Analysis of a hyperbolic geometric model for visual texture perception
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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, 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.
KeywordsNeural fields nonlinear integro-differential equations functional analysis non-Euclidean analysis stability analysis hyperbolic geometry hypergeometric functions bumps
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 , velocity and direction of motion , color . 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  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 . The question of whether some populations of neurons in V1 can represent the structure tensor is discussed in  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  and Hansel, Sompolinsky . 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  and  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
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 , 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 . The second one, defined by , 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.
where is the identity matrix and . Some easy interpretations can be made for simple examples: constant areas are characterized by , straight edges are such that , their orientation being that of , corners yield . The coherency c of the local image is measured by the ratio , large coherency reveals anisotropy in the texture.
where μ describes the stiffness of the sigmoid. is an external input.
The set can be seen as a foliated manifold by way of the set of special symmetric positive definite matrices . Indeed, we have: . Furthermore, , where is the Poincaré Disk, see, for example, . As a consequence we use the following foliation of , which allows us to write for all with . z and Δ are related by the relation and the fact that z is the representation in of with .
where we normalize to 1 the volume element for the θ coordinate.
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 . 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 .
3.1 Homogeneous solutions
Hence necessary conditions for the existence of a homogeneous solution are that:
the double integral (6) is convergent,
- does not depend upon the variable . In that case, we write instead of .
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 inputand the connectivity functionare continuous on some closed interval J containing 0, then for allin, there exists a unique solution of (7) defined on a subintervalof J containing 0 such that.
We can extend this result to the whole time real line if I and are continuous on .
Proposition 3.1.1 Ifandare continuous on, then for allin, there exists a unique solution of (7) defined onsuch that.
This implies that the maximal solution V is bounded for all , but Theorem C.0.2 of Appendix C ensures that it is impossible. Then, it follows that necessarily . □
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 , then did not depend upon the variables . 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 ofonly. Thendoes not depend upon the variable.
Mexican hat connectivity In this paragraph, we push further the computation of in the special case where W does not depend upon the time variable t and takes the special form suggested by Amari in , 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 .
with and .
In this case we can obtain a very simple closed-form formula for as shown in the following lemma.
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 . Let J be a subinterval of . We assume that:
(H1): , ,
(H2): where W is defined as for all where is the identity matrix of ,
(H3): , where .
Equivalently, we can express these hypotheses in coordinates:
(H1 bis): , ,
(H2 bis): where W is defined as for all ,
- (H3 bis): , where
3.2.1 Functional space setting
Our aim is to find a functional space where (3) is well-defined and the function maps to for all t s. A natural choice would be to choose ϕ as a -integrable function of the space variable with . Unfortunately, the homogeneous solutions (constant with respect to ) 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 . As is an open set of , is a Banach space for the norm: .
Proposition 3.2.1 Ifwithand W satisfies hypotheses (H 1bis)-(H 3bis) thenis well-defined and is fromto.
3.2.2 The existence and uniqueness of a solution of (3)
Theorem 3.2.1 If the external currentbelongs towith J an open interval containing 0 and W satisfies hypotheses (H 1bis)-(H 3bis), then fo all, there exists a unique solution of (10) defined on a subintervalof J containing 0 such thatfor all.
with . This ensures the Lipschitz continuity of 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. 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 abut 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 currentbelongs toand W satisfies hypotheses (H 1bis)-(H 3bis) with, then for all, there exists a unique solution of (10) defined onsuch thatfor all.
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 currentbelongs toand is bounded in timeand W satisfies hypotheses (H 1bis)-(H 3bis) with, then the solution of (10) is bounded for each initial condition.
The following corollary is a consequence of the previous proposition.
3.3 Semi-homogeneous solutions
We have implicitly made the assumption, that 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 . We assume that:
(C1): , ,
(C2): where is defined as for all ,
(C3): where .
Note that conditions (C1)-(C2) and Lemma 3.1.1 imply that for all , . And then, for all , the mapping is integrable on .
From now on, and the Fischer-Riesz’s theorem ensures that is a Banach space for the norm: .
Theorem 3.3.1 If the external currentbelongs towith J an open interval containing 0 andsatisfies conditions (C 1)-(C 3), then for all, there exists a unique solution of (13) defined on a subintervalof J containing 0.
This solution, defined on the subinterval J of can in fact be extended to the whole real line, and we have the following proposition.
Proposition 3.3.1 If the external currentbelongs toandsatisfies conditions (C 1)-(C 3) with, then for all, there exists a unique solution of (13) defined on.
We can also state a result on the boundedness of a solution of (13):
4 Stationary solutions
4.1 Study of the nonlinear operator
We recall that we have set for the Banach space and Proposition 3.2.1 shows that . We have the further properties:
Proposition 4.1.1satisfies the following properties:
- for all,
- is continuous on.
where H is the Heaviside function.
It is straightforward to show that both operators are well-defined on and map to . Moreover the following proposition holds.
4.2 The convolution form of the operator in the semi-homogeneous case
We recall the notation .
for a function such that this integral is well-defined.
Lemma 4.2.1 The Fourier transform in, ofdoes not depend upon the variable.
We now introduce two functions that enjoy some nice properties with respect to the Hyperbolic Fourier transform and are eigenfunctions of the linear operator .
Proposition 4.2.2 Letandthen:
By rotation, we obtain the property for all .
A consequence of this proposition is the following lemma.
Lemma 4.2.2 The linear operatoris not compact and for all, the nonlinear operatoris not compact.
Proof The previous Proposition 4.2.2 shows that has a continuous spectrum which iimplies that is not a compact operator.
If was a compact operator then its Frechet derivative would also be a compact operator, but it is impossible. As a consequence, is not a compact operator. □
4.3 The convolution form of the operator in the general case
We recall that we have set by definition: .
We next assume further that the function W is separable in z, Δ and more precisely that where and for all . The following proposition is an echo to Proposition 4.2.2.
Proposition 4.3.2 Let, andthen:
whereis the usual Fourier transform of.
A straightforward consequence of this proposition is an extension of Lemma 4.2.2 to the general case:
Lemma 4.3.1 The linear operatoris not compact and for all, the nonlinear operatoris not compact.
4.4 The set of the solutions of (14)
We have the following proposition.
Proposition 4.4.1 If the input currentis equal to a constant, that is, does not depend upon the variablesthen for all, . In the general case, if the conditionis satisfied, then.
Proof Due to the properties of the sigmoid function, there always exists a constant solution in the case where is constant. In the general case where , the statement is a direct application of the Banach fixed point theorem, as in . □
and β does not depend upon the variablesbecause of Lemma 3.1.1. Because of the property of the sigmoid function S, the equationhas always one solution.
If on the other hand the input current does depend upon these variables is invariant under the action of a subgroup ofthe group of the isometries of (see Appendix A), and the conditionis satisfied then the unique stationary solution will also be invariant under the action of the same subgroup. We refer the interested reader to our workon equivariant bifurcation of hyperbolic planforms on the subject.
When the conditionis satisfied we call primary stationary solution the unique solution in.
4.5 Stability of the primary stationary solution
In this subsection we show that the condition guarantees the stability of the primary stationary solution to (3).
Theorem 4.5.1 We suppose thatand that the conditionis satisfied, then the associated primary stationary solution of (3) is asymtotically stable.
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  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 . 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 . 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.
The theoretical study of equation (20) has been done in  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
For convenience, we note the integral with . The relation holds for all .
andis a hyperbolic disk centered at the origin of hyperbolic radius ω.
We end this subsection with the usefull and technical following formula.
withand F is the hypergeometric function of first kind.
whereis the Bessel function of the first kind andis the real Fourier transform of w. In our case instead of the Bessel function we findwhich is linked to the hypergeometric function of the first kind.
We now show that for a general monotonically decreasing weight function W, the function 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.
which implies for , since .
Consequently, for . 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 , we use the Fourier transform in which has already been introduced in Section 4. First we rewrite as a convolution product: