### 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 language | English (US) |
---|---|

Pages (from-to) | 363-382 |

Number of pages | 20 |

Journal | Journal of Mathematical Biology |

Volume | 5 |

Issue number | 4 |

DOIs | |

State | Published - 1977 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

**Repetitive activity and hopf bifurcation under point-stimulation for a simple FitzHugh-Nagumo nerve conduction model.** / Rinzel, John.

Research output: Contribution to journal › Article

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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.

KW - FitzHugh-Nagumo

KW - Hopf bifurcation

KW - Nerve conduction

KW - Repetitive activity

UR - http://www.scopus.com/inward/record.url?scp=0018195105&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0018195105&partnerID=8YFLogxK

U2 - 10.1007/BF00276107

DO - 10.1007/BF00276107

M3 - Article

C2 - 750625

AN - SCOPUS:0018195105

VL - 5

SP - 363

EP - 382

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

IS - 4

ER -