Ergodic theory of nonlinear waves in discrete and continuous excitable media

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Many spatially extended physical, chemical and biological systems, including nerve fibres and muscle tissue, form so-called excitable media. These are modelled by nonlinear dynamical systems which are spatially arranged in locally coupled cells so that the excitation can be transferred in space. In agreement with the real observations, such media support the formation and propagation of waves that interact in intricate ways and thus generates rich dynamical phenomena. The proposed project concerns the mathematical analysis and one space dimension.In the past decades, the modelling and analysis of excitable media was predominantly done by using partial differential equations (PDEs) such as FitzHugh-Nagumo (FHN) equation. However, this continuum description is often unnecessary for the overall qualitative behaviour, and rigorous analysis of wave interaction phenomena in excitable PDE is impossible with current methods -- except for a few cases. Cellular automata (CA) are an alternative type of models for a physical system in which space, time and states lie in a discrete set. The discreteness can give an enormous simplification for modelling, simulation and analysis, but lacks first principle derivation and quantitative accuracy. Nevertheless, even simple CA have a rich behaviour and work as models in many areas. Greenberg and Hastings developed a family of CA, abbreviated as GHCA, which qualitatively model, e.g., nerve fibres and support the characteristic action potential waves. Notably, the GHCA have at least 3-states, while almost all analysis thus far concerns two-state CA.This project pursues the following main research questions:Q1: What are the long term statistical (ergodic) properties of cellular automata for excitable media? What is the role of nonlinear waves in these?Q2: Can the CA and discrete dynamical systems perspective with ergodic theory help to understand the complex PDE phenomena of strong interaction of localised waves in excitable media?Previous work of the doctoral candidate and the PIs revealed that the dynamics and complexity of the 3-state GHCA stems entirely from wave interaction. Moreover, it was shown that having more than two states makes this CA amenable to analysis by symbolic dynamics. The aims of this project are on the one hand (Q1) to analyse the ergodic properties, complexity and wave phenomena of GHCA and further CA models. On the other hand it aims to transfer these results to certain PDE (Q2). The latter concerns qualitative comparison to small CA with few states, and quantitative comparison with large CA from a full discretisation with many states. This in particular concerns an extension of the theta-model from neuroscience to a scalar PDE model, which - based on numerical simulations - features qualitatively the same dynamics as the 3-state GHCA.