Numerical solution of a nonlinear advance-delay-differential equation from nerve conduction theory

Abstract

A functional differential equation which is nonlinear and involves forward and backward deviating arguments is solved numerically. The equation models conduction in a myelinated nerve axon in which the myelin completely insulates the membrane, so that the potential change jumps from node to node. The equation is of first order with boundary values given at t = +/- infinity. The problem is approximated via a difference scheme which solves the problem on a finite interval by utilizing an asymptotic representation at the endpoints, cubic interpolation and iterative techniques to approximate the delays, and a continuation method to start the procedure. The procedure is tested on a class of problems which are solvable analytically to access the scheme's accuracy and stability, then applied to the problem that models propagation in a myelinated axon. The solution's dependence on various model parameters of physical interest is studied. This is the first numerical study of myelinated nerve conduction in which the advance and delay terms are treated explicitly.

Computational methods for a mathematical model of propagation of nerve impulses in myelinated axons

“…In this section we describe an approach, based on the method of steps, to find a solution to the problem considered by Chi, Bell and Hassard in [5] and outlined in section 2 of this paper. For the convenience of the reader we repeat the four relevant equations here.…”

“…Following the approach in [5] we use the approximations v − (t) = − e λ+(t+L) for t < −L and v + (t) = 1 − + e λ−(t−L) for t > L. We set L = mτ, m ∈ N. We observe that for a specified value of τ equations (35) and (36) can be solved using the Newton-Raphson method to find λ + and λ − respectively. With λ + and λ − known, each of equations (37) and (38) involve only one unknown parameter, and a bisection method can be used to find the associated values of − and + .…”

“…An extensive analysis of this behaviour has been provided in [5], so here we will just recall the main results from that paper.…”

“…As in [5], we introduce the new variables t = t/RC, τ = τ /RC, in order to obtain a nondimensionalised model. Moreover, for the sake of simplicity, we keep the previous notation for the variables, so that we consider the following equation:…”

“…Acording to [3], neurons in the central nervous system tend to have much more variability than those in the peripheral nervous system; therefore, when we assume that the nodes are identical electrically and uniformly distributed, the model is more appropriate for describing processes in the peripheral nervous system. Under these assumptions, in [5], Chi and his co-authors obtained the following equation…”