On repetitive activity in nerve

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.