Wave propagation in excitable media with evolving boundaries

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.