Theory for a test of the electric cable model of the myelinated axon and saltatory conduction

Abstract

An experiment is proposed as a test of the usual electric cable theory and saltatory conduction theory in the myelinated nerve fiber. The experiment consists of investigating the relationship between the following quantities: 1. (1) V( x, r) the minimum height of a depolarizing rectangular voltage pulse of duration τ applied at a distance x from the nearest stimulable node such that an action potential pulse is induced at that node. 2. (2) b( x) ≡ lim V( x, τ) the rheobase voltage at a distance x from the τ → ∞ stimulable node. 3. (3) T ≡ the duration of the depolarizing pulse. Usually the theoretical analysis for the propagation of an electric signal along a nerve fiber involves a computer solution of a non-linear cable equation. In the present treatment the solution for V(x, T) b(x) is developed in the form of the real part of a complex integral; Fourier techniques allow this. The complex integral is then evaluated by computer to any desired precision. This integral method may yield greater precision for the same amount of computer time than the usual numerical methods for solving differential equations. It is also shown that V(x,τ) b(x) = 1+ D(x) τ for small enough τ or large enough τ. V b is computed for broad ranges of values of x and τ. For τ small enough the values of D( x) can be determined. The explicit dependence of D( x) on x depends on the assumptions of the electric cable model and saltatory conduction. D( x) is found to be a simple linear function of x. Appropriate electronic equipment is now available and it appears that experimental techniques have advanced to the point that this experiment can be done. The general characteristics of experimental and theoretical data should agree in order that the combined electric cable and saltatory conduction theories be consistent with reality. Of particular interest is the comparison between theory and experiment regarding the explicit dependence of D( x) on x.