Ergodic theory of nonlinear waves in discrete and continuous excitable media
Final Report Abstract
In this project we have studied and compared fully discrete and continuous models of excitable media. Such media frequently occur in spatially extended physical, chemical and biological systems. Here, a sufficiently large perturbation from a stable rest state triggers an excitation, which is transferred to its neighbours and is followed by refractory return to the rest state. Nonlinear dynamical systems can model such media and in agreement with real and experimental observations, reproduce the propagation of localized traveling waves. A paradigm for excitable media are nerve axons and myocardium in which a depolarising electrical perturbation triggers uni-directionally moving action potentials as a means of information transfer. Disfunctional transfer leads to various live threatening diseases so that an analysis of fundamental principles may ultimately help treatments. The modelling and analysis of excitable media is dominated by partial differential equations (PDEs), which fit well to an effectively space-time continuous physical situation. However, for instance myelinated nerve axons possess a discrete structure and a sufficiently general rigorous mathematical theory for interacting travelling waves in PDE is currently not available. Cellular automata (CA) provide an alternative model type in which space, time and states lie in discrete sets. In this project, we were able to fully characterize the long term dynamics of the so-called Greenberg-Hastings CA (GHCA) for excitable media in terms of specific wave dynamics, in particular ‘action potentials’ that annihilate upon collision. Moreover, we were able to determine the dynamical complexity in the sense of topological entropy. Concerning the continuum PDE perspective, we have obtained numerous results based on the so- called theta-model for neurons. On the one hand, we have verified the occurrence of annihilation sequences that roughly qualitatively correspond to those in the GHCA. On the other hand, we found that information on the initial positions is ‘forgotten’ when moving through the medium, so that the entropic complexity in the PDE based on positional data is reduced compared to GHCA. We have further corrobated this by a reduction of the dynamics to a system of so-called ordinary differential equations and analysis of its dynamics. In contrast, in the more complex FitzHugh-Nagumo PDE, we have numerically found that pulses replicate in a certain parameter regime. We have identified a subtle novel mechanism for instability in this regime that triggers this process. Moreover, we found that the stronger the fast/slow scale separation, the higher-dimensional the instability is.
Publications
- Weak and strong interaction of excitation kinks in scalar parabolic equations
A. Pauthier, J.D.M. Rademacher, D. Ulbrich
(See online at https://doi.org/10.48550/arXiv.2012.00309) - Dynamics and topological entropy of 1D Greenberg-Hastings cellular automata. Ergodic Theory and Dynamical Systems, 41(5), 1397-1430, 2021
M. Keßeböhmer, J.D.M. Rademacher, D. Ulbrich
(See online at https://doi.org/10.1017/etds.2020.18) - Pulse replication and accumulation of eigenvalues. SIAM J. Math. Anal., 53(3), 3520–357
Paul Carter, Jens D.M. Rademacher, Björn Sandstede
(See online at https://doi.org/10.1137/20m1340113)
