Traveling Wave Solutions of a Nerve Conduction Equation

J. Rinzel, J. B. Keller

Research output: Contribution to journalArticle

Abstract

We consider a pair of differential equations whose solutions exhibit the qualitative properties of nerve conduction, yet which are simple enough to be solved exactly and explicitly. The equations are of the FitzHugh-Nagumo type, with a piecewise linear nonlinearity, and they contain two parameters. All the pulse and periodic solutions, and their propagation speeds, are found for these equations, and the stability of the solutions is analyzed. For certain parameter values, there are two different pulse-shaped waves with different propagation speeds. The slower pulse is shown to be unstable and the faster one to be stable, confirming conjectures which have been made before for other nerve conduction equations. Two periodic waves, representing trains of propagated impulses, are also found for each period greater than some minimum which depends on the parameters. The slower train is unstable and the faster one is usually stable, although in some cases both are unstable.

Original languageEnglish (US)
Pages (from-to)1313-1337
Number of pages25
JournalBiophysical journal
Volume13
Issue number12
DOIs
StatePublished - 1973

ASJC Scopus subject areas

  • Biophysics

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