FAYE , Grégory

The aim of this Thesis is to give a deeper understanding of pattern formation in neural field equations with symmetry, and to understand the significance of these symmetries in modeling the visual cortex. Neural field equations are mesoscopic models that describe the spatio-temporal activity of populations of neurons. They were introduced in...

We present a reduction method to study localized solutions of an integrodifferential equation defined on the Poincaré disk. This equation arises in a problem of texture perception modeling in the visual cortex. We first derive a partial differential equation which is equivalent to the initial integrodifferential equation and then deduce that...

Neural field models with delays define a useful framework for modeling macroscopic parts of the cortex involving several populations of neurons. Nonlinear delayed integro-differential equations describe the spatio-temporal behavior of these fields. Using methods from the theory of delay differential equations, we show the existence and...

This paper completes the classification of bifurcation diagrams for H-planforms in the Poincar ́e 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...

In this paper we study neural field models with delays which define a useful framework for modeling macroscopic parts of the cortex involving several populations of neurons. Nonlinear delayed integrodifferential equations describe the spatio-temporal behavior of these fields. Using methods from the theory of delay differential equations, we...

In this paper we present an overview of pattern formation analysis for an analogue of the Swift-Hohenberg equation posed on the real hyperbolic space of dimension two, which we identify with the Poincaré disc D. Different types of patterns are considered: spatially periodicstationarysolutions,radialsolutionsandtraveling waves,howeverthereare...

The primary visual cortex (V1) can be partitioned into fundamental domains or hypercolumns consisting of one set of orientation columns arranged around a singularity or ''pinwheel'' in the orientation preference map. A recent study on the specific problem of visual textures perception suggested that textures may be represented at the population...

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...

In this poster, we present an overview of the structure tensor model. We first recall the general ideas developped in this framework: model of V1 seen as a set of hypercolumns which encode texture via the structure tensor. We present and analyse analog of Wilson-Cowan equations written in the feature space of 2x2 symmetric definite positive...

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...

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...

In many models of working memory, transient stimuli are encoded by feature-selective persistent neural activity. Such stimuli are imagined to induce the formation of a spatially localised bump of persistent activity which coexists with a stable uniform state. As an example, Camperi and Wang have proposed and studied a network model of...

The existence of spatially localized solutions in neural networks is an important topic in neuroscience as these solutions are considered to characterize work- ing (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...

We study the neural field equations introduced by Chossat and Faugeras in their article 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...

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...

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...

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...