Behaviour of Some Models of Myelinated Axons

Abstract

In this paper we discuss several approaches to modelling myelinated axons and examine the qualitative behaviour of the models. To facilitate our goal of understanding in detail the differences in mechanisms modelling myelination, we impose at the nodes the simplest nonlinear current–voltage relation which allows the models to possess appropriate threshold behaviour and propagating action potentials (travelling waves). Our type of model is a nonlinear differential-difference system and the resulting travelling wave must satisfy a nonlinear delay–differential equation of mixed type. Another type of model is a diffusion equation coupled nonlinearly to ordinary differential systems whose solutions represent boundary data for the diffusion equation. We give some threshold results and derive a relationship between conduction speed and various model parameters for a few classes of these models.