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

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

New results for delayed neural field equations

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 uniqueness of a solution of these equations. A Lyapunov analysis gives us sufficient conditions for the solutions to be asymptotically stable. We also present a study of the numerical computation of these solutions in a special case. This is, to our knowledge, the first time that a serious analysis of the problem of the existence and uniqueness of a solution of these equations has been performed. Another original contribution of ours is the definition of a Lyapunov functional and the result of stability it implies. We illustrate our work on a variety of examples that are relevant to modeling in neuroscience

Existence and stability of nonmonotone hydraulic shocks for the Saint Venant equations of inclined thin-film flow

Extending work of Yang-Zumbrun for the hydrodynamically stable case of Froude number F < 2, we categorize completely the existence and convective stability of hydraulic shock profiles of the Saint Venant equations of inclined thin-film flow. Moreover, we confirm by numerical experiment that asymptotic dynamics for general Riemann data is given in the hydrodynamic instability regime by either stable hydraulic shock waves, or a pattern consisting of an invading roll wave front separated by a finite terminating Lax shock from a constant state at plus infinity. Notably, profiles, and existence and stability diagrams are all rigorously obtained by mathematical analysis and explicit calculation

Overview of the structure tensor model

International audienceIn 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 matrices. We show that this model englobes the well-known ring model of orientation. We present some results about two classes of specific stationary solutions of the system

Hyperbolic bumps

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 visuo-spatial working memory in prefontal cortex adapted from the ring model of orientation of Ben-Yishai and colleagues. It is therefore natural to study the emergence of spatially localised bumps for the structure tensor model. This modelization was introduced by Chossat and Faugeras to describe 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. In this paper, we present an original study, based on non-Euclidean, hyperbolic, analysis, of a spatially localised bump solution