Panagiotis Kaklamanos (University of Edinburgh)
Title: Geometric singular perturbation analysis of the multiple-timescale Hodgkin-Huxley equations
Abstract: The Hodgkin-Huxley (HH) equations [1] are one of the most successful models to describe 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 talk, we consider a non-dimensionalised version of the four-dimensional Hodgkin-Huxley equations [2], and we present a novel and global three-dimensional reduction that is based on geometric singular perturbation theory (GSPT). We investigate the dynamics of the resulting reduced system in 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 (MMOs), in accordance with the geometric mechanisms introduced in [3], and we classify the various firing patterns in dependence of the external applied current [4]. Time permitting, we will demonstrate how this methodology can be applied to other systems that are expressed in similar formalisms, such as models from cardiac dynamics.
References: A. L. Hodgkin and A. F. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of physiology, 117(4) (1952): 500.
[2] J. Rubin and M. Wechselberger. Giant squid-hidden canard: the 3D geometry of the Hodgkin-Huxley model. Biological Cybernetics, 97 (2007): 5-32.
[3] P. Kaklamanos, N. Popović, and K. U. Kristiansen. Bifurcations of mixed-mode oscillations in three-timescale systems: An extended prototypical example. Chaos: An Interdisciplinary Journal of Nonlinear Science, 32 (2022): 013108.
[4] P. Kaklamanos, N. Popović, K. U. Kristiansen. Geometric singular perturbation analysis of the multiple-timescale Hodgkin-Huxley equations. arXiv preprint (2022).
May 13th 2022, 15:00, room to be defined and Zoom meeting
