38 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



For first application we choose the simplest structures, working 

 gradually to those of increasing complexity. We shall find in the 

 present chapter how a very simple mechanism will serve for the inter- 

 pretation of such superficially different sorts of data as those con- 

 cerning the occurrence and duration of a gross response, just-dis- 

 criminable intensity-differences, adaptation-times, and fusion-frequen- 

 cies in vision and perhaps other modalities. In general, even where 

 the structure is relatively simple, it is not possible to solve in closed 

 form the equations resulting from this structure. Thus certain re- 

 strictions upon the parameters may have to be introduced in order 

 to obtain a workable, even if approximate, solution. As the choice of 

 the restrictions is somewhat arbitrary, one should keep in mind that 

 other equally plausible restrictions could lead to different results and 

 increase both the accuracy and the scope of the theory. 



The simplest structure which can be studied is a single neuron, 

 and the simplest assumption that can be made about its activity is 

 that it is of the simple excitatory type. Its activity is determined 

 when we have evaluated e(t) for any 5 . However, one does not ob- 

 serve e but some response R . Thus the simplest structure in which 

 we can deal with observed quantities is a chain of two neurons, the 

 first being acted upon by some stimulus S , and the second, which may 

 be a muscular element, capable of producing some response R . The 

 response R is produced as soon as e reaches the threshold of the sec- 

 ond neuron. Hence if we set h = s(t) and solve for I , then since the 

 function s(t) depends upon S through the function $(S) (chap, i, 

 equation 1), we obtain the reaction-time t 1 (S) as a function of the 

 intensity S , this time being measured from the application of the 

 stimulus until e reaches the threshold. For this purpose we use <j> as 

 given by equation (6) of chapter i and assume 



1 h 

 tl = --\ g (1 ), (1) 



a ft log S/h, 



where h x is the threshold of the afferent neuron. 



This relationship should apply to an experiment on a simple re- 

 flex in which a stimulus of intensity S produces a response R after 

 a time t r . However, the total time t r from the application of the 

 stimulus until occurrence of the response as registered by the timing- 

 device involves, in addition to t 1 (S), also a time t which measures 

 the time for conduction plus the time required for the muscular re- 

 sponse to effect the recording instrument plus any other time of delay 

 which does not depend appreciably upon S . We may then expect the 

 equation 



