Analysis of a hyperbolic geometric model for visual texture perception

Abstract

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.

1 Introduction

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.

2 The model

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 $g_{\sigma }(x,y)=\frac{1}{2\pi {\sigma }^2}{e}^{-(x^2+y^2)/(2{\sigma }^2)}$:

$$I^{{\sigma }_1}=I\star g_{{\sigma }_1}.$$

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 $p<n$). By convolving ${\mathcal{T}}_0$ componentwise with a Gaussian $g_{{\sigma }_2}$ we finally form the tensor structure as the symmetric matrix:

$$\mathcal{T}={\mathcal{T}}_0\star g_{{\sigma }_2}=\left(\begin{array}{cc}{〈{(I_x^{{\sigma }_1})}^2〉}_{{\sigma }_2}& {〈I_x^{{\sigma }_1}{I}_y^{{\sigma }_1}〉}_{{\sigma }_2}\\ {〈I_x^{{\sigma }_1}{I}_y^{{\sigma }_1}〉}_{{\sigma }_2}& {〈{(I_y^{{\sigma }_1})}^2〉}_{{\sigma }_2}\end{array}\right),$$

where we have set for example:

$${〈{\left(I_x^{{\sigma }_1}\right)}^2〉}_{{\sigma }_2}={\left(I_x^{{\sigma }_1}\right)}^2\star g_{{\sigma }_2}.$$

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:

$$\begin{array}{rl}\mathcal{T}& ={\lambda }_1{\mathbf{e}}_1{\mathbf{e}}_1^{\mathbf{T}}+{\lambda }_2{\mathbf{e}}_2{\mathbf{e}}_2^{\mathbf{T}}=({\lambda }_1-{\lambda }_2){\mathbf{e}}_1{\mathbf{e}}_1^{\mathbf{T}}+{\lambda }_2({\mathbf{e}}_1{\mathbf{e}}_1^{\mathbf{T}}+{\mathbf{e}}_2{\mathbf{e}}_2^{\mathbf{T}})\\ =({\lambda }_1-{\lambda }_2){\mathbf{e}}_1{\mathbf{e}}_1^{\mathbf{T}}+{\lambda }_2{\mathbf{I}}_2,\end{array}$$

(1)

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:

$$\{\begin{array}{c}{\partial }_{t}V(\mathcal{T},t)=-\alpha V(\mathcal{T},t)+{\int }_{SPD(2)}W(\mathcal{T},{\mathcal{T}}^{\prime },t)S(V({\mathcal{T}}^{\prime },t))d{\mathcal{T}}^{\prime }\hfill \\ \phantom{{\partial }_{t}V(\mathcal{T},t)=}+I_{\mathrm{ext}}(\mathcal{T},t)\forall t>0,\hfill \\ V(\mathcal{T},0)=V_0(\mathcal{T}).\hfill \end{array}$$

(2)

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

$$S(x)=\frac{1}{1+e^{-\mu x}},$$

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

$$d_2(z,z^{\prime })=arctanh\frac{|z-z^{\prime }|}{|1-\overline{z}{z}^{\prime }|}.$$

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

$$d_0(\mathcal{T},{\mathcal{T}}^{\prime })=\sqrt{2{(log\Delta -log{\Delta }^{\prime })}^2+d_2^2(z,z^{\prime })}.$$

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 $z=z_1+i{z}_2$:

$$d\mathcal{T}=8\sqrt{2}\frac{d\Delta }{\Delta }\frac{d{z}_{1}d{z}_2}{{(1-{|z|}^2)}^2}.$$

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:

$$\begin{array}{rcl}{\partial }_{t}V(z,\Delta ,t)& =& -\alpha V(z,\Delta ,t)\\ +8\sqrt{2}{\int }_0^{+\infty }{\int }_{\mathbb{D}}W(z,\Delta ,z^{\prime },{\Delta }^{\prime },t)S(V(z^{\prime },{\Delta }^{\prime },t))\frac{d{\Delta }^{\prime }}{{\Delta }^{\prime }}dm(z)\\ +I_{\mathrm{ext}}(z,\Delta ,t).\end{array}$$

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

$$\{\begin{array}{c}{\partial }_{t}V(z,\Delta ,t)=-\alpha V(z,\Delta ,t)\hfill \\ \phantom{{\partial }_{t}V(z,\Delta ,t)=}+{\int }_0^{+\infty }{\int }_{\mathbb{D}}W(z,\Delta ,z^{\prime },{\Delta }^{\prime },t)S(V(z^{\prime },{\Delta }^{\prime },t))\frac{d{\Delta }^{\prime }}{{\Delta }^{\prime }}dm(z)\hfill \\ \phantom{{\partial }_{t}V(z,\Delta ,t)=}+I_{\mathrm{ext}}(z,\Delta ,t)\forall t>0,\hfill \\ V(z,\Delta ,0)=V_0(z,\Delta ).\hfill \end{array}$$

(3)

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 ${\mathcal{T}}_0$θ is equated to $\theta +k\pi$$k\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:

$$\mathbb{P}:\{\begin{array}{c}SPD(2,\mathbb{R})\to {\mathrm{S}}^{+}(1,2),\hfill \\ \mathcal{T}\mapsto \tau =({\lambda }_1-{\lambda }_2){\mathbf{e}}_1{\mathbf{e}}_1^{\mathbf{T}},\hfill \end{array}$$

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:

$$d{s}^2=2{\left(\frac{dr}{r}\right)}^2+d{\theta }^2.$$

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:

$$d_{{\mathrm{S}}^{+}(1,2)}^2({\tau }_1,{\tau }_2)=2{log}^2\left(\frac{r_1}{r_2}\right)+{|{\theta }_1-{\theta }_2|}^2,$$

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:

$$d\tau =\frac{dr}{r}\frac{d\theta }{\pi },$$

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 ${\mathrm{S}}^{+}(1,2)$:

$$\begin{array}{rl}{\partial }_{t}V(\tau ,t)=& -\alpha V(\tau ,t)+{\int }_{{\mathrm{S}}^{+}(1,2)}W(\tau ,{\tau }^{\prime },t)S(V({\tau }^{\prime },t))d{\tau }^{\prime }\\ +I_{\mathrm{ext}}(\tau ,t)\forall t>0.\end{array}$$

(4)

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

$$\begin{array}{rcl}{\partial }_{t}V(r,\theta ,t)& =& -\alpha V(r,\theta ,t)+{\int }_0^{+\infty }{\int }_0^{\pi }W(r,\theta ,r^{\prime },{\theta }^{\prime },t)S(V(r^{\prime },{\theta }^{\prime },t))\frac{d{\theta }^{\prime }}{\pi }\frac{d{r}^{\prime }}{r^{\prime }}\\ +I_{\mathrm{ext}}(r,\theta ,t).\end{array}$$

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:

$$W(\tau ,{\tau }^{\prime },t)=w(d_{{\mathrm{S}}^{+}(1,2)}(\tau ,{\tau }^{\prime }))=w\left(\sqrt{2{log}^2\left(\frac{r}{r^{\prime }}\right)+{|\theta -{\theta }^{\prime }|}^2}\right),$$

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:

$${\partial }_{t}V(\theta ,t)=-\alpha V(\theta ,t)+{\int }_0^{\pi }{w}^{\mathit{sh}}(\theta -{\theta }^{\prime })S(V({\theta }^{\prime },t))\frac{d{\theta }^{\prime }}{\pi }+I_{\mathrm{ext}}(\theta ,t),$$

(5)

where:

$$w^{\mathit{sh}}(\theta )={\int }_0^{+\infty }w\left(\sqrt{2{log}^2(r)+{\theta }^2}\right)\frac{dr}{r}.$$

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).

3 The existence and uniqueness of a solution

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:

$$\dot{V}(t)=-\alpha V(t)+\overline{W}(z,\Delta ,t)S(V(t))+I_{\mathrm{ext}}(t),$$

where:

$$\overline{W}(z,\Delta ,t)\stackrel{\mathit{def}}{=}{\int }_0^{+\infty }{\int }_{\mathbb{D}}W(z,\Delta ,z^{\prime },{\Delta }^{\prime },t)\frac{d{\Delta }^{\prime }}{{\Delta }^{\prime }}\frac{d{z}_1^{\prime }d{z}_2^{\prime }}{{(1-{|z^{\prime }|}^2)}^2}.$$

(6)

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 $(z^{\prime },{\Delta }^{\prime })$:

$$W(z,\Delta ,z^{\prime },{\Delta }^{\prime },t)\equiv w(\sqrt{2{(log\Delta -log{\Delta }^{\prime })}^2+d_2^2(z,z^{\prime })},t)$$

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:

$$\{\begin{array}{c}\dot{V}(t)=-\alpha V(t)+\overline{W}(t)S(V(t))+I_{\mathrm{ext}}(t),t>0,\hfill \\ V(0)=V_0.\hfill \end{array}$$

(7)

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 input$I_{\mathrm{ext}}(t)$and the connectivity function$\overline{W}(t)$are continuous on some closed interval J containing 0, then for all$V_0$in$\mathbb{R}$, there exists a unique solution of (7) defined on a subinterval$J_0$of J containing 0 such that$V(0)=V_0$.

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:

$$f(t,x)=-\alpha x+\overline{W}(t)S(x)+I_{\mathrm{ext}}(t).$$

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 $t\in J$:

$$|f(t,x)-f(t,y)|\le \alpha |x-y|+|\overline{W}(t)|S_m^{\prime }|x-y|,$$

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:

$$|f(t,x)-f(t,y)|\le (\alpha +C{S}_m^{\prime })|x-y|.$$

 □

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 If$I_{\mathrm{ext}}$and$\overline{W}$are continuous on${\mathbb{R}}^{+}$, then for all$V_0$in$\mathbb{R}$, there exists a unique solution of (7) defined on${\mathbb{R}}^{+}$such that$V(0)=V_0$.

Proof We have already shown the following inequality:

$$|f(t,x)-f(t,y)|\le \alpha |x-y|+|\overline{W}(t)|S_m^{\prime }|x-y|.$$

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 $\beta <+\infty$. Then we have for all $t\ge 0$:

$$\begin{array}{r}V(t)=e^{-\alpha t}{V}_0+{\int }_0^t{e}^{-\alpha (t-u)}\overline{W}(u)S(V(u))du+{\int }_0^t{e}^{-\alpha (t-u)}{I}_{\mathrm{ext}}(u)du\\ \Rightarrow |V(t)|\le |V_0|+S^m{\int }_0^{\beta }{e}^{\alpha u}|\overline{W}(u)|du+{\int }_0^{\beta }{e}^{\alpha u}|I_{\mathrm{ext}}(u)|du\forall t\in [0,\beta ],\end{array}$$

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 of$d_0(\mathcal{T},{\mathcal{T}}^{\prime })$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:

$$W(z,\Delta ,z^{\prime }{\Delta }^{\prime })=w\left(\sqrt{2{(log\Delta -log{\Delta }^{\prime })}^2+d_2^2(z,z^{\prime })}\right),$$

where:

$$w(x)=\frac{1}{\sqrt{2\pi {\sigma }_1^2}}{e}^{-\frac{x^2}{{\sigma }_1^2}}-\frac{A}{\sqrt{2\pi {\sigma }_2^2}}{e}^{-\frac{x^2}{{\sigma }_2^2}}$$

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:

$$\overline{W}=\frac{{\pi }^{\frac{3}{2}}}{2}({\sigma }_1{e}^{2{\sigma }_1^2}\mathit{erf}\left(\sqrt{2}{\sigma }_1\right)-A{\sigma }_2{e}^{2{\sigma }_2^2}\mathit{erf}\left(\sqrt{2}{\sigma }_2\right)),$$

(8)

where erf is the error function defined as:

$$\mathit{erf}(x)=\frac{2}{\sqrt{\pi }}{\int }_0^x{e}^{-u^2}du.$$

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): $\forall t\in J$, ${sup}_{t\in J}{\parallel \mathbf{W}(t)\parallel }_{L^1}<+\infty$ 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): $\forall t\in J$, ${sup}_{t\in J}{\parallel \mathbf{W}(t)\parallel }_{L^1}<+\infty$ 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:

$$f^g(t,\varphi )(z,\Delta )={\int }_{\mathbb{D}\times {\mathbb{R}}_{\ast }^{+}}W(d_2(z,z^{\prime }),|log\left(\frac{\Delta }{{\Delta }^{\prime }}\right)|,t)S(\varphi (z^{\prime },{\Delta }^{\prime }))\frac{d{\Delta }^{\prime }}{{\Delta }^{\prime }}dm(z^{\prime }).$$

(9)

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 $1\le p<+\infty$. 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 If$I_{\mathrm{ext}}\in \mathcal{C}(J,\mathcal{F})$with${sup}_{t\in J}{\parallel I_{\mathrm{ext}}(t)\parallel }_{\mathcal{F}}<+\infty$and W satisfies hypotheses (H 1bis)-(H 3bis) then$f^g$is well-defined and is from$J\times \mathcal{F}$to$\mathcal{F}$.

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

$$\begin{array}{r}|{\int }_{\mathbb{D}\times {\mathbb{R}}_{\ast }^{+}}W(d_2(z,z^{\prime }),|log\left(\frac{\Delta }{{\Delta }^{\prime }}\right)|,t)S(\varphi (z^{\prime },{\Delta }^{\prime }))\frac{d{\Delta }^{\prime }}{{\Delta }^{\prime }}dm(z^{\prime })|\\ \le S^m\underset{t\in J}{sup}{\parallel \mathbf{W}(t)\parallel }_{L^1}<+\infty .\end{array}$$

 □

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

We rewrite (3) as a Cauchy problem:

$$\{\begin{array}{c}{\partial }_{t}V(z,\Delta ,t)=-\alpha V(z,\Delta ,t)\hfill \\ \phantom{{\partial }_{t}V(z,\Delta ,t)=}+{\int }_{\mathbb{D}\times {\mathbb{R}}_{\ast }^{+}}W(d_2(z,z^{\prime }),|log\left(\frac{\Delta }{{\Delta }^{\prime }}\right)|,t)\hfill \\ \phantom{{\partial }_{t}V(z,\Delta ,t)=+{\int }_{\mathbb{D}\times {\mathbb{R}}_{\ast }^{+}}}\times S(V(z^{\prime },{\Delta }^{\prime },t))\frac{d{\Delta }^{\prime }}{{\Delta }^{\prime }}dm(z^{\prime })\hfill \\ \phantom{{\partial }_{t}V(z,\Delta ,t)=}+I_{\mathrm{ext}}(z,\Delta ,t),\hfill \\ V(z,\Delta ,0)=V_0(z,\Delta ).\hfill \end{array}$$

(10)

Theorem 3.2.1 If the external current$I_{\mathrm{ext}}$belongs to$\mathcal{C}(J,\mathcal{F})$with J an open interval containing 0 and W satisfies hypotheses (H 1bis)-(H 3bis), then fo all$V_0\in \mathcal{F}$, there exists a unique solution of (10) defined on a subinterval$J_0$of J containing 0 such that$V(z,\Delta ,0)=V_0(z,\Delta )$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

$${\parallel f^g(t,\varphi )-f^g(s,\psi )\parallel }_{\mathcal{F}}\le S_m^{\prime }\underset{t\in J}{sup}{\parallel \mathbf{W}(t)\parallel }_{L^1}{\parallel \varphi -\psi \parallel }_{\mathcal{F}}+S^m{\parallel \mathbf{W}(t)-\mathbf{W}(s)\parallel }_{L^1}.$$

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:

$${\parallel f^g(t,\varphi )-f^g(t,\psi )\parallel }_{\mathcal{F}}\le S_m^{\prime }{\mathcal{W}}_0^g{\parallel \varphi -\psi \parallel }_{\mathcal{F}}$$

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${\mathbb{R}}^n$but 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 current$I_{\mathrm{ext}}$belongs to$\mathcal{C}({\mathbb{R}}^{+},\mathcal{F})$and W satisfies hypotheses (H 1bis)-(H 3bis) with$J={\mathbb{R}}^{+}$, then for all$V_0\in \mathcal{F}$, there exists a unique solution of (10) defined on${\mathbb{R}}^{+}$such that$V(z,\Delta ,0)=V_0(z,\Delta )$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:

$${\parallel f^g(t,\varphi )-f^g(t,\psi )\parallel }_{\mathcal{F}}\le S_m^{\prime }{\mathcal{W}}_0^g{\parallel \varphi -\psi \parallel }_{\mathcal{F}}$$

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 current$I_{\mathrm{ext}}$belongs to$\mathcal{C}({\mathbb{R}}^{+},\mathcal{F})$and is bounded in time${sup}_{t\in {\mathbb{R}}^{+}}{\parallel I_{\mathrm{ext}}(t)\parallel }_{\mathcal{F}}<+\infty$and W satisfies hypotheses (H 1bis)-(H 3bis) with$J={\mathbb{R}}^{+}$, then the solution of (10) is bounded for each initial condition$V_0\in \mathcal{F}$.

Let us set:

$${\rho }^g\stackrel{\mathit{def}}{=}\frac{2}{\alpha }(S^m{\mathcal{W}}_0^g+\underset{t\in {\mathbb{R}}^{+}}{sup}{\parallel I_{\mathrm{ext}}(t)\parallel }_{\mathcal{F}}),$$

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

$${\parallel V(t)\parallel }_{\mathcal{F}}\le e^{-\alpha t}{\parallel V_0\parallel }_{\mathcal{F}}+\frac{1}{\alpha }(S^m{\mathcal{W}}_0^g+\underset{t\in {\mathbb{R}}^{+}}{sup}{\parallel I_{\mathrm{ext}}(t)\parallel }_{\mathcal{F}})(1-e^{-\alpha t}).$$

(11)

We can rewrite (11) as:

$$\begin{array}{rl}{\parallel V(t)\parallel }_{\mathcal{F}}\le & e^{-\alpha t}({\parallel V_0\parallel }_{\mathcal{F}}-\frac{1}{\alpha }(S^m{\mathcal{W}}_0^g+\underset{t\in {\mathbb{R}}^{+}}{sup}{\parallel I_{\mathrm{ext}}(t)\parallel }_{\mathcal{F}}))\\ +\frac{1}{\alpha }\left(S^m{\mathcal{W}}_0{+}^g\underset{t\in {\mathbb{R}}^{+}}{sup}{\parallel I_{\mathrm{ext}}(t)\parallel }_{\mathcal{F}}\right)\\ =& e^{-\alpha t}({\parallel V_0\parallel }_{\mathcal{F}}-\frac{{\rho }^g}{2})+\frac{{\rho }^g}{2}.\end{array}$$

(12)

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 ${\parallel V(t)\parallel }_{\mathcal{F}}<{\rho }^g$ 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

$${\rho }^g\le e^{-\alpha t_0}({\parallel V_0\parallel }_{\mathcal{F}}-\frac{{\rho }^g}{2})+\frac{{\rho }^g}{2},$$

and hence

$$t_0\le \frac{1}{\alpha }log\left(\frac{2{\parallel V_0\parallel }_{\mathcal{F}}-{\rho }^g}{{\rho }^g}\right).$$

 □

The following corollary is a consequence of the previous proposition.

Corollary 3.2.1 If$V_0\notin B_{{\rho }^g}$and$T^g=inf\{t>0\mathit{\text{such that}}V(t)\in B_{{\rho }^g}\}$then:

$$T^g\le \frac{1}{\alpha }log\left(\frac{2{\parallel V_0\parallel }_{\mathcal{F}}-{\rho }^g}{{\rho }^g}\right).$$

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:

$$\{\begin{array}{c}{\partial }_{t}V(z,t)=-\alpha V(z,t)+{\int }_{\mathbb{D}}{W}^{\mathit{sh}}(z,z^{\prime },t)S(V(z^{\prime },t))dm(z^{\prime })\hfill \\ \phantom{{\partial }_{t}V(z,t)=}+I_{\mathrm{ext}}(z,t),t>0,\hfill \\ V(z,0)=V_0(z),\hfill \end{array}$$

(13)

where

$$W^{\mathit{sh}}(z,z^{\prime },t)={\int }_0^{+\infty }W(z,\Delta ,z^{\prime },{\Delta }^{\prime },t)\frac{d{\Delta }^{\prime }}{{\Delta }^{\prime }}.$$

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): ${sup}_{t\in J}{\parallel {\mathbf{W}}^{\mathit{sh}}(t)\parallel }_{L^1}<+\infty$ 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 current$I_{\mathrm{ext}}$belongs to$\mathcal{C}(J,\mathcal{F})$with J an open interval containing 0 and$W^{\mathit{sh}}$satisfies conditions (C 1)-(C 3), then for all$V_0\in \mathcal{F}$, there exists a unique solution of (13) defined on a subinterval$J_0$of 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 current$I_{\mathrm{ext}}$belongs to$\mathcal{C}({\mathbb{R}}^{+},\mathcal{F})$and$W^{\mathit{sh}}$satisfies conditions (C 1)-(C 3) with$J={\mathbb{R}}^{+}$, then for all$V_0\in \mathcal{F}$, 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 ball$B_{\rho }$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 if$V_0\notin B_{\rho }$and$T=inf\{t>0\mathit{\text{such that}}V(t)\in B_{\rho }\}$then:

$$T\le \frac{1}{\alpha }log\left(\frac{2{\parallel V_0\parallel }_{\mathcal{F}}-\rho }{\rho }\right).$$

4 Stationary solutions

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:

$$S(x)=\frac{1}{1+e^{-x}},$$

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

$$\begin{array}{rcl}0& =& -\alpha V_{\mu }^0(z,\Delta )+{\int }_{\mathbb{D}\times {\mathbb{R}}_{\ast }^{+}}W(d_2(z,z^{\prime }),|log\left(\frac{\Delta }{{\Delta }^{\prime }}\right)|)S(\mu V_{\mu }^0(z^{\prime },{\Delta }^{\prime }))\frac{d{\Delta }^{\prime }}{{\Delta }^{\prime }}dm(z^{\prime })\\ +I_{\mathrm{ext}}(z,\Delta ).\end{array}$$

(14)

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

$${\mathcal{G}}_{\mu }(V)(z,\Delta )={\int }_{\mathbb{D}\times {\mathbb{R}}_{\ast }^{+}}W(d_2(z,z^{\prime }),|log\left(\frac{\Delta }{{\Delta }^{\prime }}\right)|)S(\mu V(z^{\prime },{\Delta }^{\prime }))\frac{d{\Delta }^{\prime }}{{\Delta }^{\prime }}dm(z^{\prime }).$$

(15)

Finally, (14) is equivalent to:

$$\alpha V_{\mu }^0(z,\Delta )={\mathcal{G}}_{\mu }(V)(z,\Delta )+I_{\mathrm{ext}}(z,\Delta ).$$

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:

$${\parallel {\mathcal{G}}_{{\mu }_1}(V)-{\mathcal{G}}_{{\mu }_2}(V)\parallel }_{\mathcal{F}}\le |{\mu }_1-{\mu }_2|W_0^g{S}_m^{\prime }{\parallel V\parallel }_{\mathcal{F}}.$$

 □

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 $(z,\Delta )\in \mathbb{D}\times {\mathbb{R}}_{\ast }^{+}$:

$${\mathcal{G}}_l(V)(z,\Delta )={\int }_{\mathbb{D}\times {\mathbb{R}}_{\ast }^{+}}W(d_2(z,z^{\prime }),|log\left(\frac{\Delta }{{\Delta }^{\prime }}\right)|)V(z^{\prime },{\Delta }^{\prime })\frac{d{\Delta }^{\prime }}{{\Delta }^{\prime }}dm(z^{\prime }),$$

(16)

and

$${\mathcal{G}}_{\infty }(V)(z,\Delta )={\int }_{\mathbb{D}\times {\mathbb{R}}_{\ast }^{+}}W(d_2(z,z^{\prime }),|log\left(\frac{\Delta }{{\Delta }^{\prime }}\right)|)H(V(z^{\prime },{\Delta }^{\prime }))\frac{d{\Delta }^{\prime }}{{\Delta }^{\prime }}dm(z^{\prime }),$$

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

$${\mathcal{G}}_{\mu }\underset{\mu \to \infty }{⟶}{\mathcal{G}}_{\infty }.$$

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

$$S(\mu y)\underset{\mu \to \infty }{⟶}H(y)\text{a.e.}y\in \mathbb{R}.$$

 □

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:

$$\alpha V_{\mu }^0(z)={\mathcal{G}}_{\mu }^{\mathit{sh}}\left(V_{\mu }^0\right)(z)+I_{\mathrm{ext}}(z),$$

(17)

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:

$${\mathcal{G}}_{\mu }^{\mathit{sh}}(V)(z)={\int }_{\mathbb{D}}{W}^{\mathit{sh}}(d_2(z,z^{\prime }))S(\mu V(z^{\prime }))dm(z^{\prime }).$$

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

$$\begin{array}{rcl}{\mathcal{G}}_l^{\mathit{sh}}(V)(z)& =& {\int }_{\mathbb{D}}{W}^{\mathit{sh}}(d_2(z,z^{\prime }))V(z^{\prime })dm(z^{\prime }),\\ {\mathcal{G}}_{\infty }^{\mathit{sh}}(V)(z)& =& {\int }_{\mathbb{D}}{W}^{\mathit{sh}}(d_2(z,z^{\prime }))H(V(z^{\prime }))dm(z^{\prime }).\end{array}$$

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:

$${\int }_{G}f(g\cdot O)dg\stackrel{\mathit{def}}{=}{\int }_{\mathbb{D}}f(z)dm(z),$$

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:

$$(f_1\ast f_2)(z)={\int }_G{f}_1(g\cdot O)f_2(g^{-1}\cdot z)dg.$$

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$\mu \ge 0$and$V\in {\mathcal{F}}^{\mathit{sh}}$we have:

$$\begin{array}{rl}{\mathcal{G}}_{\mu }^{\mathit{sh}}(V)& ={\mathbf{W}}^{\mathit{sh}}\ast S(\mu V),{\mathcal{G}}_l^{\mathit{sh}}(V)={\mathbf{W}}^{\mathit{sh}}\ast V\mathit{\text{and}}\\ {\mathcal{G}}_{\infty }^{\mathit{sh}}(V)& ={\mathbf{W}}^{\mathit{sh}}\ast H(V).\end{array}$$

(18)

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

$$\begin{array}{rcl}{\mathcal{G}}_{\mu }^{\mathit{sh}}(V)(z)& =& {\int }_{\mathbb{D}}{W}^{\mathit{sh}}(d_2(z,z^{\prime }))S(\mu V(z^{\prime }))dm(z^{\prime })\\ =& {\int }_G{W}^{\mathit{sh}}(d_2(z,g\cdot O))S(\mu V(g\cdot O))dg\\ =& {\int }_G{W}^{\mathit{sh}}\left(d_2(g{g}^{-1}\cdot z,g\cdot O)\right)S(\mu V(g\cdot O))dg\end{array}$$

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

$${\mathcal{G}}_{\mu }^{\mathit{sh}}(V)(z)={\int }_G{W}^{\mathit{sh}}\left(d_2(g^{-1}\cdot z,O)\right)S(\mu V(g\cdot O))dg={\mathbf{W}}^{\mathit{sh}}\ast S(\mu V)(z).$$

 □

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]):

$$\tilde{h}(\lambda ,b)={\int }_{\mathbb{D}}h(z)e^{(-i\lambda +1)〈z,b〉}dm(z)\forall (\lambda ,b)\in \mathbb{R}\times \partial \mathbb{D}$$

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 variable$b\in \partial \mathbb{D}$.

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

$${\tilde{\mathbf{W}}}^{\mathit{sh}}(\lambda ,b)={\int }_{\mathbb{D}}{\mathbf{W}}^{\mathit{sh}}(z)e^{(-i\lambda +1)〈z,b〉}dm(z).$$

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:

$$\begin{array}{rcl}{\tilde{\mathbf{W}}}^{\mathit{sh}}(\lambda ,b)& =& {\int }_{\mathbb{D}}{\mathbf{W}}^{\mathit{sh}}(r_{-\theta }\cdot z)e^{(-i\lambda +1)〈r_{-\theta }\cdot z,1〉}dm(z)\\ =& {\int }_{\mathbb{D}}{\mathbf{W}}^{\mathit{sh}}(z)e^{(-i\lambda +1)〈z,1〉}dm(z)\stackrel{\mathit{def}}{=}{\tilde{\mathbf{W}}}^{\mathit{sh}}(\lambda ).\end{array}$$

 □

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 Let$e_{\lambda ,b}(z)=e^{(-i\lambda +1)〈z,b〉}$and${\Phi }_{\lambda }(z)={\int }_{\partial \mathbb{D}}{e}^{(i\lambda +1)〈z,b〉}db$then:

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

  ,

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

  .

Proof We begin with $b=1\in \partial \mathbb{D}$ and use the horocyclic coordinates. We use the same changes of variables as in Lemma 3.1.1:

$$\begin{array}{rcl}{\mathcal{G}}_l^{\mathit{sh}}(e_{\lambda ,1})(n_s{a}_t\cdot O)& =& {\int }_{{\mathbb{R}}^2}{W}^{\mathit{sh}}(d_2(n_s{a}_t\cdot O,n_{s^{\prime }}{a}_{t^{\prime }}\cdot O))e^{(-i\lambda -1)t^{\prime }}d{t}^{\prime }d{s}^{\prime }\\ =& {\int }_{{\mathbb{R}}^2}{W}^{\mathit{sh}}(d_2(n_{s-s^{\prime }}{a}_t\cdot O,a_{t^{\prime }}\cdot O))e^{(-i\lambda -1)t^{\prime }}d{t}^{\prime }d{s}^{\prime }\\ =& {\int }_{{\mathbb{R}}^2}{W}^{\mathit{sh}}(d_2(a_t{n}_{-x}\cdot O,a_{t^{\prime }}\cdot O))e^{(-i\lambda -1)t^{\prime }+2t}d{t}^{\prime }dx\\ =& {\int }_{{\mathbb{R}}^2}{W}^{\mathit{sh}}(d_2(O,n_x{a}_{t^{\prime }-t}\cdot O))e^{(-i\lambda -1)t^{\prime }+2t}d{t}^{\prime }dx\\ =& {\int }_{{\mathbb{R}}^2}{W}^{\mathit{sh}}(d_2(O,n_x{a}_y\cdot O))e^{(-i\lambda -1)(y+t)+2t}dydx\\ =& e^{(-i\lambda +1)〈n_s{a}_t\cdot O,1〉}{\tilde{\mathbf{W}}}^{\mathit{sh}}(\lambda ).\end{array}$$

By rotation, we obtain the property for all $b\in \partial \mathbb{D}$.

For the second property [22], Lemma 4.7] shows that:

$${\mathbf{W}}^{\mathit{sh}}\ast {\Phi }_{\lambda }(z)={\int }_{\partial \mathbb{D}}{e}^{(i\lambda +1)〈z,b〉}{\tilde{\mathbf{W}}}^{\mathit{sh}}(\lambda )db={\Phi }_{\lambda }(z){\tilde{\mathbf{W}}}^{\mathit{sh}}(\lambda ).$$

 □

A consequence of this proposition is the following lemma.

Lemma 4.2.2 The linear operator${\mathcal{G}}_l^{\mathit{sh}}$is not compact and for all$\mu \ge 0$, the nonlinear operator${\mathcal{G}}_{\mu }^{\mathit{sh}}$is not compact.

Proof The previous Proposition 4.2.2 shows that ${\mathcal{G}}_l^{\mathit{sh}}$ has a continuous spectrum which iimplies that is not a compact operator.

Let U be in ${\mathcal{F}}^{\mathit{sh}}$, for all $V\in {\mathcal{F}}^{\mathit{sh}}$ we differentiate ${\mathcal{G}}_{\mu }^{\mathit{sh}}$ and compute its Frechet derivative:

$$D{\left({\mathcal{G}}_{\mu }^{\mathit{sh}}\right)}_U(V)(z)={\int }_{\mathbb{D}}{W}^{\mathit{sh}}(d_2(z,z^{\prime }))S^{\prime }(U(z^{\prime }))V(z^{\prime })dm(z^{\prime }).$$

If we assume further that U does not depend upon the space variable z, $U(z)=U_0$ we obtain:

$$D{\left({\mathcal{G}}_{\mu }^{\mathit{sh}}\right)}_{U_0}(V)(z)=S^{\prime }(U_0){\mathcal{G}}_l^{\mathit{sh}}(V)(z).$$

If ${\mathcal{G}}_{\mu }^{\mathit{sh}}$ was a compact operator then its Frechet derivative $D{({\mathcal{G}}_{\mu }^{\mathit{sh}})}_{U_0}$ would also be a compact operator, but it is impossible. As a consequence, ${\mathcal{G}}_{\mu }^{\mathit{sh}}$ is not a compact operator. □

4.3 The convolution form of the operator ${\mathcal{G}}_{\mu }$ in the general case

We adapt the ideas presented in the previous section in order to deal with the general case. We recall that if H is the group of positive real numbers with multiplication as operation, then the Haar measure dh is given by $\frac{dx}{x}$. For two functions $f_1$, $f_2$ in $L^1(\mathbb{D}\times {\mathbb{R}}_{\ast }^{+})$ we define the convolution ⋆ by:

$$(f_1\star f_2)(z,\Delta )\stackrel{\mathit{def}}{=}{\int }_G{\int }_H{f}_1(g\cdot O,h\cdot 1)f_2(g^{-1}\cdot z,h^{-1}\cdot \Delta )dgdh.$$

We recall that we have set by definition: $\mathbf{W}(z,\Delta )=W(d_2(z,0),|log(\Delta )|)$.

Proposition 4.3.1 For all$\mu \ge 0$and$V\in \mathcal{F}$we have:

$${\mathcal{G}}_{\mu }(V)=\mathbf{W}\star S(\mu V),{\mathcal{G}}_l(V)=\mathbf{W}\star V\mathit{\text{and}}{\mathcal{G}}_{\infty }(V)=\mathbf{W}\star H(V).$$

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Proof Let $(z,\Delta )$ be in $\mathbb{D}\times {\mathbb{R}}_{\ast }^{+}$. We follow the same ideas as in Proposition 4.2.1 and prove only the first result. We have

$$\begin{array}{rcl}{\mathcal{G}}_{\mu }(V)(z,\Delta )& =& {\int }_{\mathbb{D}\times {\mathbb{R}}_{\ast }^{+}}W(d_2(z,z^{\prime }),|log\left(\frac{\Delta }{{\Delta }^{\prime }}\right)|)S(\mu V(z^{\prime },{\Delta }^{\prime }))\frac{d{\Delta }^{\prime }}{{\Delta }^{\prime }}dm(z^{\prime })\\ =& {\int }_G{\int }_{{\mathbb{R}}_{\ast }^{+}}W(d_2(g^{-1}\cdot z,O),|log\left(\frac{\Delta }{{\Delta }^{\prime }}\right)|)S(\mu V(g\cdot O,{\Delta }^{\prime }))dg\frac{d{\Delta }^{\prime }}{{\Delta }^{\prime }}\\ =& {\int }_G{\int }_{H}W(d_2(g^{-1}\cdot z,O),|log(h^{-1}\cdot \Delta )|)S(\mu V(g\cdot O,h\cdot 1))dgdh\\ =& \mathbf{W}\star S(\mu V)(z,\Delta ).\end{array}$$

 □

We next assume further that the function W is separable in z, Δ and more precisely that $\mathbf{W}(z,\Delta )={\mathbf{W}}_1(z){\mathbf{W}}_2(log(\Delta ))$ where ${\mathbf{W}}_1(z)=W_1(d_2(z,0))$ and ${\mathbf{W}}_2(log(\Delta ))=W_2(|log(\Delta )|)$ for all $(z,\Delta )\in \mathbb{D}\times {\mathbb{R}}_{\ast }^{+}$. The following proposition is an echo to Proposition 4.2.2.

Proposition 4.3.2 Let$e_{\lambda ,b}(z)=e^{(-i\lambda +1)〈z,b〉}$, ${\Phi }_{\lambda }(z)={\int }_{\partial \mathbb{D}}{e}^{(i\lambda +1)〈z,b〉}db$and$h_{\xi }(\Delta )=e^{i\xi log(\Delta )}$then:

• $${\mathcal{G}}_l(e_{\lambda ,b}{h}_{\xi })={\tilde{\mathbf{W}}}_1(\lambda ){\hat{\mathbf{W}}}_2(\xi )e_{\lambda ,b}{h}_{\xi }$$

  ,

• $${\mathcal{G}}_l({\Phi }_{\lambda }{h}_{\xi })={\tilde{\mathbf{W}}}_1(\lambda ){\hat{\mathbf{W}}}_2(\xi ){\Phi }_{\lambda }{h}_{\xi }$$

  ,

where${\hat{\mathbf{W}}}_2$is the usual Fourier transform of${\mathbf{W}}_2$.

Proof The proof of this proposition is exactly the same as for Proposition 4.2.2. Indeed:

$$\begin{array}{rcl}{\mathcal{G}}_l(e_{\lambda ,b}{h}_{\xi })(z,\Delta )& =& {\mathbf{W}}_1\ast e_{\lambda ,b}(z){\int }_{{\mathbb{R}}_{\ast }^{+}}{\mathbf{W}}_2(log\left(\frac{\Delta }{{\Delta }^{\prime }}\right))e^{i\xi log({\Delta }^{\prime })}\frac{d{\Delta }^{\prime }}{{\Delta }^{\prime }}\\ =& {\mathbf{W}}_1\ast e_{\lambda ,b}(z)({\int }_{\mathbb{R}}{\mathbf{W}}_2(y)e^{-i\xi y}dy)e^{i\xi log(\Delta )}.\end{array}$$

 □

A straightforward consequence of this proposition is an extension of Lemma 4.2.2 to the general case:

Lemma 4.3.1 The linear operator${\mathcal{G}}_l^{\mathit{sh}}$is not compact and for all$\mu \ge 0$, the nonlinear operator${\mathcal{G}}_{\mu }^{\mathit{sh}}$is not compact.

4.4 The set of the solutions of (14)

Let ${\mathcal{B}}_{\mu }$ be the set of the solutions of (14) for a given slope parameter μ:

$${\mathcal{B}}_{\mu }=\{V\in \mathcal{F}|-\alpha V+{\mathcal{G}}_{\mu }(V)+I_{\mathrm{ext}}=0\}.$$

We have the following proposition.

Proposition 4.4.1 If the input current$I_{\mathrm{ext}}$is equal to a constant$I_{\mathrm{ext}}^0$, that is, does not depend upon the variables$(z,\Delta )$then for all$\mu \in {\mathbb{R}}^{+}$, ${\mathcal{B}}_{\mu }\ne \varnothing$. In the general case$I_{\mathrm{ext}}\in \mathcal{F}$, if the condition$\mu S_m^{\prime }{W}_0^g<\alpha$is satisfied, then$Card({\mathcal{B}}_{\mu })=1$.

Proof Due to the properties of the sigmoid function, there always exists a constant solution in the case where $I_{\mathrm{ext}}$ is constant. In the general case where $I_{\mathrm{ext}}\in \mathcal{F}$, the statement is a direct application of the Banach fixed point theorem, as in [23]. □

Remark 4.4.1 If the external input does not depend upon the variables$(z,\Delta )$and if the condition$\mu S_m^{\prime }{W}_0^g<\alpha$is satisfied, then there exists a unique stationary solution by application of Proposition 4.4.1. Moreover, this stationary solution does not depend upon the variables$(z,\Delta )$because there always exists one constant stationary solution when the external input does not depend upon the variables$(z,\Delta )$. Indeed equation (14) is then equivalent to:

$$\begin{array}{rcl}0& =& -\alpha V^0+\beta S\left(V^0\right)+I_{\mathrm{ext}}^0\text{where}\\ \beta & =& {\int }_{\mathbb{D}\times {\mathbb{R}}_{\ast }^{+}}W(d_2(z,z^{\prime }),|log\left(\frac{\Delta }{{\Delta }^{\prime }}\right)|)\frac{d{\Delta }^{\prime }}{{\Delta }^{\prime }}dm(z^{\prime })\end{array}$$

and β does not depend upon the variables$(z,\Delta )$because of Lemma 3.1.1. Because of the property of the sigmoid function S, the equation$0=-\alpha V^0+\beta S(V^0)+I_{\mathrm{ext}}^0$has always one solution.

If on the other hand the input current does depend upon these variables is invariant under the action of a subgroup of$U(1,1)$the group of the isometries of$\mathbb{D}$ (see Appendix A), and the condition$\mu S_m^{\prime }{W}_0^g<\alpha$is satisfied then the unique stationary solution will also be invariant under the action of the same subgroup. We refer the interested reader to our work[15]on equivariant bifurcation of hyperbolic planforms on the subject.

When the condition$\mu S_m^{\prime }{W}_0^g<\alpha$is satisfied we call primary stationary solution the unique solution in${\mathcal{B}}_{\mu }$.

4.5 Stability of the primary stationary solution

In this subsection we show that the condition $\mu S_m^{\prime }{W}_0^g<\alpha$ guarantees the stability of the primary stationary solution to (3).

Theorem 4.5.1 We suppose that$I\in \mathcal{F}$and that the condition$\mu S_m^{\prime }{W}_0^g<\alpha$is satisfied, then the associated primary stationary solution of (3) is asymtotically stable.

Proof Let $V_{\mu }^0$ be the primary stationary solution of (3), as $\mu S_m^{\prime }{W}_0^g<\alpha$ is satisfied. Let also $V_{\mu }$ be the unique solution of the same equation with some initial condition $V_{\mu }(0)=\varphi \in \mathcal{F}$, see Theorem 3.3.1. We introduce a new function $X=V_{\mu }-V_{\mu }^0$ which satisfies:

where $W_m(d_2(z,z^{\prime }),|log(\frac{\Delta }{{\Delta }^{\prime }})|)=S_m^{\prime }W(d_2(z,z^{\prime }),|log(\frac{\Delta }{{\Delta }^{\prime }})|)$ and the vector $\Theta (X(z,\Delta ,t))$ is given by $\Theta (X(z,\Delta ,t))=\underline{S}(\mu V_{\mu }(z,\Delta ,t))-\underline{S}(\mu V_{\mu }^0(z,\Delta ))$ with $\underline{S}={(S_m^{\prime })}^{-1}S$. We note that, because of the definition of Θ and the mean value theorem $|\Theta (X(z,\Delta ,t))|\le \mu |X(z,\Delta ,t)|$. This implies that $|\Theta (r)|\le |r|$ for all $r\in \mathbb{R}$.

If we set: $G(t)=e^{\alpha t}{\parallel X(t)\parallel }_{\infty }$, then we have:

$$G(t)\le G(0)+\mu W_0^g{S}_m^{\prime }{\int }_0^{t}G(u)du$$

and G is continuous for all $t\ge 0$. The Gronwall inequality implies that:

$$\begin{array}{r}G(t)\le G(0)e^{\mu W_0^g{S}_m^{\prime }t}\\ \Rightarrow {\parallel X(t)\parallel }_{\infty }\le e^{(\mu W_0^g{S}_m^{\prime }-\alpha )t}{\parallel X(0)\parallel }_{\infty },\end{array}$$

and the conclusion follows. □

5 Spatially localised bumps in the high gain limit

In many models of working memory, transient stimuli are encoded by feature-selective persistent neural activity. Such stimuli are imagined to induce the formation of a spatially localised bump of persistent activity which coexists with a stable uniform state. As an example, Camperi and Wang [24] have proposed and studied a network model of visuo-spatial working memory in prefontal cortex adapted from the ring model of orientation of Ben-Yishai and colleagues [1]. Many studies have emerged in the past decades to analyse these localised bumps of activity [25–29], see the paper by Coombes for a review of the domain [30]. In [25, 26, 28], the authors have examined the existence and stability of bumps and multi-bumps solutions to an integro-differential equation describing neuronal activity along a single spatial domain. In [27, 29] the study is focused on the two-dimensional model and a method is developed to approximate the integro-differential equation by a partial differential equation which makes possible the determination of the stability of circularly symmetric solutions. It is therefore natural to study the emergence of spatially localized bumps for the structure tensor model in a hypercolumn of V1. We only deal with the reduced case of equation (13) which means that the membrane activity does not depend upon the contrast of the image intensity, keeping the general case for future work.

In order to construct exact bump solutions and to compare our results to previous studies [25–29], we consider the high gain limit $\mu \to \infty$ of the sigmoid function. As above we denote by H the Heaviside function defined by $H(x)=1$ for $x\ge 0$ and $H(x)=0$ otherwise. Equation (13) is rewritten as:

$$\begin{array}{rl}{\partial }_{t}V(z,t)& =-\alpha V(z,t)+{\int }_{\mathbb{D}}W(z,z^{\prime })H(V(z^{\prime },t)-\kappa )dm(z^{\prime })+I(z,t)\\ =-\alpha V(z,t)+{\int }_{\{z^{\prime }\in \mathbb{D}|V(z^{\prime },t)\ge \kappa \}}W(z,z^{\prime })dm(z^{\prime })+I(z).\end{array}$$

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We have introduced a threshold κ to shift the zero of the Heaviside function. We make the assumption that the system is spatially homogeneous that is, the external input I does not depend upon the variables t and the connectivity function depends only on the hyperbolic distance between two points of $\mathbb{D}:W(z,z^{\prime })=W(d_2(z,z^{\prime }))$. For illustrative purposes, we will use the exponential weight distribution as a specific example throughout this section:

$$W(z,z^{\prime })=W(d_2(z,z^{\prime }))=exp(-\frac{d_2(z,z^{\prime })}{b}).$$

(21)

The theoretical study of equation (20) has been done in [21] where the authors have imposed strong regularity assumptions on the kernel function W, such as Hölder continuity, and used compactness arguments and integral equation techniques to obtain a global existence result of solutions to (20). Our approach is very different, we follow that of [25–29, 31] by proceeding in a constructive fashion. In a first part, we define what we call a hyperbolic radially symmetric bump and present some preliminary results for the linear stability analysis of the last part. The second part is devoted to the proof of a technical Theorem 5.1.1 which is stated in the first part. The proof uses results on the Fourier transform introduced in Section 4, hyperbolic geometry and hypergeometric functions. Our results will be illustrated in the following Section 6.

5.1 Existence of hyperbolic radially symmetric bumps

From equation (20) a general stationary pulse satisfies the equation:

$$\alpha V(z)={\int }_{\{z^{\prime }\in \mathbb{D}|V(z^{\prime })\ge \kappa \}}W(z,z^{\prime })dm(z^{\prime })+I_{\mathrm{ext}}(z).$$

For convenience, we note $M(z,K)$ the integral ${\int }_{K}W(z,z^{\prime })dm(z^{\prime })$ with $K=\{z\in \mathbb{D}|V(z)\ge \kappa \}$. The relation $V(z)=\kappa$ holds for all $z\in \partial K$.

Definition 5.1.1 V is called a hyperbolic radially symmetric stationary-pulse solution of (20) if V depends only upon the variable r and is such that:

$$\begin{array}{r}V(r)>\kappa ,r\in [0,\omega [,\\ V(\omega )=\kappa ,\\ V(r)<\kappa ,r\in ]\omega ,\infty [,\\ V(\infty )=0,\end{array}$$

and is a fixed point of equation (20):

$$\alpha V(r)=\mathcal{M}(r,\omega )+I_{\mathrm{ext}}(r),$$

(22)

where$I_{\mathrm{ext}}(r)=\mathcal{I}{e}^{-\frac{r^2}{2{\sigma }^2}}$is a Gaussian input and$\mathcal{M}(r,\omega )$is defined by the following equation:

$$\mathcal{M}(r,\omega )\stackrel{\mathit{def}}{=}M(z,B_h(0,\omega ))$$

and$B_h(0,\omega )$is a hyperbolic disk centered at the origin of hyperbolic radius ω.

From symmetry arguments there exists a hyperbolic radially symmetric stationary-pulse solution $V(r)$ of (20), furthermore the threshold κ and width ω are related according to the self-consistency condition

$$\alpha \kappa =\mathcal{M}(\omega )+I_{\mathrm{ext}}(\omega )\stackrel{\mathit{def}}{=}N(\omega ),$$

(23)

where

$$\mathcal{M}(\omega )\stackrel{\mathit{def}}{=}\mathcal{M}(\omega ,\omega ).$$

The existence of such a bump can then be established by finding solutions to (23) The function $N(\omega )$ is plotted in Figure 1 for a range of the input amplitude $\mathcal{I}$. The horizontal dashed lines indicate different values of ακ, the points of intersection determine the existence of stationary pulse solutions. Qualitatively, for sufficiently large input amplitude $\mathcal{I}$ we have $N^{\prime }(0)<0$ and it is possible to find only one solution branch for large ακ. For small input amplitudes $\mathcal{I}$ we have $N^{\prime }(0)>0$ and there always exists one solution branch for $\alpha \beta <{\gamma }_c\approx 0.06$. For intermediate values of the input amplitude $\mathcal{I}$, as αβ varies, we have the possiblity of zero, one or two solutions. Anticipating the stability results of Section 5.3, we obtain that when $N^{\prime }(\omega )<0$ then the corresponding solution is stable.

Fig. 1

Plot of $N(\omega )$ defined in (23) as a function of the pulse width ω for several values of the input amplitude $\mathcal{I}$ and for a fixed input width $\sigma =0.05$. The horizontal dashed lines indicate different values of ακ. The connectivity function is given in equation (21) and the parameter b is set to $b=0.2$.

We end this subsection with the usefull and technical following formula.

Theorem 5.1.1 For all$(r,\omega )\in {\mathbb{R}}^{+}\times {\mathbb{R}}^{+}$:

$$\mathcal{M}(r,\omega )=\frac{1}{4}sinh{(\omega )}^{2}cosh{(\omega )}^2{\int }_{\mathbb{R}}\tilde{\mathbf{W}}(\lambda ){\Phi }_{\lambda }^{(0,0)}(r){\Phi }_{\lambda }^{(1,1)}(\omega )\lambda tanh\left(\frac{\pi }{2}\lambda \right)d\lambda ,$$

(24)

where $\tilde{\mathbf{W}}(\lambda )$ is the Fourier Helgason transform of $\mathbf{W}(z)\stackrel{\mathit{def}}{=}W(d_2(z,0))$ and

$${\Phi }_{\lambda }^{(\alpha ,\beta )}(\omega )=F(\frac{1}{2}(\rho +i\lambda ),\frac{1}{2}(\rho -i\lambda );\alpha +1;-sinh{(\omega )}^2),$$

with$\alpha +\beta +1=\rho$and F is the hypergeometric function of first kind.

Remark 5.1.1 We recall that F admits the integral representation[32]:

$$F(\alpha ,\beta ;\gamma ;z)=\frac{\Gamma (\alpha )}{\Gamma (\beta )\Gamma (\gamma -\beta )}{\int }_0^1{t}^{\beta -1}{(1-t)}^{\gamma -\beta -1}{(1-tz)}^{-\alpha }dt$$

with$\Re (\gamma )>\Re (\beta )>0$.

Remark 5.1.2 In Section 4we introduced the function${\Phi }_{\lambda }(z)={\int }_{\partial \mathbb{D}}{e}^{(i\lambda +1)〈z,b〉}db$. In[22], it is shown that:

$${\Phi }_{\lambda }^{(0,0)}(r)={\Phi }_{\lambda }(tanh(r))\mathit{\text{if}}z=tanh(r)e^{i\theta }.$$

Remark 5.1.3 Let us point out that this result can be linked to the work of Folias and Bressloff in[31]and then used in[29]. They constructed a two-dimensional pulse for a general radially symmetric synaptic weight function. They obtain a similar formal representation of the integral of the connectivity function w over the disk$B(O,a)$centered at the origin O and of radius a. Using their notations

$$M(a,r)={\int }_0^{2\pi }{\int }_0^{a}w(|\mathbf{r}-{\mathbf{r}}^{\prime }|)r^{\prime }d{r}^{\prime }d\theta =2\pi a{\int }_0^{\infty }\breve{w}(\rho )J_0(r\rho )J_1(a\rho )d\rho ,$$

where$J_{\nu }(x)$is the Bessel function of the first kind and$\breve{w}$is the real Fourier transform of w. In our case instead of the Bessel function we find${\Phi }_{\lambda }^{(\nu ,\nu )}(r)$which is linked to the hypergeometric function of the first kind.

We now show that for a general monotonically decreasing weight function W, the function $\mathcal{M}(r,\omega )$ is necessarily a monotonically decreasing function of r. This will ensure that the hyperbolic radially symmetric stationary-pulse solution (22) is also a monotonically decreasing function of r in the case of a Gaussian input. The demonstration of this result will directly use Theorem 5.1.1.

Proposition 5.1.1 V is a monotonically decreasing function in r for any monotonically decreasing synaptic weight function W.

Proof Differentiating $\mathcal{M}$ with respect to r yields:

$$\frac{\partial \mathcal{M}}{\partial r}(r,\omega )=\frac{1}{2}{\int }_0^{\omega }{\int }_0^{2\pi }\frac{\partial }{\partial r}\left(W\left(d_2(tanh(r),tanh(r^{\prime })e^{i\theta })\right)\right)sinh(2r^{\prime })d{r}^{\prime }d\theta .$$

We have to compute

$$\begin{array}{r}\frac{\partial }{\partial r}\left(W\left(d_2(tanh(r),tanh(r^{\prime })e^{i\theta })\right)\right)\\ =W^{\prime }\left(d_2(tanh(r),tanh(r^{\prime })e^{i\theta })\right)\frac{\partial }{\partial r}\left(d_2(tanh(r),tanh(r^{\prime })e^{i\theta })\right).\end{array}$$

It is result of elementary hyperbolic trigonometry that

$$\begin{array}{r}{d}_2(tanh(r),tanh(r^{\prime })e^{i\theta })\\ ={tanh}^{-1}\left(\sqrt{\frac{tanh{(r)}^2+tanh{(r^{\prime })}^2-2tanh(r)tanh(r^{\prime })cos(\theta )}{1+tanh{(r)}^{2}tanh{(r^{\prime })}^2-2tanh(r)tanh(r^{\prime })cos(\theta )}}\right)\end{array}$$

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we let $\rho =tanh(r)$, ${\rho }^{\prime }=tanh(r^{\prime })$ and define

$$F_{{\rho }^{\prime },\theta }(\rho )=\frac{{\rho }^2+{\rho }^{\prime 2}-2\rho {\rho }^{\prime }cos(\theta )}{1+{\rho }^2{\rho }^{\prime 2}-2\rho {\rho }^{\prime }cos(\theta )}.$$

It follows that

$$\frac{\partial }{\partial \rho }{tanh}^{-1}\left(\sqrt{F_{{\rho }^{\prime },\theta }(\rho )}\right)=\frac{\frac{\partial }{\partial \rho }{F}_{{\rho }^{\prime },\theta }(\rho )}{2(1-F_{{\rho }^{\prime },\theta }(\rho ))\sqrt{F_{{\rho }^{\prime },\theta }(\rho )}},$$

and

$$\frac{\partial }{\partial \rho }{F}_{{\rho }^{\prime },\theta }(\rho )=\frac{2(\rho -{\rho }^{\prime }cos(\theta ))+2\rho {\rho }^{\prime }({\rho }^{\prime }-\rho cos(\theta ))}{{(1+{\rho }^2{\rho }^{\prime 2}-2\rho {\rho }^{\prime }cos(\theta ))}^2}.$$

We conclude that if $\rho >tanh(\omega )$ then for all $0\le {\rho }^{\prime }\le tanh(\omega )$ and $0\le \theta \le 2\pi$

$$2(\rho -{\rho }^{\prime }cos(\theta ))+2\rho {\rho }^{\prime }({\rho }^{\prime }-\rho cos(\theta ))>0,$$

which implies $\mathcal{M}(r,\omega )<0$ for $r>\omega$, since $W^{\prime }<0$.

To see that it is also negative for $r<\omega$, we differentiate equation (24) with respect to r:

$$\begin{array}{rcl}\frac{\partial \mathcal{M}}{\partial r}(r,\omega )& =& \frac{1}{4}sinh{(\omega )}^{2}cosh{(\omega )}^2\\ \times {\int }_{\mathbb{R}}\tilde{\mathbf{W}}(\lambda )\frac{\partial }{\partial r}{\Phi }_{\lambda }^{(0,0)}(r){\Phi }_{\lambda }^{(1,1)}(\omega )\lambda tanh\left(\frac{\pi }{2}\lambda \right)d\lambda .\end{array}$$

The following formula holds for the hypergeometric function (see Erdelyi in [32]):

$$\frac{d}{dz}F(a,b;c;z)=\frac{ab}{c}F(a+1,b+1;c+1;z).$$

It implies

$$\frac{\partial }{\partial r}{\Phi }_{\lambda }^{(0,0)}(r)=-\frac{1}{2}sinh(r)cosh(r)(1+{\lambda }^2){\Phi }_{\lambda }^{(1,1)}(r).$$

Substituting in the previous equation giving $\frac{\partial \mathcal{M}}{\partial r}$ we find:

$$\begin{array}{rcl}\frac{\partial \mathcal{M}}{\partial r}(r,\omega )& =& -\frac{1}{64}sinh{(2\omega )}^{2}sinh(2r)\\ \times {\int }_{\mathbb{R}}\tilde{\mathbf{W}}(\lambda )(1+{\lambda }^2){\Phi }_{\lambda }^{(1,1)}(r){\Phi }_{\lambda }^{(1,1)}(\omega )\lambda tanh\left(\frac{\pi }{2}\lambda \right)d\lambda ,\end{array}$$

implying that:

$$\mathbf{sgn}(\frac{\partial \mathcal{M}}{\partial r}(r,\omega ))=\mathbf{sgn}(\frac{\partial \mathcal{M}}{\partial r}(\omega ,r)).$$

Consequently, $\frac{\partial \mathcal{M}}{\partial r}(r,\omega )<0$ for $r<\omega$. Hence V is monotonically decreasing in r for any monotonically decreasing synaptic weight function W. □

As a consequence, for our particular choice of exponential weight function (21), the radially symmetric bump is monotonically decreasing in r, as it will be recover in our numerical experiments in Section 6.

5.2 Proof of Theorem 5.1.1

The proof of Theorem 5.1.1 goes in four steps. First we introduce some notations and recall some basic properties of the Fourier transform in the Poincaré disk. Second we prove two propositions. Third we state a technical lemma on hypergeometric functions, the proof being given in Lemma F.0.4 of Appendix F. The last step is devoted to the conclusion of the proof.

5.2.1 First step

In order to calculate $\mathcal{M}(r,\omega )$, we use the Fourier transform in $\mathbb{D}$ which has already been introduced in Section 4. First we rewrite $\mathcal{M}(r,\omega )$ as a convolution product:

Proposition 5.2.1 For all$(r,\omega )\in {\mathbb{R}}^{+}\times {\mathbb{R}}^{+}$:

$$\mathcal{M}(r,\omega )=\frac{1}{4\pi }{\int }_{\mathbb{R}}\tilde{\mathbf{W}}(\lambda ){\Phi }_{\lambda }\ast {\mathbb{1}}_{B_h(0,\omega )}(z)\lambda tanh\left(\frac{\pi }{2}\lambda \right)d\lambda .$$

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Proof We start with the definition of $\mathcal{M}(r,\omega )$ and use the convolutional form of the integral:

$$\begin{array}{rcl}\mathcal{M}(r,\omega )& =& M(z,B_h(0,\omega ))={\int }_{B_h(0,\omega )}W(z,z^{\prime })dm(z^{\prime })\\ =& {\int }_{\mathbb{D}}W(z,z^{\prime }){\mathbb{1}}_{B_h(0,\omega )}(z^{\prime })dm(z^{\prime })=\mathbf{W}\ast {\mathbb{1}}_{B_h(0,\omega )}(z).\end{array}$$

In [22], Helgason proves an inversion formula for the hyperbolic Fourier transform and we apply this result to W:

$$\begin{array}{rcl}\mathbf{W}(z)& =& \frac{1}{4\pi }{\int }_{\mathbb{R}}{\int }_{\partial \mathbb{D}}\tilde{\mathbf{W}}(\lambda ,b)e^{(i\lambda +1)〈z,b〉}\lambda tanh\left(\frac{\pi }{2}\lambda \right)d\lambda db\\ =& \frac{1}{4\pi }{\int }_{\mathbb{R}}\tilde{\mathbf{W}}(\lambda )\left({\int }_{\partial \mathbb{D}}{e}^{(i\lambda +1)〈z,b〉}db\right)\lambda tanh\left(\frac{\pi }{2}\lambda \right)d\lambda \end{array}$$

the last equality is a direct application of Lemma 4.2.1 and we can deduce that

$$\mathbf{W}(z)=\frac{1}{4\pi }{\int }_{\mathbb{R}}\tilde{\mathbf{W}}(\lambda ){\Phi }_{\lambda }(z)\lambda tanh\left(\frac{\pi }{2}\lambda \right)d\lambda .$$

(27)

Finally we have:

$$\mathcal{M}(r,\omega )=\mathbf{W}\ast {\mathbb{1}}_{B_h(0,\omega )}(z)=\frac{1}{4\pi }{\int }_{\mathbb{R}}\tilde{\mathbf{W}}(\lambda ){\Phi }_{\lambda }\ast {\mathbb{1}}_{B_h(0,\omega )}(z)\lambda tanh\left(\frac{\pi }{2}\lambda \right)d\lambda .$$

which is the desired formula. □

It appears that the study of $\mathcal{M}(r,\omega )$ consists in calculating the convolution product ${\Phi }_{\lambda }\ast {\mathbb{1}}_{B_h(0,\omega )}(z)$.

Proposition 5.2.2 For all$z=k\cdot O$for$k\in G=SU(1,1)$we have:

$${\Phi }_{\lambda }\ast {\mathbb{1}}_{B_h(0,\omega )}(z)={\int }_{B_h(0,\omega )}{\Phi }_{\lambda }(k^{-1}\cdot z^{\prime })dm(z^{\prime }).$$

Proof Let $z=k\cdot O$ for $k\in G$ we have:

$$\begin{array}{rcl}{\Phi }_{\lambda }\ast {\mathbb{1}}_{B_h(0,\omega )}(z)& =& {\int }_G{\mathbb{1}}_{B_h(0,\omega )}(g\cdot O){\Phi }_{\lambda }(g^{-1}\cdot z)dg\\ =& {\int }_G{\mathbb{1}}_{B_h(0,\omega )}(g\cdot O){\Phi }_{\lambda }(g^{-1}k\cdot O)dg\end{array}$$

for all $g,k\in G$, ${\Phi }_{\lambda }(g^{-1}k\cdot O)={\Phi }_{\lambda }(k^{-1}g\cdot O)$ so that:

$$\begin{array}{rcl}{\Phi }_{\lambda }\ast {\mathbb{1}}_{B_h(0,\omega )}(z)& =& {\int }_G{\mathbb{1}}_{B_h(0,\omega )}(g\cdot O){\Phi }_{\lambda }(k^{-1}g\cdot O)dg\\ =& {\int }_{\mathbb{D}}{\mathbb{1}}_{B_h(0,\omega )}(z^{\prime }){\Phi }_{\lambda }(k^{-1}\cdot z^{\prime })dm(z^{\prime })\\ =& {\int }_{B_h(0,\omega )}{\Phi }_{\lambda }(k^{-1}\cdot z^{\prime })dm(z^{\prime }).\end{array}$$

 □

5.2.2 Second step

In this part, we prove two results:

• the mapping $z=k\cdot O=tanh(r)e^{i\theta }\to {\Phi }_{\lambda }\ast {\mathbb{1}}_{B_h(0,\omega )}(z)$ is a radial function, that is, it depends only upon the variable r.

• the following equality holds for $z=tanh(r)e^{i\theta }$:

  $${\Phi }_{\lambda }\ast {\mathbb{1}}_{B_h(0,\omega )}(z)={\Phi }_{\lambda }(a_r\cdot O){\int }_{B_h(0,\omega )}{e}^{(i\lambda +1)〈z^{\prime },1〉}dm(z^{\prime }).$$

Proposition 5.2.3 If$z=k\cdot O$and z is written$tanh(r)e^{i\theta }$with$r=d_2(z,O)$in hyperbolic polar coordinates the function${\Phi }_{\lambda }\ast {\mathbb{1}}_{B_h(0,\omega )}(z)$depends only upon the variable r.

Proof If $z=tanh(r)e^{i\theta }$, then $z={\mathrm{rot}}_{\theta }{a}_r\cdot O$ and $k^{-1}=a_{-r}{\mathrm{rot}}_{-\theta }$. Similarly $z^{\prime }={\mathrm{rot}}_{{\theta }^{\prime }}{a}_{r^{\prime }}\cdot O$. We can write thanks to the previous Proposition 5.2.2:

$$\begin{array}{rcl}{\Phi }_{\lambda }\ast {\mathbb{1}}_{B_h(0,\omega )}(z)& =& {\int }_{B_h(0,\omega )}{\Phi }_{\lambda }(k^{-1}\cdot z^{\prime })dm(z^{\prime })\\ =& \frac{1}{2}{\int }_0^{\omega }{\int }_0^{2\pi }{\Phi }_{\lambda }(a_{-r}{\mathrm{rot}}_{{\theta }^{\prime }-\theta }{a}_{r^{\prime }}\cdot O)sinh(2r^{\prime })d{r}^{\prime }d{\theta }^{\prime }\\ =& \frac{1}{2}{\int }_0^{\omega }{\int }_0^{2\pi }{\Phi }_{\lambda }(a_{-r}{\mathrm{rot}}_{\psi }{a}_{r^{\prime }}\cdot O)sinh(2r^{\prime })d{r}^{\prime }d\psi \\ =& {\int }_{B_h(0,\omega )}{\Phi }_{\lambda }(a_{-r}\cdot z^{\prime })dm(z^{\prime }),\end{array}$$

which, as announced, is only a function of r. □

We now give an explicit formula for the integral ${\int }_{B_h(0,\omega )}{\Phi }_{\lambda }(a_{-r}\cdot z^{\prime })dm(z^{\prime })$.

Proposition 5.2.4 For all$z=tanh(r)e^{i\theta }$we have:

$${\Phi }_{\lambda }\ast {\mathbb{1}}_{B_h(0,\omega )}(z)={\Phi }_{\lambda }(a_r\cdot O){\int }_{B_h(0,\omega )}{e}^{(i\lambda +1)〈z^{\prime },1〉}dm(z^{\prime }).$$

Proof We first recall a formula from [22].

Lemma 5.2.1 For all $g\in G$the following equation holds:

$${\Phi }_{\lambda }(g^{-1}\cdot z)={\int }_{\partial \mathbb{D}}{e}^{(-i\lambda +1)〈g\cdot O,b〉}{e}^{(i\lambda +1)〈z,b〉}db.$$

Proof See [22]. □

It follows immediately that for all $z\in \mathbb{D}$ and $r\in \mathbb{R}$ we have:

$${\Phi }_{\lambda }(a_{-r}\cdot z)={\int }_{\partial \mathbb{D}}{e}^{(-i\lambda +1)〈a_r\cdot O,b〉}{e}^{(i\lambda +1)〈z,b〉}db.$$

We integrate this formula over the hyperbolic ball $B_h(0,\omega )$ which gives:

$${\int }_{B_h(0,\omega )}{\Phi }_{\lambda }(a_{-r}\cdot z^{\prime })dm(z^{\prime })={\int }_{B_h(0,\omega )}\left({\int }_{\partial \mathbb{D}}{e}^{(-i\lambda +1)〈a_r\cdot O,b〉}{e}^{(i\lambda +1)〈z^{\prime },b〉}db\right)dm(z^{\prime }),$$

and we exchange the order of integration:

We note that the integral ${\int }_{B_h(0,\omega )}{e}^{(i\lambda +1)〈z^{\prime },b〉}dm(z^{\prime })$ does not depend upon the variable $b=e^{i\varphi }$. Indeed:

$$\begin{array}{r}{\int }_{B_h(0,\omega )}{e}^{(i\lambda +1)〈z^{\prime },b〉}dm(z^{\prime })\\ =\frac{1}{2}{\int }_0^{\omega }{\int }_0^{2\pi }{\left(\frac{1-tanh{(x)}^2}{{|tanh(x)e^{i\theta }-e^{i\varphi }|}^2}\right)}^{\frac{i\lambda +1}{2}}sinh(2x)dxd\theta \\ =\frac{1}{2}{\int }_0^{\omega }{\int }_0^{2\pi }{\left(\frac{1-tanh{(x)}^2}{{|tanh(x)e^{i(\theta -\varphi )}-1|}^2}\right)}^{\frac{i\lambda +1}{2}}sinh(2x)dxd\theta \\ =\frac{1}{2}{\int }_0^{\omega }{\int }_0^{2\pi }{\left(\frac{1-tanh{(x)}^2}{{|tanh(x)e^{i{\theta }^{\prime }}-1|}^2}\right)}^{\frac{i\lambda +1}{2}}sinh(2x)dxd{\theta }^{\prime },\end{array}$$

and indeed the integral does not depend upon the variable b:

$${\int }_{B_h(0,\omega )}{e}^{(i\lambda +1)〈z^{\prime },b〉}dm(z^{\prime })={\int }_{B_h(0,\omega )}{e}^{(i\lambda +1)〈z^{\prime },1〉}dm(z^{\prime }).$$

Finally, we can write:

$$\begin{array}{rcl}{\int }_{B_h(0,\omega )}{\Phi }_{\lambda }(a_{-r}\cdot z^{\prime })dm(z^{\prime })& =& {\int }_{\partial \mathbb{D}}{e}^{(-i\lambda +1)〈a_r\cdot O,b〉}db{\int }_{B_h(0,\omega )}{e}^{(i\lambda +1)〈z^{\prime },1〉}dm(z^{\prime })\\ =& {\Phi }_{-\lambda }(a_r\cdot O){\int }_{B_h(0,\omega )}{e}^{(i\lambda +1)〈z^{\prime },1〉}dm(z^{\prime })\\ =& {\Phi }_{\lambda }(a_r\cdot O){\int }_{B_h(0,\omega )}{e}^{(i\lambda +1)〈z^{\prime },1〉}dm(z^{\prime }),\end{array}$$

because ${\Phi }_{\lambda }={\Phi }_{-\lambda }$ (as solutions of the same equation).

This completes the proof that:

$${\Phi }_{\lambda }(g^{-1}\cdot z)={\int }_{B_h(0,\omega )}{\Phi }_{\lambda }(a_{-r}\cdot z^{\prime })dm(z^{\prime })={\Phi }_{\lambda }(a_r\cdot O){\int }_{B_h(0,\omega )}{e}^{(i\lambda +1)〈z^{\prime },1〉}dm(z^{\prime }).$$

 □

5.2.3 Third step

We state a useful formula.

Lemma 5.2.2 For all$\omega >0$the following formula holds:

$${\int }_{B_h(0,\omega )}{\Phi }_{\lambda }(z)dm(z)=\pi sinh{(\omega )}^{2}cosh{(\omega )}^2{\Phi }_{\lambda }^{(1,1)}(\omega ).$$

Proof See Lemma F.0.4 of Appendix F. □

5.2.4 The main result

At this point we have proved the following proposition thanks to Propositions 5.2.1 and 5.2.4.

Proposition 5.2.5 If$z=tanh(r)e^{i\theta }\in B_h(0,\omega )$, $\mathcal{M}(r,\omega )$is given by the following formula:

$$\begin{array}{rcl}\mathcal{M}(r,\omega )& =& \frac{1}{4\pi }{\int }_{\mathbb{R}}\tilde{\mathbf{W}}(\lambda ){\Phi }_{\lambda }(a_r\cdot O){\Psi }_{\lambda }(\omega )\lambda tanh\left(\frac{\pi }{2}\lambda \right)d\lambda \\ =& \frac{1}{4\pi }{\int }_{\mathbb{R}}\tilde{\mathbf{W}}(\lambda ){\Phi }_{\lambda }^{(0,0)}(r){\Psi }_{\lambda }(\omega )\lambda tanh\left(\frac{\pi }{2}\lambda \right)d\lambda ,\end{array}$$

where

$${\Psi }_{\lambda }(\omega )\stackrel{\mathit{def}}{=}{\int }_{B_h(0,\omega )}{e}^{(i\lambda +1)〈z^{\prime },1〉}dm(z^{\prime }).$$

We are now in a position to obtain the analytic form for $\mathcal{M}(r,\omega )$ of Theorem 5.1.1. We prove that

$${\Psi }_{\lambda }(\omega )={\int }_{B_h(0,\omega )}{\Phi }_{\lambda }(z)dm(z).$$

Indeed, in hyperbolic polar coordinates, we have:

$$\begin{array}{rcl}{\Psi }_{\lambda }(\omega )& =& \frac{1}{2}{\int }_0^{\omega }{\int }_0^{2\pi }{e}^{(i\lambda +1)〈ro{t}_{\theta }{a}_r\cdot O,1〉}sinh(2r)drd\theta \\ =& \frac{1}{2}{\int }_0^{\omega }{\int }_0^{2\pi }{e}^{(i\lambda +1)〈a_r\cdot O,e^{-i\theta }〉}sinh(2r)drd\theta \\ =& \pi {\int }_0^{\omega }{\int }_{\partial \mathbb{D}}{e}^{(i\lambda +1)〈a_r\cdot O,b〉}dbsinh(2r)dr\\ =& \pi {\int }_0^{\omega }{\Phi }_{\lambda }(a_r\cdot O)sinh(2r)dr.\end{array}$$

On the other hand:

$$\begin{array}{rcl}{\int }_{B_h(0,\omega )}{\Phi }_{\lambda }(z)dm(z)& =& \frac{1}{2}{\int }_0^{\omega }{\int }_0^{2\pi }{\Phi }_{\lambda }(a_r\cdot O)sinh(2r)drd\theta \\ =& \pi {\int }_0^{\omega }{\Phi }_{\lambda }(a_r\cdot O)sinh(2r)dr.\end{array}$$

This yields

$${\Psi }_{\lambda }(\omega )={\int }_{B_h(0,\omega )}{\Phi }_{\lambda }(z)dm(z)=\pi sinh{(\omega )}^{2}cosh{(\omega )}^2{\Phi }_{\lambda }^{(1,1)}(\omega ),$$

and we use Lemma 5.2.2 to establish (24).

5.3 Linear stability analysis

We now analyse the evolution of small time-dependent perturbations of the hyperbolic stationary-pulse solution through linear stability analysis. We use classical tools already developped in [29, 31].

5.3.1 Spectral analysis of the linearized operator

Equation (20) is linearized about the stationary solution $V(r)$ by introducing the time-dependent perturbation:

$$v(z,t)=V(r)+\varphi (z,t).$$

This leads to the linear equation:

$${\partial }_t\varphi (z,t)=-\alpha \varphi (z,t)+{\int }_{\mathbb{D}}W(d_2(z,z^{\prime }))H^{\prime }(V(r^{\prime })-\kappa )\varphi (z^{\prime },t)dm(z^{\prime }).$$

We separate variables by setting $\varphi (z,t)=\varphi (z)e^{\beta t}$ to obtain the equation:

$$(\beta +\alpha )\varphi (z)={\int }_{\mathbb{D}}W(d_2(z,z^{\prime }))H^{\prime }(V(r^{\prime })-\kappa )\varphi (z^{\prime })dm(z^{\prime }).$$

Introducing the hyperbolic polar coordinates $z=tanh(r)e^{i\theta }$ and using the result:

$$H^{\prime }(V(r)-\kappa )=\delta (V(r)-\kappa )=\frac{\delta (r-\omega )}{|V^{\prime }(\omega )|}$$

we obtain:

$$\begin{array}{rcl}(\beta +\alpha )\varphi (z)& =& \frac{1}{2}{\int }_0^{\omega }{\int }_0^{2\pi }W\left(d_2(tanh(r)e^{i\theta },tanh(r^{\prime })e^{i{\theta }^{\prime }})\right)\frac{\delta (r^{\prime }-\omega )}{|V^{\prime }(\omega )|}\\ \phantom{\frac{1}{2}{\int }_0^{\omega }{\int }_0^{2\pi }}\times \varphi (tanh(r^{\prime })e^{i{\theta }^{\prime }})sinh(2r^{\prime })d{r}^{\prime }d{\theta }^{\prime }\\ =& \frac{sinh(2\omega )}{2|V^{\prime }(\omega )|}{\int }_0^{2\pi }W(d_2(tanh(r)e^{i\theta },tanh({\mathrm{r<}}^{}\end{array}$$