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

Panagiotis Kaklamanos, Nikola Popović, Kristian Uldall Kristiansen

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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.
Original languageEnglish
JournalSIAM Journal on Applied Dynamical Systems
Volume22
Issue number3
Pages (from-to)1552-1589
ISSN1536-0040
DOIs
Publication statusPublished - 2023

Keywords

  • Neuronal firing
  • Hodgkin-Huxley equations
  • Mixed-mode dynamics
  • Bifuractions
  • Multiple timescales
  • Geometric singular pertubation theory

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