Geometric Singular Perturbation Analysis of the Multiple-Timescale Hodgkin-Huxley Equations

We present a novel and global three-dimensional reduction of a nondimensionalized version of the (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. Popovic'\, and K. U. Kristiansen, Chaos, 32 (2022), 013108], and we classify the various firing patterns in terms of the external applied current. While such patterns 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.

Automated summary

We propose a novel and global reduction method for a nondimensionalized system based on geometric singular perturbation theory. By investigating the dynamics in two parameter regimes, we observe bifurcations and classify firing patterns with external current applied. Our findings reveal the underlying geometry of transitions between patterns, which has not been emphasized before despite similar patterns documented in previous studies.