The Numerical Solution of Neural Field Models Posed on Realistic Cortical Domains

Neural field equations (NFEs) are non-local and delayed dynamical systems describing the coarse grained dynamics of cortical neural activity. In this work we will show that NFEs can be numerically solved on arbitrary geometries. We will develop these schemes with a focus on a 2-manifold in R^3, modelling a realistic cortex, including gyri and sulci. The Human Connectome Project has a large collection of data available which we can incorporate into our framework. This includes real structural connectivity estimates collected from a number of subjects, as well as the cortical domain meshes. Axonal delays will also be accommodated, though require more advanced numerical techniques to be developed. This project's main aim is to develop a numerical solver in an appropriate computer programming language that handles NFEs of arbitrary dimension, posed on an arbitrary geometry, with arbitrary connectivity, and the inclusion of delays. The numerical scheme will be validated against analytical predictions from idealised systems, making use of Turing instability analysis and the construction of localised states in some singular parameter limits. This work will ultimately provide an in silico tool (based on individual human brain structure) that can inform our understanding of the clinically relevant physiological brain rhythms that are readily recorded using electro- and magneto-encephalography and functional magnetic resonance imaging.