Pattern formation for the Swift-Hohenberg equation on the hyperbolic plane

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 Poincar\'e disc D. Different types of patterns are considered: spatially periodic stationary solutions, radial solutions and traveling waves, however there are significant differences in the results with the Euclidean case. We apply equivariant bifurcation theory to the study of spatially periodic solutions on a given lattice of D also called H-planforms in reference with the "planforms" introduced for pattern formation in Euclidean space. We consider in details the case of the regular octagonal lattice and give a complete descriptions of all H-planforms bifurcating in this case. For radial solutions (in geodesic polar coordinates), we present a result of existence for stationary localized radial solutions, which we have adapted from techniques on the Euclidean plane. Finally, we show that unlike the Euclidean case, the Swift-Hohenberg equation in the hyperbolic plane undergoes a Hopf bifurcation to traveling waves which are invariant along horocycles of D and periodic in the "transverse" direction. We highlight our theoretical results with a selection of numerical simulations.Comment: Dedicated to Klaus Kirchg\"assne

Bifurcation of hyperbolic planforms

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 a bifurcation problem in the hyperbolic plane D (Poincar\'e disc). We make use of the concept of periodic lattice in D to further reduce the problem to one on a compact Riemann surface D/T, where T is a cocompact, torsion-free Fuchsian group. The knowledge of the symmetry group of this surface allows to carry out 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.Comment: 26 pages, 11 figure

Some theoretical results for a class of neural mass equations

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 is governed by nonlinear integro-differential equations defined on the Poincar\'e 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, i.e. 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.Comment: 35 pages, 7 figure

A spatialized model of textures perception using structure tensor formalism

International audienceThe 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 level in the cortex as a second-order tensor, the structure tensor, within a hypercolumn. In this paper, we present a mathematical analysis of such interacting hypercolumns that takes into account the functional geometry of local and lateral connections. The geometry of the hypercolumn is identified with that of the Poincaré disk \D. Using the symmetry properties of the connections, we investigate the spontaneous formation of cortical activity patterns. These states are characterized by tuned responses in the feature space, which are doubly-periodically distributed across the cortex

Analysis of a hyperbolic geometric model for visual texture perception

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, i.e. 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

A spatialized model of textures perception using structure tensor formalism

International audienceThe 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 level in the cortex as a second-order tensor, the structure tensor, within a hypercolumn. In this paper, we present a mathematical analysis of such interacting hypercolumns that takes into account the functional geometry of local and lateral connections. The geometry of the hypercolumn is identified with that of the Poincaré disk \D. Using the symmetry properties of the connections, we investigate the spontaneous formation of cortical activity patterns. These states are characterized by tuned responses in the feature space, which are doubly-periodically distributed across the cortex

Modèles de formation de coalitions stables dans un contexte ad-hoc et stochastique

An ad-hoc and stochastic context prevents : 1- the existence of a global view of the system that reflects a complete image of the deployment environment ; 2- the existence of a priori knowledge because of the lack of a centralized structure, the dynamic of the tasks and the random availability of the entities. We proposed different strategies to facilitate the dynamic stabilization of the interactions between the agents and the convergence towards better coordination states. Our conception of alliances and recommendations allows an agent to evolve independently, to dynamically identify reliable neighboring agents with whom to cooperate and to form Nash-stable or Core stable coalitions according to the requirements of the deployment environment. To face with the challenges of the correlation between local behavior of the agents and the properties of their environment, we use in an original way the Markovian models. We also focused on taking into account the interdependencies between the agents to increase their efficiency in order to optimize the imputed costs of the ad-hoc components where the agents are deployed. This led us to propose both mechanisms, the S-NRB (Sequential Non-return Broadcast) and the P-NRB (Parallel Non-return Broadcast) for distributed coordination seeking to maximize the social welfare of the agents. To highlight the intrinsic properties of our methods, our whole proposals have been studied theoretically and experimentally through our simulatorTravailler dans un contexte ad hoc et dynamique, pour les agents, empêche : 1- l'existence d'une vue globale du système qui reflète une image complète de l'environnement de déploiement ; 2- l'existence de connaissances a priori sur la manière de se coordonner en raison de l'absence d'une structure centralisée et de la disponibilité aléatoire des entités considérés. Nous avons proposé différentes stratégies comportementales pour faciliter la stabilisation dynamique des interactions entre les agents et la convergence vers les meilleurs états de coordination. Notre conception des alliances et des recommandations permet à un agent d'évoluer de manière autonome, d'identifier dynamiquement les agents voisins fiables avec qui coopérer et de former avec son voisinage des partitions Nash-stables selon les exigences de l'environnement de déploiement. Pour répondre à la difficulté de corrélation entre les comportements locaux des agents et les propriétés de l'environnement de déploiement des agents, nous utilisons de manière originale les modèles Markoviens. Nous nous sommes aussi focalisés sur la prise en compte des interdépendances entre les agents pour augmenter leur efficacité dans un souci d'optimisation les coûts imposés aux composants ad-hoc communicants où les agents sont déployés. Cela nous a amené à proposer le modèle S-NRB (Sequentiel Non-return Broadcast) et le modèle P-NRB (Parallel Non-return Broadcast) pour la coordination distribuée qui cherchent à maximiser le bien-être social des agents. Pour mettre en exergue les propriétés intrinsèques de nos méthodes, toutes nos propositions ont été étudiées de manière théorique et expérimentale grâce à notre simulateu