A numerical simulation of neural fields on curved geometries

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.

Keywords: Collocation method; Geodesics; Neural fields; Non-flat geometries.

Fig. 1

a The Mexican hat connectivity function b Sigmoidal firing rate function

Fig. 2

Illustration of a domain that uses Cartesian grid points as triangle vertices

Fig. 3

The error |I_m+ 1 − I_m| plotted against grid size N_m+ 1 reveals geometric convergence rates for trapezoidal rule, FFTs and linear collocation when computing the integral in (6)

Fig. 4

Illustration of the refinement procedure for a general triangulated domain in which the initial mesh is generated using the DistMesh package

Fig. 5

Convergence of linear collocation when computing the integral in (6) on a general triangulation constructed using the DistMesh package. The orange and yellow lines indicate the slopes for first and second order methods, respectively

Fig. 6

Travelling bump solutions of (2) computed on a triangulation based on a Cartesian grid, using trapezoidal, FFTs and linear collocation, respectively

Fig. 7

Travelling bump solution of (2) computed on a general DistMesh triangulation

Fig. 8

Parameterisation of a torus by coordinates (𝜃, ϕ)

Fig. 9

A plot of the error versus grid size when computing the integral in (2) on a regular polar coordinate grid of the torus using the trapezoidal rule (dashed line) and on a triangulation whose nodes coincide with the same grid using linear collocation (solid line)

Fig. 10

Snapshots of a travelling bump solution propagating clockwise on a torus with minor curvature radius r = 2.5 and major curvature radius R = 4.5 computed by solving equation (10) using linear collocation

Fig. 11

Illustration of a general triangulation of the torus generated using the DistMesh package (Persson and Strang 2004)

Fig. 12

a The tracked path of a bump solution of (2) following a non-geodesic trajectory on the torus. Toroidal regions of maximum (positive) curvature are coloured yellow while regions of minimum (negative) curvature are coloured blue. b The curvature (red line) and speed (blue line) of the bump solution along the trajectroy shown in (a)

Fig. 13

A triangulation of the left hemisphere of the rat cortex

Fig. 14

Snapshots of bump solutions of (2) propagating on the surface of the rat cortex