The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple timescales

Abstract

In recent work [J. Rubin and M. Wechselberger, Biol. Cybern. 97, 5 (2007)], we explained the appearance of remarkably slow oscillations in the classical Hodgkin-Huxley (HH) equations, modified by scaling a time constant, using recently developed theory about mixed-mode oscillations (MMOs). This theory is only rigorously valid, however, for epsilon sufficiently small, where epsilon is a parameter that arises from nondimensionalization of the HH system. Here, we illustrate how the parameter regime over which MMOs exist, and the features of the MMO patterns within this regime, vary with respect to several key parameters in the nondimensionalized HH equations, including epsilon. Moreover, we explain our findings in terms of the effects that these parameters are expected to have on certain organizing structures within the corresponding flow, generalized from analysis done previously in the singular limit.