Abstract
Despite the highly convoluted nature of the human brain, neural field models typically treat the cortex as a planar two-dimensional sheet of ne;urons. Here, we present an approach for solving neural field equations on surfaces more akin to the cortical geometries typically obtained from neuroimaging data. Our approach involves solving the integral form of the partial integro-differential equation directly using collocation techniques alongside efficient numerical procedures for determining geodesic distances between neural units. To illustrate our methods, we study localised activity patterns in a two-dimensional neural field equation posed on a periodic square domain, the curved surface of a torus, and the cortical surface of a rat brain, the latter of which is constructed using neuroimaging data. Our results are twofold: Firstly, we find that collocation techniques are able to replicate solutions obtained using more standard Fourier based methods on a flat, periodic domain, independent of the underlying mesh. This result is particularly significant given the highly irregular nature of the type of meshes derived from modern neuroimaging data. And secondly, by deploying efficient numerical schemes to compute geodesics, our approach is not only capable of modelling macroscopic pattern formation on realistic cortical geometries, but can also be extended to include cortical architectures of more physiological relevance. Importantly, such an approach provides a means by which to investigate the influence of cortical geometry upon the nucleation and propagation of spatially localised neural activity and beyond. It thus promises to provide model-based insights into disorders like epilepsy, or spreading depression, as well as healthy cognitive processes like working memory or attention.
Highlights
The nervous system consists of approximately 1011 neurons and 1014 connections all embedded within a highly constrained anatomical space
Whilst a number of recent studies have investigated the relation between surface morphology and large-scale brain connectivity of both grey and white matter (O’Dea et al 2013; Henderson and Robinson 2014; Lo et al 2015), and, to a lesser extent the effect of curvature on reactiondiffusion models of neural activity (see, for example, Kneer et al (2014), Kroos et al (2016) and references therein), the role that cortical geometry plays in non-local models of brain activity, such as the Neural field models (NFM) given by (1), is less well-studied
We have presented a computational technique for solving neural field models (NFM) on curved geometries and investigated the influence of the underlying mesh on these solutions
Summary
The nervous system consists of approximately 1011 neurons and 1014 connections all embedded within a highly constrained anatomical space. Whilst a number of recent studies have investigated the relation between surface morphology and large-scale brain connectivity of both grey and white matter (O’Dea et al 2013; Henderson and Robinson 2014; Lo et al 2015), and, to a lesser extent the effect of curvature on reactiondiffusion models of neural activity (see, for example, Kneer et al (2014), Kroos et al (2016) and references therein), the role that cortical geometry plays in non-local models of brain activity, such as the NFM given by (1), is less well-studied Some progress in this direction was recently made by Visser et al (2017), who used a time-delayed NFM to investigate the behaviour of both standing and travelling wave solutions on a sphere; their approach is restricted to geometries for which a closed-form for the distance function, d, exists.
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