Analysis of a hyperbolic geometric model for

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Journal of Mathematical Neuroscience (2011) 1:4

DOI 10.1186/2190-8567-1-4

R E S E A R C H Open Access

Received: 7 January 2011 / Accepted: 6 June 2011 / Published online: 6 June 2011 2011 Faye et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License

Abstract We study the neural eld 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 dened on the Poincar 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.

Keywords Neural elds nonlinear integro-differential equations functional

analysis non-Euclidean analysis stability analysis hyperbolic geometry

hypergeometric functions bumps

Mathematics Subject Classication 30F45 33C05 34A12 34D20 34D23

34G20 37M05 43A85 44A35 45G10 51M10 92B20 92C20

G Faye ([letter]) P Chossat O Faugeras

NeuroMathComp Laboratory, INRIA, Sophia Antipolis, CNRS, ENS Paris, Paris, France e-mail: mailto:[email protected]

Web End [email protected]

P Chossat

Dept. of Mathematics, University of Nice Sophia-Antipolis, JAD Laboratory and CNRS, Parc Valrose, 06108 Nice Cedex 02, France

Analysis of a hyperbolic geometric model for visual texture perception

Grgory Faye Pascal Chossat Olivier Faugeras

Page 2 of 51 Faye et al.

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 specic orientation [13], 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 rst 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 denite 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 inuence 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 rst 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 dened 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 ndings of the theoretical results.

Journal of Mathematical Neuroscience (2011) 1:4 Page 3 of 51

2 The model

By denition, 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 eld. These spatial derivatives are then summed nonlinearly over the receptive eld. Let I (x, y) denote the original image intensity function, where x and y are two spatial coordinates. Let I1 denote the scale-space representation of I obtained by convolution with the Gaussian kernel g (x, y) =

22 e(x2+y2)/(22):

I1 = I g1.

The gradient I1 is a two-dimensional vector of coordinates I1x, I1y which em

phasizes image edges. One then forms the 2 2 symmetric matrix of rank one T0= I1(I1)T, where T indicates the transpose of a vector. The set of 22 sym

metric positive semidenite matrices of rank one will be noted S+(1, 2) throughout the paper (see [16] for a complete study of the set S+(p, n) of n n symmetric pos

itive semidenite matrices of xed-rank p < n). By convolving T0 componentwise

with a Gaussian g2 we nally form the tensor structure as the symmetric matrix:

T = T0 g2 =

(I1x)2 2 I1xI1y 2

I1xI1y 2 (I1y)2 2

2 g2.

Since the computation of derivatives usually involves a stage of scale-space smoothing, the denition of the structure tensor requires two scale parameters. The rst one, dened by 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 1. The second one, dened by 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 elds of the neurons that may represent the structure tensor.

By construction, T is symmetric and non negative as det(T ) 0 by the inequal

ity of Cauchy-Schwarz, then it has two orthonormal eigenvectors e1, e2 and two non negative corresponding eigenvalues 1 and 2 which we can always assume to be such that 1 2 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 2 symmetric positive

denite matrices, noted SPD(2, R) throughout the paper. The distribution of these eigenvalues in the (1, 2) plane reects the local organization of the image intensity variations. Indeed, each structure tensor can be written as the linear combination:

T = 1e1eT1 + 2e2eT2 = (1 2)e1eT1 + 2

= (1 2)e1eT1 + 2I2,

where we have set for example:

x 2 2 =

I1x

e1eT1 + e2eT2

(1)

Page 4 of 51 Faye et al.

where I2 is the identity matrix and e1eT1 S+(1, 2). Some easy interpretations can be

made for simple examples: constant areas are characterized by 1 = 2 0, straight

edges are such that 1 2 0, their orientation being that of e2, corners yield

1 2 0. The coherency c of the local image is measured by the ratio c = 121+2 ,

large coherency reveals anisotropy in the texture.

We assume that a hypercolumn of V1 can represent the structure tensor in the receptive eld of its neurons as the average membrane potential values of some of its membrane populations. Let T be a structure tensor. The time evolution of the average

potential V (T , t) for a given column is governed by the following neural mass equa

tion 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 symmetric denite-positive

matrices:

tV (T , t) = V (T , t) +

SPD(2) W(T , T , t)S

V (T , t) dT

(2)

+Iext(T , t) t > 0, V (T , 0) = V0(T ).

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

S(x) =

11 + ex

where describes the stiffness of the sigmoid. Iext is an external input.

The set SPD(2) can be seen as a foliated manifold by way of the set of special symmetric positive denite matrices SSPD(2) = SPD(2) SL(2,

R). Indeed,

D, where D is the Poincar Disk, see, for example, [7]. As a consequence we use the following folia

tion of SPD(2) : SPD(2) hom=

we have: SPD(2) hom

= SSPD(2)

R+. Furthermore, SSPD(2) isom=

R+, which allows us to write for all T SPD(2), T = (z, ) with (z, )

R+. T , z and are related by the relation det(T ) = 2

and the fact that z is the representation in D of T SSPD(2) with T = T .

It is well-known [17] that D (and hence SSPD(2)) is a two-dimensional Riemannian space of constant sectional curvature equal to 1 for the distance noted d2 de

ned by

d2(z, z ) = arctanh |

z z |

|1 zz |

The isometries of D, that are the transformations that preserve the distance d2 are the elements of unitary group 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

d0 dened by

d0(T , T ) =

2(log log )2 + d22(z, z ).

Journal of Mathematical Neuroscience (2011) 1:4 Page 5 of 51

As shown in Proposition B.0.1 of Appendix B it is possible to express the volume element dT in (z1, z2, ) coordinates with z = z1 + iz2:

dT = 8

dz1 dz2 (1 |z|2)2

We note dm(z) =

dz1 dz2 (1|z|

2)2 and equation (2) can be written in (z, ) coordinates:

tV (z, , t) = V (z, , t)

+82

+ D W

z, , z , , t

V (z , , t) d dm(z)

+Iext(z, , t).

We get rid of the constant 82 by redening W as 82W .

tV (z, , t) = V (z, , t)

+ D W(z, , z , , t)S

V (z , , t) d dm(z)

+Iext(z, , t) t > 0,

V (z, , 0) = V0(z, ).

(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) and S+(1, 2). First of all, when 2 goes to zero, that is when the characteristic size of the structure becomes very small, we have T0 g2 T0, which means that the ten

sor T SPD(2,

R) degenerates to a tensor T0 S+(1, 2), which can be interpreted

as the loss of one dimension. We can write each T0 S+(1, 2) as T0 = xxT = r2uuT,

where u = (cos , sin )T and (r, ) is the polar representation of x. Since, x and x

correspond to the same T0, is equated to + k, k

Z. Thus S+(1, 2) =

R+

P1,

where P1 is the real projective space of dimension 1 (lines of R2). Then the integration scale 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 1 2 0 then

T (1 2)e1eT1 S+(1, 2) and the orientation is that of e2. We denote by

P the

projection of a 2 2 symmetric denite positive matrix on the set S+(1, 2) dened

by:

SPD(2, R) S+(1, 2),

: T = (1 2)e1eT1,

Page 6 of 51 Faye et al.

where T is as in equation (1). We can introduce a metric on the set 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 S+(1, 2) =

R+

P1, a metric on the space S+(1, 2) is given by:

ds2 = 2

r1 r2

where we normalize to 1 the volume element for the coordinate.Let now =

P(T ) be a symmetric positive semidenite matrix. The average po

tential V (, 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 S+(1, 2):

tV (, t) = V (, t) +

S+(1,2)

W(, , t)S

V ( , t) d

+Iext(, t) t > 0.

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

tV (r, , t) = V (r, , t) +

dS+(1,2)(, ) = w

2+ d2.

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

d2S+(1,2)(1, 2) = 2 log2

dr r

+ |

1 2|2,

where 1 = xT1x1, 2 = xT2x2 and (ri, i) denotes the polar coordinates of xi for i =

1, 2. The volume element in (r, ) coordinates is:

d =

dr r

(4)

+ 0 W(r, , r , , t)S

r r

V (r , , t) d dr r

+ Iext(r, , t).

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 sufcient i) to assume that the connectivity function is time-independent and has a convolutional form:

W(, , t) = w

2 log2

+ |

Journal of Mathematical Neuroscience (2011) 1:4 Page 7 of 51

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

tV (, t) = V (, t) +

0 wsh( )S

V ( , t) d + Iext(, t), (5)

where:

wsh() =

0 w

2 log2(r) + 2

drr .

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-dened and to give necessary and sufcient 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 rst 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 . This simplied 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.

3.1 Homogeneous solutions

A homogeneous solution to (2) is a solution V that does not depend upon the tensor variable T for a given homogenous input I (t) and a constant initial condition V0. In

(z, ) coordinates, a homogeneous solution of (3) is dened by:

V (t) = V (t) + W(z, , t)S

V (t)

+Iext(t),

where:

W(z, , t) def

+ D W(z, , z , , t)d dz 1 dz 2(1 |z |2)2. (6)

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

the double integral (6) is convergent, W(z, , t) does not depend upon the variable (z, ). In that case, we write W(t)

instead of W(z, , t).

Page 8 of 51 Faye et al.

In the special case where W(z, , z , , t) is a function of only the distance d0 between (z, ) and (z , ):

W(z, , z , , t) w

2(log log )2 + d22(z, z ), t

the second condition is automatically satised. 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) = V (t) + W(t)S

V (t)

+Iext(t), t > 0,

V (0) = V0.

(7)

3.1.1 A rst existence and uniqueness result

Equation (3) denes a Cauchys problem and we have the following theorem.

Theorem 3.1.1 If the external input Iext(t) and the connectivity function W(t) are continuous on some closed interval J containing 0, then for all V0 in R, there exists a unique solution of (7) dened on a subinterval J0 of J containing 0 such that

V (0) = V0.

Proof It is a direct application of Cauchys theorem on differential equations. We consider the mapping f : J

R dened by:

f (t, x) = x + W(t)S(x) + Iext(t).

It is clear that f is continuous from J

R to R. We have for all x, y

R and t J :

f (t, x) f (t, y)

|x y| +

W(t) S

|x y|,

where S m = supxR |S (x)|.

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

f (t, x) f (t, y)

( + CS m)|x y|.

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

Proposition 3.1.1 If Iext and W are continuous on R+, then for all V0 in R, there exists a unique solution of (7) dened on R+ such that V (0) = V0.

Proof We have already shown the following inequality:

f (t, x) f (t, y)

|x y|.

Then f is locally Lipschitz with respect to its second argument. Let V be a maximal solution on J0 and we denote by the upper bound of J0. We suppose that < +.

|x y| +

W(t) S

Journal of Mathematical Neuroscience (2011) 1:4 Page 9 of 51

Then we have for all t 0:

V (t) = etV0 +

0 e(tu)W(u)S

V (u) du +

0 e(tu)Iext(u) du

V (t)

|V0| + Sm

0 eu

W(u)

du +

0 eu

ext(u)

du t [0, ],

where Sm = supxR |S(x)|.

This implies that the maximal solution V is bounded for all t [0, ], but Theo

rem C.0.2 of Appendix C ensures that it is impossible. Then, it follows that necessarily = +.

3.1.2 Simplication 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+, then W(z, , t)

did not depend upon the variables (z, ). 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 d0(T , T ) only. Then W does not

depend upon the variable T .

Mexican hat connectivity In this paragraph, we push further the computation of 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 d0(T , T ).

In detail, we have:

W(z, , z ) = w

2(log log )2 + d22(z, z )

where:

w(x) =

221e x

222e x

with 0 1 2 and 0 A 1.

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

Lemma 3.1.2 When W is the specic Mexican hat function just dened then:

W =

1e221 erf

A2e2

22 erf

(8)

Page 10 of 51 Faye et al.

where erf is the error function dened as:

erf(x) =

0 eu2 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 rst introduce some hypotheses on the connectivity function W . We present them in two ways: rst on the set of structure tensors considered as the set SPD(2), then on the set of tensors seen as D

R+. Let J be a

subinterval of R. We assume that:

(H1): (T , T , t) SPD(2) SPD(2) J , W(T , T , t) W(d0(T , T ), t), (H2): W C(J, L1(SPD(2))) where W is dened as W(T , t) = W(d0(T , Id2), t)

for all (T , t) SPD(2) J where Id2 is the identity matrix of M2(

R),

(H3): t J , suptJ W(t) L

1 < + where W(t) L

1 def =

SPD(2)

|W(d0(T ,

Id2), t)| dT .

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

(H1bis): (z, z , , , t)

(

R+)2

R, W(z, , z , , t) W(d2(z, z ),

| log( ) log( )|, t),

(H2bis): W C(J, L1(

R+)) where W is dened as W(z, , t) = W(d2(z, 0), | log( )|, t) for all (z, , t)

R+ J ,

(H3bis): t J , suptJ W(t) L

1 < + where

W(t)

def

R+

W d

2(z, 0),

log( ) , t

dm(z).

3.2.1 Functional space setting

We introduce the following mapping f g : (t, ) f g(t, ) such that:

f g(t, )(z, ) =

d2(z, z ),

log

, t

dm(z ). (9)

Our aim is to nd a functional space F where (3) is well-dened and the function

f g maps F to F for all ts. A natural choice would be to choose as a Lp(

(z , )

R+

R+)-integrable function of the space variable with 1 p < +. Unfortunately, the

homogeneous solutions (constant with respect to (z, )) 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(

R+). As

R+ is an open set of

R3, F is a Banach space for the norm: F =

supzD sup R+

|(z, )|.

Journal of Mathematical Neuroscience (2011) 1:4 Page 11 of 51

Proposition 3.2.1 If Iext C(J, F) with suptJ Iext(t) F < + and W satises

hypotheses (H1bis)-(H3bis) then f g is well-dened and is from J F to F.

Proof (z, , t)

R+

R, we have:

d2(z, z ),

log

, t

(z , )

R+

dm(z )

Sm sup

W(t)

L1 < +.

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

We rewrite (3) as a Cauchy problem:

tV (z, , t) = V (z, , t)

d2(z, z ),

log

, t

R+

(10)

V (z , , t)

dm(z )

+Iext(z, , t), V (z, , 0) = V0(z, ).

Theorem 3.2.1 If the external current Iext belongs to C(J, F) with J an open interval

containing 0 and W satises hypotheses (H1bis)-(H3bis), then fo all V0 F, there

exists a unique solution of (10) dened on a subinterval J0 of J containing 0 such that V (z, , 0) = V0(z, ) for all (z, )

R+.

Proof We prove that f g is continuous on J F. We have

f g(t, ) f g(s, )

(z, )

d2(z, z ),

log

, t

(z , )

R+

(z , ) d dm(z )

d2(z, z ),

log

, t

log

d2(z, z ),

, s

R+

(z , )

dm(z ),

and therefore

g(t, ) f g(s, )

S m sup

W(t)

L1 F + Sm

W(t)

L1 .

W(s)

Because of condition (H2) we can choose |t s| small enough so that W(t)

W(s) L

1 is arbitrarily small. This proves the continuity of f g. Moreover it follows from the previous inequality that:

g(t, ) f g(t, )

S mWg0 F

Page 12 of 51 Faye et al.

with Wg0 = suptJ W(t) L

1 . This ensures the Lipschitz continuity of f g with respect to its second argument, uniformly with respect to the rst. 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, rst, we allow the connectivity function to depend upon the time variable t and, second, that our space features is no longer a

Rn 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 Iext belongs to C(

R+, F) and W satises

hypotheses (H1bis)-(H3bis) with J =

R+, then for all V0 F, there exists a unique

solution of (10) dened on R+ such that V (z, , 0) = V0(z, ) for all (z, )

R+.

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

g(t, ) f g(t, )

S mWg0 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 Iext belongs to C(

R+, F) and is bounded

in time suptR+ Iext(t) F < + and W satises hypotheses (H1bis)-(H3bis) with

J =

R+, then the solution of (10) is bounded for each initial condition V0 F.

Let us set:

g def =

SmWg0 + sup tR+

ext(t)

1 .

Proof Let V be a solution dened on R+. Then we have for all t

R+:

where Wg0 = suptR+ W(t) L

V (z, , t) = etV0(z, ) +

0 e(tu)

d2(z, z ),

log

, u

R+

V (z , , u)

dm(z ) du

+ 0 e(tu)Iext(z, , u) du.

Journal of Mathematical Neuroscience (2011) 1:4 Page 13 of 51

The following upperbound holds

V (t)

et V0 F +

SmWg0 + sup tR+

ext(t)

1 et

. (11)

We can rewrite (11) as:

V (t)

V0 F 1

SmWg0 + sup tR+

ext(t)

SmW0 +g sup tR+

ext(t)

(12)

= et

V0 F

g 2

2 .

2 (1 + et) for all t > 0 and hence V (t) F <

g for all t > 0, proving that B is stable. Now assume that V (t) F > g for all

t 0. The inequality (12) shows that for t large enough this yields a contradiction.

Therefore there exists t0 > 0 such that V (t0) F = g. At this time instant we have

g et0

If V0 Bg this implies V (t) F

V0 F

g 2

2 ,

and hence

1 log

2 V0 F g g

The following corollary is a consequence of the previous proposition.

Corollary 3.2.1 If V0 /

g and T g = inf{t > 0 such that V (t) B

} then:

T g

1 log

2 V0 F g g

3.3 Semi-homogeneous solutions

A semi-homogeneous solution of (3) is dened as a solution which does not depend upon the variable . 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 dene 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

Page 14 of 51 Faye et al.

solutions:

tV (z, t) = V (z, t) +

D Wsh(z, z , t)S

V (z , t) dm(z )

(13)

+Iext(z, t), t > 0,

V (z, 0) = V0(z),

where

Wsh(z, z , t) =

0 W(z, , z , , t)

We have implicitly made the assumption, that Wsh does not depend on the coordinate . Some conditions under which this assumption is satised 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. We assume that:

(C1): (z, z , t)

J , Wsh(z, z , t) wsh(d2(z, z ), t), (C2): Wsh C(J, L1(

D)) where Wsh is dened as Wsh(z, t) = wsh(d2(z, 0), t) for

all (z, t)

J ,

|Wsh(d2(z, 0),

t)| dm(z).

Note that conditions (C1)-(C2) and Lemma 3.1.1 imply that for all z

(C3): suptJ Wsh(t) L

1 < + where Wsh(t) L

1 def =

|Wsh(z, z , t)| dm(z ) = Wsh(t) L1 . And then, for all z

D, the mapping z

Wsh(z, z , t) is integrable on D.

From now on, F = L(

D) and the Fischer-Rieszs theorem ensures that L(D) is a Banach space for the norm: = inf{C 0, |(z)| C for almost every

Theorem 3.3.1 If the external current Iext belongs to C(J, F) with J an open interval

containing 0 and Wsh satises conditions (C1)-(C3), then for all V0 F, there exists

a unique solution of (13) dened on a subinterval J0 of J containing 0.

This solution, dened on the subinterval J of 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 Iext belongs to C(

R+, F) and Wsh satises

conditions (C1)-(C3) with J =

R+, then for all V0 F, there exists a unique solution

of (13) dened on R+.

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

Proposition 3.3.2 Let

def

2 (SmWsh0 + suptR+ I (t) F), with Wsh0 =

1 . The open ball B of F of center 0 and radius is stable un

der the dynamics of equation (13). Moreover it is an attracting set for this dynamics

suptJ Wsh(t) L

Journal of Mathematical Neuroscience (2011) 1:4 Page 15 of 51

and if V0 / B and T = inf{t > 0 such that V (t) B} then:

1 log

2 V0 F

4 Stationary solutions

We look at the equilibrium states, noted V 0 of (3), when the external input I and the connectivity W do not depend upon the time. We assume that W satises hypotheses (H1bis)-(H2bis). We redene for convenience the sigmoidal function to be:

S(x) =

11 + ex

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

0 = V 0(z, ) +

d2(z, z ),

log

V 0(z , )

R+

dm(z )

+Iext(z, ). (14)

We dene the nonlinear operator from F to F, noted G, by:

G(V )(z, ) =

d2(z, z ),

log

V (z , )

R+

dm(z ). (15)

Finally, (14) is equivalent to:

V 0(z, ) = G(V )(z, ) + Iext(z, ).4.1 Study of the nonlinear operator G

We recall that we have set for the Banach space F = L(

R+) and Proposi

tion 3.2.1 shows that G : F F. We have the further properties:

Proposition 4.1.1 G satises the following properties:

G(V1) G(V2) F Wg0S m V1 V2 F for all 0, G is continuous on

R+.

1 2|Wg0S m V F.

We denote by Gl and G the two operators from F to F dened as follows for all

V F and all (z, )

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

1(V ) G2(V )

F |

R+:

Gl(V )(z, ) =

d2(z, z ),

log

V (z , )d dm(z ), (16)

R+

Page 16 of 51 Faye et al.

and

G(V )(z, ) =

d2(z, z ),

log

V (z , )

R+

dm(z ),

where H is the Heaviside function.It is straightforward to show that both operators are well-dened on F and map F

to F. Moreover the following proposition holds.

Proposition 4.1.2 We have

G G .

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

S(y)

4.2 The convolution form of the operator G in the semi-homogeneous case

It is convenient to consider the functional space Fsh = L(

H(y) a.e. y

D) to discuss semi-homogeneous solutions. A semi-homogeneous persistent state of (3) is deduced from (14) and satises:

V 0(z) = Gsh

V 0

(z) + Iext(z), (17)

where the nonlinear operator Gsh from Fsh to Fsh is dened for all V Fsh and

D by:

Gsh(V )(z) =

D Wsh

d2(z, z )

V (z ) dm(z ).

We dene the associated operators, Gshl, Gsh:

Gshl(V )(z) =

D Wsh

d2(z, z )

V (z ) dm(z ),

V (z ) dm(z ).

We rewrite the operator Gsh in a convenient form by using the convolution in the

hyperbolic disk. First, we dene the convolution in a such space. Let O denote the center of the Poincar 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:

G f (g O) dg def=

D f (z) dm(z),

Gsh(V )(z) =

D Wsh

d2(z, z )

Journal of Mathematical Neuroscience (2011) 1:4 Page 17 of 51

for all functions of L1(D). Given two functions f1, f2 in L1(D) we dene the convolution by:

(f1 f2)(z) =

G f1(g O)f2

g1 z

dg.

We recall the notation Wsh(z) def

= Wsh(d2(z, O)).

Proposition 4.2.1 For all 0 and V Fsh we have:

Gsh(V ) = Wsh S(V ), Gshl(V ) = Wsh V and

Gsh(V ) = Wsh H(V ).

(18)

Proof We only prove the result for G. Let z

D, then:

Gsh(V )(z) =

D Wsh

d2(z, z )

V (z ) dm(z )

= G Wsh

d2(z, g O)

V (g O) dg

= G Wsh

gg1 z, g O

S V

(g O)

and for all g SU(1, 1), d2(z, z ) = d2(g z, g z ) so that:

Gsh(V )(z) = G

Wsh

g1 z, O

S V

(g O)

dg = Wsh S(V )(z).

D, we dene 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 D is dened as (see [22]):

h(, b) =

D h(z)e(i+1) z,b dm(z) (, b)

Let b be a point on the circle D. For z

for a function h :

C such that this integral is well-dened.

Lemma 4.2.1 The Fourier transform in D,

Wsh(, b) of Wsh does not depend upon

the variable b D.

Proof For all

R and b = ei

Wsh(, b) =

D Wsh(z)e(i+1) z,b dm(z).

Page 18 of 51 Faye et al.

We recall that for all

R r is the rotation of angle and we have Wsh(r z) =

Wsh(z), dm(z) = dm(r z) and z, b = r z, r b , then:

Wsh(, b) =

D Wsh(r z)e(i+1) rz,1 dm(z)

Wsh().

We now introduce two functions that enjoy some nice properties with respect to the Hyperbolic Fourier transform and are eigenfunctions of the linear operator Gshl.

Proposition 4.2.2 Let e,b(z) = e(i+1) z,b and (z) =

D e(i+1) z,b db then:

D Wsh(z)e(i+1) z,1 dm(z) def =

Wsh()e,b, Gshl( ) =

Wsh() .

Proof We begin with b = 1

D and use the horocyclic coordinates. We use the same changes of variables as in Lemma 3.1.1:

Gshl(e,1)(nsat O) =

R2 Wsh

Gshl(e,b) =

d2(nsat O, ns at O)

e(i1)t dt ds

R2 Wsh

d2(nss at O, at O)

e(i1)t dt ds

R2 Wsh

d2(atnx O, at O)

e(i1)t +2t dt dx

R2 Wsh

d2(O, nxat t O)

e(i1)t +2t dt dx

R2 Wsh

d2(O, nxay O)

e(i1)(y+t)+2t dy dx

= e(i+1) nsatO,1

Wsh().

By rotation, we obtain the property for all b

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

Wsh (z) =

D e(i+1) z,b

Wsh() db = (z)

Wsh().

A consequence of this proposition is the following lemma.

Lemma 4.2.2 The linear operator Gshl is not compact and for all 0, the nonlin

ear operator Gsh is not compact.

Proof The previous Proposition 4.2.2 shows that Gshl has a continuous spectrum

which iimplies that is not a compact operator.

Journal of Mathematical Neuroscience (2011) 1:4 Page 19 of 51

Let U be in Fsh, for all V Fsh we differentiate Gsh and compute its Frechet

derivative:

Gsh

U (V )(z) =

V (z ) dm(z ).

If we assume further that U does not depend upon the space variable z, U(z) = U0

we obtain:

Gsh

U0(V )(z) = S (U0)Gshl(V )(z).

If Gsh was a compact operator then its Frechet derivative D(Gsh)U0 would also be

a compact operator, but it is impossible. As a consequence, Gsh is not a compact

operator.

4.3 The convolution form of the operator G 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 dxx . For two functions f1, f2 in

L1(D

R+) we dene the convolution by:

(f1 f2)(z, ) def

= H f1(g O, h 1)f2

g1 z, h1

D Wsh

d2(z, z )

U(z )

dg dh.

We recall that we have set by denition: W(z, ) = W(d2(z, 0), | log( )|).

Proposition 4.3.1 For all 0 and V F we have:

G(V ) = W S(V ), Gl(V ) = W V and G(V ) = W H(V ). (19)

Proof Let (z, ) be in D

R+. We follow the same ideas as in Proposition 4.2.1 and

prove only the rst result. We have

G(V )(z, ) =

d2(z, z ),

log

V (z , )

dm(z )

R+

g1 z, O

log

V (g O, )

dg d

R+

S V

(g O, h 1)

g1 z, O

log h1

dg dh

= H W

= W S(V )(z, ).

We next assume further that the function W is separable in z, and more precisely that W(z, ) = W1(z)W2(log( )) where W1(z) = W1(d2(z, 0)) and W2(log( )) =

W2(| log( )|) for all (z, )

R+. The following proposition is an echo to Propo-

sition 4.2.2.

Page 20 of 51 Faye et al.

Proposition 4.3.2 Let e,b(z) = e(i+1) z,b , (z) =

D e(i+1) z,b db and

h ( ) = ei log( ) then:

Gl(e,bh ) =

W1() W2()e,bh , Gl( h ) =

W1()

W2() h ,

where

W2 is the usual Fourier transform of W2.

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

Gl(e,bh )(z, ) = W1 e,b(z) R+

log ei log( ) d

= W1 e,b(z)

R W2(y)eiy dy

ei log( ).

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 Gshl is not compact and for all 0, the nonlin

ear operator Gsh is not compact.

4.4 The set of the solutions of (14)

Let B be the set of the solutions of (14) for a given slope parameter :

V F| V + G(V ) + Iext = 0

We have the following proposition.

Proposition 4.4.1 If the input current Iext is equal to a constant I0ext, that is, does not depend upon the variables (z, ) then for all

R+, B = . In the general case

Iext F, if the condition S mWg0 < is satised, then Card(B) = 1.

Proof Due to the properties of the sigmoid function, there always exists a constant solution in the case where Iext is constant. In the general case where Iext F, the

statement is a direct application of the Banach xed point theorem, as in [23].

Remark 4.4.1 If the external input does not depend upon the variables (z, ) and if the condition S mWg0 < is satised, 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, ) because there always exists one constant stationary solution when the external input does not depend upon the variables (z, ). Indeed equation (14) is then equivalent to:

0 = V 0 + S

V 0

+ I0ext where

Journal of Mathematical Neuroscience (2011) 1:4 Page 21 of 51

d dm(z )

and does not depend upon the variables (z, ) because of Lemma 3.1.1. Because of the property of the sigmoid function S, the equation 0 = V 0 + S(V 0) + I0ext

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 D (see Appendix A), and the condition S mWg0 < is satised, 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 S mWg0 < is satised we call primary stationary solution the unique solution in B.

4.5 Stability of the primary stationary solution

In this subsection we show that the condition S mWg0 < guarantees the stability of the primary stationary solution to (3).

Theorem 4.5.1 We suppose that I F and that the condition S mWg0 < is satis

ed, then the associated primary stationary solution of (3) is asymtotically stable.

Proof Let V 0 be the primary stationary solution of (3), as S mWg0 < is satised. Let also V be the unique solution of the same equation with some initial condition V(0) = F, see Theorem 3.3.1. We introduce a new function X = V V 0 which

satises:

tX(z, , t) = X(z, , t)

d2(z, z ),

log

R+

d2(z, z ),

log

X(z , , t)

dm(z ),

R+

X(z, , 0) = (z, ) V 0(z, ),

where Wm(d2(z, z ), | log( )|) = S mW(d2(z, z ), | log( )|) and the vector (X(z,

, t)) is given by (X(z, , t)) = S(V(z, , t)) S(V 0(z, )) with S =

(S m)1S. We note that, because of the denition of and the mean value theorem

| (X(z, , t))| |X(z, , t)|. This implies that | (r)| |r| for all r

tX(z, , t) = X(z, , t)

d2(z, z ),

log

X(z , , t)

dm(z )

R+

etX(z, , t)

= et

d2(z, z ),

log

X(z , , t)

dm(z )

R+

Page 22 of 51 Faye et al.

X(z, , t)

= etX(z, , 0) +

0 e(tu)

d2(z, z ),

log

R+

X(z , , u)

dm(z ) du

X(z,

, t)

X(z,

, 0)

0 e(tu)

d2(z, z ),

log

R+

X(z , , u)

dm(z ) du

X(t)

X(0)

Wg0S m

0 e(tu)

X(u)

du.

If we set: G(t) = et X(t) , then we have:

G(t) G(0) + Wg0S m

0 G(u) du

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

G(t) G(0)eW

g0 S mt

X(t)

e(W

g0 S m)t

X(0)

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 [2529], 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

Journal of Mathematical Neuroscience (2011) 1:4 Page 23 of 51

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 [2529], we consider the high gain limit of the sigmoid function. As

above we denote by H the Heaviside function dened by H(x) = 1 for x 0 and

H(x) = 0 otherwise. Equation (13) is rewritten as:

tV (z, t) = V (z, t) +

D W(z, z )H

V (z , t) dm(z ) + I (z, t)

(20)

We have introduced a threshold 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(d2(z, z )).

For illustrative purposes, we will use the exponential weight distribution as a specic example throughout this section:

W(z, z ) = W

d2(z, z ) = exp

= V (z, t) +

|V (z ,t)}

W(z, z ) dm(z ) + I (z).

d2(z, z ) 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 Hlder 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 [2529, 31] by proceeding in a constructive fashion. In a rst part, we dene 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 rst 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 satises the equation:

V (z) =

|V (z )}

W(z, z ) dm(z ) + Iext(z).

For convenience, we note M(z, K) the integral

K W(z, z ) dm(z ) with K = {z

|V (z) }. The relation V (z) = holds for all z K.

Page 24 of 51 Faye et al.

Denition 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) > , r [0, [,

V () = , V (r) < , r ], [,

V () = 0,

and is a xed point of equation (20):

V (r) = M(r, ) + Iext(r), (22)

where Iext(r) = Ie

r222 is a Gaussian input and M(r, ) is dened by the following

equation:

z, Bh(0, )

and Bh(0, ) is a hyperbolic disk centered at the origin of hyperbolic radius .

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

= M() + Iext() def= N(), (23)

where

M() def= M(, ).

The existence of such a bump can then be established by nding solutions to (23) The function N() is plotted in Figure 1 for a range of the input amplitude I. The

horizontal dashed lines indicate different values of , the points of intersection determine the existence of stationary pulse solutions. Qualitatively, for sufciently large input amplitude I we have N (0) < 0 and it is possible to nd only one solution

branch for large . For small input amplitudes I we have N (0) > 0 and there al

ways exists one solution branch for < c 0.06. For intermediate values of the

input amplitude I, as varies, we have the possiblity of zero, one or two solutions.

Anticipating the stability results of Section 5.3, we obtain that when N () < 0 then the corresponding solution is stable.

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

Theorem 5.1.1 For all (r, )

M(r, ) def

= M

R+

R+:

M(r, ) =

14 sinh()2 cosh()2

W() (0,0)(r) (1,1)() tanh

d, (24)

Journal of Mathematical Neuroscience (2011) 1:4 Page 25 of 51

Fig. 1 Plot of N() dened in (23) as a function of the pulse width for several values of the input amplitude I and for a xed input width = 0.05. The horizontal dashed lines indicate different values of

. The connectivity function is given in equation (21) and the parameter b is set to b = 0.2.

where

W() is the Fourier Helgason transform of W(z) def= W(d2(z, 0)) and

(,)() = F

2( + i),

2( i); + 1; sinh()2

with + + 1 = and F is the hypergeometric function of rst kind.

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

F (, ; ; z) =

() () ( )

0 t1(1 t)1(1 tz) dt

with ( ) > () > 0.

Remark 5.1.2 In Section 4 we introduced the function (z) =

D e(i+1) z,b db.

In [22], it is shown that:

(0,0)(r) =

tanh(r) if z = tanh(r)ei.

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) =

0 w

|r r |

r dr d = 2a

w()J0(r)J1(a) d,

Page 26 of 51 Faye et al.

where J(x) is the Bessel function of the rst kind and w is the real Fourier transform

of w. In our case, instead of the Bessel function, we nd (,)(r) which is linked to the hypergeometric function of the rst kind.

We now show that for a general monotonically decreasing weight function W , the function M(r, ) 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 with respect to r yields:M

r (r, ) =

tanh(r), tanh(r )ei sinh(2r ) dr d.

We have to compute

tanh(r), tanh(r )ei

= W

tanh(r), tanh(r )ei r

tanh(r), tanh(r )ei

It is result of elementary hyperbolic trigonometry that

tanh(r), tanh(r )ei

= tanh1

tanh(r)2 + tanh(r )2 2 tanh(r) tanh(r ) cos()

1 + tanh(r)2 tanh(r )2 2 tanh(r) tanh(r ) cos()

(25)

we let = tanh(r), = tanh(r ) and dene

F , () =

2 + 2 2 cos() 1 + 2 2 2 cos()

It follows that

tanh1

F , ()

= 2(1 F , ())

F , (),

and

F , () =

2( cos()) + 2 ( cos())

(1 + 2 2 2 cos())2

We conclude that if > tanh() then for all 0 tanh() and 0 2

2 cos()

cos()

> 0,

Journal of Mathematical Neuroscience (2011) 1:4 Page 27 of 51

which implies M(r, ) < 0 for r > , since W < 0.

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

r (r, ) =

14 sinh()2 cosh()2

W() r (0,0)(r) (1,1)() tanh

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

ddzF (a, b; c; z) =

abc F (a + 1, b + 1; c + 1; z).

It implies

r (0,0)(r) =

2 sinh(r) cosh(r)

1 + 2

(1,1)(r).

Substituting in the previous equation giving Mr we nd:

r (r, ) =

164 sinh(2)2 sinh(2r)

1 + 2

(1,1)(r) (1,1)() tanh

W()

implying that:

sgn

Mr (r, )

sgn

Mr (, r)

Consequently, Mr(r, ) < 0 for r < . 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 Poincar 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, ), we use the Fourier transform in

D which has already been introduced in Section 4. First we rewrite M(r, ) as a convolution product:

Page 28 of 51 Faye et al.

Proposition 5.2.1 For all (r, )

R+

R+:

M(r, ) =

1 4

d. (26)

Proof We start with the denition of M(r, ) and use the convolutional form of the

integral:

M(r, ) = M

z, Bh(0, )

W() 1Bh(0,)(z) tanh

Bh(0,)

W(z, z ) dm(z )

D W(z, z )1Bh(0,)(z ) dm(z ) = W 1Bh(0,)(z).

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

W(z) =

1 4

d db

W(, b)e(i+1) z,b tanh

1 4

W()

D e(i+1) z,b db

tanh

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

W(z) =

1 4

d. (27)

W() (z) tanh

Finally we have:

M(r, ) = W 1Bh(0,)(z) =

1 4

W() 1Bh(0,)(z) tanh

which is the desired formula.

It appears that the study of M(r, ) consists in calculating the convolution product

1Bh(0,)(z).

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

1Bh(0,)(z) =

Bh(0,)

k1 z dm(z ).

Proof Let z = k O for k G we have:

1Bh(0,)(z) =

G 1Bh(0,)(g O)

g1 z

= G 1Bh(0,)(g O)

g1k O

Journal of Mathematical Neuroscience (2011) 1:4 Page 29 of 51

for all g, k G, (g1k O) = (k1g O) so that:

1Bh(0,)(z) = G 1Bh(0,)(g O)

k1g O dg

D 1Bh(0,)(z )

k1 z dm(z )

Bh(0,)

k1 z dm(z ).

5.2.2 Second step

In this part, we prove two results:

the mapping z = k O = tanh(r)ei 1Bh(0,)(z) is a radial function, that

is, it depends only upon the variable r.

the following equality holds for z = tanh(r)ei :

1Bh(0,)(z) = (ar O)

Bh(0,)e(i+1) z ,1 dm(z ).

Proposition 5.2.3 If z = k O and z is written tanh(r)ei with r = d2(z, O) in hy

perbolic polar coordinates the function 1Bh(0,)(z) depends only upon the vari

able r.

Proof If z = tanh(r)ei , then z = rotar O and k1 = arrot . Similarly z =

rot ar O. We can write thanks to the previous Proposition 5.2.2:

1Bh(0,)(z) = Bh(0,)

k1 z dm(z )

0 (arrot

ar O) sinh(2r ) dr d

0 (arrotar O) sinh(2r ) dr d

Bh(0,) (ar z ) dm(z ),

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

We now give an explicit formula for the integral

Bh(0,) (ar z ) dm(z ).

Proposition 5.2.4 For all z = tanh(r)ei we have:

1Bh(0,)(z) = (ar O)

Bh(0,)e(i+1) z ,1 dm(z ).

Page 30 of 51 Faye et al.

Proof We rst recall a formula from [22].

Lemma 5.2.1 For all g G the following equation holds:

g1 z

= D e(i+1) gO,b e(i+1) z,b db.

Proof See [22].

It follows immediately that for all z

D and r

R we have:

D e(i+1) arO,b e(i+1) z,b db.

We integrate this formula over the hyperbolic ball Bh(0, ) which gives:

Bh(0,) (ar z ) dm(z ) = Bh(0,)

D e(i+1) arO,b e(i+1) z ,b db

dm(z ),

(ar z) =

and we exchange the order of integration:

= D e(i+1) arO,b Bh(0,)e(i+1) z ,b dm(z )

db.

We note that the integral

Bh(0,) e(i+1) z ,b dm(z ) does not depend upon the variable b = ei. Indeed:

Bh(0,)e(i+1) z ,b dm(z )

i+1

2 sinh(2x) dx d

1 tanh(x)2| tanh(x)ei ei|2

i+1

2 sinh(2x) dx d

1 tanh(x)2| tanh(x)ei() 1|2

i+1

2 sinh(2x) dx d ,

1 tanh(x)2| tanh(x)ei 1|2

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

Bh(0,)e(i+1) z ,b dm(z ) = Bh(0,)e(i+1) z ,1 dm(z ).

Finally, we can write:

Bh(0,) (ar z ) dm(z ) =

D e(i+1) arO,b db Bh(0,)e(i+1) z ,1 dm(z )

Journal of Mathematical Neuroscience (2011) 1:4 Page 31 of 51

= (ar O)

Bh(0,)e(i+1) z ,1 dm(z )

= (ar O)

Bh(0,)e(i+1) z ,1 dm(z ),

because = (as solutions of the same equation).

This completes the proof that:

g1 z

Bh(0,) (ar z ) dm(z ) = (ar O) Bh(0,)e(i+1) z ,1 dm(z ).

5.2.3 Third step

We state a useful formula.

Lemma 5.2.2 For all > 0 the following formula holds:

Bh(0,) (z) dm(z) = sinh()2 cosh()2 (1,1)().

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)ei Bh(0, ), M(r, ) is given by the following

formula:

M(r, ) =

1 4

W() (ar O) () tanh

1 4

W() (0,0)(r) () tanh

where

Bh(0,)e(i+1) z ,1 dm(z ).

We are now in a position to obtain the analytic form for M(r, ) of Theorem 5.1.1.

We prove that

() =

Bh(0,) (z) dm(z).

() def

Page 32 of 51 Faye et al.

Indeed, in hyperbolic polar coordinates, we have:

() =

0 e(i+1) rotarO,1 sinh(2r) dr d

0 e(i+1) arO,ei sinh(2r) dr d

D e(i+1) arO,b db sinh(2r) dr

0 (ar O) sinh(2r) dr.

On the other hand:

Bh(0,) (z) dm(z) =1 2

0 (ar O) sinh(2r) dr d

0 (ar O) sinh(2r) dr.

This yields

Bh(0,) (z) dm(z) = sinh()2 cosh()2 (1,1)(),

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) + (z, t).

This leads to the linear equation:

t(z, t) = (z, t) +

D W

() =

(z , t) dm(z ).

We separate variables by setting (z, t) = (z)et to obtain the equation:

( + )(z) =

D W

d2(z, z )

V (r )

d2(z, z )

V (r )

(z ) dm(z ).

Journal of Mathematical Neuroscience (2011) 1:4 Page 33 of 51

Introducing the hyperbolic polar coordinates z = tanh(r)ei and using the result:

V (r) = V (r) =(r )

|V ()|

we obtain:

( + )(z) =

0 W

tanh(r)ei, tanh(r )ei (r )

|V ()|

tanh(r )ei

sinh(2r ) dr d

sinh(2) 2|V ()|

0 W

tanh(r)ei, tanh(r )ei tanh()ei d .

Note that we have formally differentiated the Heaviside function, which is permissible since it arises inside a convolution. One could also develop the linear stability analysis by considering perturbations of the threshold crossing points along the lines of Amari [20]. Since we are linearizing about a stationary rather than a traveling pulse, we can analyze the spectrum of the linear operator without the recourse to Evans functions.

With a slight abuse of notation we are led to study the solutions of the integral equation:

( + )(r, ) =

sinh(2) 2|V ()|

W(r, ; )(, ) d , (28)

where the following equality derives from the denition of the hyperbolic distance in equation (25):

W(r, ; ) def= W tanh1

tanh(r)2 + tanh()2 2 tanh(r) tanh() cos()

1 + tanh(r)2 tanh()2 2 tanh(r) tanh() cos()

Essential spectrum If the function satises the condition

W(r, ; )(, ) d = 0 r,

then equation (28) reduces to:

+ = 0

yielding the eigenvalue:

= < 0.

This part of the essential spectrum is negative and does not cause instability.

Page 34 of 51 Faye et al.

Discrete spectrum If we are not in the previous case we have to study the solutions of the integral equation (28).

This equation shows that (r, ) is completely determined by its values (, ) on the circle of equation r = . Hence, we need only to consider r = , yielding the

integral equation:

( + )(, ) =

sinh(2) 2|V ()|

W(, ; )(, ) d .

The solutions of this equation are exponential functions e , where satises:

( + ) =

sinh(2) 2|V ()|

W(, ; )e d .

By the requirement that is 2-periodic in , it follows that = in, where n

Thus the integral operator with kernel W has a discrete spectrum given by:(n + )

sinh(2) 2|V ()|

W(, ; )ein d

sinh(2) 2|V ()|

= 0 W tanh1

2 tanh()2(1 cos( ))

1 + tanh()4 2 tanh()2 cos( )

ein d

sinh(2)

= |V ()|

0 W tanh1

2 tanh() sin( )

(1 tanh()2)2 + 4 tanh()2 sin( )2 ei2n d .

n is real since:

(n)

sinh(2)

= |V ()|

0 W tanh1

2 tanh() sin( )

(1 tanh()2)2 + 4 tanh()2 sin( )2 sin(2n ) d

= 0.

Hence,

n = (n)

= +

sinh(2)

|V ()|

Journal of Mathematical Neuroscience (2011) 1:4 Page 35 of 51

2 tanh() sin( )

(1 tanh()2)2 + 4 tanh()2 sin( )2 cos(2n ) d .

We can state the folliwing proposition:

Proposition 5.3.1 Provided that for all n 0, n < 0 then the hyperbolic stationary

pulse is stable.

We now derive a reduced condition linking the parameters for the stability of hyperbolic stationary pulse.

Reduced condition Since W tanh1(r) is a positive function of r, it follows that: n 0.

Stability of the hyperbolic stationary pulse requires that for all n 0, n < 0. This

can be rewritten as:

sinh(2)

|V ()|

0 W tanh1

2 tanh() sin( )

(1 tanh()2)2 + 4 tanh()2 sin( )2

cos(2n ) d < , n 0.

Using the fact that n 0 for all n 1, we obtain the reduced stability condition:

W0()

|V ()|

< ,

where

W0() def= sinh(2)

0 W tanh1

2 tanh() sin( )

(1 tanh()2)2 + 4 tanh()2 sin( )2 d .

From (22) we have:

V () =

Mr() + I ()

where

Mr () def=

r (, )

164 sinh(2)3

W()

1 + 2

(1,1)() (1,1)() tanh

We have previously established that Mr() > 0 and I () is negative by denition.

Hence, letting D() = |I ()|, we have

()

Mr() + D()

Page 36 of 51 Faye et al.

By substitution we obtain another form of the reduced stability condition:

D() > W0() Mr(). (29)

We also have:

M () =

dd M(, ) =

r (, ) +

(, ) = W0() Mr(),

and

N () = M () + I () = W0() Mr() D(),showing that the stability condition (29) is satised when N () < 0 and is not satised when N () > 0.

Proposition 5.3.2 (Reduced condition) If N () < 0 then for all n 0, n < 0 and

the hyperbolic stationary pulse is stable.

6 Numerical results

The aim of this section is to numerically solve (13) for different values of the parameters. This implies developing a numerical scheme that approaches the solution of our equation, and proving that this scheme effectively converges to the solution.

Since equation (13) is dened on D, computing the solutions on the whole hyperbolic disk has the same complexity as computing the solutions of usual Euclidean neural eld equations dened on R2. As most authors in the Euclidean case [26, 27, 29, 31], we reduce the domain of integration to a compact region of the hyperbolic disk. Practically, we work in the Euclidean ball of radius a = 0.5 and center 0. Note

that a Euclidean ball centered at the origin is also a centered hyperbolic ball, their radii being different.

We have divided this section into four parts. The rst part is dedicated to the study of the discretization scheme of equation (13). In the following two parts, we study the solutions for different connectivity functions: an exponential function, Section 6.2, and a difference of Gaussians, Section 6.3.

6.1 Numerical schemes

Let us consider the modied equation of (13):

tV (z, t) = V (z, t)

+ B(0,a) W(z, z )S

V (z , t) dm(z ) + I (z, t), t J,

V (z, 0) = V0(z).

(30)

We assume that the connectivity function satises the conditions (C1)-(C2). Moreover we express z in (Euclidean) polar coordinates such that z = rei , V (z, t) =

Journal of Mathematical Neuroscience (2011) 1:4 Page 37 of 51

V (r, , t) and W(z, z ) = W(r, , r , ). The integral in equation (30) is then:

B(0,a) W(z, z )S

V (z , t) dm(z )

0 W(r, , r , )S

V (r , , t) r dr d (1 r 2)2.

We dene R to be the rectangle R def= [0, a] [0, 2].

6.1.1 Discretization scheme

We discretize R in order to turn (30) into a nite number of equations. For this

purpose we introduce h1 = aN , N

N =

\{0} and h2 = 2M , M

i [[1, N + 1]] ri = (i 1)h1,

j [[1, M + 1]] j = (j 1)h2,

and obtain the (N + 1)(M + 1) equations: dVdt (ri, j, t) = V (ri, j , t) +

W(ri, j , r , )S

V (r , , t) r dr d (1 r 2)2

+ I (ri, j , t)

which dene the discretization of (30):

d Vdt (t) = V (t) + W S( V )(t) +(t), t J,

V (0) = V0,

(31)

where V (t) MN+1,M+1(

R), V (t)i,j = V (ri, j, t). Similar denitions apply to

and V0. Moreover:

W S( V )(t)i,j = R

W(ri, j, r , )S

V (r , , t) r dr d (1 r 2)2.

Mn,p(R) is the space of the matrices of size n p with real coefcients. It remains

to discretize the integral term. For this as in [33], we use the rectangular rule for the quadrature so that for all (r, ) R we have:

0 W(r, , r , )S

V (r , , t) r dr d (1 r 2)2

= h1h2

N+1

k=1

M+1

l=1 W(r, , rk, l)S

V (rk, l, t) rk(1 r2k)2.

Page 38 of 51 Faye et al.

We end up with the following numerical scheme, where Vi,j (t) (resp. Ii,j (t)) is an

approximation of Vi,j (t) (resp.i,j ), (i, j) [[1, N + 1]] [[1, M + 1]]:

dVi,jdt (t) = Vi,j (t) + h1h2

N+1

k=1

M+1

l=1

Wi,jk,lS(Vk,l)(t) + Ii,j (t)

with

Wi,jk,l def= W(ri, j, rk, l)

rk (1r2k)

2 .

6.1.2 Discussion

We discuss the error induced by the rectangular rule for the quadrature. Let f be a function which is C2 on a rectangular domain [a, b] [c, d]. If we denote by Ef this

error, then |Ef | (ba)

2(dc)2

4mn

f C2 where m and n are the number of subintervals

2 ||2 sup[a,b][c,d] |f | where, as usual, is a multi-index. As a consequence, if we want to control the error, we have to impose that the solution is, at least, C2 in space.

Four our numerical experiments we use the specic function ode45 of Matlab which is based on an explicit Runge-Kutta formula (see [34] for more details on Runge-Kutta methods).

We can also establish a proof of the convergence of the numerical scheme which is exactly the same as in [33] excepted that we use the theorem of continuous dependence of the solution for ordinary differential equations.

6.2 Purely excitatory exponential connectivity function

In this subsection, we give some numerical solutions of (13) in the case where the connectivity function is an exponential function, w(x) = e

|x|

used and f C

b , with b a positive parameter. Only excitation is present in this case. In all the experiments we set = 0.1

and S(x) =

1 1+e

x with = 10.

6.2.1 Constant input

We x the external input I (z) to be of the form:

I (z) = Ie

d2(z,0)2

2 .

In all experiments we set I = 0.1 and = 0.05, this means that the input has a sharp

prole centered at 0.

We show in Figure 2 plots of the solution at time T = 2,500 for three different

values of the width b of the exponential function. When b = 1, the whole network

is highly excited, whereas as b changes from 1 to 0.1 the amplitude of the solution decreases, and the area of high excitation becomes concentrated around the external input.

Journal of Mathematical Neuroscience (2011) 1:4 Page 39 of 51

6.2.2 Variable input

In this paragraph, we allow the external current to depend upon the time variable. We have:

I (z, t) = Ie

d2(z,z0(t))2

2 ,

where z0(t) = r0ei 0t. This is a bump rotating with angular velocity 0 around the

circle of radius r0 centered at the origin. In our numerical experiments we set r0 =

0.4, 0 = 0.01, I = 0.1 and = 0.05. We plot in Figure 3 the solution at different

times T = 100, 150, 200, 250.

6.2.3 High gain limit

We consider the high gain limit of the sigmoid function and we propose to

illustrate Section 5 with a numerical simulation. We set = 1, = 0.04, = 0.18.

We x the input to be of the form:

I (z) = Ie

with I = 0.04 and = 0.05. Then the condition of existence of a stationary pulse

(23) is satised, see Figure 1. We plot a bump solution according to (23) in Figure 4.

Fig. 2 Plots of the solution of equation (13) at T = 2,500 for the values = 10, = 0.1 and for decreas

ing values of the width b of the connectivity, see text.

d2(z,0)2 2

Page 40 of 51 Faye et al.

Fig. 3 Plots of the solution of equation (13) in the case of an exponential connectivity function with b = 0.1 at different times with a time-dependent input, see text.

6.3 Excitatory and inhibitory connectivity function

We give some numerical solutions of (13) in the case where the connectivity function is a difference of Gaussians, which features an excitatory center and an inhibitory surround:

w(x) =

1 1+e

12 so that it is equal to 0 at the origin

and we choose the external input equal to zero, I (z, t) = 0. In this case the constant

function equal to 0 is a solution of (13).

Fig. 4 Plot of a bump solution of equation (22) for the values = 1, = 0.04, = 0.18 and

for b = 0.2 for the width of the

connectivity, see text.

221e x221

222e x222 with 1 = 0.1, 2 = 0.2 and A = 1.

We illustrate the behaviour of the solutions when increasing the slope of the sigmoid. We set the sigmoid S(x) =

Journal of Mathematical Neuroscience (2011) 1:4 Page 41 of 51

For small values of the slope , the dynamics of the solution is trivial: every solution asymptotically converges to the null solution, as shown in top left hand corner of Figure 5 with = 1. When increasing , the stability bound, found in Section 4.5

is no longer satised and the null solution may no longer be stable. In effect this solution may bifurcate to other, more interesting solutions. We plot in Figure 5, some solutions at T = 2,500 for different values of ( = 3, 5, 10, 20 and 30). We can see

exotic patterns which feature some interesting symmetries. The formal study of these bifurcated solutions is left for future work.

7 Conclusion

In this paper, we have studied the existence and uniqueness of a solution of the evolution equation for a smooth neural mass model called the structure tensor model. This model is an approach to the representation and processing of textures and edges in the visual area V1 which contains as a special case the well-known ring model of orientations (see [1, 2, 19]). We have also given a rigorous functional framework for the study and computation of the stationary solutions to this nonlinear integro-differential equation. This work sets the basis for further studies beyond the spatially periodic case studied in [15], where the hypothesis of spatial periodicity allows one to replace the unbounded (hyperbolic) domain by a compact one, hence making the functional analysis much simpler.

We have completed our study by constructing and analyzing spatially localised bumps in the high-gain limit of the sigmoid function. It is true that networks with Heaviside nonlinearities are not very realistic from the neurobiological perspective and lead to difcult mathematical considerations. However, taking the high-gain limit is instructive since it allows the explicit construction of stationary solutions which is impossible with sigmoidal nonlinearities. We have constructed what we called a hyperbolic radially symmetric stationary-pulse and presented a linear stability analysis adapted from [31]. The study of stationary solutions is very important as it conveys information for models of V1 that is likely to be biologically relevant. Moreover our study has to be thought of as the analog in the case of the structure tensor model to the analysis of tuning curves of the ring model of orientations (see [1, 2, 19, 35]). However, these solutions may be destabilized by adding lateral spatial connections in a spatially organized network of structure tensor models; this remains an area of future investigation. As far as we know, only Bressloff and coworkers looked at this problem (see [3, 4, 1114]).

Finally, we illustrated our theoretical results with numerical simulations based on rigorously dened numerical schemes. We hope that our numerical experiments will lead to new and exciting investigations such as a thorough study of the bifurcations of the solutions of our equations with respect to such parameters as the slope of the sigmoid and the width of the connectivity function.

Appendix A: Isometries of D

We briey descrbies the isometries of D, that is, the transformations that preserve the distance d2. We refer to the classical textbooks in hyperbolic goemetry for details,

Page 42 of 51 Faye et al.

Fig. 5 Plots of the solutions of equation (13) in the case where the connectivity function is the difference of two Gaussians at time T = 2,500 for = 0.1 and for increasing values of the slope of the sigmoid,

see text.

Journal of Mathematical Neuroscience (2011) 1:4 Page 43 of 51

for example, [17]. The direct isometries (preserving the orientation) in D are the elements of the special unitary group, noted SU(1, 1), of 2 2 Hermitian matrices

with determinant equal to 1. Given:

such that ||2 ||2 = 1,

an element of SU(1, 1), the corresponding isometry in D is dened by:

z =

z +

D. (32)

Orientation reversing isometries of D are obtained by composing any transformation (32) with the reection : z z. The full symmetry group of the Poincar disc is

therefore:

U(1, 1) = SU(1, 1) SU(1, 1).

Let us now describe the different kinds of direct isometries acting in D. We rst dene the following one parameter subgroups of SU(1, 1):

K def =

, z

rot =

ei2 0 0 ei 2

A def =

ar =

cosh r sinh r sinh r cosh r

, r

N def =

ns =

1 + is is is 1 is

, s

Note that rot z = eiz and also ar O = tanh r, with O being the center of the

Poincar disk that is the point represented by z = 0.

The group K is the orthogonal group O(2). Its orbits are concentric circles. It is possible to express each point z

D in hyperbolic polar coordinates: z = rotar O =

tanh rei and r = d2(z, 0).

The orbits of A converge to the same limit points of the unit circle D, b1 = 1

when r . They are circular arcs in

D going through the points b1 and b1.

The orbits of N are the circles inside D and tangent to the unit circle at b1. These circles are called horocycles with base point b1. N is called the horocyclic group. It is also possible to express each point z

D in horocyclic coordinates: z = nsar O,

where ns are the transformations associated with the group N (s

R) and ar the

transformations associated with the subroup A (r

R).

A.1 Iwasawa decomposition

The following decomposition holds, see [36]:

SU(1, 1) = KAN.

This theorem allows us to decompose any isometry of D as the product of at most three elements in the groups, K, A and N.

Page 44 of 51 Faye et al.

Appendix B: Volume element in structure tensor space

Let T be a structure tensor

T =

!x1 x3 x3 x2

2 its determinant, 0. T can be written

T = T ,

where T has determinant 1. Let z = z1 + iz2 be the complex number representation

of T in the Poincar disk

D. In this part of the appendix, we present a simple form for the volume element in full structure tensor space, when parametrized as ( , z).

Proposition B.0.1 The volume element in ( , z1, z2) coordinates is

dV = 82

dz1 dz2 (1 |z|2)2

. (33)

Proof In order to compute the volume element in ( , z1, z2) space, we need to express the metric gT in these coordinates. This is obtained from the inner product

in the tangent space TT at point T of SDP(2). The tangent space is the set S(2) of

symmetric matrices and the inner product is dened by:

gT (A, B) = tr(T 1AT 1B), A, B S(2).

We note that gT (A, B) = g

(A, B)/ 2. We write g instead of g

. A basis of TT

for that matter) is given by:

x1 = !

1 0 0 0

(or T

x2 = !

0 0 0 1

x3 = !

0 1

1 0

and the metric is given by:

gij = g

xi ,

, i, j = 1, 2, 3.

The determinant GT of gT is equal to G/ 6, where G is the determinant of g = g

G is found to be equal to 2. The volume element is thus:

dV =

3 dx1 dx2 dx3.

We then use the relations:

x1 = x1, x2 = x2, x3 = x3,

Journal of Mathematical Neuroscience (2011) 1:4 Page 45 of 51

where xi, i = 1, 2, 3, is given by:

x1 =

(1 + z1)2 + z22 1 z21 z22

x2 =

(1 z1)2 + z22 1 z21 z22

x3 =

2z21 z21 z22

The determinant of the Jacobian of the transformation (x1, x2, x3) ( , z1, z2)

is found to be equal to:

8 2(1 |z|2)2

Hence, the volume element in ( , z1, z2) coordinates is

dV = 82

dz1 dz2 (1 |z|2)2

Appendix C: Global existence of solutions

Theorem C.0.1 Let O be an open connected set of a real Banach space F and J be

an open interval of R. We consider the initial value problem:

V (t) = f

t, V(t)

, V(t0) = V0.

(34)

We suppose that f C(J O, F) and is locally Lipschitz with respect to its second

argument. Then for all (t0, V0) J O, there exists > 0 and V C1(]t , t0 +[, O)

unique solution of (34).

Lemma C.0.1 Under hypotheses of Theorem C.0.1, if V1 C1(J1, O) and V2 C1(J2, O) are two solutions and if there exists t0 J1 J2 such that V1(t0) = V2(t0)

then:

V1(t) = V2(t) for all t J1 J2.

This lemma shows the existence of a larger interval J0 on which the initial value problem (34) has a unique solution. This solution is called the maximal solution.

Theorem C.0.2 Under hypotheses of Theorem C.0.1, let V C1(J0, O) be a max

imal solution. We denote by b the upper bound of J and the upper bound of J0. Then either = b or for all compact set K O, there exists < such that:

V(t) O|K, for all t with t J0

We have the same result with the lower bounds.

Page 46 of 51 Faye et al.

Theorem C.0.3 We suppose f C(J F, F) and is globally Lipschitz with respect

to its second argument. Then for all (t0, V0) J F, there exists a unique V C1(J, F) solution of (34).

Appendix D: Proof of Lemma 3.1.1

Lemma D.0.2 When W is only a function of d0(T , T ), then W does not depend

upon the variable T .

Proof We work in (z, ) coordinates and we begin by rewriting the double integral (6) for all (z, )

R+

W(z, , t) =

+ D W

2(log log )2 + d22(z, z ), t

dz 1 dz 2 (1 |z |2)2

The change of variable yields:

W(z, , t) =

+ D W

2(log )2 + d22(z, z ), t

dz 1 dz 2 (1 |z |2)2

And it establishes that W does not depend upon the variable . To nish the proof, we show that the following integral does not depend upon the variable z

d2(z, z ) dz 1 dz 2(1 |z |2)2, (35)

where f is a real-valued function such that (z) is well dened.We express z in horocyclic coordinates: z = nsar O (see Appendix A) and (35)

becomes:

(z) =

D f

R f

d2(nsar O, ns ar O)

e2r ds dr

d2(nss ar O, ar O)

e2r ds dr .

R f

With the change of variable s s = xe2r, this becomes:

(z) =

R f

d2(nxe2r ar O, ar O)

e2(r r) dx dr .

The relation nxe

2r ar O = arnx O (proved, for example, in [22]) yields:

(z) =

d2(arnx O, ar O)

e2(rr) dx dr

R f

d2(O, nxar r O)

e2(r r) dx dr

R f

Journal of Mathematical Neuroscience (2011) 1:4 Page 47 of 51

d2(O, nxay O)

e2y dx dy

R f

D f

d2(O, z ) dm(z )

2)2 , which shows that (z) does not depend upon the variable z, as announced.

Appendix E: Proof of Lemma 3.1.2

In this section we prove the following lemma.

Lemma E.0.3 When W is the following Mexican hat function:

W(z, , z ) = w

2(log log )2 + d22(z, z )

with z = z 1 + iz 2 and dm(z ) =

dz 1 dz 2 (1|z |

where:

w(x) =

221e x

222e x

with 0 1 2 and 0 A 1.

Then:

W =

1e221 erf

A2e2

22 erf

where erf is the error function dened as:

erf(x) =

0 eu2 du.

Proof We consider the following double integrals:

i =

22ie (log log

)2 2

i e

2 (z,z )

i d

dz 1 dz 2 (1 |z |2)2

, i = 1, 2, (36)

so that:

W = 1 A 2.

Since the variables are separable, we have:

i =

2 (z,z )

0 22ie (log log

)2 2

i d

D e

i dz 1 dz 2

(1 |z |2)2

Page 48 of 51 Faye et al.

One can easily see that:

0 22ie (log log

)2 2

i d

1 2.

We now give a simplied expression for i. We set fi(x) = ex2/(22

i ) and then

we have, because of Lemma 3.1.1:

i =

1 2

D fi

d2(O, z )

dm(z ) =

1 2

D fi

arctanh

|z |

dm(z )

1 2

1 0 fi

arctanh(r) r dr d(1 r2)2 =

1 arctanh(r) r dr(1 r2)2

= 2

1 arctanh

2(r) 22

i r dr

(1 r2)2

The change of variable x = arctanh(r) implies dx =

dr 1r

2 and yields:

i = 2

0 e

x222

i tanh(x)1 tanh2(x)

dx = 2

0 e

x222

i sinh(x) cosh(x) dx

0 e

i sinh(2x) dx

x222

+2x

0 e

x222

= 22e2

2 i

(x2

2i )2 22

+ i dx

0 e

(x+2

2i )2 22

i dx

2 ie2

2 i

+ eu2 du

2 ie2

2 i

eu2 du

then we have a simplied expression for i:

i =

2 ie2

2i erf

Appendix F: Proof of Lemma 5.2.2

Lemma F.0.4 For all > 0 the following formula holds:

Bh(0,) (z) dm(z) = sinh()2 cosh()2 (1,1)().

Journal of Mathematical Neuroscience (2011) 1:4 Page 49 of 51

Proof We write z in hyperbolic polar coordinates, z = tanh(r)ei (see Appendix A).

We have:

Bh(0,) (z) dm(z) =1 2

tanh(r)ei sinh(2r) dr d.

Because of the above denition of , this reduces to

tanh(r) sinh(2r) dr.

In [22] Helgason proved that:

tanh(r) = F

, 1 ; 1; sinh(r)2

with = 12(1 + i). We then use the formula obtained by Erdelyi in [32]:

F (, 1 ; 1; z) =

d dz

zF (, 1 ; 2; z)

Using some simple hyperbolic trigonometry formulae we obtain:

sinh(2r)F

, 1 ; 1; sinh(r)2

d dr

sinh(r)2F

, 1 ; 2; sinh(r)2

from which we deduce

Bh(0,) (z) dm(z) = sinh()2F

, 1 ; 2; sinh()2

Finally we use the equality shown in [32]:

F (a, b; c; z) = (1 z)cabF (c a, c b; c; z).

In our case we have: a = , b = 1 , c = 2 and z = sinh()2, so 2 = 12(3

i), 1 + = 12(3 + i). We obtain

Bh(0,) (z) dm(z)

= sinh()2 cosh()2F

2(3 i),

2(3 + i); 2; sinh()2

Since Hypergeometric functions are symmetric with respect to the rst two variables:

F (a, b; c; z) = F (b, a; c; z),

we write

12(3 i),12(3 + i); 2; sinh()2

Page 50 of 51 Faye et al.

12(3 + i),12(3 i); 2; sinh()2 = (1,1)(),

which yields the announced formula

Bh(0,) (z) dm(z) = sinh()2 cosh()2 (1,1)().

Competing interests

The authors declare that they have no competing interests.

Acknowledgements This work was partially funded by the ERC advanced grant NerVi number 227747.

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We study the neural field equations introduced by Chossat and Faugeras to model the representation and the processing of image edges and textures in the hypercolumns of the cortical area V1. The key entity, the structure tensor, intrinsically lives in a non-Euclidean, in effect hyperbolic, space. Its spatio-temporal behaviour is governed by nonlinear integro-differential equations defined on the Poincaré disc model of the two-dimensional hyperbolic space. Using methods from the theory of functional analysis we show the existence and uniqueness of a solution of these equations. In the case of stationary, i.e. time independent, solutions we perform a stability analysis which yields important results on their behavior. We also present an original study, based on non-Euclidean, hyperbolic, analysis, of a spatially localised bump solution in a limiting case. We illustrate our theoretical results with numerical simulations.

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

Details

Title

Analysis of a hyperbolic geometric model for visual texture perception

Author

Faye, Gregory; Chossat, Pascal; Faugeras, Olivier

Pages

1-51

Publication year

2011

Publication date

Jun 2011

Publisher

Springer Nature B.V.

e-ISSN

21908567

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1186/2190-8567-1-4

ProQuest document ID

1654015258

Springer-Verlag Berlin Heidelberg 2011

Analysis of a hyperbolic geometric model for visual texture perception

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