Ergodic theory of nonlinear waves in discrete and continuous excitable media

许多空间延伸的物理、化学和生物系统，包括神经纤维和肌肉组织，形成所谓的可兴奋介质。这些模型的非线性动力系统，在空间上安排在本地耦合的细胞，使激励可以在空间中转移。与真实的观测一致，这种介质支持波的形成和传播，这些波以复杂的方式相互作用，从而产生丰富的动力学现象。在过去的几十年里，可激发介质的建模和分析主要是使用偏微分方程（PDE），如FitzHugh-Nagumo（FHN）方程。然而，这种连续描述往往是不必要的整体定性行为，和严格的波相互作用现象的分析，在可激发偏微分方程是不可能与目前的方法-除了少数情况下。元胞自动机（CA）是物理系统的一种替代类型的模型，其中空间，时间和状态位于离散集中。离散性可以为建模、仿真和分析提供极大的简化，但缺乏第一性原理推导和定量准确性。然而，即使是简单的CA也有丰富的行为，并在许多领域作为模型工作。 Greenberg和Hastings开发了一个CA家族，缩写为GHCA，其定性地建模，例如，神经纤维和支持特征动作电位波。值得注意的是，GHCA至少有3个状态，而迄今为止几乎所有的分析都涉及两个状态的CA。该项目追求以下主要研究问题：Q1：什么是长期的统计（遍历）性能的细胞自动机的可激发媒体？非线性波在其中的作用是什么？问题2：CA和离散动力系统的观点与遍历理论是否有助于理解可激发介质中局域波强相互作用的复杂PDE现象？博士候选人和PI先前的工作表明，三态GHCA的动力学和复杂性完全源于波的相互作用。此外，它表明，有两个以上的状态，使CA适合分析的符号动力学。这个项目的目的是一方面（Q1）分析的遍历性，复杂性和波动现象的GHCA和进一步的CA模型。另一方面，它旨在将这些结果转移到某些PDE（Q2）。后者涉及定性比较小CA与少数国家，和定量比较大CA从一个完整的离散化与许多国家。这特别涉及从神经科学到标量PDE模型的θ模型的扩展，该模型基于数值模拟，定性地具有与3状态GHCA相同的动力学特性。

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.