Continuation of localised coherent structures in nonlocal neural field equations

James Rankin 1 Daniele Avitabile 2 Javier Baladron 1 Gregory Faye 1 David J. B. Lloyd
1 NEUROMATHCOMP - Mathematical and Computational Neuroscience
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR6621
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.
Type de document :
Pré-publication, Document de travail
21 pages, 13 figures, submitted for peer review. 2013
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Contributeur : Pierre Kornprobst <>
Soumis le : mardi 6 août 2013 - 13:47:19
Dernière modification le : vendredi 12 janvier 2018 - 11:32:01


  • HAL Id : hal-00850408, version 1
  • ARXIV : 1304.7206



James Rankin, Daniele Avitabile, Javier Baladron, Gregory Faye, David J. B. Lloyd. Continuation of localised coherent structures in nonlocal neural field equations. 21 pages, 13 figures, submitted for peer review. 2013. 〈hal-00850408〉



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