Non-local interactions in cell populations on time-dependent spatial domains

The collective migration of embryonic neural crest (NC) cells is an important part of embryonic development. NC cells migrate in streams away from the neural tube along well-defined paths to reach their destination. This results in an essentially one-dimensional spatial domain.

In this talk we focus on two aspects which have been implicated in NC cell migration: spatial domain growth and contact-induced repulsion. The latter effect is represented by a non-local repulsion term in our time-dependent PDE model.

We first introduce our basic model on a fixed spatial domain. This is followed by discussing time-dependent domains and conservation laws on them. We then formulate our model on a time-dependent domain and consider its numerical simulation. The bottleneck of the numerical scheme arises from the non-local term in our model. We focus attention on how to efficiently approximate this. The talk concludes with some instructing simulations and discusses observations. We also outline open problems and future work.