Continuation of localised coherent structures in nonlocal neural field equations

Rankin, James; Avitabile, Daniele; Baladron, Javier; Faye, Gregory; Lloyd, David J.B.

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.

Citation

Rankin, J., Avitabile, D., Baladron, J., Faye, G., & Lloyd, D. J. (2014). Continuation of localised coherent structures in nonlocal neural field equations. SIAM Journal on Scientific Computing, 36(1), Article B70-B93. https://doi.org/10.1137/130918721

• Journal Article Type: Article
• Publication Date: Jan 1, 2014
• Deposit Date: Mar 17, 2014
• Publicly Available Date: Mar 17, 2014
• Journal: SIAM Journal on Scientific Computing
• Print ISSN: 1064-8275
• Electronic ISSN: 1064-8275
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 36
• Issue: 1
• Article Number: B70-B93
• DOI: https://doi.org/10.1137/130918721
• Public URL: https://nottingham-repository.worktribe.com/output/999658
• Publisher URL: http://epubs.siam.org/doi/abs/10.1137/130918721