Hamilton's principle of least action in nervous excitation

Abstract

In contrast to the general membrane potential as found by Teorell and by Meyer and Sievers, the nerve potential represents a real potential function resulting from the singular solution of Nernst's differential equation. Planck–Goldman's potential is found to be valid in the nerve. It furnishes the minimum value of the line integral of the force and satisfies the principle of least constraint. Beneath this minimum principle with respect to the path according to Hamilton there exists a minimum principle with respect to time. It is the principle of least action. In the present case it is given by the time integral of Planck–Goldman's equation. Both principles are contained in Hamilton's principal function. Our investigations deal with the question of whether nervous excitation is controlled by the principal function or not. Because according to calculus of variations the minimum principles result from a variation of a definite integral, and regarding that the experimentally found potential difference Δϕ= 70 –(–55)= 125 mV represents the definite line integral of the force we take it as the integrand of the principal function. Then the time integration yields the definite value of the action W= 1.05h where h is Planck's constant. This means that a single quantum can produce a definite electric peak of 125 mV. Resulting from the definite value of the action this peak is independent of the strength or the quality of the stimulus. Thus, a sequence of n quanta gives rise to a sequence of m(⩽n)definite electric peaks.