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- Pascal Chossat, Grégory Faye, Olivier P. Faugeras
- J. Nonlinear Science
- 2011

Motivated by a model for the perception of textures by the visual cortex in primates, we analyse the bifurcation of periodic patterns for nonlinear equations describing the state of a system defined on the space of structure tensors, when these equations are further invariant with respect to the isometries of this space. We show that the problem reduces to… (More)

- Grégory Faye, Pascal Chossat, Olivier P. Faugeras
- Journal of mathematical neuroscience
- 2011

We study the neural field equations introduced by Chossat and Faugeras to model the representation and the processing of image edges and textures in the hypercolumns of the cortical area V1. The key entity, the structure tensor, intrinsically lives in a non-Euclidean, in effect hyperbolic, space. Its spatio-temporal behaviour is governed by nonlinear… (More)

- Grégory Faye, Pascal Chossat
- J. Nonlinear Science
- 2012

This paper completes the classification of bifurcation diagrams for H-planforms in the Poincaré disc D whose fundamental domain is a regular octagon. An H-planform is a steady solution of a PDE or integro-differential equation in D, which is invariant under the action of a lattice subgroup Γ of U(1, 1), the group of isometries of D. In our case Γ generates… (More)

- Grégory Faye, James Rankin, Pascal Chossat
- Journal of mathematical biology
- 2013

The existence of spatially localized solutions in neural networks is an important topic in neuroscience as these solutions are considered to characterize working (short-term) memory. We work with an unbounded neural network represented by the neural field equation with smooth firing rate function and a wizard hat spatial connectivity. Noting that stationary… (More)

- James Rankin, Daniele Avitabile, Javier Baladron, Grégory Faye, David J. B. Lloyd
- SIAM J. Scientific Computing
- 2014

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… (More)

- Grégory Faye, Arnd Scheel
- 2013

We establish Fredholm properties for a class of nonlocal differential operators. Using mild convergence and localization conditions on the nonlocal terms, we also show how to compute Fredholm indices via a generalized spectral flow, using crossing numbers of generalized spatial eigenvalues. We illustrate possible applications of the results in a nonlinear… (More)

- Grégory Faye, Arnd Scheel
- 2013

We prove the existence of fast traveling pulse solutions in excitable media with non-local coupling. Existence results had been known, until now, in the case of local, diffusive coupling and in the case of a discrete medium, with finite-range, non-local coupling. Our approach replaces methods from geometric singular perturbation theory, that had been… (More)

We investigate pinning regions and unpinning asymptotics in nonlocal equations. We show that phenomena are related to but different from pinning in discrete and inhomogeneous media. We establish unpinning asymptotics using geometric singular perturbation theory in several examples. We also present numerical evidence for the dependence of unpinning… (More)

- Zachary P. Kilpatrick, Grégory Faye
- SIAM J. Applied Dynamical Systems
- 2014

We study the effects of additive noise on traveling pulse solutions in spatially extended neural fields with linear adaptation. Neural fields are evolution equations with an integral term characterizing synaptic interactions between neurons at different spatial locations of the network. We introduce an auxiliary variable to model the effects of local… (More)

- Grégory Faye, Jonathan Touboul
- SIAM Journal of Applied Mathematics
- 2014

Neural field equations are integro-differential systems describing the macroscopic activity of spatially extended pieces of cortex. In such cortical assemblies, the propagation of information and the transmission machinery induce communication delays, due to the transport of information (propagation delays) and to the synaptic machinery (constant delays).… (More)