Pascal Chossat
email: pascal.chossat@unice.fr

Keywords: Primary visual cortex, neural field equations, texture perception, structure tensor, pattern formation, hyperbolic geometry

The modelling 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 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 stucture 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 substantially the analysis. 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 modelling of other features such as color vision, or other brain functions.

The hyperbolic model for edge and texture detection in the primary visual cortex

Since the discovery by David Hubel and Torsten Wiesel (Nobel prize 1981) of the tiling of the primary visual cortex V1 into columns of orientations, or hypercolumns, tremendous progress has been made in the physiological and functional exploration of this brain area, which is mainly dedicated to the integration of basic visual features of images in the visual field, such as contours, contrast, size (spatial frequency), ocular dominance, color. A series of remarkable experiments have unveiled significant part of the functional architecture of hypercolumns and numerous works based on these results have been dedicated to give a coherent description of the way in which the visual informations are treated in V1 [START_REF] Fitzpatrick | The functional organization of local circuits in visual cortex: insights from the study of tree shrew striate cortex[END_REF].

An extensive survey can be found in [START_REF] Petitot | Elements of Neurogeometry[END_REF] . This in turn has led to suggest that this neuronal architecture implements a specific geometry, which was optimized through natural selection to efficiently process images at the basic level of V1. Although correlations seem to exist between hypercolumns of orientation, columns of ocular dominance and color perception (blobs of cytochrome oxydase), a mathematical representation of these correlations is still unclear [START_REF] Bartfeld | Relationships between orientation-preference pinwheels, cytochrome oxidase blobs, and ocular-dominance columns in primate striate cortex[END_REF] and we concentrate on texture detection in the following.

The best understood feature is contour detection. Nearly 30 years after the discovery of the columnar structure of V1, evidence arose that columns themselves are organized around "pinwheels", which are singular points at which all orientations are equally detected [START_REF] Bonhoeffer | Iso-orientation domains in cat visual cortex are arranged in pinwheel-like patterns[END_REF]. In hypercolumns pinwheels come by pairs of clockwise and anticlockwise variation of the orientation when turning around the pinwheel, which gives the lattice of pinwheels a cristalline structure reminiscent of spin glasses in physics [START_REF] Petitot | Elements of Neurogeometry[END_REF]. It was also discovered that hypercolumns are interconnected through long range "lateral" fibers, with the property that neurons which detect a specific orientation in one hypercolumn are preferentially connected to neurons with the same orientation in other columns. Moreover this set of connections is not isotropic. Instead, the neural fibers tend to be aligned along a specific direction which is strongly correlated with the orientation itself [START_REF] Bosking | Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex[END_REF]. These facts being put together allowed to identify V1, seen as a contour detector, with a fiber bundle whose base is the visual field and the fiber is the set of orientations "above" each small region in the visual field. In addition the lateral connections endow this bundle with a natural sub-Riemannian structure which allows V1 to compute "global" contours from local informations. This geometrical approach of contour detection in V1 has been exposed in [START_REF] Sarti | Functional geometry of the horizontal connectivity in the primary visual cortex[END_REF][START_REF] Petitot | Elements of Neurogeometry[END_REF].

In parallel integro-differential neural field models of activity in cortical tissues (grey matter) have been proposed [START_REF] Wilson | Excitatory and inhibitory interactions in localized populations of model neurons[END_REF][START_REF] Wilson | A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue[END_REF]. Indeed in the context of large populations of neurons it would be prohibitive to apply the classical Hodgkin-Huxley equations or even a simplified version like the integrate and fire equations [START_REF] Gerstner | Neural Dynamics[END_REF].

Instead, averaged membrane potentials over "small" populations are considered in the continuous limit, meaning that the fiber bundle of orientations is identified with R 2 × P 1 where R 2 is the base plane (the "horizontal" extension of V1) and P 1 is the projective line (set of orientations) sitting "above" each point in R 2 . For convenience P 1 is further identified with the circle S 1 (mod π). Then the evolution of the neural field in this domain is described by a phenomenological integro-differential equation which we refer to as Wilson-Cowan equation, with the hope that it will capture the main qualitative behavior of the neuronal activity in V1. This equation was applied to the problem of orientation tuning, which is the enhancement within the columns of orientation of the orientation's selectivity [START_REF] Ben-Yishai | Theory of orientation tuning in visual cortex[END_REF][START_REF] Bressloff | Dynamical Mechanism for Sharp Orientation Tuning in an Integrate-and-Fire Model of a Cortical Hypercolumn[END_REF]. The analysis here relies on bifurcation theory for neural fields defined on S 1 (the "fiber" of orientation angles). By taking account of the interactions between columns of orientation, this extends to a problem of pattern formation in the full fiber bundle R 2 × S 1 . This spatial extension was considered by Bressloff, Cowan, Golubitsky et al. [START_REF] Bressloff | Geometric visual hallucinations, euclidean symmetry and the functional architecture of striate cortex[END_REF] who developed a theory for geometric hallucinations reported by patients under various circumstances (like psychotropic drug ingestion). Spontaneous pattern formation as a Turing instability for neural fields in R 2 , hence not taking into account the orientation, had been first considered by Ermentrout [START_REF] Ermentrout | A mathematical theory of visual hallucination patterns[END_REF] but the work in [START_REF] Bressloff | Geometric visual hallucinations, euclidean symmetry and the functional architecture of striate cortex[END_REF], which allows to build contours of the images, extended it considerably. More recently Citti and Sarti [START_REF] Sarti | The constitution of visual perceptual units in the functional architecture of V1[END_REF] considered the same model as in [START_REF] Bressloff | Geometric visual hallucinations, euclidean symmetry and the functional architecture of striate cortex[END_REF] but switching on an input in the equation to simulate the response of V1 to a stimulus. Wilson-Cowan equation and the orientation tuning as a bifurcation problem will be recalled in Section 3 where basic principles of bifurcation theory with symmetry (or "equivariant bifurcation theory") are also recalled. Although the algebraic tools in this theory are superfluous for the tuning of orientation problem, they simplify and help organizing the analysis in the spatially extended case [START_REF] Bressloff | Geometric visual hallucinations, euclidean symmetry and the functional architecture of striate cortex[END_REF] and become unavoidable when more "exotic" symmetries are considered, like those appearing in the models introduced in Section 4.

The challenge is to integrate more features in the model, in a way consistent with the orientation model and with the known experimental facts about the functional organization of V1. This comes back to replacing the fiber bundle R 2 × P 1 by R 2 × Ξ where Ξ is a "feature space" containing the set of orientations and endowed with a "nice enough" geometry to allow for analysis and computations. Several theories have been proposed to extend the orientation model. In [START_REF] Sarti | The symplectic structure of the primary visual cortex[END_REF][START_REF] Sarti | Functional geometry of the horizontal connectivity in the primary visual cortex[END_REF] the scale of the receptive profile of each neuron is taken into account. Then Ξ ∼ S 1 ×R where the first component is the circle of orientation angles (mod π) and the second component is the scale. Therefore the fiber bundle is 4-dimensional and the nice thing is that it inherits a natural symplectic structure (which extends the contact structure of the orientation model). This structure gives an elegant explanation of shape recognition by the brain [START_REF] Sarti | The symplectic structure of the primary visual cortex[END_REF]. Another model was proposed in [START_REF] Bressloff | A spherical model for orientation and spatial frequency tuning in a cortical hypercolumn[END_REF] to take account of the spatial frequency. Indeed experiments have shown that neurons in V1 are also sentitive to the frequency of the gratings which are presented to the animal in the experimentations [START_REF] Valois | Spatial Vision[END_REF]. Some experimental studies suggest that in the hypercolumn the spatial frequency goes to a minimum at one pinwheel and to a maximum value at the other pinwheel. Under a suitable change of variables the authors expressed the feature space Ξ as a sphere where the polar angle is the orientation and the azimuthal angle is a normalized logarithmic function of the scale. Therefore the poles represent the pinwheels. This sphere is endowed with its natural Riemannian metric, so that Ξ ∼ S 2 with its isometry group O(3). Now the orientation tuning problem is extended to account for frequency tuning and falls into the class of pattern formation on the sphere, which has been widely studied in other contexts (see [START_REF] Chossat | Methods in Equivariant Bifurcations and Dynamical Systems[END_REF][START_REF] Golubitsky | Singularities and Groups in Bifurcation Theory[END_REF]).

The aim of this paper is to review an alternative way to extend the orientation model of hypercolumns, which was proposed ten years ago by [START_REF] Chossat | Hyperbolic planforms in relation to visual edges and textures perception[END_REF]. The idea is to replace the orientation angle by a "Gaussian average" at a specifed scale of the matrix ∇I t (∇I) where ∇I is the gradient of the image intensity and t indicates the transpose matrix [START_REF] Bigun | Optimal orientation detection of linear symmetry[END_REF][START_REF] Knutsson | Representing local structure using tensors[END_REF]. This symmetric, positive definite matrix is called in computer vision the structure tensor or second-moment matrix. It accurately detects edges and corners, on the contrary to the gradient itself which leads to ambiguous interpretations in certain cases. This will be explained in Section 4.1. The space SPD(2) of structure tensors may therefore be a good candidate for the feature space Ξ when restricted to edge and texture detection. The space SPD(2) possesses a natural Riemannian structure with isometry group GL(2, R), the group of real invertible matrices. As we shall see, this allows to further identify Ξ with R + * ×D, D being the Poincaré disc, or pseudo-sphere, equipped with its hyperbolic structure. The analysis of the Wilson-Cowan equation and its bifurcations in this hyperbolic geometry setting has been undertaken in a series of publications [START_REF] Chossat | Hyperbolic planforms in relation to visual edges and textures perception[END_REF][START_REF] Chossat | Bifurcations of hyperbolic planforms[END_REF][START_REF] Faye | Analysis of a hyperbolic geometric model for visual texture perception[END_REF][START_REF] Faye | Bifurcation diagrams and heteroclinic networks of octagonal h-planforms[END_REF][START_REF] Faye | A spatialized model of visual texture perception using the structure tensor formalism Networks and[END_REF]. In this paper I give a synthetic presentation of these studies. The concept of structure tensor is introduced in Section 4, as well as the geometry of the space of structure tensors and its reduction to the hyperbolic surface, of which the Poincaré disc is the most suitable representation for our purpose. Section 5 is devoted to pattern formation, that is bifurcation analysis from an homogeneous basic state, of the Wilson-Cowan equation in the space of structure tensors. Some elements about spectral and Fourier analysis in the Poincaré disc are first introduced. Then it is shown that bifurcation analysis can be performed in the same spirit as in the Euclidian case, but it is technically more involved. Three types of solutions are presented: bifurcation of periodic waves, bifurcation of periodic patterns (in a sense which will be explained) and bifurcation in a bounded subdomain of the Poincaré disc. Localized solutions (bumps) have also been studied in [START_REF] Faye | Localized states in an unbounded neural field equation with smooth firing rate function : a multi-parameter analysis[END_REF][START_REF] Faye | Localized radial bumps of a neural field equation on the Euclidean plane and the Poincaré disk[END_REF] but the techniques are different and quite involved and will not be presented here.

The paper ends with a Conclusion where the main results are summarized and commented.

2 Hypercolumns of orientation and the fuctional architecture of V1

The signal (spikes of action potential) which is generated on the retina is preprocessed in an intermediate region of the brain called the Lateral Geniculate Nucleus (LGN). Then the outgoing neural fibers are projected to the occipital part of the brain, in a region called the primary visual cortex V1. This region is dedicated to a first treatment of the basic features of the image, such as orientations, contrast, spatial frequency, colour. Higher level visual zones V 2, V 3 etc, which surround V1, refine further this process and a complex and not fully understood interaction between these successive zones (including the LGN) is constantly operating. Nevertheless in order to understand the basic functioning of the visual cortex it is relevant to simplify the landscape by assuming that V1 is disconnected from its higher level neighbors.

The following experimental facts have been well established. There have been many important contributions to the anatomical and functional exploration of V1. An extensive bibliography accompanies the description of these results in [START_REF] Petitot | Elements of Neurogeometry[END_REF] (chapters 3 and 4), which has inspired the present section.

• V1 is layered in six " horizontal sheets" of neural tissue. Most of the incoming fibers from the LGN arrive in layer 4. In the following we shall not differentiate these zones and we shall consider V1 as a 2D area.

• The fibers projecting the retina to V1 map the visual field in a conformal manner, called the retinocortical map (figure 1). A good approximation is given by the complex logarithm function r = log z.

• Each neuron in V1 responds to a specific small zone in the retinal plane called the receptive field of the neuron. This zone is generally elongated in a direction which defines the orientation to which the neuron is sensitive and the response of the neuron, called its receptive profile (RP), is well approximated by Gabor functions or derivatives of Gaussian functions [START_REF] Petitot | Elements of Neurogeometry[END_REF]. Figure Figure 2 shows an example of a RP. The typical receptive profile of a simple neuron in V1 (from [START_REF] Petitot | Elements of Neurogeometry[END_REF]). Green color corresponds to positive signal and red color to negative signal.

• The position and orientation to which neurons are responding in V1 are approximately constant in the "vertical" direction. This defines the columns of orientation. Since V1 is identified with a 2D area, each column of orientation corresponds to a point on this surface. Orientations vary continuously on the surface V1 except at singular points, called pinwheels, at which neurons respond to all orientations (their RP is circular). Along a small circle around a pinwheel all orientations that are detected within a small patch of the visual field are represented by steps of about 10 • . This property gives the distribution of pinwheels and orientations on V1 a crystalline structure as shown on Figure 2. The elementary "cells" in this lattice is what we call hypercolumns, which incorportae pairs of pinwheels with reverse orientations. Iso-orientation lines converge to the pinwheels. Note that opposite Figure 3: The crystalline structure of pinwheels and orientations in V1 (from [START_REF] Petitot | Elements of Neurogeometry[END_REF]). Magnification shows a typical hypercolumn. Orientations are represented by colors. Iso-orientation lines (rays) are shown.

rays correspond to orientations which differ by angle π/2.

• Long range connections between distant hypercolumns are selective in the following way. A neuron selective to one orientation θ in the hypercolumn projects fibers preferentially to neurons which are selective to the same orientation. Moreover these connections tend to be aligned along a direction which, in a suitable frame centered on the pinwheel, is defined by the angle θ. This property is called coaxiality.

The lattice of pinwheels in V1 shows a remarkable regularity in first approximation [START_REF] Schnabel | Random waves in the brain: symmetries and defect generation in the visual cortex[END_REF]. If V1 is identified to a plane (in the "horizontal" directions), it can be well approximated by a periodic or a quasi-periodic lattice [START_REF] Bressloff | Dynamical Mechanism for Sharp Orientation Tuning in an Integrate-and-Fire Model of a Cortical Hypercolumn[END_REF]. It is however a common procedure to replace this discrete structure by a continuous one, so that the "horizontal" extension of V1, which is homeomorphic to the visual field by retinotopy, is idealized by R 2 . Above each point in this plane the hypercolumn of orientation is modeled by the projective line P 1 or equivalently, by the circle S 1 (mod π). The relevance of this idealization is discussed in [START_REF] Petitot | Elements of Neurogeometry[END_REF]. Therefore V1 is now a fiber bundle R 2 × S 1 (S 1 is more convenient than P 1 for our purpose). It is a remarkable fact that the action of the group of rigid displacements

SE(2) := R 2 SO(2) given by (r, θ) → (R ϕ r + a, θ + ϕ), (a, ϕ) ∈ SE(2) (1)
defines a connection (in geometrical sense) in R 2 × S 1 , which is well-adapted to the detection of contours of images and which moreover fits well with the neural connectivity in V1 that has been described above! There is a rich geometric structure associated to this, which has been exploited to analyze problems of visual perception such as contour continuation, illusory contours, visual hallucinations, detection of shapes [START_REF] Citti | A cortical based model of perceptual completion in the rototranslation space[END_REF][START_REF] Sarti | The symplectic structure of the primary visual cortex[END_REF][START_REF] Sarti | Functional geometry of the horizontal connectivity in the primary visual cortex[END_REF][START_REF] Citti | Neurogeometry of perception: Isotropic and Anisotropic Aspects Axiomathes[END_REF][START_REF] Bressloff | Geometric visual hallucinations, euclidean symmetry and the functional architecture of striate cortex[END_REF][START_REF] Sarti | The constitution of visual perceptual units in the functional architecture of V1[END_REF].

3 Bifurcation of neural fields in V1: the tuning of orientations

The neural field equations

In a celebrated paper [START_REF] Wilson | Excitatory and inhibitory interactions in localized populations of model neurons[END_REF], H.R. Wilson and J.D. Cowan stipulated more than 45 years ago that the cortex is fractionned into spatially localized zones into which neurons are densely connected to each other and dedicated to perform the same "task". On the basis of a coarse-graining argument they replace each of these populations by a couple of differential equations, one for the mean firing rate v E (t) of excitatory neurons and the other for the mean firing rate v I (t) of inhibitory neurons. At a larger scale such local populations can be interconnected, so that the activity is represented by two scalar fields, one for the excitatory populations of neurons and one for the inhibitory populations. In the case of V1 the spatially extended domain is identified with the fiber bundle R 2 ×Ξ where Ξ is a Riemannian space where each point represents a set of features of the image, for example an angle θ (mod π) if one is exclusively interested in the orientation. The system can be further reduced to a single equation by assuming short-range excitation and longer range inhibition, which seems consistant with observations. Then the system of Wilson and Cowan reduces to a single integro-differential equation

∂v ∂t (r, ξ, t) = -αv(r, ξ, t) + R 2 Ξ
w(r, ξ; r , ξ )S µ (v(r , ξ , t))dξ dr + I ext [START_REF] Bartfeld | Relationships between orientation-preference pinwheels, cytochrome oxidase blobs, and ocular-dominance columns in primate striate cortex[END_REF] where the connectivity function w(r, ξ; r , ξ ) models the synaptic strength (weight) from "neuron" (r , ξ ) to "neuron" (r, ξ), so that w(r, ξ; r , ξ ) takes a maximal and positive value when (r, ξ) = (r , ξ ), negative values when the two points are further apart, and is integrable at each point (r, ξ). An important and natural assumption is that the form of w is consistent with the geometrical organization of the connectivity. This claim will be made more precise in Section 3.3. The value α > 0 is the time constant of the units (localized population) whose activity decays in the absence of stimuli. It can be set to 1 by rescaling time, so that we will take α = 1 in all the following. The function S µ (x) is a sigmoid with limiting values 0 and 1. Roughly speaking it represents the fraction of neurons that are excited with activity x. A usual choice is S µ (x) = (1 + e -µ(x-κ) ) -1 where µ ≥ 0 is the "gain" of the transfer function S µ and κ is a threshold value. Finally the term I ext is the resultant of external inputs to the system, which may be a function of r and ξ (but we always assume that it is time-independant). Despite its highly simplified form this equation has proven to be efficient in modeling the evolution of the coarse grained activity in V1, as we will see in section 3.3. Before this a short introduction to bifurcation theory is provided without proofs or technical details.

A short overview of bifurcation theory

Let us rewrite Eq. ( 2) for neural fields in a domain D which is not specified for the moment but which is assumed to be a Riemannian space with volume element dx:

∂v ∂t (x, t) = -v(x, t) + D w(x; x )S µ (v(x , t))dx + I ext (3)
In this section we set I ext = 0 (for simplicity). We assume that the connectivity kernel w is an integrable function in D and that it is invariant under a group of isometric transformations G: if g ∈ G then w(gx, gx ) = w(x, x ). For example if D = R 2 × S1 as in Seciton 2, then G SE(2) with the action defined by (1) (or E(2) if reflections are included in the symmetries). Equation ( 3) is well-defined and is Lipschitz continuous in the space L ∞ (D) of bounded functions in D, which is a Banach space with the sup norm. Then the Cauchy problem is well-posed [START_REF] Faye | Analysis of a hyperbolic geometric model for visual texture perception[END_REF] (other functional settings with more regularity can be proposed, see e.g. [START_REF] Potthast | Existence and properties of solutions for neural field equations[END_REF] when

D = R n ).
The invariance property of the kernel w implies that Eq. ( 3) is invariant under isometric transformations: for a function f let T g f (x) = f (g -1 x), then g → T g defines a linear representation of the group G in the space L ∞ (D). Applying T g to (3) does not change this equation, which is said G-invariant.

Let's now assume there exists a smooth family v µ of basic states of (3): at each value of µ, v µ is a stationnary and homogeneous solution of (3). Homogeneous means that v µ is invariant under the action of G: T g v µ = v µ . In the cases we will consider here, this is equivalent to say that v µ does not depend on x. The basic state corresponds to the state of rest of the neural netwok. It is sometimes set to v µ = 0 by setting S µ (0) = 0, possibly after a suitable translation [START_REF] Bressloff | Geometric visual hallucinations, euclidean symmetry and the functional architecture of striate cortex[END_REF]. It is asymptotically stable when the parameter µ is lower than a critical value µ 0 [START_REF] Faye | Analysis of a hyperbolic geometric model for visual texture perception[END_REF]. Then the question is, what happens when µ exceeds µ 0 . The couple (µ 0 , v µ 0 ) is a bifurcation point.

The characterization of asymptotic stability of v µ is that the linearized operator at v µ

(L µ u)(x) = -u(x) + D w(x; x )S µ (v µ (x ))u(x , t)dx (4)
has its spectrum Σ(L µ ) included in the half-plane {λ ∈ C ; Re(λ) ≤ ξ < 0}.

In order to apply classical bifurcation theory we now require that at µ = µ 0 , the set Σ c = Σ(L µ 0 ) ∩ iR consists of a finite number of isolated eigenvalues of finite multiplicity. When the isometry group G is non compact these conditions are in general not satisfied and the analysis should be restricted to some subclass of functions in D in which G operates as a compact group [START_REF] Chossat | Methods in Equivariant Bifurcations and Dynamical Systems[END_REF]. Σ c is the center part of the spectrum. The corresponding solutions of the linearized equation at

v µ 0 ∂u ∂t (x, t) = -u(x, t) + D w(x; x )S µ 0 (v µ 0 (x ))u(x )dx (5)
are called critical or marginal modes. They do not increase nor decay exponentially with time. By the center manifold theorem the dynamics in a neighborhood of the bifurcation point is driven by the dynamics on a finite dimensional submanifold of D, the center manifold, the tangent space of which at v µ 0 is spanned by the critical modes [START_REF] Haragus | Local bifurcations, center manifolds, and normal forms in infinite dimensional systems[END_REF][START_REF] Chossat | Methods in Equivariant Bifurcations and Dynamical Systems[END_REF]. Let V 0 denote the space of these critical modes. The finite dimensional dynamics is then governed by an ODE in V 0 , ẋ = f (x, µ), the asymptotic expansion of which can be computed by solving a sequence of Fredholm alternatives and changes of variables which put the system in its simplest form (called a normal form). Note that, f (0, µ 0 ) = 0 and the Jacobian matrix f x (0, µ 0 ) has spectrum Σ c . This ODE is the bifurcation equation. Its µ-dependent bounded solutions in a neighborhood of the bifurcation point define the bifurcated branches of solutions.

In general, with one single varying parameter one would expect that Σ c consists of simple eigenvalues and either Σ c = {0} (steady-state bifurcation) or Σ c = {±iω 0 } (Hopf bifurcation). However symmetry can force higher multiplicity of the critical eigenvalues. Indeed the G-invariance of (3) implies that the operator L µ is equivariant: for all g ∈ G, L µ T g = T g L µ (the same holds for the nonlinear part of ( 3)). Therefore if ζ is an eigenvector (possibly a generalized eigenvector) for an eigenvalue in Σ c , the same is true for T g ζ for any g ∈ G: the space V 0 is G-invariant and its dimension is forced by the representation T g restricted to V 0 . In this short overview of bifurcation theory we focuse on steady-state bifurcation. In this case the counterpart for 0 to be a simple eigenvalue of the G-equivariant linear operator L µ 0 is that the representation T g restricted to V 0 is absolutely irreducible: it is not decomposable into lower dimensional representations and moreover the only matrices that commute with T g in V 0 are real scalar multiples of the identity matrix. This property is generic for bifurcation problems with a scalar parameter [START_REF] Golubitsky | Singularities and Groups in Bifurcation Theory[END_REF]. It implies in particular that V 0 ker(L µ 0 ) (0 is a semisimple eigenvalue). Another way to say it is that the critical modes do not depend on time.

The G-invariance property propagates to the center manifold and to the map f (x, µ), which is defined in V 0 . This is a strong constraint on the structure of f and on its bifurcated solutions. One straightforward but tremendous consequence of this invariance is that branches of solutions can be classified by their isotropy type. Let H be an isotropy subgroup of G, that is the largest group that fixes an element x ∈ V 0 : T h x = x, all h ∈ H. Then the set of elements in V 0 which are fixed by H is a linear subspace V H ⊂ V 0 . Moreover if x 0 ∈ V H , then the trajectory x(t) with that initial condition for the bifurcation equation lies entirely in V H . The isotropy type or orbit type of H is the conjugacy class of H in G, which is further identified with the set G/H. If a solution x has isotropy H, then its G-orbit, that is the set of solutions T g x, g ∈ G, is isomorphic to G/H. As an example of application of these algebraic properties one has the Theorem 1. (Equivariant Branching Lemma [START_REF] Golubitsky | Singularities and Groups in Bifurcation Theory[END_REF]) For a generic one-parameter bifurcation problem with G symmetry, there exists a branch of bifurcated solutions with isotropy H for any H such that dim V H = 1.

Indeed the equation ẋ = f (x, µ) restricted to V H is a scalar equation and classical bifurcation theory at a simple eigenvalue applies to it. Moreover f x (0, µ) = σ(µ)Id (Id is the identity matrix in V 0 ) by absolute irreducibility of the representation of G, hence the generic condition (for

µ ∈ R) σ (µ 0 ) = 0 is valid in any axis V H ⊂ V 0 .
Remark. The normalizer of H in G, noted N (H), is the largest subgroup which acts in V H . In the case when dim V H = 1, either N (H)/H {Id} and the branch of solutions can transcritical (two-dided)) or N (H)/H {±Id} and the branch is always a pitchfork: the bifurcated solutions come by pairs of opposite signs in V H , for either µ > µ 0 or µ < µ 0 .

Much more can be said on this theory, especially when time dependent solutions are considered. In any case the most important information to collect for systems which have symmetry is the way in which the symmetry group acts on critical modes. Then the Equivariant Branching Lemma allows in many cases to determine the full bifurcation diagram. Note however that bifurcated branches may also exist which have isotropy group not statisfying dim V H = 1, however being computable by other means (like Morse lemma or degree theory, see [START_REF] Chossat | Methods in Equivariant Bifurcations and Dynamical Systems[END_REF] for a thorough exposition of this case).

The tuning of orientations

The model and bifurcation methods introduced in the previous section have been applied to the tuning of orientation and spontaneous activation of patterns in V1 leading to visual geometric hallucinations [START_REF] Ermentrout | A mathematical theory of visual hallucination patterns[END_REF][START_REF] Bressloff | Geometric visual hallucinations, euclidean symmetry and the functional architecture of striate cortex[END_REF]. In this case D = R 2 × Ξ with Ξ S 1 . In this section I survey these two problems, refering the interested reader to the bibliography for further details and discussion.

The tuning problem adresses the question of the sharpening in the hypercolumns of the signal arriving from the LGN with respect to orientation. There seems to be such a mechanism acting in the local cortical circuits (columns of orientation) and a simple model of this mechanism was proposed by [START_REF] Ben-Yishai | Theory of orientation tuning in visual cortex[END_REF] and [START_REF] Bressloff | Dynamical Mechanism for Sharp Orientation Tuning in an Integrate-and-Fire Model of a Cortical Hypercolumn[END_REF]. Equation (2) should now be considered in its local version, that is within a hypercolumn: the connectivity function w now depends only on the orientation angles (this is called the ring model) and (2) reduces to

∂v ∂t (θ, t) = -v(θ, t) + S 1 w(θ; θ )S(v(θ , t))dθ + I ext (6)
where the angular dependance of v and w is π-periodic. The question is: if the orientation input I ext is broad, does the output v show a sharper distribuiton of orientations? When the spatial extension is neglected the isometry group SE(2) reduces to SO(2) acting in the fiber S 1 , with the constraint that angles 0 and π are identified: any function in (6) must satisfy f (θ + π) = f (θ). Moreover w should only depend on the distance |θ-θ | because there should not be an a priori preferred orientation (in fact the connectivity within the hypercolumns shows a high degree of isotropy). It is usually assumed that the strength of the excitation is dominant when θ is closer to θ while the inhibition becomes dominant when |θ -θ | is larger. A model for this consists in taking w(θ; θ ) = h(θ -θ ) where h is the "Mexican hat" function

h(x) = e -x 2 /2σ 1 -Ae -x 2 /2σ 2 , -π/2 ≤ x ≤ π/2 (7)
and constants are suitably chosen (typically A < 1 and σ 1 < σ 2 , see Fig. 4). In order for a tuning effect to exist it is necessary that in the limit when I ext = 0 solutions to ( 6)-( 7) exist, which have broken θ invariance (they are not isotropic) and moreover are stable under small perturbations. This is a typical bifurcation problem with symmetry group G = O(2) because the fact that w only depends on the distance |θ -θ | implies that Equation ( 6) is invariant with respect to the transformations v(θ) → v(θ + ϕ) and v(θ) → v(-θ).

The first step is to consider an isolated column of orientation: I ext = 0. Then the form of w and S µ implies that homogeneous steady states, solutions of the equation v = wS µ (v) where w = 2π 0 h(s)ds, always exist and are stable when µ is small enough. Such solutions, which we note v µ , are O(2)-invariant and since this group is compact the method described in 3.3 applies. The linearized operator at v µ reads

L µ u = -u + S µ (v µ ) 2π 0 h(θ -θ ))u(θ ))dθ (8)
Expanding h(s) = ĥn e 2ins , ĥn is real because h is an even function. Applying Fourier analysis we obtain the real eigenvalues

λ n (µ) = -1 + S µ (v µ ) ĥn (9)
with associated eigenfunctions cos(2nθ) and sin(2nθ). Therefore the eigenvalues are double except at n = 0 for which it is simple (there is no symmetry-breaking in this case). Note that S µ (v µ ) = µe -µ(vµ-κ) /(1 + e -µ(vµ-κ) ) 2 . Hence when µ is small enough all eigenvalues are negative and there exists a value µ = µ c at which the largest eigenvalue is λ p = 0 for a certain integer p while all other eigenvalues remain negative.

Assuming p > 0, this is an O(2) symmetry-breaking steady-state bifurcation and V 0 is the space of critical modes {x cos(2pθ)+y sin(2pθ); (x, y) ∈ R 2 }. The bifurcation equation is an ODE in (x, y). Instead of computing explicitely this equation and its bifurcated solutions let us apply Theorem 1. Writing z = x + iy, the action of O(2) in V 0 is spanned by rotations R ϕ : z → e iϕ z and reflection κ : z → z.

Clearly H = {Id, κ} is an isotropy subgroup such that V H = R (the horizontal axis), hence the Equivariant branching lemma applies. Moreover rotations by π/2 transform the eigenvectors in their opposite, hence x in -x, which implies a pitchfork bifurcation. Remark that any axis in R 2 is also an invariant axis and its orbit type is G/H S 1 . Therefore bifurcated solutions form circular orbits by the action of SO( 2) and moreover we have obtained all possible bifurcated solutions (if no additional degeneracy occurs in the problem).

The bifurcated branches can be either supercritical (i.e. for µ > µ c ) or subcritical. This depends on a coefficient which can be computed (at least numerically) from the bifurcation equation in V H . The subcritical case gives rise to a hysteresis phenomenon: the branches bend back towards larger µ and the states of orientation tuning jump from the baic state v µ to a well-developed solution "far" from bifurcation, see [START_REF] Bressloff | Dynamical Mechanism for Sharp Orientation Tuning in an Integrate-and-Fire Model of a Cortical Hypercolumn[END_REF] for a discussion. From the orientation tuning point of view the most relevant case is when p = 1, which corresponds to an unambiguously tuned response indicating the presence of a local contour. When the external input is switched on with a specific orientation but a weak signal, the rotational symmetry is broken and it can be shown that the phase with that orientation is selected and the signal is sharpened [START_REF] Bressloff | Dynamical Mechanism for Sharp Orientation Tuning in an Integrate-and-Fire Model of a Cortical Hypercolumn[END_REF], which explains the phenomenon of orientation tuning.

Spontaneous activation of patterns in V1

The extension of the previous analysis to the spontaneous activation of the lattice of interconnected hypercolumns was undertaken by [START_REF] Bressloff | Geometric visual hallucinations, euclidean symmetry and the functional architecture of striate cortex[END_REF]. The following is a short exposition of the method which was used by [START_REF] Bressloff | Geometric visual hallucinations, euclidean symmetry and the functional architecture of striate cortex[END_REF]. Spontaneous activity means "in the absence of external input", so we set I ext = 0 in Eq. ( 2). Now the variables are r ∈ R 2 and θ ∈ S 1 (mod π). Everything lies in the expression of the connectivity function w(r, θ; r , θ ), which we decompose into the sum of a local term w loc (θ, θ ) (the ring model) and a term accounting for the long-range "lateral" connection βw lat (r, θ; r , θ ). The coefficient β fixes the relative strength of the lateral connections. When β = 0 (disconnected hypercolumns), w does not depend on spatial variables and the symmetry group of Eq. ( 2) is E(2) × O(2). It is biologically plausible to assume β 1. It is also natural to assume that w lat is invariant under the isometry group of the plane R 2 but we have to account for the coaxiality property [START_REF] Balazs | Chaos on the pseudosphere[END_REF]. Under these conditions the problem can be seen as a perturbation of the ring model, which has full symmetry E(2) × O(2), by a term which is invariant under coaxial transformations v(r, θ) → v(R ϕ r + a, θ + ϕ). We also expect invariance by the reflection v(r, θ) → v(r, -θ) where r = (r cos θ, -r sin θ). Hence the symmetry group is now G = E(2) acting in R 2 × S 1 via these transformations. A model of function w lat satisfying these constraints was introduced in [START_REF] Bressloff | Geometric visual hallucinations, euclidean symmetry and the functional architecture of striate cortex[END_REF].

The bifurcation analysis proceeds along the same lines as exposed in Section 3.2 with the important difference that now the group G is not compact. If β = 0 the spectral analysis is similar to that of the ring model with the additional property that since in this case the equation does not depend on r any distribution of orientations amongst R 2 (i.e. on V1) is possible. When β > 0 the spectral analysis of the linear operator L µ , which now incoroporates the kernel w lat , proceeds in looking for linear modes of the form

ζ(r, θ) = u(θ -ϕ)e ik•r + c.c. (10)
with k = k(cos ϕ, sin ϕ). Because of translational invariance any wave vector k is allowed. Moreover rotational invariance implies that the functions u do only depend on the wave number k = k and not on the direction ϕ. Therefore not only the spectrum is continuous but the eigenvalues have infinite multiplicity. The computation of the critical modes and the determination of the bifurcation point are not straightforward. They were undertaken in [START_REF] Bressloff | Geometric visual hallucinations, euclidean symmetry and the functional architecture of striate cortex[END_REF] by perturbation analysis with respect to β and I leave the interested reader to consult this reference for the details.

In the perspective of this review what is important is the way to reduce the problem to one which can be handled with the bifurcation method introduced in 3.2. The idea, which goes back to Turing for pattern formation [START_REF]Turing The chemical basis of morphogenesis[END_REF], is to look for solutions which are doubly periodic in R 2 , that is which satisfy the condition f (r + a + b) = f (r) with a and b not colinear. Let L = {ma + nb; m, n ∈ Z} be the lattice subgroup of R 2 spanned by a and b. For [START_REF] Bressloff | Scalar and pseudoscalar bifurcations with application to pattern formation on the visual cortex[END_REF] this comes back to retain only those wave vectors k such that k • a = 2mπ and k • b = 2nπ with m, n ∈ Z. Now doubly periodic functions can be seen as functions which are defined on the torus R 2 /L, so that now the domain of the Wilson-Cowan equation becomes D = R 2 /L × S 1 . The symmetry group acting on this space is G = R 2 /L D p where D p is the dihedral group of order 2p (called the holohedry of the lattice, see [START_REF] Miller | Symmetry Groups and their Applications[END_REF]). Here there are only three possibilities, depending on the angle between a and b: p = 2 (rhombic lattice), p = 4 (square lattice) and p = 6 (hexagonal lattice). These groups are compact and the spectral analysis leads to isolated eigenvalues with finite multiplicities. In each case several absolutely irreducible representations of G exist which lead to different critical spaces V 0 and different sets of isotropy types. The overall picture is rather complicated but the remarkable result is that the bifurcated patterns, once tranformed to retina images through the inverse of the retinotopic map, show a striking similarity with geometric hallucinations drawn by patients under various types of non visual stimulations (psychotropic drugs...). The group theoretical point of view of this problem has led to further developments, see [START_REF] Bressloff | Scalar and pseudoscalar bifurcations with application to pattern formation on the visual cortex[END_REF], [START_REF] Golubitsky | Bifurcation on the Visual Cortex with Weakly Anisotropic Lateral Coupling[END_REF]. [START_REF] Bigun | Optimal orientation detection of linear symmetry[END_REF] The structure tensor model 4.1 A texture detector and its implementation for V1

Despite its success the ring model and its spatial extension described in the previous sections are incomplete in the sense that they account for only one (although essential) feature of images, namely the orientation of contours. In order to enhance the image texture processing specialists of computer vision use a mathematical object named structure tensor, which is a 2 × 2 symmetric, positive definite matrix. Its interest lies in the fact that not only the eigendirection associated with the smallest eigenvalue gives the orientation of the image, but also its two positive eigenvalues characterize the degree of inhomogeneity and contrast of the image at a given scale. The structure tensor is defined as follows. In all the following we note x = (x, y) in canonical coordinates in R 2 and

g σ (x) = 1 2πσ 2 e -x 2 +y 2 2σ 2
the 2D Gaussian function with standard deviation σ. Let I be the image intensity and I σ 1 = g σ 1 * I (convolution product) be the intensity function filtered (smoothed) at the scale σ 1 . Then the structure tensor at x is the matrix

T (x) = g σ 2 * (I σ 1 x ) 2 g σ 2 * I σ 1 x I σ 1 y g σ 2 * I σ 1 x I σ 1 y g σ 2 * (I σ 1 y ) 2 (11)
where subscripts x and y denote the partial derivatives w.r.t. these variables.

The parameter σ 2 relates to the size of the receptive field of neurons and to the characteristic scale at which the texture is seen [START_REF] Chossat | Hyperbolic planforms in relation to visual edges and textures perception[END_REF]. Note that T has the form

T = a c c b , a > 0, ab -c 2 > 0.
T = (λ 1 -λ 2 ) t e 1 e 1 + λ 2 I 2 . ( 12
)
where t e 1 e 1 is the tensor product of e 1 by its transpose. This has the following informative consequences for the image at scale σ 2 :

• The size of λ j defines the variation of the intensity, hence the contrast, along the direction e j ;

• if λ 1 ≈ λ 2 the image is isotropic;

• if λ 1 λ 2 ≈ 0 an edge is detected along the direction e 2 ;

• if λ 1 ≥ λ 2 0 a corner is likely to be present;

• λ 1 -λ 2 measures the degree of anisotropy of the image. The coefficient κ = λ 1 -λ 2 λ 1 +λ 2 is called the coherence of the image.

These properties are synthetized by drawing the locus of points t xT x = 1 (Figure 5). In [START_REF] Chossat | Hyperbolic planforms in relation to visual edges and textures perception[END_REF] it was hypothethized that the structure tensors at one point x of the surface of V1 is encoded in the corresponding hypercolumn. Although there has been no experimental testing of this hypothesis yet, this idea is supported by the strong evidence that quantities related to image derivatives are represented in the neural activity [START_REF] Chalupa | The Visual Neurosciences[END_REF][START_REF] Petitot | Elements of Neurogeometry[END_REF].

The aim now is to take Ξ = SP D(2) in Eq. [START_REF] Wilson | Excitatory and inhibitory interactions in localized populations of model neurons[END_REF] as a generalization of the orientation model. The first idea would be to consider SP D(2) as the cone embedded in R 3 with its Euclidean metric. It would then be difficult to conciliate this choice with the natural assumption that neurons do not know in which coordinate system are expressed the components of the structure tensor. In other words Eq.

(2) should be invariant under coordinate changes in the space of positive definite quadratic forms, in particular the connectivity function should share this property. In mathematical terms SP D( 2) is isomorphic to the homogeneous space GL(2)/O(2) where GL( 2) is the group of invertible real matrices, which acts on SP D(2) by G • T = t GT G. This induces a metric on the tangent space, defined by g T (A, B) = tr(T -1 AT -1 B), and the corresponding distance is

d 0 (T , T ) = log 2 σ 1 + log 2 σ 2 (13)
where σ 1 , σ 2 are the eigenvalues of T -1 T , see [START_REF] Pennec | A Riemannian Framework for Tensor Computing[END_REF][START_REF] Moakher | A differential geometric approach to the geometric mean of symmetric positie-definite matrices[END_REF] for details. Hence GL( 2) is the group of isometries of SP D(2).

If we now restrict the analysis to the activity inside one hypercolumn, hence discarding lateral intercations like in the ring model, then S 1 must be replaced by GL( 2) in ( 6) and the connectivity function must satisfy the invariance relation w(T ; T ) = w(g • T , g • T ) for all g ∈ GL(2). Extending the domain to V1 requires to take account of the anisotropy of lateral connections. Like in the planar case (see Sec. 3.3) w(x, T ; x , T ) = w loc (T , T ) + βw lat (x, T ; x , T ) where β 1 and w lat satisfies the coaxiality requirement plus the fact that lateral fibers should preferentially connect neurons associated to same or similar features. This point has been discussed in [START_REF] Faye | A spatialized model of visual texture perception using the structure tensor formalism Networks and[END_REF] who proposed a general form for this function, with an additional small parameter measuring the degree of anisotropy in w lat .

Remark 1. suppose T has eigenvalues λ 1 λ 2 ≈ 0. Then by [START_REF] Chossat | Hyperbolic planforms in relation to visual edges and textures perception[END_REF] T is wellapproximated by the symmetric semi-definite tensor λ 1 t e 1 e 1 and the set of these tensors is isomorphic to R + * × P 1 where P 1 is the real projective line. This shows how the ring model is embedded in the structure tensor model. This correspondence was discussed in [START_REF] Faye | Analysis of a hyperbolic geometric model for visual texture perception[END_REF]. This also comes back to take the limit σ 2 = 0 in [START_REF] Chalupa | The Visual Neurosciences[END_REF], that is to consider the limit of very small image scale. The isometry group acting on this reduced model is R + * ×O(2). Despite the formal similarity between this formulation and the model which was considered in [START_REF] Sarti | The symplectic structure of the primary visual cortex[END_REF] to take account of the scale, the two are different. Here the R + * component refers to a measure of local contrast rather than scale.

In the rest of this paper we forget the spatial extension of V1 and concentrate on the structure tensor model in a single hypercolumn. The Wilson-Cowan equation

w(T ; T )S µ (v(T , t))dT + I ext ( 14
)
where dT is the volume element for the Riemannian metric g T , which we shall explicit in the next section, and w(T ; T ) depends only on the distance in SP D(2).

Our aim is now to describe the bifurcation of patterns from an homogeneous state of rest, thereby extending the orientation tuning analysis of Sec. 3.3.

The Poincaré disc formulation of Eq. (14)

In [START_REF] Chossat | Hyperbolic planforms in relation to visual edges and textures perception[END_REF] the equation ( 14

d SP D(2) (∆, z; ∆ , z ) = 2 log ∆ ∆ 2 + 2d D (z, z ) 2 (15)
where

d D (z, z ) = 2atanh|(z -z )(1 -zz ) -2 | is the distance in D. The volume element is dT = 2 √ 2 d∆ ∆ dm(z ) ( 16
)
where dm(z) = 4(1 -|z| 2 ) -1 dz ∧ dz is the surface element in D.

We can now reformulate Eq. ( 14)

∂v ∂t (∆,z,t) = (∆,z,t) + R * + D w(∆,z; ∆ ,z )S µ (v(∆ ,z ,t)) d∆ ∆ dm(z ) + I ext (17)
The coefficient 2 √ 2 in ( 16) has been incorporated in w, which now is a function of d SP D(2) (eq. ( 15)). The existence and unicity of solutions of ( 17) was studied in [START_REF] Faye | Analysis of a hyperbolic geometric model for visual texture perception[END_REF], which gave similar results to the classical analysis of Wilson-Cowan equations in the Euclidean setting. As in Section 3.3 we concentrate on the problem of spontaneous tuning when I ext = 0.

Remark 2. The identification of SP D(2) with R + * × D is mathematically convenient as we shall see in the next section, but it can also be interpreted in the context of image processing by V1. It was noted in [START_REF] Faye | A spatialized model of visual texture perception using the structure tensor formalism Networks and[END_REF] that the modulus of z ∈ D is closely related to the distance from the pinwheel center in the hypercolumn. Indeed it has been measured experimentally that near pinwheel centers (at which all orientations are equally detected) the strength of orientation tuning is low, while the tuning curve response it much sharper (high selectivity) "far" from the pinwheel [START_REF] Nauhaus | Building maps from maps in primary vi-sual cortex[END_REF]. Now recall the coherence κ = λ 1 -λ 2 λ 1 +λ 2 , which measures the selectivity of a structure tensor with eigenvalues λ 1 and λ 2 (Section 4). Expressed in z coordinates for a structure tensor in SSP D(2), it gives κ = 2|z| 1+|z| 2 . Hence z = 0 corresponds to the pinwheel (no orientation) whereas z = |z|e iθ close to the boundary of D corresponds to a point in the hypercolumn with high selectivity in the orientation θ. About the ∆ coordinate, it can be noted that the difference between images seen at a given scale with structure tensors with same component z ∈ D and determinants ∆ and ∆ , relates just to a difference of contrast between the two at scale σ 2 .

Before considering the bifurcation problem for [START_REF] Waves | bumps, and patterns in neural field theories[END_REF] we need to recall some important facts about the geometry of D and in particular its isometries.

Isometries of the Poincaré disc

The direct isometries (preserving the orientation) in D are the elements of the special unitary group, noted SU(1, 1), which are the matrices

α β β α such that |α| 2 -|β| 2 = 1.
These elements act in D by [START_REF] Valois | Spatial Vision[END_REF]. The corresponding isometry in the space of structure tensors is given by

γ • z = αz + β βz + α , z ∈ D ( 18
) Let α = α 1 + iα 2 , β = β 1 + iβ 2 in
γ • T = t γ T γ, ( 19
)
where

γ = α 1 + β 1 α 2 + β 2 β 2 -α 2 α 1 -β 1 ∈ SL(2, R). (20)
Because isometries are conformal maps, they preserve angles. However they do not transform straight lines into straight lines. Given two points z = z in D, there is a unique geodesic passing through them, which is the arc of circle containing z and z and intersecting the unit circle at right angles. This circle degenerates to a straight line when the two points lie on the same diameter. Any geodesic uniquely defines a reflection through it. Reflections are orientation reversing, one representative is the complex conjugation κ (reflection through the geodesic R):

κ • z = z.
The full symmetry group of the Poincaré disc is therefore

U(1, 1) = SU(1, 1) ∪ κ • SU(1, 1) (21)
A fundamental property of SU (1, 1) is that any element is the product of three "simple" transformations belonging respectively to three one-parameter subgroups K, A and N , the definition of which is given below. Hence one can write SU (1, 1) = KAN (Iwasawa decomposition, see [START_REF] Iwaniec | Spectral methods of automorphic forms[END_REF]). The corresponding subgroups of SSP D(2) can be computed using [START_REF] Faye | Bifurcation diagrams and heteroclinic networks of octagonal h-planforms[END_REF].

Definition 1.                        K = {r ϕ = e iϕ/2 0 0 e -iϕ/2 , ϕ ∈ S 1 } A = {a τ = cosh τ 2 sinh τ 2 sinh τ 2 cosh τ 2 , τ 2 ∈ R} N = {n s = 1 + is -is is 1 -is , s ∈ R}
We describe the action of these three groups, see Fig. 7 for an illustration.

• K is isomorphic to SO(2): r ϕ • z = e iϕ z for z ∈ D. Its orbits are concentric circles. In SSP D(2) it acts by by rotating with an angle ϕ/2 the orthonormal basis (e 1 , e 2 ) in which the coordinates of T are expressed.

• The elements of A are sometimes called "boosts" in the theoretical Physics literature [START_REF] Balazs | Chaos on the pseudosphere[END_REF]. All orbits converge to the limits points b ±1 = ±1 on the unit circle (boundary of the Poincaré disc). In particular the diameter b 1 b -1 is an orbit. The action of a τ ∈ A on SSP D(2) consists in scaling the first vector of the orthonormal basis (e 1 , e 2 ) by e τ and the second one by e -τ .

• The orbits of N are the circles tangent to the unit circle at the point b 1 . These orbits are called horocycles. There is no obvious interpretation of these transformations in the space of structure tensors. Any non trivial direct isometry falls into one of the following classes (representatives of which being given by elements of K, A and N respectively):

(i) Elliptic elements: one fixed point in D;

(ii) Hyperbolic elements: two (and only two) fixed points on the unit circle;

(iii) Parabolic elements: one and only one fixed point on the unit circle.

Tuning of textures in D

The aim of this section is to provide the tools which are necessary to study the bifurcation of solutions of 17 when I ext = 0 and to describe (without proofs) the main results which were obtained in [START_REF] Chossat | Hyperbolic planforms in relation to visual edges and textures perception[END_REF][START_REF] Chossat | Bifurcations of hyperbolic planforms[END_REF]. Compared to previous models in which the domain Ξ was Euclidean or spherical, the novelty here is that Ξ = R + * ×D and D is equipped with hyperbolic geometry. Fourier analysis is more involved than in R 2 or S 2 and is shortly described in Section 5.1. Since the R + * part of the domain does not pose a new challenge we shall for the sake of simplicity consider solutions which are independent of the variable ∆. Then Equation 17 reduces to

∂v ∂t (z, t) = -v(z, t) + D w(z; z )S µ (v(z , t))dm(z ) (22)
and w(z; z ) is a function of distance d D (z; z ) only: w(z, z ) = h(d D (z, z )) where h is the function defined in [START_REF] Bressloff | Dynamical Mechanism for Sharp Orientation Tuning in an Integrate-and-Fire Model of a Cortical Hypercolumn[END_REF]. The integral part of this equation can be written in a convolution form as follows.

Let O be the center of D and dg represent the Harr measure on the group G = SU (1, 1) (see [START_REF] Helgason | Groups and Geometric Analysis[END_REF]), normalized by

G f (g • O)dg = D f (z)dm(z)
for all functions in L 1 (D).

Definition 2. Given f 1 , f 2 in L 1 (D) we define their convolution by

(f 1 * f 2 )(z) = G f 1 (g • O)f 2 (g -1 • z)dg.
Proposition 2. Let W (z) = w(z, O), then for all bounded function v in D the following equality holds

D w(z; z )S µ (v(z , t))dm(z ) = W * S µ (v)(z).
Proof. This is a consequence of the G-invariance of w: w(g • z, g • z ) = w(z, z ). Writing z = g • O and integrating over G instead of D, the result follows easily.

Spectral and Fourier analysis in D

Classical Fourier analysis doesn't apply in D. Nevertheless a similar theory exists, however with notable differences. In the following only some basic notions are exposed and I refer to [START_REF] Helgason | Groups and Geometric Analysis[END_REF] for a thorough exposition and to [START_REF] Balazs | Chaos on the pseudosphere[END_REF] for a theoretical physics point of view. Given a point z ∈ D there is a unique couple of hyperbolic and parabolic orbits which intersect at z. Hence we may write

z = n s a τ • O, (s, τ ) ∈ R 2
∆ D = (1 -x 2 -y 2 ) 2 4 (∂ 2 x + ∂ 2 y ).
By construction this operator is invariant under the transformation group U (1, 1).

Proposition 3. The functions

e ρ,b (z) = e (iρ+ 1 2 ) z,b , ρ ∈ C, (23)
are eigenfunctions of ∆ D with eigenvalues -(ρ 2 + 1 4 ).

Real eigenvalues of ∆ D correspond to taking ρ real or purely imaginary in 23. The latter case is irrelevant for us as it corresponds to exponentially diverging eigenfunctions when |z| → 1. Therefore the real spectrum of -∆ D is continuous and bounded from below by 1/4. The corresponding elementary eigenfunctions are the analogue of the Euclidean plane waves. When b = b 1 they are expressed as e ρ,b (z) = e (iρ+1/2)τ with τ being the spatial coordinate of the propagation. These waves propagate along the geodesics emanating from b 1 , which cross orthogonally the horocycles. The point b 1 is the "source", which makes a big difference with Euclidean case where there is no prefered direction (reflectional symmetry). Because of the factor e τ /2 these plane waves are not periodic. However this term balances the growth of the surface element in the energy density, which in horocyclic coordinates is |e ρ,b (s, τ )| 2 e -τ dsdτ .

It follows from Helgason's theory that any eigenfunction of ∆ D can be expressed as an integral over a distribution of the elements b ∈ ∂D: where T ρ is a distribution defined on the unit circle ∂D. Definition 3. Given a function f on D, its Helgason-Fourier transform is defined by

f (ρ, b) = D f (z)e -ρ,b (z)dm(z) (24)
for all ρ ∈ C, b ∈ ∂D for which the integral exists.

Theorem 3 (inverse Helgason-Fourier transform). For f a differentiable function with compact support in D, the following formula hold

f (z) = 1 2π R ∂D f (ρ, b))e ρ,b (z)ρ tanh(πρ)dρdb (25)
where db is normalized by ∂D db = 1.

The following proposition is useful when dealing with integral equations in D. There is another way to express the eigenfunctions of the Laplace-Beltrami operator, by choosing, instead of the horocyclic coordinates, the geodesic coordinates (τ, θ) such that z = tanh(τ /2)e iθ , 0 ≤ τ < +∞, θ ∈ S 1 . This will be useful when looking for states in a bounded circular subdomain of D in Section 5.2.3. An arbitrary eigenfunction can be expanded as

Ψ l (z) = +∞ -∞ a m P m l (cosh(τ ))e imθ (26)
where P m l are associated Legendre polynomials, a m ∈ C and l = -

1 2 ± iρ [1].

Bifurcation of patterns in D

As in the Euclidean case (Sec. 3.4), the hyperbolic domain D is not compact and the spectrum of ∆ D is continuous. Moreover the eigenvalues have in general infinite multiplicity because they are independant of the direction b ∈ ∂D. In order to compute bifurcated branches of solutions of ( 22) one would like to apply the same idea as in 3.4, that is to look for periodic patterns. Periodic patterns in D are functions invariant under the action of a discrete subgroup Γ of transformations in G = U (1, 1). Thanks to the G-invariance of ( 22), one can project this equation on the quotient domain D/Γ. If this domain happens to be compact, which is not always the case, the restricted spectrum will be discrete with eigenvalues of finite multiplicity, so that the method of 3.2 applies. In Section 5.2.1 a simple case of bifurcation of periodic patterns is presented, which is the hyperbolic counter part of the bifurcation of stripes in Turing-like instability. These periodic waves are traveling at constant speed while in the Euclidean case they are stationnary. Unfortunately they do not appear to be physically relevant because their energy density growth to infinity towards the boundary of D.

The case of "true" periodic patterns was discussed in [START_REF] Chossat | Hyperbolic planforms in relation to visual edges and textures perception[END_REF][START_REF] Chossat | Bifurcations of hyperbolic planforms[END_REF][START_REF] Faye | Bifurcation diagrams and heteroclinic networks of octagonal h-planforms[END_REF], with detailed computations when the fundamental domain of the pattern is a regular octagon. This is presented in Section 5.2.2. Finally I also present briefly in Section 5.2.3 the simpler case of a compact subdomain inside D with prescribed boundary conditions, which gets rid of the high degeneracy induced by the SU (1, 1) invariance of the problem. It can indeed be argued that since natural images can only produce a bounded set of structure tensors, it is relevant to take for the domain a disc in D, invariant by the action of the rotation group K SO(2) in order to incorporate the ring model in this hyperbolic model. From the group theoretic point of view the analysis is then similar to the ring model and the bifurcated branches of solutions can be computed analytically. Compared to the ring model the additional information lies in the "radial" modes of the critical eigenvalue problem. Let's mention for completeness the bifurcation of localized solutions (bumps) in [START_REF] Faye | Localized states in an unbounded neural field equation with smooth firing rate function : a multi-parameter analysis[END_REF]. Localized solutions decay to 0 as z → ∞. In the case of a planar domain such states are known to exist and play an important role in cortex modeling, see [START_REF] Waves | bumps, and patterns in neural field theories[END_REF] and references therein. No general method exists to compute these solutions and further restrictions, both on the equation and on the type of solutions, are needed. Already in the Euclidean case the technical difficulties are quite involved. Roughly speaking the method consists in considering the radial variable as a time variable and treating the equation as a dynamical system, which requires to choose the connectivity function such that it is the kernel of a differential operator (this idea was already applied by [START_REF] Laing | PDE Methods for Nonlocal Models[END_REF] to numerically compute localized solutions in a planar domain). I refer to [START_REF] Faye | Localized states in an unbounded neural field equation with smooth firing rate function : a multi-parameter analysis[END_REF][START_REF] Faye | Localized radial bumps of a neural field equation on the Euclidean plane and the Poincaré disk[END_REF] for further details.

Periodic hyperbolic waves

The elementary waves e ρ,b (z) = e (iρ+1/2) z,b are not periodic except when ρ = α + i/2, α ∈ R. Indeed in horocyclic coordinates, if we take b = b 1 (see Fig. 7), e ρ,b 1 (n s a τ •O) = e iατ . Fig. 8 illustrates how these waves look like. Any point on the unit circle is a source of similar waves rotated by the corresponding angle. Suppose we look for solutions of [START_REF] Faye | Analysis of a hyperbolic geometric model for visual texture perception[END_REF] which have the same properties as these periodic waves: (i) invariance under the action of the horocyclic group N , (ii) invariance by a discrete subgroup of the one-parameter group A: v((τ

+ k) • O) = v(τ • O), all τ ∈ R, k ∈ R
fixed (the "period"). By periodicity the action of A in the space of such functions projects onto the 1-torus R/kZ S 1 , which is compact. By condition (i) Eq. ( 22) becomes

∂ t v = -v(t, τ ) + R R w(n s a τ • O, n s a τ • O)S(v(t, a τ )ds e -τ dτ = -v(t, τ ) + R R w(a τ -τ • O, n s • O)ds S(v(t, τ ))dτ
where for the second equality we have used the formula a τ n s = n se τ a τ (see [START_REF] Helgason | Groups and Geometric Analysis[END_REF]).

The integral in parenthesis is a function χ(τ -τ ), so that we may rewrite the equation with the usual convolution product where now v = v(t, τ ), τ ∈ R and the equation is invariant by space-time translations (t, τ ) → (t -t 0 , τ -τ 0 ). We now look for the bifurcation of periodic hyperbolic waves from a basic state using the method of 3.2. By definition basic state solutions v(µ) of ( 27 ±iω 0 (in the example µ c ≈ 0.65 and ω 0 ≈ 0.04). This is Hopf bifurcation, however it reduces to steady-state bifurcation thanks to space and time translational invariance. Indeed let us look for solutions in the form v(t, τ ) = ṽ(t, ηt + τ ), η being an unknown speed of propagation of waves. In the new variable s = ηt -τ , Eq. ( 27) reads ∂ t ṽ = -η∂ s ṽ -ṽ + χ * S µ (ṽ)

∂ t v = -v + χ * S(v) (27)
The above linear analysis shows that bifurcation occurs for this equation with a 0 double eigenvalue at µ 0 , η = η c = ω 0 /α c and critical modes ζ = e iαcs = e i(αcτ +ω 0 t) and ζ. Moreover, since we are now looking for solutions with 2π/α c periodicity in s, the symmetry group is R/α c Z SO(2), which is compact. Therefore the center manifold reduction theorem applies. This is now a steady-state bifurcation with a double 0 eigenvalue as a consequence of the SO(2) invariance of ṽ. Hence dim V 0 = 2, however there is an additional variable η. Let us write points in V 0 as r[e iϕ ζ + e -iϕ ζ], then the real and imaginary parts of the bifurcation equation for stationnary solutions read (as a consequence of the SO(2) invariance, see [START_REF] Chossat | Methods in Equivariant Bifurcations and Dynamical Systems[END_REF]):

0 = rg(µ, r 2 ) 0 = -ηα c + ω 0 + h(µ, r 2 )
where g and h are real functions of order at least one in |µ| + r 2 [START_REF] Chossat | Methods in Equivariant Bifurcations and Dynamical Systems[END_REF]. The pase ϕ is arbitrary. The second equation solves for η = η c + o(1) and the first one gives pitchfork branches of solutions r = 0 (basic state) and r = ±[ √ cµ + O(µ)]

(if c = 0 one must go to higher order). Coming back to [START_REF] Golubitsky | Singularities and Groups in Bifurcation Theory[END_REF] the solutions have the form v = v µ (ηt + τ ). The time dependance corresponds to a uniform speed of translation along the "space" variable τ (traveling waves), and this speed tends to η c as µ tends to 0. It is interesting to note that these solutions are the hyperbolic version of the "rolls" or "stripes" solutions which are well-known in physics, like convection rolls in fluids, stripes in reaction-diffusion systems. There is however a notable difference: in our case the solutions depend on time and are traveling at constant speed from a "source" situated on the boundary of the Poincaré disc. The occurence of stationnary waves is non generic. This is because in hyperbolic geometry these waves have a prefered direction given by the "source" on ∂D.

Periodic patterns

The bifurcation the periodic waves in the previous section has the advantage of simplicity, however their density of energy is not bounded in D as it grows like e -τ dsdτ , which doesn't look physically meaningful. This drawback would disappear with solutions which are periodic in D, that is which are invariant under the action of a discrete subgroup of SU (1, 1) with a compact fundamental domain (or tile). Subgroups sharing these conditions are called cocompact Fuchsian groups.

Their study was initially developed by Poincaré (who gave the name "Fuchsian group"), see [START_REF] Katok | Fuchsian Groups[END_REF]. We quickly review some important general features of these groups, before passing to a specific example. Let Γ be a cocompact Fuchsian group. Its fundamental domain is a compact set

F Γ ⊂ D such that (i) if γ = id ∈ Γ, then γF Γ ∩ Fγ ; (ii) γ∈Γ γF Γ = D.
The fundamental domain of a cocompact Fuchsian group can always be built as a (hyperbolic) polygon, called Dirichlet region [START_REF] Katok | Fuchsian Groups[END_REF].

A 22), suppose that we look for solutions which are invariant under a lattice group Γ. Since we assume that the connectivity function depends only on the distance d D (z, z ), the equation is invariant under the action of SU (1, 1), and therefore we can reduce the analysis to the compact quotient domain D/Γ:

∂v ∂t (z, t) = (z, t) + D/Γ w(z; z )dm(z ) (29)
It follows that the spectrum of the linearized operator is discrete, the eigenvalues have finite multiplicities and ( 29) is invariant under the action of G Γ . Therefore one can apply equivariant bifurcation theory to compute the solution branches and classify them according to their isotropy, as exposed in 3.2. The leading part of the bifurcated solutions are eigenfunctions of the Laplace-Beltrami operator which satisfy the periodicity and isotropy conditions. They were named H-planforms in [START_REF] Chossat | Hyperbolic planforms in relation to visual edges and textures perception[END_REF]. Unfortunately there are several drawbacks to this program. The first one is that the eigenvalues and eigenfunctions are difficult to compute, as there exist no explicit formula to express them. More precisely, for Γ-periodic eigenfunctions the distribution T ρ in Theorem 2 does not admit any explicit expression. Efficient numerical algorithms are necessary to compute these eigenvalues and the corresponding H-planforms. The second drawback is that unlike the Euclidean case the size of the fundamental domain is fixed by its type, which implies that the eigenvalues depend on F Γ , hence on Γ. Whether a bifurcation of Γ-periodic states would correspond to a destabilization of the basic state under random perturbations, or whether another bifurcation with another lattice group would have already occured, is not known. Nevertheless a thorough analysis was performed by [START_REF] Chossat | Bifurcations of hyperbolic planforms[END_REF][START_REF] Faye | Bifurcation diagrams and heteroclinic networks of octagonal h-planforms[END_REF] in the case when the fundamental domain is the regular octagon O shown in Figure 10. To end this section I will quickly review this study, which is also interesting for its own. The lattice group of O is generated by four hyperbolic translations

g 0 = 1 + √ 2 2 + 2 √ 2 2 + 2 √ 2 1 + √ 2 (30)
and g j = r jπ/4 g 0 r -jπ/4 , j = 1, 2, 3, where r ϕ denotes the rotation of angle ϕ around the origin in D. The opposite sides of the octagon are identified by periodicity, so that the corresponding quotient surface D/Γ is isomorphic to a "double doughnut" (genus g = 2), which is the simplest case of periodic tiling [START_REF] Balazs | Chaos on the pseudosphere[END_REF]. The full symmetry group of this fundamental domain, including reflections, contains 96 elements and is noted G * . It can be generated as follows. Let T be a hyperbolic triangle with angles π/8, π/3 and π/2 at the vertices A, B and C respectively. Let Λ be the group generated by reflections through the edges of T . It can be shown that G * Γ/Λ. This means that O is tesselated by 96 copies of T as illustrated in Figure 11. Furthermore G * G ∪ κG where G is the group of direct isometries generated repectively by the rotation by π/2 around the vertex B and π/3 around the vertex C of T . The element κ is the reflection through the real axis in D. It can be shown that G is isomorphic to the group GL(2, 3) of invertible 2 × 2 matrices over the field Z/3Z. Once the group G * is known its absolutely irreducible representations can be determined, which was done in [START_REF] Chossat | Bifurcations of hyperbolic planforms[END_REF] with the help of the computer algebra software programm GAP. A total of 27 different types of solutions have been identified with the equivariant branching lemma and the corresponding H-planforms have been computed numerically. Table 1 gives a synthetic view of these results; Table 1: The number of absolutely irreducible representations and the corresponding number of isotropy types of bifurcated solutions.

Figure 12 shows four examples of such patterns, which correspond to the cases in which dim V 0 = 1 (the 4 absolutely irreducible representations of dimension 1). The physical relevance of these patterns is an open question. Note that identifying the isotropies of an H-planform from its picture is an interesting exercise... For those in upper right and lower left pictures, the answer is in [START_REF] Chossat | Bifurcations of hyperbolic planforms[END_REF]. To end this section, Fig. 13 illustrates the tiling of D with the H-planform on the lower left corner of Fig. 12.

In the 4-dimensional representation spaces, solutions with submaximal isotropy (that is, with dim V H = 2) do also exist, and in one case time dependent solutions (related to the presence of heteroclinic cycles) can also bifurcate [START_REF] Faye | Bifurcation diagrams and heteroclinic networks of octagonal h-planforms[END_REF].

A simpler approach: restricting to a bounded domain in D

This approach originates from the need to incorporate the structure tensor model in the spatially extended V1 in a tractable way [START_REF] Faye | A spatialized model of visual texture perception using the structure tensor formalism Networks and[END_REF]. A more direct method, starting from the octagonal-periodic patterns presented in the previous section, was also investigated, but this leads to substantially more complicated computations. It sounds physically relevant to assume that the structure tensors which are detected by neurons in hypercolumns are bounded and therefore belong to some compact domain in D. Since orientation is part of the features represented by the structure tensor it is also relevant to assume that the domain is a disc. Let therefore Ω ⊂ D be the disc of radius tanh(ω/2) < 1. We use geodesic coordinates (τ, θ) for points in Ω: z = tanh(τ /2)e imθ (see the last paragraph of Section 5.1). Note that, of all the isometries of D only rotations r ϕ around the centre of the disc and reflections through its diameters remain. Hence when D is replaced by Ω the symmetry group of Eq. ( 22) reduces to O(2), which is the same symmetry group as in the ring model of Section 6. We also assume that the structure tensors vanish on the boundary of Ω. where µ = S (0). Since Ω is compact we expect the spectrum consists in isolated eigenvalues with finite multiplicity. Then, due to the invariance of Ω by rotations and reflections, the bifurcation problem enters into the classical framework of bifurcation with O(2) symmetry.

In order to compute the bifurcation point and corresponding eigenvectors, we need to determine a suitable basis of square integrable eigenfunctions of the Laplace-Beltrami operator in Ω with Dirichlet boundary conditions. By formula (26), we look for eigenfunctions of the form e imθ P m l (cosh(τ )) with l = -1/2 + iρ (ρ real) and satisfying the bouundary condition P m l (cosh(ω)) = 0. Here P m l are associated Legendre functions. It was shown in [START_REF] Faye | A spatialized model of visual texture perception using the structure tensor formalism Networks and[END_REF] that the boundary conditions are satisfied for a discrete set of values l m,n = -1/2 + ρ m,n of l such that n ∈ N * , 0 < l m,n < l m,n+1 and lim n→+∞ l m,n = +∞. The corresponding eigenvalues of the Laplace-Beltrami operator are λ m,n = -l m,n (l m,n + 1). Moreover the functions P m lm,n are orthogonal for the measure sinh(τ )dτ . We write Y m n (τ ) the normalized eigenfunctions. In analogy with [START_REF] Bressloff | A spherical model for orientation and spatial frequency tuning in a cortical hypercolumn[END_REF] for the spherical model, we may expend the connectivity function as with corresponding eigenvectors Y m n (τ ) cos(mθ) and Y m n (τ ) sin(mθ). The basic state becomes unstable at the critical value µ c = w(m 0 , n 0 ) -1 where w(m 0 , n 0 ) = max m,n { w(m, n)}. The relevant cases from the biological point of view are (m 0 , n 0 ) = (0, 1) and (1, 1). In the first case the resulting bifurcated state behaves like Y 0 1 (τ ) (radial solution, hence no orientation preference) while in the second case it behaves like Y 1 1 (τ ) cos(θ -ϕ) where ϕ is an arbitrary phase (unimodal orientation). See Figure 14 for a representation of these two states. The bifurcation analysis and the integration into the "spatialized" model including the long range interactions are provided in [START_REF] Faye | A spatialized model of visual texture perception using the structure tensor formalism Networks and[END_REF]. It was concluded there that for the spatially extended problem no new solution was obtained with regard to the "classical" model of [START_REF] Bressloff | Geometric visual hallucinations, euclidean symmetry and the functional architecture of striate cortex[END_REF]. This is not a surprise since the symmetries are the same. Nevertheless the structure tensor model in Ω contains more information due to the additional, radial variable τ which features selectivity of the signal.

Conclusion

I conclude this survey by some remarks about the structure tensor model. First, although some arguments can be found to support the biological relevance of this model, no experimental protocol has yet been designed to test the hypothesis. Next, one is still far from understanding in a satisfactory manner the bifurcation of solutions in the hyperbolic plane or Poincaré disc. In particular there is no evidence that the solutions which have been computed so far are physically or physiologically observable. These difficulties were the reason why the model has not been further studied. Nevertheless the methods which have been developed, not only are interesting for their own, but could also be applied to other contexts of neurogeometry or physics. This was the main motivation for writing this review paper. A final remark is that approximating the Poincaré disc with a disc Ω of finite size as in Section 5.2.3, simplifies considerably the bifurcation and stability analysis. The spatialized problem R 2 ×Ξ was tackled in [START_REF] Faye | A spatialized model of visual texture perception using the structure tensor formalism Networks and[END_REF] in both cases: 1) with Ξ = Ω, 2) with Ξ = D and periodic octagonal tiling. Case 1 allows to extend the framework of spontaneous hallucinatory patterns as studied by [START_REF] Bressloff | Geometric visual hallucinations, euclidean symmetry and the functional architecture of striate cortex[END_REF] but it does not predict new hallucinatory patterns, on the contrary to case 2 in which new patterns have been found. However this work is quite technical and goes beyond the scope of the present survey.

Figure 1 :

 1 Figure 1: The retino-cortical map: left: the retina, right: the primary visual cortex. Colored regions on the left are projected to same color regions on the right (taken from [46], colored version).

Figure 2 :

 2 Figure2: The typical receptive profile of a simple neuron in V1 (from[START_REF] Petitot | Elements of Neurogeometry[END_REF]). Green color corresponds to positive signal and red color to negative signal.

Figure 4 :

 4 Figure 4: Example of a "Mexican hat" connectivity function.

Figure 5 : 2 .

 52 Figure 5: The ellipse xT t x = 1. Semi-axes a = 1/ √ λ 1 , b = 1/ √ λ 2 .

Figure 6 :

 6 Figure 6: The hyperboloid SSP D(2) and projection on the disc D.

Proposition 1 .

 1 ) was reformulated in a particularly convenient form. First remark that any matrix in SP D(2) can be written as T = ∆ T where ∆ = √ det T > 0 and therefore det T = 1. The subspace SSP D(2) = {T / det T = 1} is a hyperboloid which we further identify with the disc D = {z ∈ C / |z| < 1} by the stereographic projection z = a -b + 2ic 2 + a + b (see Fig. 6), where we have noted T = a c c b . The metric induced by SP D(2) on SSP D(2) is the Minkovski metric of the hyperbolic surface of constant curvature +1 and the identification with D is the Poincaré disc of metric g z (dz, dz) = 4(1 -|z| 2 ) -2 dzdz. Then the following proposition holds. As a Riemannian manifold with metric g T , SP D(2) R + * × D. The distance in these coordinates is expressed as

Figure 7 :

 7 Figure 7: The orbits of the subgroups K, A, N . The green segment on the right panel illustrates the "inner product" z, b when b = b 1 (see Sec. 5.1).

  and O the center of the disc. The values s and τ are the horocyclic coordinates of z. Now let b ∈ ∂D be a point on the unit circle. We define the "inner product" z, b as the smallest signed distance from O to the horocycle based at b and passing by z, see the picture on the right of Fig. 7. This value is clearly invariant under rotations in K. By a suitable rotation we take b = b 1 = 1, then by (18) and the expression of the distance d D (see Prop. 1), one has z, b 1 = τ . Note that z, b is also invariant by the action of the parabolic group N . Finally elementary calculus shows that that in horocyclic coordinates, the surface element is dm(z) = e -τ dsdτ . Let ∆ D be the Laplace-Beltrami operator in D, which in Cartesian coordinates z = x + iy reads

Theorem 2 .

 2 Any eigenfunction of the operator ∆ D admits a decomposition of the form ∂D e ρ,b (z)dT ρ (b)

Proposition 4 .

 4 Let f 1 , f 2 ∈ L 1 (D) and suppose that f 2 is invariant under the rotation group K. Then f 1 * f 2 = f1 f2 where f indicates the Helgason-Fourier transform of f .

Figure 8 :

 8 Figure 8: An example of hyperbolic waves with source at b 1 .

Figure 9 :

 9 Figure 9: Real (blue) and imaginary (red) parts of χ (with σ 1 = 0.9, σ 2 = 1, A = 0.6 in (7)) for periodic waves.

  ) are independent of time and are translation invariant in τ , hence constants. By a suitable change of variable we can set wlog v µ = 0 and S µ (0) = µ . The bifurcation point is obtained by solving the eigenvalue equation λu = -u + µχ * u with u(τ ) = e iατ , α ∈ R. In Fourier space it gives λ(α) = -1 + µ χ(α) where χ is the Fourier transform of χ in the usual sense. Numerical computation shows that χ has real and imaginary parts varying with α as in Figure9for a typical "Mexican hat" function w(z, z ) = h(d D (z, z )). The most unstable eigenvalues correspond to the maximum of the real part of χ, which in Figure9occurs at |α| = α c ≈ 0.76. The critical value µ 0 of the bifurcation parameter is the value at which the real part of λ(α c ) vanishes and the corresponding eigenvalues are simple and purely imaginary:

Figure 10 :

 10 Figure 10: The fundamental octagonal domain O and its generators (redrawn from [1])

Figure 11 :

 11 Figure 11: Tesselation of the regular hyperbolic octagon by successive reflections of the triangle colored in purple in the plot. dimension of absolutely irred. representation 1 2 3 4 number of nonequivalent representations 4 2 4 3 number of isotropy types with dim V H = 1 4 3 10 10

Figure 12 :

 12 Figure 12: The four H-planforms associated with the four irreductible representations of dimension 1, see text. Upper left: H-planform with full G * symmetry. Lower right : H-planform with symmetry G.

Figure 13 :

 13 Figure 13: An example of tiling with a H-planform (see text). The polygon marked with black line edges on the right of O is the copy of this octagon by the hyperbolic translation g 0 (see (30)).

+∞

  m=0 n∈N * w m,n Y m n (τ )Y m n (τ ) cos(m(θ -θ )).Then the eigenvalue problem reduces toλ m,n = -1 + µ w m,n

  (a) Mode (0, 1) with ρ 0,1 = 0.2798. (b) Mode (1, 1) with ρ 1,1 = 0.334.

Figure 14 :

 14 Figure 14: Plot of bifurcated solutions of (31), of the form P m lm,n (cosh(τ )) cos(mθ) with (m, n) = (0, 1) and (1, 1).

  cocompact Fuchsian group contains no parabolic element, that is no element conjugate to an element in the parabolic subgroup N . When it does not contain elliptic elements (conjugate to a rotation in K) as well, we call this group a lattice group of D. The action of a lattice group Γ contains no fixed point in D. Therefore the quotient surface D/Γ is a compact Riemann surface, which can be seen as the fundamental domain with "opposite" sides being identified by periodicity. Conversely a compact Riemann surface is the fundamental domain of a lattice group of D if and only if it has genus g ≥ 2. The case g = 1 corresponds to a lattice in the Euclidean plane, for which the quotient R 2 /Γ is a torus. Lattice groups are classified according to the group G Γ of automorphisms of D/Γ F Γ . In the Euclidean case G Γ = H R 2 /Z 2 where H is the holohedry of the lattice and there are only 4 holohedries. By contrast in the hyperbolic case there are infinitely many groups G Γ , however they always have finite order.Coming back to equation (

Acknowledgements

This review paper is the aftermath of a talk given by the author at the workshop "Géométrie et représentation de la couleur", Paris 2018-11-21, which was organized by the research network Information, Signal, Image et ViSion (GdR 720 Cnrs).

Funding

This work was partially funded by Université Côte d'Azur, Mathneuro project, Inria & Cnrs.

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