Geometric Singular Perturbation Analysis of the Multiple-Timescale Hodgkin–Huxley Equations
Panagiotis Kaklamanos, Nikola Popović, Kristian Uldall Kristiansen
Abstract
We present a novel and global three-dimensional reduction of a nondimensionalized version of the four-dimensional Hodgkin-Huxley equations [J. Rubin and M. Wechselberger, Biol. Cybernet., 97(2007), pp. 5-32] that is based on geometric singular perturbation theory. We investigate the dynamics of the resulting three-dimensional system in two parameter regimes in which the flow evolves on three distinct timescales. Specifically, we demonstrate that the system exhibits bifurcations of oscillatory dynamics and complex mixed-mode oscillations, in accordance with the geometric mechanisms introduced in [P. Kaklamanos, N. Popovi\'c, and K. U. Kristiansen, Chaos, 32 (2022), 013108],and we classify the various firing patterns in terms of the external applied current. While such pat-terns have been documented in [S. Doi, S. Nabetani, and S. Kumagai, Biol. Cybernet., 85 (2001),pp. 51-64] for the multiple-timescale Hodgkin-Huxley equations, we elucidate the geometry that underlies the transitions between them, which had not been previously emphasized.