Repetitive activity and hopf bifurcation under point-stimulation for a simple FitzHugh-Nagumo nerve conduction model

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Abstract

In response to point-stimulation with a constant current, a neuron may propagate a repetitive train of action potentials along its axon. For maintained repetitive activity, the current strength I must be, typically, neither too small nor too large. For I outside some range, time independent steady behavior is observed following a transient phase just after the current is applied. We present analytical results for a piecewise linear FitzHugh-Nagumo model for a point-stimulated (non-space-clamped) nerve which are consistent with this qualitative experimental picture. For each value of I there is a unique, spatially nonuniform, steady state solution. We show that this solution is stable except for an interval (I*, I*) of I values. Stability for I too small or too large corresponds to experiments with sub-threshold I or with excessive I which leads to 'nerve block'. For I = I*, I* we find Hopf bifurcation of spatially nonuniform, time periodic solutions. We conclude that (I*, I*) lies interior to the range of I values for repetitive activity. The values of I* and I* and their dependence on the model parameters are determined. Qualitative differences between results for the point-stimulated configuration and the space-clamped case are discussed.

Original languageEnglish (US)
Pages (from-to)363-382
Number of pages20
JournalJournal Of Mathematical Biology
Volume5
Issue number4
DOIs
StatePublished - Dec 1977

Keywords

  • FitzHugh-Nagumo
  • Hopf bifurcation
  • Nerve conduction
  • Repetitive activity

ASJC Scopus subject areas

  • Modeling and Simulation
  • Agricultural and Biological Sciences (miscellaneous)
  • Applied Mathematics

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