



[1 



-&2< 



t'-h)] =hm ( 2 ) 



^(1 - e-^') + & [1 - e^»(*'-*.)] - xp 2 



The value of t' depends on S x , S 2 and £> • 



To simplify the problem further, let <£ 2 = y*2 and let o- 2 be always 

 less than h . This can be done readily by restricting S 2 or requiring 

 that cj> 2 < h . Then no response can occur prior to the moment t = , 

 even if t 2 < . If we set t w = t' — t 2 we obtain by solving equation (2) 



