PHYSICOMATHEMATICAL THEORY 229 



Physicomathematical Aspects of Excitation and Propagation of 

 Nerve Impulses 



Throughout the development of any physical experiment, one may 

 observe the accumulated data presented in the form of tables. One 

 column usually contains the independent variable and a parallel column 

 the dependent variable. Another way of presenting the data is by 

 drawing a graph or by expressing the results in a mathematical formula. 



If, as in the problem of nerve conduction, one suspects that a basic 

 physical law is involved, statements of relations are generally proposed 

 between the rate of change of some quantity and other quantities 

 developed by the experimental evidence. Such a relation presented in 

 rigid mathematical terms is called a differential equation; it will contain 

 derivatives of functions and also the functions themselves. 



For instance, Newton's second law of motion states that the force 

 equals the time rate of change of the momentum. Written in mathe- 

 matical terms, this law is 



d(mv) 



f = ' — i — 

 dt 



where mv is the momentum. 



If the mass is constant we can rewrite the equation as 



„ dv 



F = m — 



dt 



which may be solved by direct integration. 



On the other hand, if an effort is to be made to find relations between 

 two phenomena, as, for instance, between the local excitatory process of a 

 nerve and the stimulus, then the method of approach is as follows : 



1. We start by assuming a working hypothesis. The data provide 

 the clue. 



2. We next set up a mathematical expression representing the rate of 

 change of the two variables to be examined. 



3. Then we integrate the equation in order to reproduce the working 

 hypothesis in a mathematical form suitable for experimental verification. 



Should subsequent experimental data force one to reject the solution, 

 then the fundamental assumption upon which the solution rests must be 

 modified so as to include the new data. As a result, we gradually build 

 up a more comprehensive theory which may eventually give the clue 

 to the fundamental law governing the phenomenon under investigation. 



In this way the several mathematical versions of the phenomena 

 involved in pulse propagation might have originated. The two simplest 



