Spatial stability of traveling wave solutions of a nerve conduction equation

Biophysical Journal
J Rinzel

Abstract

A simplified FitzHugh-Nagumo nerve conduction equation with known traveling wave solutions is considered. The spatial stability of these solutions is analyzed to determine which solutions should occur in signal transmission along such a nerve model. It is found that the slower of the two pulse solutions is unstable while the faster one is stable, so the faster one should occur. This agrees with conjectures which have been made about the solutions of other nerve conduction equations. Furthermore for certain parameter values the equation has two periodic wave solutions, each representing a train of impulses, at each frequency less than a maximum frequency wmax. The slower one is found to be unstable and the faster one to be stable, while that at wmax is found to be neutrally stable. These spatial stability results complement the previous results of Rinzel and Keller (1973. Biophys. J. 13: 1313) on temporal stability, which are applicable to the solutions of initial value problems.

Citations

Jan 1, 1977·Bulletin of Mathematical Biology·P Rissman
Jan 1, 1978·Pharmacology & Therapeutics. Part B: General & Systematic Pharmacology·V I Krinsky
Sep 21, 1990·Journal of Theoretical Biology·C ElphickE A Spiegel
Jun 5, 2010·PLoS Computational Biology·Henry C Tuckwell, Jürgen Jost
Mar 10, 2012·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Patrick M BoyleEdward J Vigmond

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