Solving neural field equations on curved surfaces with isogeometric collocation

The aim of this study is to develop a new numerical algorithm for simulating neural fields on curved geometries, and eventually human brain surfaces, using Computer-Aided Design (CAD) techniques. We present a CAD-integrated analysis approach, referred to as isogeometric collocation, for solving neural field equations on surfaces that closely resemble cortical geometries typically derived from neuroimaging data. Our methodology involves solving partial integro-differential equations directly using isogeometric collocation techniques, combined with efficient numerical procedures such as heat methods for determining geodesic distances between neural units. To demonstrate the effectiveness of our approach, we initially investigate localised activity patterns in a two-dimensional neural field equation posed on a torus, with the eventual goal of extending the analysis to human brain geometries derived directly from neuroimaging point cloud data. We aim to establish a comprehensive methodology that seamlessly integrates realistic geometries with the analysis of partial integro-differential equations in computational neuroscience. This study is particularly significant for two reasons. Firstly, it addresses the highly irregular nature of point clouds derived from modern neuroimaging data while mitigating the limitations of current time-consuming numerical methods. Secondly, by employing efficient geodesic computation schemes, this approach not only models pattern formation on realistic cortical geometries but can also accommodate cortical architectures of greater physiological relevance.