Continuation of localised coherent structures in nonlocal neural field equations

TL;DR

This paper investigates localized activity patterns in neural field equations on the Euclidean plane, using numerical continuation methods to explore bifurcations and pattern organization relevant to cortical activity.

Contribution

It introduces a matrix-free Newton-Krylov solver for direct continuation of localized patterns in integral neural field models, enabling analysis without PDE reformulation.

Findings

Support for patterns with varying spatial extent via homoclinic snaking

Patterns organized by spatial interactions at specific scales

Framework applicable to studying localized cortical activity

Abstract

We study localised activity patterns in neural field equations posed on the Euclidean plane; such models are commonly used to describe the coarse-grained activity of large ensembles of cortical neurons in a spatially continuous way. We employ matrix-free Newton-Krylov solvers and perform numerical continuation of localised patterns directly on the integral form of the equation. This opens up the possibility to study systems whose synaptic kernel does not lead to an equivalent PDE formulation. We present a numerical bifurcation study of localised states and show that the proposed models support patterns of activity with varying spatial extent through the mechanism of homoclinic snaking. The regular organisation of these patterns is due to spatial interactions at a specific scale associated with the separation of excitation peaks in the chosen connectivity function. The results presented form a basis for the general study of localised cortical activity with inputs and, more specifically, for investigating the localised spread of orientation selective activity that has been observed in the primary visual cortex with local visual input.