Wave propagation in excitable media with evolving boundaries

本项目旨在确定具有时变域的非线性偏微分方程中行波的存在性和稳定性条件。行波是生物系统中传输信号的一种常见形式。在许多情况下，例如在发育生物学或肿瘤生长中观察到的情况，由于结构域的生长或组织的重新排列，运输发生的结构域随着时间的推移而演变。虽然已有许多关于生长区域模式形成的研究，但关于区域形状的时间演变如何影响波传播的结果却很少。当考虑在可激发介质上传播时，由于这些系统固有的非线性，可用的结果甚至更少。该项目旨在通过寻找可以在具有不断变化的域形状的非线性PDE模型中建立波的条件来解决这一差距。PDE模型中的行脉冲解可以理解为将原模型转化为共运动坐标系后与鞍点之间的同斜连接。这种观点有助于构建色散曲线，将波的特性（如速度）与底层动力学特性联系起来。我们最近（通过Evans函数方法）计算了在无限一维域[1][2]上的非局部耦合、可激发PDE模型中行进脉冲的稳定性条件，现在寻求将这些结果扩展到有限但不断发展的域。本分析将首先考虑域经历收敛扩展的情况，其中域沿着一个轴扩展，同时在正交轴上收缩，而面积没有总体变化。这种结构域进化通常见于发育中的生物系统（即受精后的早期阶段）。在这样的系统中，生化信号的传输对于确保组织的正常发育至关重要。与以前一样，该项目将确定在这种情况下传播解决方案的存在性和稳定性条件，在近似的情况下，波的传播发生在比域增长更快的时间尺度上，与许多生物系统一致。这一假设将有助于对系统进行混合时间尺度的分析，以便通过在较慢的时间尺度上研究系统来理解随着域的发展而变化的波的剖面和速度。一旦完成，分析将扩展到更一般类型的域形状演化。这个项目中的数学分析将与发育中的斑马鱼胚胎（与生物科学的Steffen Scholpp博士合作）联系在一起，斑马鱼胚胎是发育生物学中的一个典型系统，它经历了收敛扩展，通常被用作具有非局部信号的系统的范例。总之，本项目旨在建立一个数学框架来理解具有动态边界的PDE系统中的波传播。这些动态可以作为时变输入施加到固定的域，也可以通过域形状的缓慢演变来合并。这种系统通常在广泛的生物环境中观察到。该项目将通过与UoE的专家发育生物学家合作，确保所使用的模型适用于范例生物系统。

This project aims to identify existence and stability conditions for travelling waves in nonlinear PDEs with time-dependent domains. Travelling waves are a common modality for transporting signals in biological systems. In many scenarios, such as those observed in developmental biology or tumour growth, the domain over which the transport takes place evolves over time, either due to domain growth, or to re-arrangement of the tissue. Whilst there exists a number of studies of pattern formation on growing domains, there is paucity of results on how temporal evolution of the domain shape affects wave propagation. Even fewer results are available when considering propagation over excitable media, due to the inherent nonlinearity of these systems. This project aims to address this gap by finding conditions under which waves can be established in nonlinear PDE models with evolving domain shapes.Travelling pulse solutions in PDE models may be understood as homoclinic connections to and from a saddle point after transforming the original modelinto a co-moving coordinate system. This perspective facilitates the construction of dispersion curvesthat link wave properties, such as speed, to properties of the underlying dynamics. We have recently computed stability conditions (via an Evans function approach) for travelling pulses in non-locally coupled, excitable PDE models posed over infinite one-dimensional domains[1][2] and now seek to expand these results to a finite but evolving domain.This analysis will begin by considering the case in which the domain undergoes convergent extension, in which a domain expands along one axis whilst shrinkingin the orthogonal axis with no overall change in area. Such domain evolution is commonly seen in developing biology systems (i.e., early stage post fertilisation). In such systems, transport of biochemical signals is crucial to ensure that the tissue develops correctly[3]. As before, the project will identify existence and stability conditions for propagating solutions in this scenario, under the approximation that the wave propagation takes place on a faster timescale than the domain growth, consistent with many biological systems. This assumption will facilitate a mixed-timescale analysis of the system so that the changes to the profile and speed of the wave as the domain evolves can be understood by studying the system on the slower timescale. Once complete, the analysis will be extended to more general types of domain shape evolution.The mathematical analysis in this project will be linked to the developing zebrafish embryo (in collaboration with Dr Steffen Scholpp, Bioscience, whichis a prototypical system in developmental biology that undergoes convergent extension that is commonly used as an exemplar of a system with non-local signalling[4].In summary, this project aims to establish a mathematical framework for understanding wave propagation in PDE systems with dynamic boundaries. These dynamics may either be imposed as time-varying inputs to fixed domains, or may be incorporated via slow evolution of the domain shape. Such systems are commonly observed across a wide range of biological contexts. This project will ensure that the models used are appropriate to an exemplar biological system through collaboration with expert development biologists at UoE.