A note on the asymptotic reduction of the Hodgkin–Huxley equations for nerve impulses

Abstract

The (standard) FitzHugh reduction of the Hodgkin–Huxley equations for the propagation of nerve impulses ignores the dynamics of the activation gates. This assumption is invalid and leads to an over-estimation of the wave speed by a factor of 5 and the wrong dependence of wave speed on sodium channel conductance. The error occurs because a non-dimensional parameter, which is assumed to be small in the FitzHugh reduction, is in fact large (≈18). We analyse the Hodgkin–Huxley equations for propagating nerve impulses in the limit that this non-dimensional parameter is large, and show that the analytical results are consistent with numerical simulations of the Hodgkin–Huxley equations.