138 research outputs found
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
Hyperbolic planforms in relation to visual edges and textures perception
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 to test these hypotheses. We study
the specific problem of visual edges and textures perception and suggest that
these features may be represented at the population level in the visual cortex
as a specific second-order tensor, the structure tensor, perhaps within a
hypercolumn. We then extend the classical ring model to this case and show that
its natural framework is the non-Euclidean hyperbolic geometry. This brings in
the beautiful structure of its group of isometries and certain of its subgroups
which have a direct interpretation in terms of the organization of the neural
populations that are assumed to encode the structure tensor. By studying the
bifurcations of the solutions of the structure tensor equations, the analog of
the classical Wilson and Cowan equations, under the assumption of invariance
with respect to the action of these subgroups, we predict the appearance of
characteristic patterns. These patterns can be described by what we call
hyperbolic or H-planforms that are reminiscent of Euclidean planar waves and of
the planforms that were used in [1, 2] to account for some visual
hallucinations. If these patterns could be observed through brain imaging
techniques they would reveal the built-in or acquired invariance of the neural
organization to the action of the corresponding subgroups.Comment: 34 pages, 11 figures, 2 table
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
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
Coverage, Continuity and Visual Cortical Architecture
The primary visual cortex of many mammals contains a continuous
representation of visual space, with a roughly repetitive aperiodic map of
orientation preferences superimposed. It was recently found that orientation
preference maps (OPMs) obey statistical laws which are apparently invariant
among species widely separated in eutherian evolution. Here, we examine whether
one of the most prominent models for the optimization of cortical maps, the
elastic net (EN) model, can reproduce this common design. The EN model
generates representations which optimally trade of stimulus space coverage and
map continuity. While this model has been used in numerous studies, no
analytical results about the precise layout of the predicted OPMs have been
obtained so far. We present a mathematical approach to analytically calculate
the cortical representations predicted by the EN model for the joint mapping of
stimulus position and orientation. We find that in all previously studied
regimes, predicted OPM layouts are perfectly periodic. An unbiased search
through the EN parameter space identifies a novel regime of aperiodic OPMs with
pinwheel densities lower than found in experiments. In an extreme limit,
aperiodic OPMs quantitatively resembling experimental observations emerge.
Stabilization of these layouts results from strong nonlocal interactions rather
than from a coverage-continuity-compromise. Our results demonstrate that
optimization models for stimulus representations dominated by nonlocal
suppressive interactions are in principle capable of correctly predicting the
common OPM design. They question that visual cortical feature representations
can be explained by a coverage-continuity-compromise.Comment: 100 pages, including an Appendix, 21 + 7 figure
The hyperbolic model for edge and texture detection in the primary visual cortex
International audienceThe 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. Ten years ago an attempt was made to extend these models by replacing orientation (an angle) with a second-order tensor built from the gradient of the image intensity, and it was named the structure tensor. This assumption does not follow from biological observations (experimental evidence is still lacking) but from the idea that the effectiveness of texture processing with the structure tensor in computer vision may well be exploited by the brain itself. The drawback is that in this case the geometry is not Euclidean but hyperbolic instead, which complicates the analysis substantially. The purpose of this review is to present the methodology that was developed in a series of papers to investigate this quite unusual problem, specifically from the point of view of tuning and pattern formation. These methods, which rely on bifurcation theory with symmetry in the hyperbolic context, might be of interest for the modeling of other features such as color vision or other brain functions
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
Forced symmetry breaking of Euclidean equivariant partial differential equations, pattern formation and Turing instabilities
Many natural phenomena may be modelled using systems of differential equations that possess symmetry. Often the modelling process introduces additional symmetries that are only approximately present in the real physical system. This thesis investigates how the inclusion of small symmetry breaking effects changes the behaviour of the original solutions, such a process is called forced symmetry breaking.
Part I introduces the general equivariant bifurcation theory required for the rest of this work. In particular, we generalise previous techniques used to study forced symmetry breaking to a certain class of Euclidean invariant problems. This allows the study of the effects of forced symmetry breaking on spatially periodic solutions to differential equations.
Part II considers spatially periodic solutions in two dimensions that are supported by the square or hexagonal lattices. The methods of Part I are applied to investigate how the translation free solutions, supported by these lattices, are altered when the perturbation term possesses certain symmetries. This leads to a partial classification theorem, describing the behaviour of these solutions.
This classification is extended in Part III to three-dimensional solutions. In particular, the cubic lattices: simple, face centred, and body centred cubic, are considered. The analysis follows the same lines as Part II, but is necessarily more complex. This complexity is also present in the results, there are much richer dynamical possibilities.
Parts II and III lead to a partial classification of the behaviour of spatially periodic solutions to differential equations in two and three dimensions.
Finally in Part IV the results of Part III, concerning the body centred cubic lattice, are applied to the black-eye Turing instability. In particular, the model of Gomes [39] is cast in a new light where forced symmetry breaking is present, leading to several qualitative predictions. Nonlinear optical systems and the Polyacrylamide-Methylene Blue-Oxygen reaction are also discussed
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
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