4 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



Now ^ is a function of S , and S may well be a function of t , in 

 which case $ is a function of t . The complete solution of equation 

 (1) is then given by 



e = e- at [e + a( t e aT ct>(T)dT], (7) 



J 



where £ is the initial value of e . When, in particular, S , and there- 

 fore also </> , is constant with respect to time, this becomes 



e = e-** e + <£(1 - e- at ). (8) 



Thus £ approaches the value <f> asymptotically, the approach being in 

 all cases monotonic, and either increasing or decreasing according to 

 whether </> exceeds or is exceeded by e. . 



If we were now to introduce an assumption to relate the muscu- 

 lar contraction with the applied e , we should have a system of for- 

 mulae to be evaluated sequentially along any neural pathway from 

 receptor to effector, for relating the time and the intensity of the re- 

 sponse to the temporal form of the stimulus. But this would obvious- 

 ly provide only a very incomplete picture. A given stimulus not only 

 leads to the contraction of one set of muscles ; it leads also to the re- 

 laxation of the antagonistic muscles. Any effective movement involves 

 both components, of contraction and the inhibition of contraction. 

 Thus we are inevitably led to extend our picture to include the phe- 

 nomenon of inhibition. 



There are many ways in which such a phenomenon could be in- 

 troduced into our schematic picture, but the simplest way seems to be 

 to suppose that at least some neurons have the property of creating, 

 as the result of their activity, an inhibitory state of intensity j , 

 (briefly, an inhibition j) , antagonistic to the excitatory state e , and to 

 suppose that the production of j follows the same formal law as that 

 for e: 



dj/dt = b{y>-j). (9) 



The function y> is of the same type as <£ and it is only as a matter of 

 convenience that we introduce a separate symbol. 



Rashevsky (1938, 1940) commonly assumes that in general the 

 activity of any neuron leads to the production of both s and j , al- 

 though for particular neurons, the one or the other may be negligible 

 in amount. Evidently we may always replace a single neuron devel- 

 oping both e and j by a pair, one developing £ alone and one j alone. 

 It is useful, however, to consider some of the characteristics of a 

 "mixed" neuron of the Rashevsky type. 



Since £ and j are antagonistic, we are now supposing that 



<r = s-j (10) 



