TRANS-SYNAPTIC DYNAMICS 3 



assume it to be representable by the term ae . If we then take cuf> , 

 proportional to the frequency, to represent this rate of increase of e 

 by the impulses, we obtain the equation (Rashevsky, 1938) 



de/dt = a(<j> — e) (1) 



which we assume to describe the development of e . We take e to be 

 a measure of the stimulus acting upon any neuron which originates 

 at the synapse in question. Note that by equation (1) we pass, in a 

 sense, directly from origin to terminus of the neuron, compressing 

 into the function 6 our only reference to the intra- neuronal dynamics. 

 When <f> = the impulses have zero frequency, i.e. do not occur, and 

 we shall say the neuron is at rest. Nevertheless, £ is not necessarily 

 zero, and in fact, after the neuron has been active, e vanishes asymp- 

 totically only according to equation (1) in which </> = . 



Now $ is proportional to the frequency of the generating im- 

 pulses, and this is, as implied, an increasing function of the applied 

 stimulus with, however, a finite asymptotic value. Hence we may 

 write 



$=<j>(S), (2) 



where S is a measure of the applied stimulus. However, in order for 

 the impulses to occur, S must exceed a certain minimal value, called 

 the threshold, which is characteristic of the neuron in question and 

 which we shall denote by h . Hence <£ (S) is zero for S = h , and for 

 S > h , </> (<S) is positive, is monotonically increasing but with a de- 

 creasing slope, and approaches a finite asymptotic value for large 

 values of 5 . 



Relatively simple analytic functions possessing these properties 

 for S > h are the following (Rashevsky, 1938; in this connection cf. 

 Hartline and Graham, 1932, and Matthews, 1933) : 



<£ = 4>o[l-e- Q < s -»>], (3) 



«/> (S-h)d + h 



</> = log , (4) 



log<5 S 



where 6 is small and </> is the asymptotic value of </> . For not too large 

 values of S either function may be approximated by an expression of 

 the form 



<l> = a(S-h), (5) 



and the second by 



4> = fi\og(S/h). (6) 



In any case, for S ^ h , <f> = . 



