### Abstract

Theoretical results for repetitive activity characteristics of the space and current-clamped Hodgkin-Huxley equations are reviewed and extended. Mathematical solutions are related to stimulus-response curves of steady (adapted) temporal frequency ω versus strength I of a depolarizing current step. These correspond to stable time-periodic solutions and are found for I(nu)<I<I_{2}. We find additional periodic solutions for I(nu)<I<I_{1}, where I_{1}<I_{2}, which are unstable. Existence of such solutions and their instability was previously conjectured by other investigators. We compute periodic solutions by a numerical method outlined here, which is insensitive to the stability of the solution. The equations also have a unique steady-state solution for all I. It is unstable for I_{1}<I<I_{2} and stable otherwise. These different solution branches are each represented on solution amplitude versus I and ω versus I curves to illustrate a complete and consistent picture of the solution structure. Temperature dependence of these features and the critical current values is determined. Because there are two stable states, a repetitive one and a steady one, for I(nu)<I<I_{1} the model exhibits hysteresis. Experimental implications are discussed. For small current steps the model shows oscillations of frequency ω which damp to a steady state. We illustrate how a systemic scaling and perturbation procedure leads to a reduced two-variable, V-n, model from which ω, I_{1}, and their temperature dependence may be estimated.

Original language | English (US) |
---|---|

Pages (from-to) | 2793-2802 |

Number of pages | 10 |

Journal | Federation Proceedings |

Volume | 37 |

Issue number | 14 |

State | Published - 1978 |

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### ASJC Scopus subject areas

- Medicine(all)

### Cite this

**On repetitive activity in nerve.** / Rinzel, J.

Research output: Contribution to journal › Article

*Federation Proceedings*, vol. 37, no. 14, pp. 2793-2802.

}

TY - JOUR

T1 - On repetitive activity in nerve

AU - Rinzel, J.

PY - 1978

Y1 - 1978

N2 - Theoretical results for repetitive activity characteristics of the space and current-clamped Hodgkin-Huxley equations are reviewed and extended. Mathematical solutions are related to stimulus-response curves of steady (adapted) temporal frequency ω versus strength I of a depolarizing current step. These correspond to stable time-periodic solutions and are found for I(nu)<I<I2. We find additional periodic solutions for I(nu)<I<I1, where I1<I2, which are unstable. Existence of such solutions and their instability was previously conjectured by other investigators. We compute periodic solutions by a numerical method outlined here, which is insensitive to the stability of the solution. The equations also have a unique steady-state solution for all I. It is unstable for I1<I<I2 and stable otherwise. These different solution branches are each represented on solution amplitude versus I and ω versus I curves to illustrate a complete and consistent picture of the solution structure. Temperature dependence of these features and the critical current values is determined. Because there are two stable states, a repetitive one and a steady one, for I(nu)<I<I1 the model exhibits hysteresis. Experimental implications are discussed. For small current steps the model shows oscillations of frequency ω which damp to a steady state. We illustrate how a systemic scaling and perturbation procedure leads to a reduced two-variable, V-n, model from which ω, I1, and their temperature dependence may be estimated.

AB - Theoretical results for repetitive activity characteristics of the space and current-clamped Hodgkin-Huxley equations are reviewed and extended. Mathematical solutions are related to stimulus-response curves of steady (adapted) temporal frequency ω versus strength I of a depolarizing current step. These correspond to stable time-periodic solutions and are found for I(nu)<I<I2. We find additional periodic solutions for I(nu)<I<I1, where I1<I2, which are unstable. Existence of such solutions and their instability was previously conjectured by other investigators. We compute periodic solutions by a numerical method outlined here, which is insensitive to the stability of the solution. The equations also have a unique steady-state solution for all I. It is unstable for I1<I<I2 and stable otherwise. These different solution branches are each represented on solution amplitude versus I and ω versus I curves to illustrate a complete and consistent picture of the solution structure. Temperature dependence of these features and the critical current values is determined. Because there are two stable states, a repetitive one and a steady one, for I(nu)<I<I1 the model exhibits hysteresis. Experimental implications are discussed. For small current steps the model shows oscillations of frequency ω which damp to a steady state. We illustrate how a systemic scaling and perturbation procedure leads to a reduced two-variable, V-n, model from which ω, I1, and their temperature dependence may be estimated.

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M3 - Article

VL - 37

SP - 2793

EP - 2802

JO - Federation Proceedings

JF - Federation Proceedings

SN - 0014-9446

IS - 14

ER -