Travelling wave front solutions of a differential-difference equation arising in the modelling of myelinated nerve axon

Abstract

Most nerves in higher animals are myelinated, that is they have an insulating sheath made up of myelin cells. Between these cells are short gaps, known as nodes of Ranvier. A simple model of myelinated nerve gives an ordinary differential equation for the potential at the n-th node in terms of the potential at adjacent nodes. Looking for travelling wave front solutions of such a system this becomes a differential-difference equation with both advanced and retarded arguments. The analysis of such an equation is undertaken in the (u,u’) plane, and exhibits some interesting differences from the corresponding unmyelinated model, where the equation is simply an ordinary differential equation.