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, that is, 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.

Mathematics Subject Classification:30F45, 33C05, 34A12, 34D20, 34D23, 34G20, 37M05, 43A85, 44A35, 45G10, 51M10, 92B20, 92C20.

The selectivity of the responses of individual neurons to external features is often the basis of neuronal representations of the external world. For example, neurons in the primary visual cortex (V1) respond preferentially to visual stimuli that have a specific orientation [1–3], spatial frequency [4], velocity and direction of motion [5], color [6]. A local network in the primary visual cortex, roughly 1 mm² of cortical surface, is assumed to consist of subgroups of inhibitory and excitatory neurons each of which is tuned to a particular feature of an external stimulus. These subgroups are the so-called Hubel and Wiesel hypercolumns of V1. We have introduced in [7] a new approach to model the processing of image edges and textures in the hypercolumns of area V1 that is based on a nonlinear representation of the image first order derivatives called the structure tensor [8, 9]. We suggested that this structure tensor was represented by neuronal populations in the hypercolumns of V1. We also suggested that the time evolution of this representation was governed by equations similar to those proposed by Wilson and Cowan [10]. The question of whether some populations of neurons in V1 can represent the structure tensor is discussed in [7] but cannot be answered in a definite manner. Nevertheless, we hope that the predictions of the theory we are developing will help deciding on this issue.

Our present investigations were motivated by the work of Bressloff, Cowan, Golubitsky, Thomas and Wiener [11, 12] on the spontaneous occurence of hallucinatory patterns under the influence of psychotropic drugs, and its extension to the structure tensor model. A further motivation was the following studies of Bressloff and Cowan [4, 13, 14] where they study a spatial extension of the ring model of orientation of Ben-Yishai [1] and Hansel, Sompolinsky [2]. To achieve this goal, we first have to better understand the local model, that is the model of a ‘texture’ hypercolumn isolated from its neighbours.

The aim of this paper is to present a rigorous mathematical framework for the modeling of the representation of the structure tensor by neuronal populations in V1. We would also like to point out that the mathematical analysis we are developing here, is general and could be applied to other integro-differential equations defined on the set of structure tensors, so that even if the structure tensor were found to be not represented in a hypercolumn of V1, our framework would still be relevant. We then concentrate on the occurence of localized states, also called bumps. This is in contrast to the work of [7] and [15] where ‘spatially’ periodic solutions were considered. The structure of this paper is as follows. In Section 2 we introduce the structure tensor model and the corresponding equations. We also link our model to the ring model of orientations. In Section 3 we use classical tools of evolution equations in functional spaces to analyse the problem of the existence and uniqueness of the solutions of our equations. In Section 4 we study stationary solutions which are very important for the dynamics of the equation by analysing a nonlinear convolution operator and making use of the Haar measure of our feature space. In Section 5, we push further the study of stationary solutions in a special case and we present a technical analysis involving hypergeometric functions of what we call a hyperbolic radially symmetric stationary-pulse in the high gain limit. Finally, in Section 6, we present some numerical simulations of the solutions to verify the findings of the theoretical results.

By definition, the structure tensor is based on the spatial derivatives of an image in a small area that can be thought of as part of a receptive field. These spatial derivatives are then summed nonlinearly over the receptive field. Let I(x,y) denote the original image intensity function, where x and y are two spatial coordinates. Let I^{{\sigma }_1} denote the scale-space representation of I obtained by convolution with the Gaussian kernel g_{\sigma }(x,y)=\frac{1}{2\pi {\sigma }^2}{e}^{-(x^2+y^2)/(2{\sigma }^2)}:

The gradient \nabla I^{{\sigma }_1} is a two-dimensional vector of coordinates I_x^{{\sigma }_1}I_y^{{\sigma }_1} which emphasizes image edges. One then forms the 2\times 2 symmetric matrix of rank one {\mathcal{T}}_0=\nabla I^{{\sigma }_1}{(\nabla I^{{\sigma }_1})}^{\mathbf{T}}, where ^T indicates the transpose of a vector. The set of 2\times 2 symmetric positive semidefinite matrices of rank one will be noted {\mathrm{S}}^{+}(1,2) throughout the paper (see [16] for a complete study of the set {\mathrm{S}}^{+}(p,n) of n\times n symmetric positive semidefinite matrices of fixed-rank ). By convolving {\mathcal{T}}_0 componentwise with a Gaussian g_{{\sigma }_2} we finally form the tensor structure as the symmetric matrix:

where we have set for example:

Since the computation of derivatives usually involves a stage of scale-space smoothing, the definition of the structure tensor requires two scale parameters. The first one, defined by {\sigma }_1, is a local scale for smoothing prior to the computation of image derivatives. The structure tensor is insensitive to noise and details at scales smaller than {\sigma }_1. The second one, defined by {\sigma }_2, is an integration scale for accumulating the nonlinear operations on the derivatives into an integrated image descriptor. It is related to the characteristic size of the texture to be represented, and to the size of the receptive fields of the neurons that may represent the structure tensor.

By construction, \mathcal{T} is symmetric and non negative as det(\mathcal{T})\ge 0 by the inequality of Cauchy-Schwarz, then it has two orthonormal eigenvectors {\mathbf{e}}_1, {\mathbf{e}}_2 and two non negative corresponding eigenvalues {\lambda }_1 and {\lambda }_2 which we can always assume to be such that {\lambda }_1\ge {\lambda }_2\ge 0. Furthermore the spatial averaging distributes the information of the image over a neighborhood, and therefore the two eigenvalues are always positive. Thus, the set of the structure tensors lives in the set of 2\times 2 symmetric positive definite matrices, noted SPD(2,\mathbb{R}) throughout the paper. The distribution of these eigenvalues in the ({\lambda }_1,{\lambda }_2) plane reflects the local organization of the image intensity variations. Indeed, each structure tensor can be written as the linear combination:

where {\mathbf{I}}_2 is the identity matrix and {\mathbf{e}}_1{\mathbf{e}}_1^{\mathbf{T}}\in {\mathrm{S}}^{+}(1,2). Some easy interpretations can be made for simple examples: constant areas are characterized by {\lambda }_1={\lambda }_2\approx 0, straight edges are such that {\lambda }_1\gg {\lambda }_2\approx 0, their orientation being that of {\mathbf{e}}_2, corners yield {\lambda }_1\ge {\lambda }_2\gg 0. The coherency c of the local image is measured by the ratio c=\frac{{\lambda }_1-{\lambda }_2}{{\lambda }_1+{\lambda }_2}, large coherency reveals anisotropy in the texture.

We assume that a hypercolumn of V1 can represent the structure tensor in the receptive field of its neurons as the average membrane potential values of some of its membrane populations. Let \mathcal{T} be a structure tensor. The time evolution of the average potential V(\mathcal{T},t) for a given column is governed by the following neural mass equation adapted from [7] where we allow the connectivity function W to depend upon the time variable t and we integrate over the set of 2\times 2 symmetric definite-positive matrices:

The nonlinearity S is a sigmoidal function which may be expressed as:

where μ describes the stiffness of the sigmoid. I_{\mathrm{ext}} is an external input.

The set SPD(2) can be seen as a foliated manifold by way of the set of special symmetric positive definite matrices SSPD(2)=SPD(2)\cap SL(2,\mathbb{R}). Indeed, we have: SPD(2)\stackrel{\mathit{hom}}{=}SSPD(2)\times {\mathbb{R}}_{\ast }^{+}. Furthermore, SSPD(2)\stackrel{\mathit{isom}}{=}\mathbb{D}, where \mathbb{D} is the Poincaré Disk, see, for example, [7]. As a consequence we use the following foliation of SPD(2):SPD(2)\stackrel{\mathit{hom}}{=}\mathbb{D}\times {\mathbb{R}}_{\ast }^{+}, which allows us to write for all \mathcal{T}\in SPD(2)\mathcal{T}=(z,\Delta ) with (z,\Delta )\in \mathbb{D}\times {\mathbb{R}}_{\ast }^{+}. \mathcal{T}z and Δ are related by the relation det(\mathcal{T})={\Delta }^2 and the fact that z is the representation in \mathbb{D} of \tilde{\mathcal{T}}\in SSPD(2) with \mathcal{T}=\Delta \tilde{\mathcal{T}}.

It is well-known [17] that \mathbb{D} (and hence SSPD(2)) is a two-dimensional Riemannian space of constant sectional curvature equal to −1 for the distance noted d_2 defined by

The isometries of \mathbb{D}, that are the transformations that preserve the distance d_2 are the elements of unitary group \mathrm{U}(1,1). In Appendix A we describe the basic structure of this group. It follows, for example, [7, 18], that SDP(2) is a three-dimensional Riemannian space of constant sectional curvature equal to −1 for the distance noted d_0 defined by

As shown in Proposition B.0.1 of Appendix B it is possible to express the volume element d\mathcal{T} in (z_1,z_2,\Delta ) coordinates with z=z_1+i{z}_2:

We note dm(z)=\frac{d{z}_{1}d{z}_2}{{(1-{|z|}^2)}^2} and equation (2) can be written in (z,\Delta ) coordinates:

We get rid of the constant 8\sqrt{2} by redefining W as 8\sqrt{2}W.

In [7], we have assumed that the representation of the local image orientations and textures is richer than, and contains, the local image orientations model which is conceptually equivalent to the direction of the local image intensity gradient. The richness of the structure tensor model has been expounded in [7]. The embedding of the ring model of orientation in the structure tensor model can be explained by the intrinsic relation that exists between the two sets of matrices SPD(2,\mathbb{R}) and {\mathrm{S}}^{+}(1,2). First of all, when {\sigma }_2 goes to zero, that is when the characteristic size of the structure becomes very small, we have T_0\star g_{{\sigma }_2}\to T_0, which means that the tensor \mathcal{T}\in SPD(2,\mathbb{R}) degenerates to a tensor {\mathcal{T}}_0\in {\mathrm{S}}^{+}(1,2), which can be interpreted as the loss of one dimension. We can write each {\mathcal{T}}_0\in {\mathrm{S}}^{+}(1,2) as {\mathcal{T}}_0=x{x}^{\mathbf{T}}=r^{2}u{u}^{\mathbf{T}}, where u={(cos\theta ,sin\theta )}^{\mathbf{T}} and (r,\theta ) is the polar representation of x. Since, x and −x correspond to the same {\mathcal{T}}_0θ is equated to \theta +k\pik\in \mathbb{Z}. Thus {\mathrm{S}}^{+}(1,2)={\mathbb{R}}_{\ast }^{+}\times {\mathbb{P}}^1, where {\mathbb{P}}^1 is the real projective space of dimension 1 (lines of {\mathbb{R}}^2). Then the integration scale {\sigma }_2, at which the averages of the estimates of the image derivatives are computed, is the link between the classical representation of the local image orientations by the gradient and the representation of the local image textures by the structure tensor. It is also possible to highlight this explanation by coming back to the interpretation of straight edges of the previous paragraph. When {\lambda }_1\gg {\lambda }_2\approx 0 then \mathcal{T}\approx ({\lambda }_1-{\lambda }_2){\mathbf{e}}_1{\mathbf{e}}_1^{\mathbf{T}}\in {\mathrm{S}}^{+}(1,2) and the orientation is that of {\mathbf{e}}_2. We denote by \mathbb{P} the projection of a 2\times 2 symmetric definite positive matrix on the set {\mathrm{S}}^{+}(1,2) defined by:

where \mathcal{T} is as in equation (1). We can introduce a metric on the set {\mathrm{S}}^{+}(1,2) which is derived from a well-chosen Riemannian quotient geometry (see [16]). The resulting Riemannian space has strong geometrical properties: it is geodesically complete and the metric is invariant with respect to all transformations that preserve angles (orthogonal transformations, scalings and pseudoinversions). Related to the decomposition {\mathrm{S}}^{+}(1,2)={\mathbb{R}}_{\ast }^{+}\times {\mathbb{P}}^1, a metric on the space {\mathrm{S}}^{+}(1,2) is given by:

The space {\mathrm{S}}^{+}(1,2) endowed with this metric is a Riemannian manifold (see [16]). Finally, the distance associated to this metric is given by:

where {\tau }_1=x_1^{\mathbf{T}}{x}_1{\tau }_2=x_2^{\mathbf{T}}{x}_2 and (r_i,{\theta }_i) denotes the polar coordinates of x_i for i=1,2. The volume element in (r,\theta ) coordinates is:

where we normalize to 1 the volume element for the θ coordinate.

Let now \tau =\mathbb{P}(\mathcal{T}) be a symmetric positive semidefinite matrix. The average potential V(\tau ,t) of the column has its time evolution that is governed by the following neural mass equation which is just a projection of equation (2) on the subspace {\mathrm{S}}^{+}(1,2):

In (r,\theta ) coordinates, (4) is rewritten as:

This equation is richer than the ring model of orientation as it contains an additional information on the contrast of the image in the orthogonal direction of the prefered orientation. If one wants to recover the ring model of orientation tuning in the visual cortex as it has been presented and studied by [1, 2, 19], it is sufficient i) to assume that the connectivity function is time-independent and has a convolutional form:

and ii) to look at semi-homogeneous solutions of equation (4), that is, solutions which do not depend upon the variable r. We finally obtain:

where:

It follows from the above discussion that the structure tensor contains, at a given scale, more information than the local image intensity gradient at the same scale and that it is possible to recover the ring model of orientations from the structure tensor model.

The aim of the following sections is to establish that (3) is well-defined and to give necessary and sufficient conditions on the different parameters in order to prove some results on the existence and uniqueness of a solution of (3).

In this section we provide theoretical and general results of existence and uniqueness of a solution of (2). In the first subsection (Section 3.1) we study the simpler case of the homogeneous solutions of (2), that is, of the solutions that are independent of the tensor variable \mathcal{T}. This simplified model allows us to introduce some notations for the general case and to establish the useful Lemma 3.1.1. We then prove in Section 3.2 the main result of this section, that is the existence and uniqueness of a solution of (2). Finally we develop the useful case of the semi-homogeneous solutions of (2), that is, of solutions that depend on the tensor variable but only through its z coordinate in \mathbb{D}.

3.1 Homogeneous solutions

A homogeneous solution to (2) is a solution V that does not depend upon the tensor variable \mathcal{T} for a given homogenous input I(t) and a constant initial condition V_0. In (z,\Delta ) coordinates, a homogeneous solution of (3) is defined by:

where:

Hence necessary conditions for the existence of a homogeneous solution are that:

• the double integral (6) is convergent,

• \overline{W}(z,\Delta ,t)

  does not depend upon the variable (z,\Delta ). In that case, we write \overline{W}(t) instead of \overline{W}(z,\Delta ,t).

In the special case where W(z,\Delta ,z^{\prime },{\Delta }^{\prime },t) is a function of only the distance d_0 between (z,\Delta ) and (z^{\prime },{\Delta }^{\prime }):

the second condition is automatically satisfied. The proof of this fact is given in Lemma D.0.2 of Appendix D. To summarize, the homogeneous solutions satisfy the differential equation:

3.1.1 A first existence and uniqueness result

Equation (3) defines a Cauchy’s problem and we have the following theorem.

Theorem 3.1.1 If the external inputI_{\mathrm{ext}}(t)and the connectivity function\overline{W}(t)are continuous on some closed interval J containing 0, then for allV_0in\mathbb{R}, there exists a unique solution of (7) defined on a subintervalJ_0of J containing 0 such thatV(0)=V_0.

Proof It is a direct application of Cauchy’s theorem on differential equations. We consider the mapping f:J\times \mathbb{R}\to \mathbb{R} defined by:

It is clear that f is continuous from J\times \mathbb{R} to \mathbb{R}. We have for all x,y\in \mathbb{R} and t\in J:

where S_m^{\prime }={sup}_{x\in \mathbb{R}}|S^{\prime }(x)|.

Since, \overline{W} is continuous on the compact interval J, it is bounded there by C>0 and:

 □

We can extend this result to the whole time real line if I and \overline{W} are continuous on \mathbb{R}.

Proposition 3.1.1 IfI_{\mathrm{ext}}and\overline{W}are continuous on{\mathbb{R}}^{+}, then for allV_0in\mathbb{R}, there exists a unique solution of (7) defined on{\mathbb{R}}^{+}such thatV(0)=V_0.

Proof We have already shown the following inequality:

Then f is locally Lipschitz with respect to its second argument. Let V be a maximal solution on J_0 and we denote by β the upper bound of J_0. We suppose that . Then we have for all t\ge 0:

where S^m={sup}_{x\in \mathbb{R}}|S(x)|.

This implies that the maximal solution V is bounded for all t\in [0,\beta ], but Theorem C.0.2 of Appendix C ensures that it is impossible. Then, it follows that necessarily \beta =+\infty. □

3.1.2 Simplification of (6) in a special case

Invariance In the previous section, we have stated that in the special case where W was a function of the distance between two points in \mathbb{D}\times {\mathbb{R}}_{\ast }^{+}, then \overline{W}(z,\Delta ,t) did not depend upon the variables (z,\Delta ). As already said in the previous section, the following result holds (see proof of Lemma D.0.2 of Appendix D).

Lemma 3.1.1 Suppose that W is a function ofd_0(\mathcal{T},{\mathcal{T}}^{\prime })only. Then\overline{W}does not depend upon the variable\mathcal{T}.

Mexican hat connectivity In this paragraph, we push further the computation of \overline{W} in the special case where W does not depend upon the time variable t and takes the special form suggested by Amari in [20], commonly referred to as the ‘Mexican hat’ connectivity. It features center excitation and surround inhibition which is an effective model for a mixed population of interacting inhibitory and excitatory neurons with typical cortical connections. It is also only a function of d_0(\mathcal{T},{\mathcal{T}}^{\prime }).

In detail, we have:

where:

with 0\le {\sigma }_1\le {\sigma }_2 and 0\le A\le 1.

In this case we can obtain a very simple closed-form formula for \overline{W} as shown in the following lemma.

Lemma 3.1.2 When W is the specific Mexican hat function just defined then:

where erf is the error function defined as:

Proof The proof is given in Lemma E.0.3 of Appendix E. □

3.2 General solution

We now present the main result of this section about the existence and uniqueness of solutions of equation (2). We first introduce some hypotheses on the connectivity function W. We present them in two ways: first on the set of structure tensors considered as the set SPD(2), then on the set of tensors seen as D\times {\mathbb{R}}_{\ast }^{+}. Let J be a subinterval of \mathbb{R}. We assume that:

• (H1): \forall (\mathcal{T},{\mathcal{T}}^{\prime },t)\in SPD(2)\times SPD(2)\times J, W(\mathcal{T},{\mathcal{T}}^{\prime },t)\equiv W(d_0(\mathcal{T},{\mathcal{T}}^{\prime }),t),

• (H2): \mathbf{W}\in \mathcal{C}(J,L^1(SPD(2))) where W is defined as \mathbf{W}(\mathcal{T},t)=W(d_0(\mathcal{T},{\mathrm{Id}}_2),t) for all (\mathcal{T},t)\in SPD(2)\times J where {\mathrm{Id}}_2 is the identity matrix of {\mathcal{M}}_2(\mathbb{R}),

• (H3): \forall t\in J, where {\parallel \mathbf{W}(t)\parallel }_{L^1}\stackrel{\mathit{def}}{=}{\int }_{SPD(2)}|W(d_0(\mathcal{T},{\mathrm{Id}}_2),t)|d\mathcal{T}.

Equivalently, we can express these hypotheses in (z,\Delta ) coordinates:

• (H1 bis): \forall (z,z^{\prime },\Delta ,{\Delta }^{\prime },t)\in {\mathbb{D}}^2\times {({\mathbb{R}}_{\ast }^{+})}^2\times \mathbb{R}, W(z,\Delta ,z^{\prime },{\Delta }^{\prime },t)\equiv W(d_2(z,z^{\prime }),|log(\Delta )-log({\Delta }^{\prime })|,t),

• (H2 bis): \mathbf{W}\in \mathcal{C}(J,L^1(\mathbb{D}\times {\mathbb{R}}_{\ast }^{+})) where W is defined as \mathbf{W}(z,\Delta ,t)=W(d_2(z,0),|log(\Delta )|,t) for all \forall (z,\Delta ,t)\in \mathbb{D}\times {\mathbb{R}}_{\ast }^{+}\times J,

• (H3 bis): \forall t\in J, where

  {\parallel \mathbf{W}(t)\parallel }_{L^1}\stackrel{\mathit{def}}{=}{\int }_{\mathbb{D}\times {\mathbb{R}}_{\ast }^{+}}\left|W(d_2(z,0),|log(\Delta )|,t)\right|\frac{d\Delta }{\Delta }dm(z).

3.2.1 Functional space setting

We introduce the following mapping f^g:(t,\varphi )\to f^g(t,\varphi ) such that:

Our aim is to find a functional space \mathcal{F} where (3) is well-defined and the function f^g maps \mathcal{F} to \mathcal{F} for all t s. A natural choice would be to choose ϕ as a L^p(\mathbb{D}\times {\mathbb{R}}_{\ast }^{+})-integrable function of the space variable with . Unfortunately, the homogeneous solutions (constant with respect to (z,\Delta )) do not belong to that space. Moreover, a valid model of neural networks should only produce bounded membrane potentials. That is why we focus our choice on the functional space \mathcal{F}=L^{\infty }(\mathbb{D}\times {\mathbb{R}}_{\ast }^{+}). As \mathbb{D}\times {\mathbb{R}}_{\ast }^{+} is an open set of {\mathbb{R}}^3, \mathcal{F} is a Banach space for the norm: {\parallel \varphi \parallel }_{\mathcal{F}}={sup}_{z\in \mathbb{D}}{sup}_{\Delta \in {\mathbb{R}}_{\ast }^{+}}|\varphi (z,\Delta )|.

Proposition 3.2.1 IfI_{\mathrm{ext}}\in \mathcal{C}(J,\mathcal{F})withand W satisfies hypotheses (H 1bis)-(H 3bis) thenf^gis well-defined and is fromJ\times \mathcal{F}to\mathcal{F}.

Proof\forall (z,\Delta ,t)\in \mathbb{D}\times {\mathbb{R}}_{\ast }^{+}\times \mathbb{R}, we have:

 □

3.2.2 The existence and uniqueness of a solution of (3)

We rewrite (3) as a Cauchy problem:

Theorem 3.2.1 If the external currentI_{\mathrm{ext}}belongs to\mathcal{C}(J,\mathcal{F})with J an open interval containing 0 and W satisfies hypotheses (H 1bis)-(H 3bis), then fo allV_0\in \mathcal{F}, there exists a unique solution of (10) defined on a subintervalJ_0of J containing 0 such thatV(z,\Delta ,0)=V_0(z,\Delta )for all(z,\Delta )\in \mathbb{D}\times {\mathbb{R}}_{\ast }^{+}.

Proof We prove that f^g is continuous on J\times \mathcal{F}. We have

and therefore

Because of condition (H2) we can choose |t-s| small enough so that {\parallel \mathbf{W}(t)-\mathbf{W}(s)\parallel }_{L^1} is arbitrarily small. This proves the continuity of f^g. Moreover it follows from the previous inequality that:

with {\mathcal{W}}_0^g={sup}_{t\in J}{\parallel \mathbf{W}(t)\parallel }_{L^1}. This ensures the Lipschitz continuity of f^g with respect to its second argument, uniformly with respect to the first. The Cauchy-Lipschitz theorem on a Banach space yields the conclusion. □

Remark 3.2.1 Our result is quite similar to those obtained by Potthast and Graben in[21]. The main differences are that first we allow the connectivity function to depend upon the time variable t and second that our space features is no longer a{\mathbb{R}}^nbut a Riemanian manifold. In their article Potthast and Graben also work with a different functional space by assuming more regularity for the connectivity function W and then obtain more regularity for their solutions.

Proposition 3.2.2 If the external currentI_{\mathrm{ext}}belongs to\mathcal{C}({\mathbb{R}}^{+},\mathcal{F})and W satisfies hypotheses (H 1bis)-(H 3bis) withJ={\mathbb{R}}^{+}, then for allV_0\in \mathcal{F}, there exists a unique solution of (10) defined on{\mathbb{R}}^{+}such thatV(z,\Delta ,0)=V_0(z,\Delta )for all(z,\Delta )\in \mathbb{D}\times {\mathbb{R}}_{\ast }^{+}.

Proof We have just seen in the previous proof that f^g is globally Lipschitz with respect to its second argument:

then Theorem C.0.3 of Appendix C gives the conclusion. □

3.2.3 The intrinsic boundedness of a solution of (3)

In the same way as in the homogeneous case, we show a result on the boundedness of a solution of (3).

Proposition 3.2.3 If the external currentI_{\mathrm{ext}}belongs to\mathcal{C}({\mathbb{R}}^{+},\mathcal{F})and is bounded in timeand W satisfies hypotheses (H 1bis)-(H 3bis) withJ={\mathbb{R}}^{+}, then the solution of (10) is bounded for each initial conditionV_0\in \mathcal{F}.

Let us set:

where {\mathcal{W}}_0^g={sup}_{t\in {\mathbb{R}}^{+}}{\parallel \mathbf{W}(t)\parallel }_{L^1}.

Proof Let V be a solution defined on {\mathbb{R}}^{+}. Then we have for all t\in {\mathbb{R}}^{+\ast }:

The following upperbound holds

We can rewrite (11) as:

If V_0\in B_{\rho }^g this implies {\parallel V(t)\parallel }_{\mathcal{F}}\le \frac{{\rho }^g}{2}(1+e^{-\alpha t}) for all t>0 and hence for all t>0, proving that B_{\rho } is stable. Now assume that {\parallel V(t)\parallel }_{\mathcal{F}}>{\rho }^g for all t\ge 0. The inequality (12) shows that for t large enough this yields a contradiction. Therefore there exists t_0>0 such that {\parallel V(t_0)\parallel }_{\mathcal{F}}={\rho }^g. At this time instant we have

and hence

 □

The following corollary is a consequence of the previous proposition.

Corollary 3.2.1 IfV_0\notin B_{{\rho }^g}andT^g=inf\{t>0\mathit{\text{such that}}V(t)\in B_{{\rho }^g}\}then:

3.3 Semi-homogeneous solutions

A semi-homogeneous solution of (3) is defined as a solution which does not depend upon the variable Δ. In other words, the populations of neurons is not sensitive to the determinant of the structure tensor, that is to the contrast of the image intensity. The neural mass equation is then equivalent to the neural mass equation for tensors of unit determinant. We point out that semi-homogeneous solutions were previously introduced in [7] where a bifurcation analysis of what they called H-planforms was performed. In this section, we define the framework in which their equations make sense without giving any proofs of our results as it is a direct consequence of those proven in the general case. We rewrite equation (3) in the case of semi-homogeneous solutions:

where

We have implicitly made the assumption, that W^{\mathit{sh}} does not depend on the coordinate Δ. Some conditions under which this assumption is satisfied are described below and are the direct transductions of those of the general case in the context of semi-homogeneous solutions.

Let J be an open interval of \mathbb{R}. We assume that:

• (C1): \forall (z,z^{\prime },t)\in {\mathbb{D}}^2\times J, W^{\mathit{sh}}(z,z^{\prime },t)\equiv w^{\mathit{sh}}(d_2(z,z^{\prime }),t),

• (C2): {\mathbf{W}}^{\mathit{sh}}\in \mathcal{C}(J,L^1(\mathbb{D})) where {\mathbf{W}}^{\mathit{sh}} is defined as {\mathbf{W}}^{\mathit{sh}}(z,t)=w^{\mathit{sh}}(d_2(z,0),t) for all (z,t)\in \mathbb{D}\times J,

• (C3): where {\parallel {\mathbf{W}}^{\mathit{sh}}(t)\parallel }_{L^1}\stackrel{\mathit{def}}{=}{\int }_{\mathbb{D}}|W^{\mathit{sh}}(d_2(z,0),t)|dm(z).

Note that conditions (C1)-(C2) and Lemma 3.1.1 imply that for all z\in \mathbb{D}, {\int }_{\mathbb{D}}|W^{\mathit{sh}}(z,z^{\prime },t)|dm(z^{\prime })={\parallel {\mathbf{W}}^{\mathit{sh}}(t)\parallel }_{L^1}. And then, for all z\in \mathbb{D}, the mapping z^{\prime }\to W^{\mathit{sh}}(z,z^{\prime },t) is integrable on \mathbb{D}.

From now on, \mathcal{F}=L^{\infty }(\mathbb{D}) and the Fischer-Riesz’s theorem ensures that L^{\infty }(\mathbb{D}) is a Banach space for the norm: {\parallel \psi \parallel }_{\infty }=inf\{C\ge 0,|\psi (z)|\le C\text{for almost every}z\in \mathbb{D}\}.

Theorem 3.3.1 If the external currentI_{\mathrm{ext}}belongs to\mathcal{C}(J,\mathcal{F})with J an open interval containing 0 andW^{\mathit{sh}}satisfies conditions (C 1)-(C 3), then for allV_0\in \mathcal{F}, there exists a unique solution of (13) defined on a subintervalJ_0of J containing 0.

This solution, defined on the subinterval J of \mathbb{R} can in fact be extended to the whole real line, and we have the following proposition.

Proposition 3.3.1 If the external currentI_{\mathrm{ext}}belongs to\mathcal{C}({\mathbb{R}}^{+},\mathcal{F})andW^{\mathit{sh}}satisfies conditions (C 1)-(C 3) withJ={\mathbb{R}}^{+}, then for allV_0\in \mathcal{F}, there exists a unique solution of (13) defined on{\mathbb{R}}^{+}.

We can also state a result on the boundedness of a solution of (13):

Proposition 3.3.2 Let\rho \stackrel{\mathit{def}}{=}\frac{2}{\alpha }(S^m{\mathcal{W}}_0^{\mathit{sh}}+{sup}_{t\in {\mathbb{R}}^{+}}{\parallel I(t)\parallel }_{\mathcal{F}}), with{\mathcal{W}}_0^{\mathit{sh}}={sup}_{t\in J}{\parallel {\mathbf{W}}^{\mathit{sh}}(t)\parallel }_{L^1}. The open ballB_{\rho }of\mathcal{F}of center 0 and radius ρ is stable under the dynamics of equation (13). Moreover it is an attracting set for this dynamics and ifV_0\notin B_{\rho }andT=inf\{t>0\mathit{\text{such that}}V(t)\in B_{\rho }\}then:

We look at the equilibrium states, noted V_{\mu }^0 of (3), when the external input I and the connectivity W do not depend upon the time. We assume that W satisfies hypotheses (H1 bis)-(H2 bis). We redefine for convenience the sigmoidal function to be:

so that a stationary solution (independent of time) satisfies:

We define the nonlinear operator from \mathcal{F} to \mathcal{F}, noted {\mathcal{G}}_{\mu }, by:

Finally, (14) is equivalent to:

4.1 Study of the nonlinear operator {\mathcal{G}}_{\mu }

We recall that we have set for the Banach space \mathcal{F}=L^{\infty }(\mathbb{D}\times {\mathbb{R}}_{\ast }^{+}) and Proposition 3.2.1 shows that {\mathcal{G}}_{\mu }:\mathcal{F}\to \mathcal{F}. We have the further properties:

Proposition 4.1.1{\mathcal{G}}_{\mu }satisfies the following properties:

• {\parallel {\mathcal{G}}_{\mu }(V_1)-{\mathcal{G}}_{\mu }(V_2)\parallel }_{\mathcal{F}}\le \mu W_0^g{S}_m^{\prime }{\parallel V_1-V_2\parallel }_{\mathcal{F}}

  for all\mu \ge 0,

• \mu \to {\mathcal{G}}_{\mu }

  is continuous on{\mathbb{R}}^{+}.

Proof The first property was shown to be true in the proof of Theorem 3.3.1. The second property follows from the following inequality:

 □

We denote by {\mathcal{G}}_l and {\mathcal{G}}_{\infty } the two operators from \mathcal{F} to \mathcal{F} defined as follows for all V\in \mathcal{F} and all (z,\Delta )\in \mathbb{D}\times {\mathbb{R}}_{\ast }^{+}:

and

where H is the Heaviside function.

It is straightforward to show that both operators are well-defined on \mathcal{F} and map \mathcal{F} to \mathcal{F}. Moreover the following proposition holds.

Proposition 4.1.2 We have

Proof It is a direct application of the dominated convergence theorem using the fact that:

 □

4.2 The convolution form of the operator {\mathcal{G}}_{\mu } in the semi-homogeneous case

It is convenient to consider the functional space {\mathcal{F}}^{\mathit{sh}}=L^{\infty }(\mathbb{D}) to discuss semi-homogeneous solutions. A semi-homogeneous persistent state of (3) is deduced from (14) and satisfies:

where the nonlinear operator {\mathcal{G}}_{\mu }^{\mathit{sh}} from {\mathcal{F}}^{\mathit{sh}} to {\mathcal{F}}^{\mathit{sh}} is defined for all V\in {\mathcal{F}}^{\mathit{sh}} and z\in \mathbb{D} by:

We define the associated operators, {\mathcal{G}}_l^{\mathit{sh}},{\mathcal{G}}_{\infty }^{\mathit{sh}}:

We rewrite the operator {\mathcal{G}}_{\mu }^{\mathit{sh}} in a convenient form by using the convolution in the hyperbolic disk. First, we define the convolution in a such space. Let O denote the center of the Poincaré disk that is the point represented by z=0 and dg denote the Haar measure on the group G=SU(1,1) (see [22] and Appendix A), normalized by:

for all functions of L^1(\mathbb{D}). Given two functions f_1,f_2 in L^1(\mathbb{D}) we define the convolution ∗ by:

We recall the notation {\mathbf{W}}^{\mathit{sh}}(z)\stackrel{\mathit{def}}{=}{W}^{\mathit{sh}}(d_2(z,O)).

Proposition 4.2.1 For all\mu \ge 0andV\in {\mathcal{F}}^{\mathit{sh}}we have:

Proof We only prove the result for {\mathcal{G}}_{\mu }. Let z\in \mathbb{D}, then:

and for all g\in SU(1,1), d_2(z,z^{\prime })=d_2(g\cdot z,g\cdot z^{\prime }) so that:

 □

Let b be a point on the circle \partial \mathbb{D}. For z\in \mathbb{D}, we define the ‘inner product’ 〈z,b〉 to be the algebraic distance to the origin of the (unique) horocycle based at b through z (see [7]). Note that 〈z,b〉 does not depend on the position of z on the horocycle. The Fourier transform in \mathbb{D} is defined as (see [22]):

for a function h:\mathbb{D}\to \mathbb{C} such that this integral is well-defined.

Lemma 4.2.1 The Fourier transform in\mathbb{D}, {\tilde{\mathbf{W}}}^{\mathit{sh}}(\lambda ,b)of{\mathbf{W}}^{\mathit{sh}}does not depend upon the variableb\in \partial \mathbb{D}.

Proof For all \lambda \in \mathbb{R} and b=e^{i\theta }\in \partial \mathbb{D},

We recall that for all \varphi \in \mathbb{R}r_{\varphi } is the rotation of angle ϕ and we have {\mathbf{W}}^{\mathit{sh}}(r_{\varphi }\cdot z)={\mathbf{W}}^{\mathit{sh}}(z), dm(z)=dm(r_{\varphi }\cdot z) and 〈z,b〉=〈r_{\varphi }\cdot z,r_{\varphi }\cdot b〉, then:

 □

We now introduce two functions that enjoy some nice properties with respect to the Hyperbolic Fourier transform and are eigenfunctions of the linear operator {\mathcal{G}}_l^{\mathit{sh}}.

Proposition 4.2.2 Lete_{\lambda ,b}(z)=e^{(-i\lambda +1)〈z,b〉}and{\Phi }_{\lambda }(z)={\int }_{\partial \mathbb{D}}{e}^{(i\lambda +1)〈z,b〉}dbthen:

• {\mathcal{G}}_l^{\mathit{sh}}(e_{\lambda ,b})={\tilde{\mathbf{W}}}^{\mathit{sh}}(\lambda )e_{\lambda ,b}

  ,

• {\mathcal{G}}_l^{\mathit{sh}}({\Phi }_{\lambda })={\tilde{\mathbf{W}}}^{\mathit{sh}}(\lambda ){\Phi }_{\lambda }

  .