Wave propagation in excitable media with evolving boundaries

边界不断变化的可激发介质中的波传播

• 批准号: 2583485
• 金额: --
• 依托单位: UNIVERSITY OF EXETER
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2583485
• 关键词: Wave; propagation; excitable; media; evolving

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.

该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和稳定性条件。行波是生物系统中传输信号的常见方式。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，发生运输的域会随着时间的推移而演变，这要么是由于域的生长，要么是由于组织的重新排列。虽然存在许多关于生长域上图案形成的研究，但关于域形状的时间演化如何影响波传播的结果却很少。由于这些系统固有的非线性，当考虑可激励介质上的传播时，可用的结果甚至更少。该项目旨在通过寻找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 模型中的行进脉冲解可以理解为在将原始模型转换为共动坐标系后与鞍点之间的同宿连接。这种观点有利于构建将波的特性（例如速度）与基础动力学特性联系起来的频散曲线。我们最近计算了在无限一维域上提出的非局部耦合、可激励 PDE 模型中的行进脉冲的稳定性条件（通过埃文斯函数方法）[1][2]，现在寻求将这些结果扩展到有限但不断演变的域。这一分析将首先考虑域经历收敛扩展的情况，其中域沿一个轴扩展，同时沿一个轴收缩。 正交轴，面积没有总体变化。这种结构域的进化在生物系统的发育过程中很常见（即受精后的早期阶段）。在此类系统中，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，该项目将在这种情况下确定传播解决方案的存在和稳定性条件，近似认为波传播发生在比域增长更快的时间尺度上，这与许多生物系统一致。这种假设将有助于对系统进行混合时间尺度分析，以便通过在较慢的时间尺度上研究系统来理解波的轮廓和速度随着域演化而发生的变化。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联系起来（与生物科学的 Steffen Scholpp 博士合作，这是发育生物学中的一个原型系统，它经历收敛扩展，通常用作非局部信号传导系统的范例[4]。总而言之，该项目旨在建立一个 用于理解具有动态边界的偏微分方程系统中的波传播的数学框架。这些动态可以作为时变输入施加到固定域，也可以通过域形状的缓慢演化来合并。这种系统在广泛的生物环境中普遍存在。该项目将通过与伦敦大学的专家发育生物学家合作，确保所使用的模型适用于示范生物系统。