Waves and patterns in heterogeneous neural media

Project description

Neural field models have long been used to study the waves and patterns of electrical activity that can spread across the cortex of the human brain. They are typically cast as integro-differential equations and have been extensively studied in idealised scenarios that encompass assumptions that i) firing rates rather than spike times are used to encode interactions, ii) tissue interactions (strength and delay) only depend on distance, iii) cortical geometry is flat. This PhD project will develop new mathematical approaches for dealing with more formal connections between spike and rate based modelling by using so-called next generation neural field models [1], develop perturbative techniques for handling heterogeneous neural media by extending ideas in [2], and develop a new suite of pattern formation methodologies building on Turing and weakly nonlinear analysis, interface dynamics, and scientific computation suitable for describing the spread and scatter of spatio-temporal patterns through through folded cortex with regions of both negative and positive curvature at gyri and sulci respectively. The latter will build on recent ideas developed for the study of reaction-diffusion equations on curved surfaces, e.g., [3].