Wave propagation in excitable media with evolving boundaries.

本计画的目的是探讨非线性时变偏微分方程中行波的存在性与稳定性条件。行波是用于在生物系统中传输信号的常见模态。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，转运发生的区域随着时间的推移而演变，这是由于区域生长或组织的重新排列。虽然有一些研究的图案形成的增长领域，有缺乏的结果，如何随时间演变的域形状影响波的传播。甚至更少的结果时，考虑到可激发介质的传播，由于这些系统的固有的非线性。本项目的目标是通过寻找在具有不断变化的区域形状的非线性偏微分方程模型中可以建立波的条件来解决这一问题。偏微分方程模型中的行波脉冲解可以理解为在将原始模型转换到一个共同运动的坐标系中之后，从鞍点到鞍点的同宿连接。这种观点有利于构建色散曲线，将波的特性（如速度）与基础动力学特性联系起来。我们最近计算了稳定性条件（通过Evans函数方法）在无限一维域上提出的非局部耦合、可激发PDE模型中的行波脉冲[1][2]，现在试图将这些结果扩展到有限但不断发展的域。该分析将开始考虑域进行收敛扩展的情况，其中畴沿沿着一个轴扩展，同时沿正交轴收缩，而面积没有总体变化。这样的域进化在发展中的生物系统中常见（即，受精后早期）。在这样的系统中，生化信号的传输对于确保组织正确发育至关重要[3]。和以前一样，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，近似波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。这一假设将有助于系统的混合时标分析，以便通过在较慢的时标上研究系统来理解波的轮廓和速度随着域的演变而发生的变化。一旦完成，分析将扩展到更一般类型的域形状进化。该项目中的数学分析将与发育中的斑马鱼胚胎相关联（与生物科学的Steffen Scholpp博士合作，其是发育生物学中的原型系统，其经历了会聚延伸，通常用作具有非局部信号传导的系统的范例[4]。本计画旨在建立一个数学架构，以了解具有动态边界的偏微分方程系统中的波传播。这些动态可以作为时变输入施加到固定域，或者可以通过域形状的缓慢演变来并入。这种系统通常在广泛的生物背景下观察到。该项目将通过与UoE的专家发展生物学家合作，确保所使用的模型适用于示范生物系统。

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.