Grégory Faye

Continuation of Localized Coherent Structures in Nonlocal Neural Field Equations

We study localized 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 localized patterns directly on the integral form of the equation. This opens up the possibility of studying systems whose synaptic kernel does not lead to an equivalent PDE formulation. We present a numerical bifurcation study of localized states and show that the proposed models support patterns of activity with varying spatial extent through the mechanism of homoclinic snaking. The regular organization 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 localized cortical activity with inputs and, more specifically, for the investi...

Bifurcation of Hyperbolic Planforms

Motivated by a model for the perception of textures by the visual cortex in primates, we analyze 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 a bifurcation problem in the hyperbolic plane

Analysis of a hyperbolic geometric model for visual texture perception

\mathcal {D}

Localized states in an unbounded neural field equation with smooth firing rate function: a multi-parameter analysis

(Poincaré disc). We make use of the concept of a periodic lattice in

Bifurcation Diagrams and Heteroclinic Networks of Octagonal H-Planforms

\mathcal {D}

Existence and Stability of Traveling Pulses in a Neural Field Equation with Synaptic Depression

to further reduce the problem to one on a compact Riemann surface

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

\mathcal {D}/\varGamma

Pulse Bifurcations in Stochastic Neural Fields

, where Γ is a cocompact, torsion-free Fuchsian group. The knowledge of the symmetry group of this surface allows us to use the machinery of equivariant bifurcation theory. Solutions which generically bifurcate are called “H-planforms”, by analogy with the “planforms” introduced for pattern formation in Euclidean space. This concept is applied to the case of an octagonal periodic pattern, where we are able to classify all possible H-planforms satisfying the hypotheses of the Equivariant Branching Lemma. These patterns are, however, not straightforward to compute, even numerically, and in the last section we describe a method for computation illustrated with a selection of images of octagonal H-planforms.

Linear spreading speeds from nonlinear resonant interaction

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 integro-differential equations defined on the Poincaré disc model of the two-dimensional hyperbolic space. Using methods from the theory of functional analysis we show the existence and uniqueness of a solution of these equations. In the case of stationary, that is, time independent, solutions we perform a stability analysis which yields important results on their behavior. We also present an original study, based on non-Euclidean, hyperbolic, analysis, of a spatially localised bump solution in a limiting case. We illustrate our theoretical results with numerical simulations.Mathematics Subject Classification:30F45, 33C05, 34A12, 34D20, 34D23, 34G20, 37M05, 43A85, 44A35, 45G10, 51M10, 92B20, 92C20.

Monotone traveling waves for delayed neural field equations

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 solutions of our neural field equation are equivalent to homoclinic orbits in a related fourth order ordinary differential equation, we apply normal form theory for a reversible Hopf bifurcation to prove the existence of localized solutions; further, we present results concerning their stability. Numerical continuation is used to compute branches of localized solution that exhibit snaking-type behaviour. We describe in terms of three parameters the exact regions for which localized solutions persist.