Abstract. The Hodgkin-Huxley (HH) equations [A. L. Hodgkin and A. F. Huxley, The Journal of Physiology, 117 (1952)] are one of the most successful models for the propagation of action potentials in neurons. For their work, Hodgkin and Huxley received the 1963 Nobel Prize in Physiology and Medicine. The original HH system is four-dimensional, with dynamics evolving on at least three distinct timescales.
In this work, we present a novel and global three-dimensional reduction of a modified version of the HH equations [J. Rubin and M. Wechselberger, Biological cybernetics, 97 (2007)] that is based on geometric singular perturbation theory. We investigate the dynamics of our reduction in two distinct parameter regimes. We demonstrate that in the first regime, the system exhibits bifurcations of oscillatory dynamics and complex mixed-mode oscillations, in accordance with the geometric mechanism introduced in [P. Kaklamanos, N. Popovic, and K. U. Kristiansen, (2020)], while in the second regime, it displays characteristics of a slow-fast system in non-standard form. We relate our results to previous work [S. Doi et al., Biological cybernetics, 85 (2001)] where such dynamics in the corresponding regimes have been documented but the underlying geometry was not emphasised.
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