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

Research output: Contribution to journalArticle

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<inf>*</inf>, I<sup>*</sup>) 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<inf>*</inf>, I<sup>*</sup> we find Hopf bifurcation of spatially nonuniform, time periodic solutions. We conclude that (I<inf>*</inf>, I<sup>*</sup>) lies interior to the range of I values for repetitive activity. The values of I<inf>*</inf> and I<sup>*</sup> 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 - 1977

Fingerprint

FitzHugh-Nagumo
Hopf bifurcation
Neural Conduction
Nerve
Conduction
Hopf Bifurcation
nerve tissue
Nerve Block
action potentials
axons
Action Potentials
Neurons
Axons
Time-periodic Solutions
Action Potential
neurons
Steady-state Solution
Piecewise Linear
Range of data
Neuron

Keywords

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

ASJC Scopus subject areas

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

Cite this

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title = "Repetitive activity and hopf bifurcation under point-stimulation for a simple FitzHugh-Nagumo nerve conduction model",
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.",
keywords = "FitzHugh-Nagumo, Hopf bifurcation, Nerve conduction, Repetitive activity",
author = "John Rinzel",
year = "1977",
doi = "10.1007/BF00276107",
language = "English (US)",
volume = "5",
pages = "363--382",
journal = "Journal of Mathematical Biology",
issn = "0303-6812",
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TY - JOUR

T1 - Repetitive activity and hopf bifurcation under point-stimulation for a simple FitzHugh-Nagumo nerve conduction model

AU - Rinzel, John

PY - 1977

Y1 - 1977

N2 - 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.

AB - 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.

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KW - Hopf bifurcation

KW - Nerve conduction

KW - Repetitive activity

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