CHOSSAT , Pascal

The modeling of neural fields in the visual cortex involves geometrical structures which describe in mathematical formalism the functional architecture of this cortical area. The case of contour detection and orientation tuning has been extensively studied and has become a paradigm for the mathematical analysis of image processing by the brain....

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...

The onset of convection for spherically invariant Rayleigh-Bénard fluid flow is driven by marginal modes associated with spherical harmonics of a certain degree $\ell$, which depends upon the aspect ratio of the spherical shell. At certain critical values of the aspect ratio, marginal modes of degrees $\ell$ and $\ell+1$ coexist. Initially...

Robust heteroclinic cycles in equivariant dynamical systems in $\mathbb{R}^4$ have been a subject of intense scientific investigation because, unlike heteroclinic cycles in $\mathbb{R}^3$, they can have an intricate geometric structure and complex asymptotic stability properties that are not yet completely understood. In a recent work, we have...

We review and extend the previous work where a model was introduced for Hopfield-type neural networks, which allows for the existence of heteroclinic dynamics between steady patterns. This dynamics is a mathematical model of periodic or aperiodic switching between stored information items in the brain, in particular, in the context of...

In generic dynamical systems heteroclinic cycles are invariant sets of codimension at least one, but they can be structurally stable in systems which are equivariant under the action of a symmetry group, due to the existence of flow-invariant subspaces. For dynamical systems in the minimal dimension for which such robust heteroclinic cycles...

The onset of convection for spherically invariant Rayleigh-Bénard fluid flow is driven by marginal modes associated with spherical harmonics of a certain degree $\ell$, which depends upon the aspect ratio of the spherical shell. At certain critical values of the aspect ratio, marginal modes of degrees $\ell$ and $\ell+1$ coexist. Initially...

Learning or memory formation are associated with the strengthening of the synaptic connections between neurons according to a pattern reflected by the input. According to this theory a retained memory sequence is associated to a dynamic pattern of the associated neural circuit. In this work we consider a class of network neuron models, known as...

The onset of convection for spherically invariant Rayleigh–Bénard fluid flow is driven by marginal modes associated with spherical harmonics of a certain degree , which depends upon the aspect ratio of the spherical shell. At certain critical values of the aspect ratio, marginal modes of degrees and coexist. Initially motivated by an experiment...

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...

We propose to use bifurcation theory and pattern formation as theoretical probes for various hypotheses about the neural organization of the brain. This allows us to make predictions about the kinds of patterns that should be observed in the activity of real brains through, e.g., optical imaging, and opens the door to the design of experiments...

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...

This paper challenges and significantly extends the seminal work in \cite{ermentrout-cowan:79b,bressloff-cowan-etal:01}. In this work, we improve on the following points the seminal work as in \cite{ermentrout-cowan:79b,bressloff-cowan-etal:01}. We consider the problem of describing mathematically the spontaneous activity of V1 by combining...

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...

International audience

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...

Adaptation of behavior to a changing environment requires the brain to change the probability of selecting actions based on their reward or punishment. The present study proposes a simple neural mechanism of gain modulation that makes possible rapid changes in the probability of selecting actions after different levels of negative feedback. The...

Prediction is the ability of the brain to quickly activate a target concept in response to arelated stimulus (prime). Experiments point to the existence of an overlap between the populationsof the neurons coding for different stimuli, and other experiments show that prime-targetrelations arise in the process of long term memory formation. The...

The relative equilibria of a symmetric Hamiltonian dynamical system are the critical points of the so-called augmented Hamiltonian. The underlying geometric structure of the system is used to decompose the critical point equations and construct a collection of implicitly defined functions and reduced equations describing the set of relative...

An important function of the brain is to predict which stimulus is likely to occur based on the perceived cues. The present research studied the branching behavior of a computational network model of populations of excitatory and inhibitory neurons, both analytically and through simulations. Results show how synaptic efficacy, retroactive...