Neutrally stable traveling wave solutions of nerve conduction equations

Research output: Contribution to journalArticle

Abstract

The equations studied by Hodgkin and Huxley, FitzHugh, Nagumo, and others belong to a general class of nerve conduction equations. Each has a family of periodic traveling wave solutions and typically more than one solitary pulse wave solution. Instability has been conjectured for certain of these solutions. Here stability is studied formally and neutral stability transitions are characterized parametrically. Two notions of linear stability are formulated. For initial value problems, temporal stability is applicable. Spatial stability is appropriate for boundary value problems like signal transmission along a nerve in response to a spatially localized time-dependent stimulus. Here it is shown that periodic wave trains with maximum or minimum frequency have neutral spatial stability. For solitary pulse solutions, propagation speed versus a typical parameter usually describes a double branched curve. It is shown that the speed curve knee corresponds to neutral spatial and temporal stability.

Original languageEnglish (US)
Pages (from-to)205-217
Number of pages13
JournalJournal of Mathematical Biology
Volume2
Issue number3
DOIs
StatePublished - 1975

Fingerprint

Neural Conduction
Nerve
Traveling Wave Solutions
Conduction
nerve tissue
Convergence of numerical methods
knees
Knee
Periodic Travelling Wave Solution
Initial value problems
Propagation Speed
Curve
FitzHugh-Nagumo
Periodic Wave
Boundary value problems
Linear Stability
Initial Value Problem
Boundary Value Problem

ASJC Scopus subject areas

  • Agricultural and Biological Sciences (miscellaneous)
  • Mathematics (miscellaneous)

Cite this

Neutrally stable traveling wave solutions of nerve conduction equations. / Rinzel, J.

In: Journal of Mathematical Biology, Vol. 2, No. 3, 1975, p. 205-217.

Research output: Contribution to journalArticle

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