Symmetric and Antisymmetric Pulses in Parallel Coupled Nerve Fibres

Abstract

Travelling wave solutions for equations that model two parallel coupled nerve fibres are found. The travelling wave here represents the action potential. It is shown that the introduction of weak coupling between the fibres induces either symmetry or antisymmetry of the action potentials. A symmetric pulse is a solution where both fibres fire simultaneously and the action potentials propagate locked in phase at the same wave speed along the length of each fibre; an antisymmetric pulse is a solution where one fibre fires, resulting in an action potential propagating along it, while the other remains at rest. Geometric singular perturbation theory and the exchange lemma are used to prove the existence of solutions. In addition, a technique which involves the use of differential forms for detecting transversalities of small order is introduced.