Two and three dimensional reductions of the Hodgkin-Huxley system: separation of time scales and bifurcation schemes

We study two different two-dimensional reductions of the Hodgkin-Huxley equations. We show that they display the same qualitative bifurcation scheme as the original equations but overestimate the current range where periodic emission occurs. This is essentially due to the assumption that the evolution of the sodium activation variable m is instantaneous with respect to the dynamics of the variables h and n, an hypothesis that breaks down at high values of the injected current. To prove this point we compare the current-amplitude relation, the current-frequency relation, and the shapes of individual spikes for the two reduced models to the results obtained for the original Hodgkin-Huxley model and for a three dimensional model with instantaneous sodium activation. We show that a more satisfying agreement with the original Hodgkin-Huxley equations is obtained if we modify the evolution equation for the potential by incorporating the prominent features of the dynamics of m.