Ergodic theory of nonlinear waves in discrete and continuous excitable media

许多空间扩展的物理、化学和生物系统，包括神经纤维和肌肉组织，形成了所谓的可兴奋介质。这些模型是由空间上排列在局部耦合单元中的非线性动力系统来建模的，以便激励可以在空间中传递。与实际观测一致的是，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组(PDE)，如Fitzhugh-Nagumo(FHN)方程。然而，对于整个定性行为，这种连续描述通常是不必要的，并且用目前的方法不可能对可激发PDE中的波相互作用现象进行严格的分析--除了少数情况。元胞自动机(CA)是空间、时间和状态位于一个离散集合中的物理系统的另一种类型的模型。这种离散性可以极大地简化建模、仿真和分析，但缺乏第一原理推导和定量精度。然而，即使是简单的CA也有丰富的行为，在许多领域都是模范。Greenberg和Hastings开发了CA家族，缩写为GHCA，它定性地模拟神经纤维，并支持特征动作电位波。值得注意的是，GHCA至少有3个状态，而到目前为止几乎所有的分析都是关于两个状态的CA。这个项目追求以下主要研究问题：Q1：对于可激发介质，细胞自动机的长期统计(遍历)特性是什么？非线性波在其中的作用是什么？Q2：CA和离散动力系统的遍历理论能帮助理解可激发介质中局域波强相互作用的复杂PDE现象吗？博士生和PI以前的工作揭示了三态GHCA的动力学和复杂性完全源于波的相互作用。此外，还表明具有两个以上的状态使得这种CA易于用符号动力学来分析。本项目的目的是一方面(Q1)分析GHCA和进一步的CA模型的遍历性、复杂性和波动现象。另一方面，它的目的是将这些结果转移到特定的偏微分方程(Q2)。后者涉及与状态较少的小型CA的定性比较，以及与大型CA的完全离散化多状态的定量比较。这尤其涉及到从神经科学的theta模型到标量PDE模型的扩展，该模型基于数值模拟，定性地具有与三态GHCA相同的动力学特征。

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.