In many models of working memory, transient stimuli are encoded by feature-selective persistent neural activity. Such stimuli are imagined to induce the formation of a spatially localised bump of persistent activity which coexists with a stable uniform state. As an example, Camperi and Wang [24] have proposed and studied a network model of visuo-spatial working memory in prefontal cortex adapted from the ring model of orientation of Ben-Yishai and colleagues [1]. Many studies have emerged in the past decades to analyse these localised bumps of activity [25–29], see the paper by Coombes for a review of the domain [30]. In [25, 26, 28], the authors have examined the existence and stability of bumps and multi-bumps solutions to an integro-differential equation describing neuronal activity along a single spatial domain. In [27, 29] the study is focused on the two-dimensional model and a method is developed to approximate the integro-differential equation by a partial differential equation which makes possible the determination of the stability of circularly symmetric solutions. It is therefore natural to study the emergence of spatially localized bumps for the structure tensor model in a hypercolumn of V1. We only deal with the reduced case of equation (13) which means that the membrane activity does not depend upon the contrast of the image intensity, keeping the general case for future work.
In order to construct exact bump solutions and to compare our results to previous studies [25–29], we consider the high gain limit of the sigmoid function. As above we denote by H the Heaviside function defined by for and otherwise. Equation (13) is rewritten as:
(20)
We have introduced a threshold κ to shift the zero of the Heaviside function. We make the assumption that the system is spatially homogeneous that is, the external input I does not depend upon the variables t and the connectivity function depends only on the hyperbolic distance between two points of . For illustrative purposes, we will use the exponential weight distribution as a specific example throughout this section:
(21)
The theoretical study of equation (20) has been done in [21] where the authors have imposed strong regularity assumptions on the kernel function W, such as Hölder continuity, and used compactness arguments and integral equation techniques to obtain a global existence result of solutions to (20). Our approach is very different, we follow that of [25–29, 31] by proceeding in a constructive fashion. In a first part, we define what we call a hyperbolic radially symmetric bump and present some preliminary results for the linear stability analysis of the last part. The second part is devoted to the proof of a technical Theorem 5.1.1 which is stated in the first part. The proof uses results on the Fourier transform introduced in Section 4, hyperbolic geometry and hypergeometric functions. Our results will be illustrated in the following Section 6.
5.1 Existence of hyperbolic radially symmetric bumps
From equation (20) a general stationary pulse satisfies the equation:
For convenience, we note the integral with . The relation holds for all .
Definition 5.1.1 V is called a hyperbolic radially symmetric stationary-pulse solution of (20) if V depends only upon the variable r and is such that:
and is a fixed point of equation (20):
(22)
whereis a Gaussian input andis defined by the following equation:
andis a hyperbolic disk centered at the origin of hyperbolic radius ω.
From symmetry arguments there exists a hyperbolic radially symmetric stationary-pulse solution of (20), furthermore the threshold κ and width ω are related according to the self-consistency condition
(23)
where
The existence of such a bump can then be established by finding solutions to (23) The function is plotted in Figure 1 for a range of the input amplitude . The horizontal dashed lines indicate different values of ακ, the points of intersection determine the existence of stationary pulse solutions. Qualitatively, for sufficiently large input amplitude we have and it is possible to find only one solution branch for large ακ. For small input amplitudes we have and there always exists one solution branch for . For intermediate values of the input amplitude , as αβ varies, we have the possiblity of zero, one or two solutions. Anticipating the stability results of Section 5.3, we obtain that when then the corresponding solution is stable.
We end this subsection with the usefull and technical following formula.
Theorem 5.1.1 For all:
(24)
where
is the Fourier Helgason transform of
and
withand F is the hypergeometric function of first kind.
Remark 5.1.1 We recall that F admits the integral representation[32]:
with.
Remark 5.1.2 In Section 4we introduced the function. In[22], it is shown that:
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 diskcentered at the origin O and of radius a. Using their notations
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.
Proof Differentiating with respect to r yields:
We have to compute
It is result of elementary hyperbolic trigonometry that
(25)
we let , and define
It follows that
and
We conclude that if then for all and
which implies for , since .
To see that it is also negative for , we differentiate equation (24) with respect to r:
The following formula holds for the hypergeometric function (see Erdelyi in [32]):
It implies
Substituting in the previous equation giving we find:
implying that:
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:
Proposition 5.2.1 For all:
(26)
Proof We start with the definition of and use the convolutional form of the integral:
In [22], Helgason proves an inversion formula for the hyperbolic Fourier transform and we apply this result to W:
the last equality is a direct application of Lemma 4.2.1 and we can deduce that
(27)
Finally we have:
which is the desired formula. □
It appears that the study of consists in calculating the convolution product .
Proposition 5.2.2 For allforwe have:
Proof Let for we have:
for all , so that:
□
5.2.2 Second step
In this part, we prove two results:
-
the mapping is a radial function, that is, it depends only upon the variable r.
-
the following equality holds for :
Proposition 5.2.3 Ifand z is writtenwithin hyperbolic polar coordinates the functiondepends only upon the variable r.
Proof If , then and . Similarly . We can write thanks to the previous Proposition 5.2.2:
which, as announced, is only a function of r. □
We now give an explicit formula for the integral .
Proposition 5.2.4 For allwe have:
Proof We first recall a formula from [22].
Lemma 5.2.1 For all the following equation holds:
Proof See [22]. □
It follows immediately that for all and we have:
We integrate this formula over the hyperbolic ball which gives:
and we exchange the order of integration:
We note that the integral does not depend upon the variable . Indeed:
and indeed the integral does not depend upon the variable b:
Finally, we can write:
because (as solutions of the same equation).
This completes the proof that:
□
5.2.3 Third step
We state a useful formula.
Lemma 5.2.2 For allthe following formula holds:
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, is given by the following formula:
where
We are now in a position to obtain the analytic form for of Theorem 5.1.1. We prove that
Indeed, in hyperbolic polar coordinates, we have:
On the other hand:
This yields
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 by introducing the time-dependent perturbation:
This leads to the linear equation:
We separate variables by setting to obtain the equation:
Introducing the hyperbolic polar coordinates and using the result:
we obtain:
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:
(28)
where the following equality derives from the definition of the hyperbolic distance in equation (