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Localized radial bumps of a neural field equation on the Euclidean plane and the Poincaré disk

Grégory Faye 1, 2 James Rankin 1 David J. B. Lloyd 3 
1 NEUROMATHCOMP - Mathematical and Computational Neuroscience
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR6621
Abstract : We analyse radially symmetric localized bump solutions of an integro-differential neural field equation posed in Euclidean and hyperbolic geometry. The connectivity function and the nonlinear firing rate function are chosen such that radial spatial dynamics can be considered. Using integral transforms, we derive a partial differential equation for the neural field equation in both geometries and then prove the existence of small amplitude radially symmetric spots bifurcating from the trivial state. Numerical continuation is then used to path follow the spots and their bifurcations away from onset in parameter space. It is found that the radial bumps in Euclidean geometry are linearly stable in a larger parameter region than bumps in the hyperbolic geometry. We also find and path follow localized structures that bifurcate from branches of radially symmetric solutions with D6-symmetry and D8-symmetry in the Euclidean and hyperbolic cases, respectively. Finally, we discuss the applications of our results in the context of neural field models of short term memory and edges and textures selectivity in a hypercolumn of the visual cortex.
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Submitted on : Wednesday, July 17, 2013 - 2:13:21 PM
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Grégory Faye, James Rankin, David J. B. Lloyd. Localized radial bumps of a neural field equation on the Euclidean plane and the Poincaré disk. Nonlinearity, 2012, 26 (2), pp.437-478. ⟨10.1088/0951-7715/26/2/437⟩. ⟨hal-00845586⟩



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