The numerical solution of neural field models posed on realistic cortical domains

The mathematical modelling of neural activity is a hugely complex and prominent area of exploration that has been the focus of many researchers since the mid 1900s. Although many advancements and scientific breakthroughs have been made, there is still a great deal that is not yet understood about the brain. There have been a considerable amount of studies in mathematical neuroscience that consider the brain as a simple one-dimensional or two-dimensional domain; however, this is not biologically realistic and is primarily selected as the domain of choice to aid analytical progress. The primary aim of this thesis is to develop and provide a novel suite of codes to facilitate the computationally efficient numerical solution of large-scale delay differential equations, and utilise this to explore both neural mass and neural field models with space-dependent delays. Through this, we seek to widen the scope of models of neural activity by posing them on realistic cortical domains and incorporating real brain data to describe non-local cortical connections. The suite is validated using a selection of examples that compare numerical and analytical results, along with recreating existing results from the literature. The relationship between structural connectivity and functional connectivity is then analysed as we use an eigenmode fitting approach to inform the desired stability regimes of a selection of neural mass models with delays. Here, we explore a next-generation neural mass model developed by Coombes and Byrne [36], and compare results to the more traditional Wilson-Cowan formulation [180, 181]. Finally, we examine a variety of solutions to three different neural field models that incorporate real structural connectivity, path length, and geometric surface data, using our NFESOLVE library to efficiently compute the numerical solutions. We demonstrate how the field version of the next-generation model can yield intricate and detailed solutions which push us closer to recreating observed brain dynamics