## 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 language | English (US) |
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Pages (from-to) | 363-382 |

Number of pages | 20 |

Journal | Journal Of Mathematical Biology |

Volume | 5 |

Issue number | 4 |

DOIs | |

State | Published - 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