PHYSICOMATHEMATICAL ASPECTS OF EXCITATION 231 



The second hypothesis is a consequence of the observations that the 

 local excitatory process p diminishes with time for all values of p. The 

 simplest mathematical form of this statement is dp/dt decreases directly 

 with p. Thus the process follows the natural law of depreciation of 

 values with time, and 



dp 



where k is the constant of proportionality or the coefficient of diminution 

 of the excitatory process. The negative sign indicates that the activity 

 diminishes as time goes on. 



The third hypothesis is that the local excitatory process must attain a 

 critical limiting value b in order that it may be adequate to initiate the 

 nerve impulse. That is, p must reach a threshold value 6 such that 



p = b or p > b 



to start the nerve excitation. 



Thus the whole phenomenon is described by the following differential 

 equation 



d -f = KS-kp 

 dt y 



with excitation occurring when p is equal to or greater than b. 



Since no stipulation was made as to what form S shall take, the stimu- 

 lus may, therefore, be a type of electric current. The choice will be 

 limited to the voltage derived from a direct or an alternating current, or 

 from the discharge from a condenser. For simplicity's sake a direct 

 current of potential V is chosen; then a statement for the time taken to 

 build up an excitatory process to its threshold value is formulated by 

 assuming that the excitatory process p attains its threshold value b in t\ 

 seconds. The mathematical equivalence of this statement is to inte- 

 grate the excitatory process from its initial zero value to its final value b 

 in time zero to t\ seconds. 



J KV - kp = J dt 



Integrating this expression with the aid of A Short Table of Integrals, by 

 Peirce, and solving for the time, one obtains 



1 KV 



h = j log 



A k ° KV - kp 



Thus a latent period of t\ seconds must elapse between the moment of 

 establishing the current and the moment of release or spread of the exci- 



