54 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



is the case for single nerve fibers is illustrated in the data by C. 

 Pecher (Landahl, 1941c). 



If for a particular neuron the mean value of the threshold is h , 

 the coefficient of variability is v , and if p(C) represents a normal 

 curve of unit area, then the probability P of a response in the absence 

 of a stimulus is given by the integral of p(C) from 1/v to oo. If t is 

 the least time for a fluctuation to have effect, then after a time t/P 

 one could reasonably expect a chance response. The mean frequency 

 of such responses would then be given by P/r per second. These re- 

 sponses would not be periodic. If t is taken to be of the order of mag- 

 nitude of 10~ 3 seconds, then for v = 30% the mean time between re- 

 sponses would be a few seconds, while for v = 20%, the mean time be- 

 tween responses would be a number of hours. From this we see that 

 the probability of a chance response, even over a considerable period 

 of time, becomes negligible rapidly as v becomes much smaller than 

 one-fifth. But one should consider also the slower changes in the en- 

 vironment of the neuron which not only changes the threshold but 

 also its degree of variation. 



Variations in the threshold would cause the response-times and 

 other measurable variables to be distributed in some manner about 

 the value corresponding to the mean value of the threshold. As an 

 illustration, let us estimate the dependence of a measure of the varia- 

 tion in response-times on the intensity of the stimulus. We consider 

 the case of a simple excitatory afferent stimulated by a constant stim- 

 ulus S and acting on an efferent of threshold h . Suppose that t x is 

 the value of t for which e = h . Then if h is decreased by an amount 

 vh , e = h is satisfied by t = U. — a. Then 



1 / vh \ 



a = -log[ 1+ (D 



a V <£ — h J 



is an average variation in the reaction-time due to the variation in 

 threshold. In general, since one must consider more than one chain, 

 one may suppose that variations, essentially independent of <f> , are 

 introduced at other synapses and at the end-organ. Let a be a meas- 

 ure of the total effect of this variation. Then the measured variation 

 in the response-time will be given by the square root of the sum of 

 the squares of these two variations. As v is generally quite small 

 a = vh/((p — h)a and thus 



a t = Veto 2 + vya*{4>/h - l) 2 , (2) 



and we have a relation between a measure of the variation in re- 

 sponse-times and the stimulus-intensity in terms of ^ . We may com- 



