THE ASYMPTOTIC STRUCTURE OF THE HODGKIN–HUXLEY EQUATIONS

Abstract

We analyze the asymptotic structure of the Hodgkin–Huxley system of equations, in terms of the concepts of slow manifold and fast foliation, based on Tikhonov's theorem on asymptotics of solutions of slow–fast systems of differential equations. We test Zeeman's conjecture that the jump onset–slow return structure of the action potential in realistic equations of biological excitability may be due to a cusp singularity of the slow manifold with respect to the fast foliation. We find that although the cusp singularity can appear in such equations, the characteristic features in question cannot be reproduced within the Tikhonov scheme and require development of different asymptotic approaches.