Analysis of a hyperbolic geometric model for visual texture perception

Open Access
Research

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.

Keywords

Neural fields nonlinear integro-differential equations functional analysis non-Euclidean analysis stability analysis hyperbolic geometry hypergeometric functions bumps 

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, 2, 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 ) Open image in new window denote the original image intensity function, where x and y are two spatial coordinates. Let I σ 1 Open image in new window denote the scale-space representation of I obtained by convolution with the Gaussian kernel g σ ( x , y ) = 1 2 π σ 2 e ( x 2 + y 2 ) / ( 2 σ 2 ) Open image in new window:
I σ 1 = I g σ 1 . Open image in new window
The gradient I σ 1 Open image in new window is a two-dimensional vector of coordinates I x σ 1 Open image in new window I y σ 1 Open image in new window which emphasizes image edges. One then forms the 2 × 2 Open image in new window symmetric matrix of rank one T 0 = I σ 1 ( I σ 1 ) T Open image in new window, where T indicates the transpose of a vector. The set of 2 × 2 Open image in new window symmetric positive semidefinite matrices of rank one will be noted S + ( 1 , 2 ) Open image in new window throughout the paper (see [16] for a complete study of the set S + ( p , n ) Open image in new window of n × n Open image in new window symmetric positive semidefinite matrices of fixed-rank p < n Open image in new window). By convolving T 0 Open image in new window componentwise with a Gaussian g σ 2 Open image in new window we finally form the tensor structure as the symmetric matrix:
T = T 0 g σ 2 = ( ( I x σ 1 ) 2 σ 2 I x σ 1 I y σ 1 σ 2 I x σ 1 I y σ 1 σ 2 ( I y σ 1 ) 2 σ 2 ) , Open image in new window
where we have set for example:
( I x σ 1 ) 2 σ 2 = ( I x σ 1 ) 2 g σ 2 . Open image in new window

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 σ 1 Open image in new window, 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 σ 1 Open image in new window. The second one, defined by σ 2 Open image in new window, 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, T Open image in new window is symmetric and non negative as det ( T ) 0 Open image in new window by the inequality of Cauchy-Schwarz, then it has two orthonormal eigenvectors e 1 Open image in new window, e 2 Open image in new window and two non negative corresponding eigenvalues λ 1 Open image in new window and λ 2 Open image in new window which we can always assume to be such that λ 1 λ 2 0 Open image in new window. 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 × 2 Open image in new window symmetric positive definite matrices, noted SPD ( 2 , R ) Open image in new window throughout the paper. The distribution of these eigenvalues in the ( λ 1 , λ 2 ) Open image in new window plane reflects the local organization of the image intensity variations. Indeed, each structure tensor can be written as the linear combination:
T = λ 1 e 1 e 1 T + λ 2 e 2 e 2 T = ( λ 1 λ 2 ) e 1 e 1 T + λ 2 ( e 1 e 1 T + e 2 e 2 T ) = ( λ 1 λ 2 ) e 1 e 1 T + λ 2 I 2 , Open image in new window
(1)

where I 2 Open image in new window is the identity matrix and e 1 e 1 T S + ( 1 , 2 ) Open image in new window. Some easy interpretations can be made for simple examples: constant areas are characterized by λ 1 = λ 2 0 Open image in new window, straight edges are such that λ 1 λ 2 0 Open image in new window, their orientation being that of e 2 Open image in new window, corners yield λ 1 λ 2 0 Open image in new window. The coherency c of the local image is measured by the ratio c = λ 1 λ 2 λ 1 + λ 2 Open image in new window, 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 T Open image in new window be a structure tensor. The time evolution of the average potential V ( T , t ) Open image in new window 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 × 2 Open image in new window symmetric definite-positive matrices:
{ t V ( T , t ) = α V ( T , t ) + SPD ( 2 ) W ( T , T , t ) S ( V ( T , t ) ) d T t V ( T , t ) = + I ext ( T , t ) t > 0 , V ( T , 0 ) = V 0 ( T ) . Open image in new window
(2)
The nonlinearity S is a sigmoidal function which may be expressed as:
S ( x ) = 1 1 + e μ x , Open image in new window

where μ describes the stiffness of the sigmoid. I ext Open image in new window is an external input.

The set SPD ( 2 ) Open image in new window can be seen as a foliated manifold by way of the set of special symmetric positive definite matrices SSPD ( 2 ) = SPD ( 2 ) SL ( 2 , R ) Open image in new window. Indeed, we have: SPD ( 2 ) = hom SSPD ( 2 ) × R + Open image in new window. Furthermore, SSPD ( 2 ) = isom D Open image in new window, where D Open image in new window is the Poincaré Disk, see, for example, [7]. As a consequence we use the following foliation of SPD ( 2 ) : SPD ( 2 ) = hom D × R + Open image in new window, which allows us to write for all T SPD ( 2 ) Open image in new window T = ( z , Δ ) Open image in new window with ( z , Δ ) D × R + Open image in new window. T Open image in new windowz and Δ are related by the relation det ( T ) = Δ 2 Open image in new window and the fact that z is the representation in D Open image in new window of T ˜ SSPD ( 2 ) Open image in new window with T = Δ T ˜ Open image in new window.

It is well-known [17] that D Open image in new window (and hence SSPD(2)) is a two-dimensional Riemannian space of constant sectional curvature equal to −1 for the distance noted d 2 Open image in new window defined by
d 2 ( z , z ) = arctanh | z z | | 1 z ¯ z | . Open image in new window
The isometries of D Open image in new window, that are the transformations that preserve the distance d 2 Open image in new window are the elements of unitary group U ( 1 , 1 ) Open image in new window. 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 Open image in new window defined by
d 0 ( T , T ) = 2 ( log Δ log Δ ) 2 + d 2 2 ( z , z ) . Open image in new window
As shown in Proposition B.0.1 of Appendix B it is possible to express the volume element d T Open image in new window in ( z 1 , z 2 , Δ ) Open image in new window coordinates with z = z 1 + i z 2 Open image in new window:
d T = 8 2 d Δ Δ d z 1 d z 2 ( 1 | z | 2 ) 2 . Open image in new window
We note dm ( z ) = d z 1 d z 2 ( 1 | z | 2 ) 2 Open image in new window and equation (2) can be written in ( z , Δ ) Open image in new window coordinates:
t V ( z , Δ , t ) = α V ( z , Δ , t ) + 8 2 0 + D W ( z , Δ , z , Δ , t ) S ( V ( z , Δ , t ) ) d Δ Δ dm ( z ) + I ext ( z , Δ , t ) . Open image in new window
We get rid of the constant 8 2 Open image in new window by redefining W as 8 2 W Open image in new window.
{ t V ( z , Δ , t ) = α V ( z , Δ , t ) t V ( z , Δ , t ) = + 0 + D W ( z , Δ , z , Δ , t ) S ( V ( z , Δ , t ) ) d Δ Δ dm ( z ) t V ( z , Δ , t ) = + I ext ( z , Δ , t ) t > 0 , V ( z , Δ , 0 ) = V 0 ( z , Δ ) . Open image in new window
(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 , R ) Open image in new window and S + ( 1 , 2 ) Open image in new window. First of all, when σ 2 Open image in new window goes to zero, that is when the characteristic size of the structure becomes very small, we have T 0 g σ 2 T 0 Open image in new window, which means that the tensor T SPD ( 2 , R ) Open image in new window degenerates to a tensor T 0 S + ( 1 , 2 ) Open image in new window, which can be interpreted as the loss of one dimension. We can write each T 0 S + ( 1 , 2 ) Open image in new window as T 0 = x x T = r 2 u u T Open image in new window, where u = ( cos θ , sin θ ) T Open image in new window and ( r , θ ) Open image in new window is the polar representation of x. Since, x and −x correspond to the same T 0 Open image in new windowθ is equated to θ + k π Open image in new window k Z Open image in new window. Thus S + ( 1 , 2 ) = R + × P 1 Open image in new window, where P 1 Open image in new window is the real projective space of dimension 1 (lines of R 2 Open image in new window). Then the integration scale σ 2 Open image in new window, 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 λ 1 λ 2 0 Open image in new window then T ( λ 1 λ 2 ) e 1 e 1 T S + ( 1 , 2 ) Open image in new window and the orientation is that of e 2 Open image in new window. We denote by P Open image in new window the projection of a 2 × 2 Open image in new window symmetric definite positive matrix on the set S + ( 1 , 2 ) Open image in new window defined by:
P : { SPD ( 2 , R ) S + ( 1 , 2 ) , T τ = ( λ 1 λ 2 ) e 1 e 1 T , Open image in new window
where T Open image in new window is as in equation (1). We can introduce a metric on the set S + ( 1 , 2 ) Open image in new window 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 S + ( 1 , 2 ) = R + × P 1 Open image in new window, a metric on the space S + ( 1 , 2 ) Open image in new window is given by:
d s 2 = 2 ( d r r ) 2 + d θ 2 . Open image in new window
The space S + ( 1 , 2 ) Open image in new window endowed with this metric is a Riemannian manifold (see [16]). Finally, the distance associated to this metric is given by:
d S + ( 1 , 2 ) 2 ( τ 1 , τ 2 ) = 2 log 2 ( r 1 r 2 ) + | θ 1 θ 2 | 2 , Open image in new window
where τ 1 = x 1 T x 1 Open image in new window τ 2 = x 2 T x 2 Open image in new window and ( r i , θ i ) Open image in new window denotes the polar coordinates of x i Open image in new window for i = 1 , 2 Open image in new window. The volume element in ( r , θ ) Open image in new window coordinates is:
d τ = d r r d θ π , Open image in new window

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

Let now τ = P ( T ) Open image in new window be a symmetric positive semidefinite matrix. The average potential V ( τ , t ) Open image in new window 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 S + ( 1 , 2 ) Open image in new window:
t V ( τ , t ) = α V ( τ , t ) + S + ( 1 , 2 ) W ( τ , τ , t ) S ( V ( τ , t ) ) d τ + I ext ( τ , t ) t > 0 . Open image in new window
(4)
In ( r , θ ) Open image in new window coordinates, (4) is rewritten as:
t V ( r , θ , t ) = α V ( r , θ , t ) + 0 + 0 π W ( r , θ , r , θ , t ) S ( V ( r , θ , t ) ) d θ π d r r + I ext ( r , θ , t ) . Open image in new window
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 ( τ , τ , t ) = w ( d S + ( 1 , 2 ) ( τ , τ ) ) = w ( 2 log 2 ( r r ) + | θ θ | 2 ) , Open image in new window
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:
t V ( θ , t ) = α V ( θ , t ) + 0 π w sh ( θ θ ) S ( V ( θ , t ) ) d θ π + I ext ( θ , t ) , Open image in new window
(5)
where:
w sh ( θ ) = 0 + w ( 2 log 2 ( r ) + θ 2 ) d r r . Open image in new window

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 T Open image in new window. 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 D Open image in new window.

3.1 Homogeneous solutions

A homogeneous solution to (2) is a solution V that does not depend upon the tensor variable T Open image in new window for a given homogenous input I ( t ) Open image in new window and a constant initial condition V 0 Open image in new window. In ( z , Δ ) Open image in new window coordinates, a homogeneous solution of (3) is defined by:
V ˙ ( t ) = α V ( t ) + W ¯ ( z , Δ , t ) S ( V ( t ) ) + I ext ( t ) , Open image in new window
where:
W ¯ ( z , Δ , t ) = def 0 + D W ( z , Δ , z , Δ , t ) d Δ Δ d z 1 d z 2 ( 1 | z | 2 ) 2 . Open image in new window
(6)

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

In the special case where W ( z , Δ , z , Δ , t ) Open image in new window is a function of only the distance d 0 Open image in new window between ( z , Δ ) Open image in new window and ( z , Δ ) Open image in new window:
W ( z , Δ , z , Δ , t ) w ( 2 ( log Δ log Δ ) 2 + d 2 2 ( z , z ) , t ) Open image in new window
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:
{ V ˙ ( t ) = α V ( t ) + W ¯ ( t ) S ( V ( t ) ) + I ext ( t ) , t > 0 , V ( 0 ) = V 0 . Open image in new window
(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 ext ( t ) Open image in new windowand the connectivity function W ¯ ( t ) Open image in new windoware continuous on some closed interval J containing 0, then for all V 0 Open image in new windowin R Open image in new window, there exists a unique solution of (7) defined on a subinterval J 0 Open image in new windowof J containing 0 such that V ( 0 ) = V 0 Open image in new window.

Proof It is a direct application of Cauchy’s theorem on differential equations. We consider the mapping f : J × R R Open image in new window defined by:
f ( t , x ) = α x + W ¯ ( t ) S ( x ) + I ext ( t ) . Open image in new window
It is clear that f is continuous from J × R Open image in new window to R Open image in new window. We have for all x , y R Open image in new window and t J Open image in new window:
| f ( t , x ) f ( t , y ) | α | x y | + | W ¯ ( t ) | S m | x y | , Open image in new window

where S m = sup x R | S ( x ) | Open image in new window.

Since, W ¯ Open image in new window is continuous on the compact interval J, it is bounded there by C > 0 Open image in new window and:
| f ( t , x ) f ( t , y ) | ( α + C S m ) | x y | . Open image in new window

 □

We can extend this result to the whole time real line if I and W ¯ Open image in new window are continuous on R Open image in new window.

Proposition 3.1.1 If I ext Open image in new windowand W ¯ Open image in new windoware continuous on R + Open image in new window, then for all V 0 Open image in new windowin R Open image in new window, there exists a unique solution of (7) defined on R + Open image in new windowsuch that V ( 0 ) = V 0 Open image in new window.

Proof We have already shown the following inequality:
| f ( t , x ) f ( t , y ) | α | x y | + | W ¯ ( t ) | S m | x y | . Open image in new window
Then f is locally Lipschitz with respect to its second argument. Let V be a maximal solution on J 0 Open image in new window and we denote by β the upper bound of J 0 Open image in new window. We suppose that β < + Open image in new window. Then we have for all t 0 Open image in new window:
V ( t ) = e α t V 0 + 0 t e α ( t u ) W ¯ ( u ) S ( V ( u ) ) d u + 0 t e α ( t u ) I ext ( u ) d u | V ( t ) | | V 0 | + S m 0 β e α u | W ¯ ( u ) | d u + 0 β e α u | I ext ( u ) | d u t [ 0 , β ] , Open image in new window

where S m = sup x R | S ( x ) | Open image in new window.

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

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 D × R + Open image in new window, then W ¯ ( z , Δ , t ) Open image in new window did not depend upon the variables ( z , Δ ) Open image in new window. 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 ( T , T ) Open image in new windowonly. Then W ¯ Open image in new windowdoes not depend upon the variable T Open image in new window.

Mexican hat connectivity In this paragraph, we push further the computation of W ¯ Open image in new window 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 ( T , T ) Open image in new window.

In detail, we have:
W ( z , Δ , z Δ ) = w ( 2 ( log Δ log Δ ) 2 + d 2 2 ( z , z ) ) , Open image in new window
where:
w ( x ) = 1 2 π σ 1 2 e x 2 σ 1 2 A 2 π σ 2 2 e x 2 σ 2 2 Open image in new window

with 0 σ 1 σ 2 Open image in new window and 0 A 1 Open image in new window.

In this case we can obtain a very simple closed-form formula for W ¯ Open image in new window as shown in the following lemma.

Lemma 3.1.2 When W is the specific Mexican hat function just defined then:
W ¯ = π 3 2 2 ( σ 1 e 2 σ 1 2 erf ( 2 σ 1 ) A σ 2 e 2 σ 2 2 erf ( 2 σ 2 ) ) , Open image in new window
(8)
where erf is the error function defined as:
erf ( x ) = 2 π 0 x e u 2 d u . Open image in new window

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 × R + Open image in new window. Let J be a subinterval of R Open image in new window. We assume that:

Equivalently, we can express these hypotheses in ( z , Δ ) Open image in new window coordinates:

3.2.1 Functional space setting

We introduce the following mapping f g : ( t , ϕ ) f g ( t , ϕ ) Open image in new window such that:
f g ( t , ϕ ) ( z , Δ ) = D × R + W ( d 2 ( z , z ) , | log ( Δ Δ ) | , t ) S ( ϕ ( z , Δ ) ) d Δ Δ dm ( z ) . Open image in new window
(9)

Our aim is to find a functional space F Open image in new window where (3) is well-defined and the function f g Open image in new window maps F Open image in new window to F Open image in new window for all t s. A natural choice would be to choose ϕ as a L p ( D × R + ) Open image in new window-integrable function of the space variable with 1 p < + Open image in new window. Unfortunately, the homogeneous solutions (constant with respect to ( z , Δ ) Open image in new window) 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 F = L ( D × R + ) Open image in new window. As D × R + Open image in new window is an open set of R 3 Open image in new window, F Open image in new window is a Banach space for the norm: ϕ F = sup z D sup Δ R + | ϕ ( z , Δ ) | Open image in new window.

Proposition 3.2.1 If I ext C ( J , F ) Open image in new windowwith sup t J I ext ( t ) F < + Open image in new windowand W satisfies hypotheses (H 1bis)-(H 3bis) then f g Open image in new windowis well-defined and is from J × F Open image in new windowto F Open image in new window.

Proof ( z , Δ , t ) D × R + × R Open image in new window, we have:
| D × R + W ( d 2 ( z , z ) , | log ( Δ Δ ) | , t ) S ( ϕ ( z , Δ ) ) d Δ Δ dm ( z ) | S m sup t J W ( t ) L 1 < + . Open image in new window

 □

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

We rewrite (3) as a Cauchy problem:
{ t V ( z , Δ , t ) = α V ( z , Δ , t ) t V ( z , Δ , t ) = + D × R + W ( d 2 ( z , z ) , | log ( Δ Δ ) | , t ) t V ( z , Δ , t ) = + D × R + × S ( V ( z , Δ , t ) ) d Δ Δ dm ( z ) t V ( z , Δ , t ) = + I ext ( z , Δ , t ) , V ( z , Δ , 0 ) = V 0 ( z , Δ ) . Open image in new window
(10)

Theorem 3.2.1 If the external current I ext Open image in new windowbelongs to C ( J , F ) Open image in new windowwith J an open interval containing 0 and W satisfies hypotheses (H 1bis)-(H 3bis), then fo all V 0 F Open image in new window, there exists a unique solution of (10) defined on a subinterval J 0 Open image in new windowof J containing 0 such that V ( z , Δ , 0 ) = V 0 ( z , Δ ) Open image in new windowfor all ( z , Δ ) D × R + Open image in new window.

Proof We prove that f g Open image in new window is continuous on J × F Open image in new window. We have
and therefore
f g ( t , ϕ ) f g ( s , ψ ) F S m sup t J W ( t ) L 1 ϕ ψ F + S m W ( t ) W ( s ) L 1 . Open image in new window
Because of condition (H2) we can choose | t s | Open image in new window small enough so that W ( t ) W ( s ) L 1 Open image in new window is arbitrarily small. This proves the continuity of f g Open image in new window. Moreover it follows from the previous inequality that:
f g ( t , ϕ ) f g ( t , ψ ) F S m W 0 g ϕ ψ F Open image in new window

with W 0 g = sup t J W ( t ) L 1 Open image in new window. This ensures the Lipschitz continuity of f g Open image in new window 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 R n Open image in new windowbut 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 ext Open image in new windowbelongs to C ( R + , F ) Open image in new windowand W satisfies hypotheses (H 1bis)-(H 3bis) with J = R + Open image in new window, then for all V 0 F Open image in new window, there exists a unique solution of (10) defined on R + Open image in new windowsuch that V ( z , Δ , 0 ) = V 0 ( z , Δ ) Open image in new windowfor all ( z , Δ ) D × R + Open image in new window.

Proof We have just seen in the previous proof that f g Open image in new window is globally Lipschitz with respect to its second argument:
f g ( t , ϕ ) f g ( t , ψ ) F S m W 0 g ϕ ψ F Open image in new window

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 ext Open image in new windowbelongs to C ( R + , F ) Open image in new windowand is bounded in time sup t R + I ext ( t ) F < + Open image in new windowand W satisfies hypotheses (H 1bis)-(H 3bis) with J = R + Open image in new window, then the solution of (10) is bounded for each initial condition V 0 F Open image in new window.

Let us set:
ρ g = def 2 α ( S m W 0 g + sup t R + I ext ( t ) F ) , Open image in new window

where W 0 g = sup t R + W ( t ) L 1 Open image in new window.

Proof Let V be a solution defined on R + Open image in new window. Then we have for all t R + Open image in new window:
The following upperbound holds
V ( t ) F e α t V 0 F + 1 α ( S m W 0 g + sup t R + I ext ( t ) F ) ( 1 e α t ) . Open image in new window
(11)
We can rewrite (11) as:
V ( t ) F e α t ( V 0 F 1 α ( S m W 0 g + sup t R + I ext ( t ) F ) ) + 1 α ( S m W 0 + g sup t R + I ext ( t ) F ) = e α t ( V 0 F ρ g 2 ) + ρ g 2 . Open image in new window
(12)
If V 0 B ρ g Open image in new window this implies V ( t ) F ρ g 2 ( 1 + e α t ) Open image in new window for all t > 0 Open image in new window and hence V ( t ) F < ρ g Open image in new window for all t > 0 Open image in new window, proving that B ρ Open image in new window is stable. Now assume that V ( t ) F > ρ g Open image in new window for all t 0 Open image in new window. The inequality (12) shows that for t large enough this yields a contradiction. Therefore there exists t 0 > 0 Open image in new window such that V ( t 0 ) F = ρ g Open image in new window. At this time instant we have
ρ g e α t 0 ( V 0 F ρ g 2 ) + ρ g 2 , Open image in new window
and hence
t 0 1 α log ( 2 V 0 F ρ g ρ g ) . Open image in new window

 □

The following corollary is a consequence of the previous proposition.

Corollary 3.2.1 If V 0 B ρ g Open image in new windowand T g = inf { t > 0 such that V ( t ) B ρ g } Open image in new windowthen:
T g 1 α log ( 2 V 0 F ρ g ρ g ) . Open image in new window

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:
{ t V ( z , t ) = α V ( z , t ) + D W sh ( z , z , t ) S ( V ( z , t ) ) dm ( z ) t V ( z , t ) = + I ext ( z , t ) , t > 0 , V ( z , 0 ) = V 0 ( z ) , Open image in new window
(13)
where
W sh ( z , z , t ) = 0 + W ( z , Δ , z , Δ , t ) d Δ Δ . Open image in new window

We have implicitly made the assumption, that W sh Open image in new window 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 R Open image in new window. We assume that:

Note that conditions (C1)-(C2) and Lemma 3.1.1 imply that for all z D Open image in new window, D | W sh ( z , z , t ) | dm ( z ) = W sh ( t ) L 1 Open image in new window. And then, for all z D Open image in new window, the mapping z W sh ( z , z , t ) Open image in new window is integrable on D Open image in new window.

From now on, F = L ( D ) Open image in new window and the Fischer-Riesz’s theorem ensures that L ( D ) Open image in new window is a Banach space for the norm: ψ = inf { C 0 , | ψ ( z ) | C for almost every z D } Open image in new window.

Theorem 3.3.1 If the external current I ext Open image in new windowbelongs to C ( J , F ) Open image in new windowwith J an open interval containing 0 and W sh Open image in new windowsatisfies conditions (C 1)-(C 3), then for all V 0 F Open image in new window, there exists a unique solution of (13) defined on a subinterval J 0 Open image in new windowof J containing 0.

This solution, defined on the subinterval J of R Open image in new window 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 ext Open image in new windowbelongs to C ( R + , F ) Open image in new windowand W sh Open image in new windowsatisfies conditions (C 1)-(C 3) with J = R + Open image in new window, then for all V 0 F Open image in new window, there exists a unique solution of (13) defined on R + Open image in new window.

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

Proposition 3.3.2 Let ρ = def 2 α ( S m W 0 sh + sup t R + I ( t ) F ) Open image in new window, with W 0 sh = sup t J W sh ( t ) L 1 Open image in new window. The open ball B ρ Open image in new windowof F Open image in new windowof 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 B ρ Open image in new windowand T = inf { t > 0 such that V ( t ) B ρ } Open image in new windowthen:
T 1 α log ( 2 V 0 F ρ ρ ) . Open image in new window

4 Stationary solutions

We look at the equilibrium states, noted V μ 0 Open image in new window 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 ) = 1 1 + e x , Open image in new window
so that a stationary solution (independent of time) satisfies:
0 = α V μ 0 ( z , Δ ) + D × R + W ( d 2 ( z , z ) , | log ( Δ Δ ) | ) S ( μ V μ 0 ( z , Δ ) ) d Δ Δ dm ( z ) + I ext ( z , Δ ) . Open image in new window
(14)
We define the nonlinear operator from F Open image in new window to F Open image in new window, noted G μ Open image in new window, by:
G μ ( V ) ( z , Δ ) = D × R + W ( d 2 ( z , z ) , | log ( Δ Δ ) | ) S ( μ V ( z , Δ ) ) d Δ Δ dm ( z ) . Open image in new window
(15)
Finally, (14) is equivalent to:
α V μ 0 ( z , Δ ) = G μ ( V ) ( z , Δ ) + I ext ( z , Δ ) . Open image in new window

4.1 Study of the nonlinear operator G μ Open image in new window

We recall that we have set for the Banach space F = L ( D × R + ) Open image in new window and Proposition 3.2.1 shows that G μ : F F Open image in new window. We have the further properties:

Proposition 4.1.1 G μ Open image in new windowsatisfies the following properties:

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:
G μ 1 ( V ) G μ 2 ( V ) F | μ 1 μ 2 | W 0 g S m V F . Open image in new window

 □

We denote by G l Open image in new window and G Open image in new window the two operators from F Open image in new window to F Open image in new window defined as follows for all V F Open image in new window and all ( z , Δ ) D × R + Open image in new window:
G l ( V ) ( z , Δ ) = D × R + W ( d 2 ( z , z ) , | log ( Δ Δ ) | ) V ( z , Δ ) d Δ Δ dm ( z ) , Open image in new window
(16)
and
G ( V ) ( z , Δ ) = D × R + W ( d 2 ( z , z ) , | log ( Δ Δ ) | ) H ( V ( z , Δ ) ) d Δ Δ dm ( z ) , Open image in new window

where H is the Heaviside function.

It is straightforward to show that both operators are well-defined on F Open image in new window and map F Open image in new window to F Open image in new window. Moreover the following proposition holds.

Proposition 4.1.2 We have
G μ μ G . Open image in new window
Proof It is a direct application of the dominated convergence theorem using the fact that:
S ( μ y ) μ H ( y ) a.e. y R . Open image in new window

 □

4.2 The convolution form of the operator G μ Open image in new window in the semi-homogeneous case

It is convenient to consider the functional space F sh = L ( D ) Open image in new window to discuss semi-homogeneous solutions. A semi-homogeneous persistent state of (3) is deduced from (14) and satisfies:
α V μ 0 ( z ) = G μ sh ( V μ 0 ) ( z ) + I ext ( z ) , Open image in new window
(17)
where the nonlinear operator G μ sh Open image in new window from F sh Open image in new window to F sh Open image in new window is defined for all V F sh Open image in new window and z D Open image in new window by:
G μ sh ( V ) ( z ) = D W sh ( d 2 ( z , z ) ) S ( μ V ( z ) ) dm ( z ) . Open image in new window
We define the associated operators, G l sh , G sh Open image in new window:
G l sh ( V ) ( z ) = D W sh ( d 2 ( z , z ) ) V ( z ) dm ( z ) , G sh ( V ) ( z ) = D W sh ( d 2 ( z , z ) ) H ( V ( z ) ) dm ( z ) . Open image in new window
We rewrite the operator G μ sh Open image in new window 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 Open image in new window and dg denote the Haar measure on the group G = SU ( 1 , 1 ) Open image in new window (see [22] and Appendix A), normalized by:
G f ( g O ) d g = def D f ( z ) dm ( z ) , Open image in new window
for all functions of L 1 ( D ) Open image in new window. Given two functions f 1 , f 2 Open image in new window in L 1 ( D ) Open image in new window we define the convolution ∗ by:
( f 1 f 2 ) ( z ) = G f 1 ( g O ) f 2 ( g 1 z ) d g . Open image in new window

We recall the notation W sh ( z ) = def W sh ( d 2 ( z , O ) ) Open image in new window.

Proposition 4.2.1 For all μ 0 Open image in new windowand V F sh Open image in new windowwe have:
G μ sh ( V ) = W sh S ( μ V ) , G l sh ( V ) = W sh V and G sh ( V ) = W sh H ( V ) . Open image in new window
(18)
Proof We only prove the result for G μ Open image in new window. Let z D Open image in new window, then:
G μ sh ( V ) ( z ) = D W sh ( d 2 ( z , z ) ) S ( μ V ( z ) ) dm ( z ) = G W sh ( d 2 ( z , g O ) ) S ( μ V ( g O ) ) d g = G W sh ( d 2 ( g g 1 z , g O ) ) S ( μ V ( g O ) ) d g Open image in new window
and for all g SU ( 1 , 1 ) Open image in new window, d 2 ( z , z ) = d 2 ( g z , g z ) Open image in new window so that:
G μ sh ( V ) ( z ) = G W sh ( d 2 ( g 1 z , O ) ) S ( μ V ( g O ) ) d g = W sh S ( μ V ) ( z ) . Open image in new window

 □

Let b be a point on the circle D Open image in new window. For z D Open image in new window, we define the ‘inner product’ z , b Open image in new window to be the algebraic distance to the origin of the (unique) horocycle based at b through z (see [7]). Note that z , b Open image in new window does not depend on the position of z on the horocycle. The Fourier transform in D Open image in new window is defined as (see [22]):
h ˜ ( λ , b ) = D h ( z ) e ( i λ + 1 ) z , b dm ( z ) ( λ , b ) R × D Open image in new window

for a function h : D C Open image in new window such that this integral is well-defined.

Lemma 4.2.1 The Fourier transform in D Open image in new window, W ˜ sh ( λ , b ) Open image in new windowof W sh Open image in new windowdoes not depend upon the variable b D Open image in new window.

Proof For all λ R Open image in new window and b = e i θ D Open image in new window,
W ˜ sh ( λ , b ) = D W sh ( z ) e ( i λ + 1 ) z , b dm ( z ) . Open image in new window
We recall that for all ϕ R Open image in new window r ϕ Open image in new window is the rotation of angle ϕ and we have W sh ( r ϕ z ) = W sh ( z ) Open image in new window, dm ( z ) = dm ( r ϕ z ) Open image in new window and z , b = r ϕ z , r ϕ b Open image in new window, then:
W ˜ sh ( λ , b ) = D W sh ( r θ z ) e ( i λ + 1 ) r θ z , 1 dm ( z ) = D W sh ( z ) e ( i λ + 1 ) z , 1 dm ( z ) = def W ˜ sh ( λ ) . Open image in new window

 □

We now introduce two functions that enjoy some nice properties with respect to the Hyperbolic Fourier transform and are eigenfunctions of the linear operator G l sh Open image in new window.

Proposition 4.2.2 Let e λ , b ( z ) = e ( i λ + 1 ) z , b Open image in new windowand Φ λ ( z ) = D e ( i λ + 1 ) z , b d b Open image in new windowthen:

Proof We begin with b = 1 D Open image in new window and use the horocyclic coordinates. We use the same changes of variables as in Lemma 3.1.1:
G l sh ( e λ , 1 ) ( n s a t O ) = R 2 W sh ( d 2 ( n s a t O , n s a t O ) ) e ( i λ 1 ) t d t d s = R 2 W sh ( d 2 ( n s s a t O , a t O ) ) e ( i λ 1 ) t d t d s = R 2 W sh ( d 2 ( a t n x O , a t O ) ) e ( i λ 1 ) t + 2 t d t d x = R 2 W sh ( d 2 ( O , n x a t t O ) ) e ( i λ 1 ) t + 2 t d t d x = R 2 W sh ( d 2 ( O , n x a y O ) ) e ( i λ 1 ) ( y + t ) + 2 t d y d x = e ( i λ + 1 ) n s a t O , 1 W ˜ sh ( λ ) . Open image in new window

By rotation, we obtain the property for all b D Open image in new window.

For the second property [22], Lemma 4.7] shows that:
W sh Φ λ ( z ) = D e ( i λ + 1 ) z , b W ˜ sh ( λ ) d b = Φ λ ( z ) W ˜ sh ( λ ) . Open image in new window

 □

A consequence of this proposition is the following lemma.

Lemma 4.2.2 The linear operator G l sh Open image in new windowis not compact and for all μ 0 Open image in new window, the nonlinear operator G μ sh Open image in new windowis not compact.

Proof The previous Proposition 4.2.2 shows that G l sh Open image in new window has a continuous spectrum which iimplies that is not a compact operator.

Let U be in F sh Open image in new window, for all V F sh Open image in new window we differentiate G μ sh Open image in new window and compute its Frechet derivative:
D ( G μ sh ) U ( V ) ( z ) = D W sh ( d 2 ( z , z ) ) S ( U ( z ) ) V ( z ) dm ( z ) . Open image in new window
If we assume further that U does not depend upon the space variable z, U ( z ) = U 0 Open image in new window we obtain:
D ( G μ sh ) U 0 ( V ) ( z ) = S ( U 0 ) G l sh ( V ) ( z ) . Open image in new window

If G μ sh Open image in new window was a compact operator then its Frechet derivative D ( G μ sh ) U 0 Open image in new window would also be a compact operator, but it is impossible. As a consequence, G μ sh Open image in new window is not a compact operator. □

4.3 The convolution form of the operator G μ Open image in new window 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 d x x Open image in new window. For two functions f 1 Open image in new window, f 2 Open image in new window in L 1 ( D × R + ) Open image in new window we define the convolution ⋆ by:
( f 1 f 2 ) ( z , Δ ) = def G H f 1 ( g O , h 1 ) f 2 ( g 1 z , h 1 Δ ) d g d h . Open image in new window

We recall that we have set by definition: W ( z , Δ ) = W ( d 2 ( z , 0 ) , | log ( Δ ) | ) Open image in new window.

Proposition 4.3.1 For all μ 0 Open image in new windowand V F Open image in new windowwe have:
G μ ( V ) = W S ( μ V ) , G l ( V ) = W V and G ( V ) = W H ( V ) . Open image in new window
(19)
Proof Let ( z , Δ ) Open image in new window be in D × R + Open image in new window. We follow the same ideas as in Proposition 4.2.1 and prove only the first result. We have
G μ ( V ) ( z , Δ ) = D × R + W ( d 2 ( z , z ) , | log ( Δ Δ ) | ) S ( μ V ( z , Δ ) ) d Δ Δ dm ( z ) = G R + W ( d 2 ( g 1 z , O ) , | log ( Δ Δ ) | ) S ( μ V ( g O , Δ ) ) d g d Δ Δ = G H W ( d 2 ( g 1 z , O ) , | log ( h 1 Δ ) | ) S ( μ V ( g O , h 1 ) ) d g d h = W S ( μ V ) ( z , Δ ) . Open image in new window

 □

We next assume further that the function W is separable in z, Δ and more precisely that W ( z , Δ ) = W 1 ( z ) W 2 ( log ( Δ ) ) Open image in new window where W 1 ( z ) = W 1 ( d 2 ( z , 0 ) ) Open image in new window and W 2 ( log ( Δ ) ) = W 2 ( | log ( Δ ) | ) Open image in new window for all ( z , Δ ) D × R + Open image in new window. The following proposition is an echo to Proposition 4.2.2.

Proposition 4.3.2 Let e λ , b ( z ) = e ( i λ + 1 ) z , b Open image in new window, Φ λ ( z ) = D e ( i λ + 1 ) z , b d b Open image in new windowand h ξ ( Δ ) = e i ξ log ( Δ ) Open image in new windowthen:

where W ˆ 2 Open image in new windowis the usual Fourier transform of W 2 Open image in new window.

Proof The proof of this proposition is exactly the same as for Proposition 4.2.2. Indeed:
G l ( e λ , b h ξ ) ( z , Δ ) = W 1 e λ , b ( z ) R + W 2 ( log ( Δ Δ ) ) e i ξ log ( Δ ) d Δ Δ = W 1 e λ , b ( z ) ( R W 2 ( y ) e i ξ y d y ) e i ξ log ( Δ ) . Open image in new window

 □

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 G l sh Open image in new windowis not compact and for all μ 0 Open image in new window, the nonlinear operator G μ sh Open image in new windowis not compact.

4.4 The set of the solutions of (14)

Let B μ Open image in new window be the set of the solutions of (14) for a given slope parameter μ:
B μ = { V F | α V + G μ ( V ) + I ext = 0 } . Open image in new window

We have the following proposition.

Proposition 4.4.1 If the input current I ext Open image in new windowis equal to a constant I ext 0 Open image in new window, that is, does not depend upon the variables ( z , Δ ) Open image in new windowthen for all μ R + Open image in new window, B μ Open image in new window. In the general case I ext F Open image in new window, if the condition μ S m W 0 g < α Open image in new windowis satisfied, then Card ( B μ ) = 1 Open image in new window.

Proof Due to the properties of the sigmoid function, there always exists a constant solution in the case where I ext Open image in new window is constant. In the general case where I ext F Open image in new window, 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 , Δ ) Open image in new windowand if the condition μ S m W 0 g < α Open image in new windowis 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 , Δ ) Open image in new windowbecause there always exists one constant stationary solution when the external input does not depend upon the variables ( z , Δ ) Open image in new window. Indeed equation (14) is then equivalent to:
0 = α V 0 + β S ( V 0 ) + I ext 0 where β = D × R + W ( d 2 ( z , z ) , | log ( Δ Δ ) | ) d Δ Δ dm ( z ) Open image in new window

and β does not depend upon the variables ( z , Δ ) Open image in new windowbecause of Lemma 3.1.1. Because of the property of the sigmoid function S, the equation 0 = α V 0 + β S ( V 0 ) + I ext 0 Open image in new windowhas 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 ) Open image in new windowthe group of the isometries of D Open image in new window (see Appendix A), and the condition μ S m W 0 g < α Open image in new windowis 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 μ S m W 0 g < α Open image in new windowis satisfied we call primary stationary solution the unique solution in B μ Open image in new window.

4.5 Stability of the primary stationary solution

In this subsection we show that the condition μ S m W 0 g < α Open image in new window guarantees the stability of the primary stationary solution to (3).

Theorem 4.5.1 We suppose that I F Open image in new windowand that the condition μ S m W 0 g < α Open image in new windowis satisfied, then the associated primary stationary solution of (3) is asymtotically stable.

Proof Let V μ 0 Open image in new window be the primary stationary solution of (3), as μ S m W 0 g < α Open image in new window is satisfied. Let also V μ Open image in new window be the unique solution of the same equation with some initial condition V μ ( 0 ) = ϕ F Open image in new window, see Theorem 3.3.1. We introduce a new function X = V μ V μ 0 Open image in new window which satisfies:
where W m ( d 2 ( z , z ) , | log ( Δ Δ ) | ) = S m W ( d 2 ( z , z ) , | log ( Δ Δ ) | ) Open image in new window and the vector Θ ( X ( z , Δ , t ) ) Open image in new window is given by Θ ( X ( z , Δ , t ) ) = S ̲ ( μ V μ ( z , Δ , t ) ) S ̲ ( μ V μ 0 ( z , Δ ) ) Open image in new window with S ̲ = ( S m ) 1 S Open image in new window. We note that, because of the definition of Θ and the mean value theorem | Θ ( X ( z , Δ , t ) ) | μ | X ( z , Δ , t ) | Open image in new window. This implies that | Θ ( r ) | | r | Open image in new window for all r R Open image in new window.
If we set: G ( t ) = e α t X ( t ) Open image in new window, then we have:
G ( t ) G ( 0 ) + μ W 0 g S m 0 t G ( u ) d u Open image in new window
and G is continuous for all t 0 Open image in new window. The Gronwall inequality implies that:
G ( t ) G ( 0 ) e μ W 0 g S m t X ( t ) e ( μ W 0 g S m α ) t X ( 0 ) , Open image in new window

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, 26, 27, 28, 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, 26, 27, 28, 29], we consider the high gain limit μ Open image in new window of the sigmoid function. As above we denote by H the Heaviside function defined by H ( x ) = 1 Open image in new window for x 0 Open image in new window and H ( x ) = 0 Open image in new window otherwise. Equation (13) is rewritten as:
t V ( z , t ) = α V ( z , t ) + D W ( z , z ) H ( V ( z , t ) κ ) dm ( z ) + I ( z , t ) = α V ( z , t ) + { z D | V ( z , t ) κ } W ( z , z ) dm ( z ) + I ( z ) . Open image in new window
(20)
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 D : W ( z , z ) = W ( d 2 ( z , z ) ) Open image in new window. For illustrative purposes, we will use the exponential weight distribution as a specific example throughout this section:
W ( z , z ) = W ( d 2 ( z , z ) ) = exp ( d 2 ( z , z ) b ) . Open image in new window
(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, 26, 27, 28, 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:
α V ( z ) = { z D | V ( z ) κ } W ( z , z ) dm ( z ) + I ext ( z ) . Open image in new window

For convenience, we note M ( z , K ) Open image in new window the integral K W ( z , z ) dm ( z ) Open image in new window with K = { z D | V ( z ) κ } Open image in new window. The relation V ( z ) = κ Open image in new window holds for all z K Open image in new window.

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:
V ( r ) > κ , r [ 0 , ω [ , V ( ω ) = κ , V ( r ) < κ , r ] ω , [ , V ( ) = 0 , Open image in new window
and is a fixed point of equation (20):
α V ( r ) = M ( r , ω ) + I ext ( r ) , Open image in new window
(22)
where I ext ( r ) = I e r 2 2 σ 2 Open image in new windowis a Gaussian input and M ( r , ω ) Open image in new windowis defined by the following equation:
M ( r , ω ) = def M ( z , B h ( 0 , ω ) ) Open image in new window

and B h ( 0 , ω ) Open image in new windowis 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 ) Open image in new window of (20), furthermore the threshold κ and width ω are related according to the self-consistency condition
α κ = M ( ω ) + I ext ( ω ) = def N ( ω ) , Open image in new window
(23)
where
M ( ω ) = def M ( ω , ω ) . Open image in new window
The existence of such a bump can then be established by finding solutions to (23) The function N ( ω ) Open image in new window is plotted in Figure 1 for a range of the input amplitude I Open image in new window. 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 I Open image in new window we have N ( 0 ) < 0 Open image in new window and it is possible to find only one solution branch for large ακ. For small input amplitudes I Open image in new window we have N ( 0 ) > 0 Open image in new window and there always exists one solution branch for α β < γ c 0.06 Open image in new window. For intermediate values of the input amplitude I Open image in new window, 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 ( ω ) < 0 Open image in new window then the corresponding solution is stable.
Fig. 1

Plot of N ( ω ) Open image in new window defined in (23) as a function of the pulse width ω for several values of the input amplitude I Open image in new window and for a fixed input width σ = 0.05 Open image in new window. 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 Open image in new window.

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

Theorem 5.1.1 For all ( r , ω ) R + × R + Open image in new window:
M ( r , ω ) = 1 4 sinh ( ω ) 2 cosh ( ω ) 2 R W ˜ ( λ ) Φ λ ( 0 , 0 ) ( r ) Φ λ ( 1 , 1 ) ( ω ) λ tanh ( π 2 λ ) d λ , Open image in new window
(24)
where W ˜ ( λ ) Open image in new window is the Fourier Helgason transform of W ( z ) = def W ( d 2 ( z , 0 ) ) Open image in new window and
Φ λ ( α , β ) ( ω ) = F ( 1 2 ( ρ + i λ ) , 1 2 ( ρ i λ ) ; α + 1 ; sinh ( ω ) 2 ) , Open image in new window

with α + β + 1 = ρ Open image in new windowand F is the hypergeometric function of first kind.

Remark 5.1.1 We recall that F admits the integral representation[32]:
F ( α , β ; γ ; z ) = Γ ( α ) Γ ( β ) Γ ( γ β ) 0 1 t β 1 ( 1 t ) γ β 1 ( 1 t z ) α d t Open image in new window

with ( γ ) > ( β ) > 0 Open image in new window.

Remark 5.1.2 In Section 4we introduced the function Φ λ ( z ) = D e ( i λ + 1 ) z , b d b Open image in new window. In[22], it is shown that:
Φ λ ( 0 , 0 ) ( r ) = Φ λ ( tanh ( r ) ) if z = tanh ( r ) e i θ . Open image in new window
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 ) Open image in new windowcentered at the origin O and of radius a. Using their notations
M ( a , r ) = 0 2 π 0 a w ( | r r | ) r d r d θ = 2 π a 0 w ˘ ( ρ ) J 0 ( r ρ ) J 1 ( a ρ ) d ρ , Open image in new window

where J ν ( x ) Open image in new windowis the Bessel function of the first kind and w ˘ Open image in new windowis the real Fourier transform of w. In our case instead of the Bessel function we find Φ λ ( ν , ν ) ( r ) Open image in new windowwhich is linked to the hypergeometric function of the first kind.

We now show that for a general monotonically decreasing weight function W, the function M ( r , ω ) Open image in new window 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 M Open image in new window with respect to r yields:
M r ( r , ω ) = 1 2 0 ω 0 2 π r ( W ( d 2 ( tanh ( r ) , tanh ( r ) e i θ ) ) ) sinh ( 2 r ) d r d θ . Open image in new window
We have to compute
r ( W ( d 2 ( tanh ( r ) , tanh ( r ) e i θ ) ) ) = W ( d 2 ( tanh ( r ) , tanh ( r ) e i θ ) ) r ( d 2 ( tanh ( r ) , tanh ( r ) e i θ ) ) . Open image in new window
It is result of elementary hyperbolic trigonometry that
d 2 ( tanh ( r ) , tanh ( r ) e i θ ) = tanh 1 ( tanh ( r ) 2 + tanh ( r ) 2 2 tanh ( r ) tanh ( r ) cos ( θ ) 1 + tanh ( r ) 2 tanh ( r ) 2 2 tanh ( r ) tanh ( r ) cos ( θ ) ) Open image in new window
(25)
we let ρ = tanh ( r ) Open image in new window, ρ = tanh ( r ) Open image in new window and define
F ρ , θ ( ρ ) = ρ 2 + ρ 2 2 ρ ρ cos ( θ ) 1 + ρ 2 ρ 2 2 ρ ρ cos ( θ ) . Open image in new window
It follows that
ρ tanh 1 ( F ρ , θ ( ρ ) ) = ρ F ρ , θ ( ρ ) 2 ( 1 F ρ , θ ( ρ ) ) F ρ , θ ( ρ ) , Open image in new window
and
ρ F ρ , θ ( ρ ) = 2 ( ρ ρ cos ( θ ) ) + 2 ρ ρ ( ρ ρ cos ( θ ) ) ( 1 + ρ 2 ρ 2 2 ρ ρ cos ( θ ) ) 2 . Open image in new window
We conclude that if ρ > tanh ( ω ) Open image in new window then for all 0 ρ tanh ( ω ) Open image in new window and 0 θ 2 π Open image in new window
2 ( ρ ρ cos ( θ ) ) + 2 ρ ρ ( ρ ρ cos ( θ ) ) > 0 , Open image in new window

which implies M ( r , ω ) < 0 Open image in new window for r > ω Open image in new window, since W < 0 Open image in new window.

To see that it is also negative for r < ω Open image in new window, we differentiate equation (24) with respect to r:
M r ( r , ω ) = 1 4 sinh ( ω ) 2 cosh ( ω ) 2 × R W ˜ ( λ ) r Φ λ ( 0 , 0 ) ( r ) Φ λ ( 1 , 1 ) ( ω ) λ tanh ( π 2 λ ) d λ . Open image in new window
The following formula holds for the hypergeometric function (see Erdelyi in [32]):
d d z F ( a , b ; c ; z ) = a b c F ( a + 1 , b + 1 ; c + 1 ; z ) . Open image in new window
It implies
r Φ λ ( 0 , 0 ) ( r ) = 1 2 sinh ( r ) cosh ( r ) ( 1 + λ 2 ) Φ λ ( 1 , 1 ) ( r ) . Open image in new window
Substituting in the previous equation giving M r Open image in new window we find:
M r ( r , ω ) = 1 64 sinh ( 2 ω ) 2 sinh ( 2 r ) × R W ˜ ( λ ) ( 1 + λ 2 ) Φ λ ( 1 , 1 ) ( r ) Φ λ ( 1 , 1 ) ( ω ) λ tanh ( π 2 λ ) d λ , Open image in new window
implying that:
sgn ( M r ( r , ω ) ) = sgn ( M r ( ω , r ) ) . Open image in new window

Consequently, M r ( r , ω ) < 0 Open image in new window for r < ω Open image in new window. 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 M ( r , ω ) Open image in new window, we use the Fourier transform in D Open image in new window which has already been introduced in Section 4. First we rewrite M ( r , ω ) Open image in new window as a convolution product:

Proposition 5.2.1 For all ( r , ω ) R + × R + Open image in new window:
M ( r , ω ) = 1 4 π R W ˜ ( λ ) Φ λ 1 B h ( 0 , ω ) ( z ) λ tanh ( π 2 λ ) d λ . Open image in new window
(26)
Proof We start with the definition of M ( r , ω ) Open image in new window and use the convolutional form of the integral:
M ( r , ω ) = M ( z , B h ( 0 , ω ) ) = B h ( 0 , ω ) W ( z , z ) dm ( z ) = D W ( z , z ) 1 B h ( 0 , ω ) ( z ) dm ( z ) = W 1 B h ( 0 , ω ) ( z ) . Open image in new window
In [22], Helgason proves an inversion formula for the hyperbolic Fourier transform and we apply this result to W:
W ( z ) = 1 4 π R D W ˜ ( λ , b ) e ( i λ + 1 ) z , b λ tanh ( π 2 λ ) d λ d b = 1 4 π