Olivier Faugeras, Romain Veltz and François Grimbert- Persistent neural states: stationary...

8/3/2019 Olivier Faugeras, Romain Veltz and Franois Grimbert- Persistent neural states: stationary localized activity patterns

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apport

de rec her che

ISSN

0249-6399

ISRN

INRIA/RR--????--FR+ENG

Thme BIO

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Persistent neural states: stationary localized activity

patterns

in nonlinear continuous n-population, q-dimensional

neural networks

Olivier Faugeras Romain Veltz Franois Grimbert

N ????December 10, 2007


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Unit de recherche INRIA Sophia Antipolis2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France)

Tlphone : +33 4 92 38 77 77 Tlcopie : +33 4 92 38 77 65

Persistent neural states: stationary localized activity patterns

in nonlinear continuous n-population, q-dimensional neural

networks

Olivier Faugeras, Romain Veltz, Franois Grimbert

Thme BIO Systmes biologiquesProjet Odysse

Rapport de recherche n ???? December 10, 2007 41 pages

Abstract: Neural continuum networks are an important aspect of the modeling of macroscopicparts of the cortex. Two classes of such networks are considered: voltage- and activity-based. Inboth cases our networks contain an arbitrary number, n, of interacting neuron populations. Spatialnon-symmetric connectivity functions represent cortico-cortical, local, connections, external inputsrepresent non-local connections. Sigmoidal nonlinearities model the relationship between (average)membrane potential and activity. Departing from most of the previous work in this area we do not

assume the nonlinearity to be singular, i.e., represented by the discontinuous Heaviside function.Another important difference with previous work is our relaxing of the assumption that the domainof definition where we study these networks is infinite, i.e. equal to R orR2. We explicitely considerthe biologically more relevant case of a bounded subset ofRq, q = 1, 2, 3, a better model of apiece of cortex. The time behaviour of these networks is described by systems of integro-differentialequations. Using methods of functional analysis, we study the existence and uniqueness of a sta-tionary, i.e., time-independent, solution of these equations in the case of a stationary input. Thesesolutions can be seen as persistent, they are also sometimes called bumps. We show that un-der very mild assumptions on the connectivity functions and because we do not use the Heavisidefunction for the nonlinearities, such solutions always exist. We also give sufficient conditions onthe connectivity functions for the solution to be absolutely stable, that is to say independent of theinitial state of the network. We then study the sensitivity of the solution(s) to variations of suchparameters as the connectivity functions, the sigmoids, the external inputs, and, last but not least,

the shape of the domain of existence of the neural continuum networks. These theoretical results

This work was partially supported by the EC project FACETS and the Fondation dEntreprise EADS.

INRIA/ENS/ENPC, Odysse Team, 2004 route des Lucioles, Sophia-Antipolis, France INRIA/ENS/ENPC, Odysse Team, 2004 route des Lucioles, Sophia-Antipolis, France INRIA/ENS/ENPC, Odysse Team, 2004 route des Lucioles, Sophia-Antipolis, France


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2 Faugeras, Veltz, Grimbert

are illustrated and corroborated by a large number of numerical experiments in most of the cases2 n 3, 2 q 3.

Key-words: Neural masses, persistent states, integro-differential operators, compact operators,fixed points, stability, Lyapunov function

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Etats neuronaux persistants : activit stationnaire localisedans des rsaux de neurones non linaires continus de dimension

q n populations

Rsum : Les rseaux continus neuronaux sont un aspect important de la modlisation de partiesmacroscopiques du cortex. Deux classes de tels rseaux sont considrs : ceux bass sur une des-cription du voltage et ceux bass sur une description de lactivit. Dans les deux cas, nos rseauxcontiennent un nombre arbitraire, n de populations de neurones qui intragissent. Les fonctionsde connectivit spatiale non-symtrique reprsentent les connections locales cortico-corticales, lesentres extrieures reprsentent les connections non locales.

Des non-linarits sigmodales modlisent les relations entre les potentiels de membrane (moyen)

et lactivit (moyenne). Se dpartageant des prcdents travaux dans ce domaine, nous ne supposonspas que la non-linarit est singulire, i.e., reprsente par une fonction de Heaviside discontinue.Une autre diffrence importante avec les travaux prcdents est notre abandon de la supposition quele domaine o on tudie ces rseaux est infini, i.e. gal R ou R2. Nous considrons de faon expli-cite les cas plus biologiquement plausibles densembles borns deRq, q = 1, 2, 3, correspondant un meilleur modle de morceau de cortex. Le comportement temporel de ces rseaux est dcrit parun systme dquations intgro-diffrentielles. En utilisant des mthodes danalyse fonctionnelle, ontudie lexistence et lunicit des solutions stationnaires, i.e., indpendantes du temps, de ces qua-tions dans le cas dune entre stationnaire. Ces solutions peuvent tre vues comme persistantes,elles sont aussi parfois appeles bumps. Nous montrons que sous de trs faibles hypothses surles fonctions de connectivit et parce que nous nutilisons pas la fonction de Heaviside pour les non-linrits, de telles solutions existent toujours. Nous donnons aussi des conditions suffisantes sur lesfonctions de connectivit pour que la solution soit absolument stable, cest--dire quelle ne dpende

pas de ltat initial du rseau. Nous tudions ensuite la sensibilit de(s) solution(s) aux variationsde paramtres comme les fonctions de connectivit, les sigmodes, les entres externes, et, aussi, laforme du domaine dexistence du rseau neuronal continu. Ces rsultats thoriques sont illustrset corrobors par un grand nombre dexpriences numriques dans les cas 2 n 3, 2 q 3.

Mots-cls : Masses neurales, tats persistants, oprateurs intgro-diffrentiels, oprateurs com-pacts, points fixes, stabilit, fonction de Lyapunov


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Contents

1 Introduction 5

2 The models 5

2.1 The local models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 The voltage-based model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 The activity-based model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 The continuum models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Existence of stationary solutions 11

4 Stability of the stationary solutions 13

4.1 The voltage-based model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 The activity-based model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5 Numerical experiments 17

5.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2 Examples of bumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6 Sensitivity of the bump to variations of the parameters 22

6.1 The finite dimensional parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.1.1 Sensitivity of the bump to the exterior current . . . . . . . . . . . . . . . . . 236.1.2 Sensitivity of the bump to the weights . . . . . . . . . . . . . . . . . . . . 236.1.3 Sensitivity of the bump to the thresholds . . . . . . . . . . . . . . . . . . . . 266.1.4 Sensitivity of the bump to the slope of the sigmoid . . . . . . . . . . . . . . 26

6.2 Sensitivity of the bump to variations of the shape of the domain . . . . . . . . . . 276.2.1 Numerical application for the domain derivatives . . . . . . . . . . . . . . . 30

7 Conclusion and perspectives 31

A Notations and background material 34

A.1 Matrix norms and spaces of functions . . . . . . . . . . . . . . . . . . . . . . . . . 34A.2 Choice of the quadrature method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36A.3 Shape derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

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Persistent neural states 5

1 Introduction

We analyze the ability of neuronal continuum networks to display localized persistent activity orbumps. This type of activity is related for example to working memory which involves the hold-ing and processing of information on the time scale of seconds. Experiments in primates have shownthat there exist neurons in the prefrontal cortex that have high firing rates during the period the animalis "remembering" the spatial location of an event before using the information being remembered[7, 19, 41]. Realistic models for this type of activity have involved spatially extended networks ofcoupled neural elements or neural masses and the study of spatially localized areas of high activityin these systems. A neuronal continuum network is first built from a local description of the dy-namics of a number of interacting neuron populations where the spatial structure of the connectionsis neglected. This local description can be thought of as representing such a structure as a cortical

column [42, 43, 5]. We call it a neural mass [18]. Probably the most well-known neural mass modelis that of Jansen and Rit [32] based on the original work of Lopes Da Silva and colleagues [38, 39]and of Van Rotterdam and colleagues [52]. A complete analysis of the bifurcation diagram of thismodel can be found in [21]. The model has been used to simulate evoked potentials, i.e., EEG ac-tivities in normal [31] and epileptic patients [54, 53]. In a similar vein, David and Friston [10] haveused an extension of this model to simulate a large variety of cerebral rhythms ( , , , , and )in MEG/EEG simulations. Another important class of such models is the one introduced by Wilsonand Cowan [56, 30].

These local descriptions are then assembled spatially to form the neuronal continuum network.This continuum network is meant to represent a macroscopic part of the neocortex, e.g. a visual areasuch as V1. The spatial connections are models of cortico-cortical connections. Other, non-localconnections with, e.g., such visual areas as the LGN or V2, are also considered. Other researchershave used several interconnected neural masses to simulate epileptogenic zones [54, 53, 37] or to

study the connectivity between cortical areas [9]. In this paper we consider a continuum of neuralmasses.

2 The models

We briefly discuss local and spatial models.

2.1 The local models

We consider n interacting populations of neurons such as those shown in figure 1. The figure isinspired by the work of Alex Thomson [50] and Wolfgang Maass [26]. It shows six populationsof neurons. Red indicates excitation, blue inhibition. The thickness of the arrows pertain to thestrength of the interaction. The six populations are located in layers 2/3, 4 and 5 of the neo-cortex.The following derivation follows closely that of Ermentrout [15]. We consider that each neuralpopulation i is described by its average membrane potential Vi(t) or by its average instantaneousfiring rate i(t), the relation between the two quantities being of the form i(t) = Si(Vi(t)) [20, 11],

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L2/3-E

L5-E

L4-E

L5-I

L4-I

L2/3-I

Figure 1: A model with six interacting neural populations.

where Si is sigmoidal and smooth. The functions Si, i = 1, , n satisfy the following propertiesintroduced in the

Definition 2.1 For all i = 1, , n, |Si| Sim (boundedness). We note Sm = maxi Sim. Forall i = 1, , n , the derivative Si ofSi is positive and bounded by S

im > 0 (boundedness of the

derivatives). We note DSm = maxi Sim andDSm the diagonal matrix diag(Sim).

A typical example of a function Si is given in equation (15) below. Its shape is shown in figure 2for the values of the parameters = 0 and s = 0.5, 1, 10. We have Sim = 1 and Sim = s. Whens , Sconverges to the Heaviside function H defined by

H(v) =

0 if v < 01 otherwise

Neurons in population j are connected to neurons in population i. A single action potential fromneurons in population j is seen as a post-synaptic potential P SPij(t s) by neurons in populationi, where s is the time of the spike hitting the terminal and t the time after the spike. We neglect the

delays due to the distance travelled down the axon by the spikes.Assuming that they sum linearly, the average membrane potential of population i due to actionpotentials of population j is

Vi(t) =k

P SPij(t tk),

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s = .5s = 1s = 10

0

0.2

0.4

0.6

0.8

1

S(v)=1/(1+exp(-s*v))

10 8 6 4 2 2 4 6 8 10

v

Figure 2: Three examples of sigmoid functions for different values of the parameter s, see text.

where the sum is taken over the arrival times of the spikes produced by the neurons in population j.

The number of spikes arriving between t and t + dt is j(t)dt. Therefore we have

Vi(t) =j

t0

P SPij(t s)j(s) ds =j

t0

P SPij(t s)Sj(Vj(s)) ds,

or, equivalently

i(t) = Si

j

t0

P SPij(t s)j(s) ds

(1)

The P SPij can depend on several variables in order to account for adaptation, learning, etc . . .There are two main simplifying assumptions that appear in the literature [15] and produce two dif-ferent models.

2.1.1 The voltage-based model

The assumption, [29], is that the post-synaptic potential has the same shape no matter which presy-naptic population caused it, the sign and amplitude may vary though. This leads to the relation

P SPij(t) = wijP SPi(t).

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Ifwij > 0 the population j excites population i whereas it inhibits it when wij < 0.Finally, if we assume that P SPi(t) = Aiet/iY(t), or equivalently that

idPSPi(t)

dt+ P SPi(t) = Ai(t), (2)

we end up with the following system of ordinary differential equations

idVi(t)

dt+ Vi(t) =

j

wijSj(Vj(t)) + Iiext(t), (3)

that describes the dynamic behaviour of a cortical column. We have incoporated the constant Aiin the weights wij and added an external current Iext(t) to model the non-local connections of

population i. We introduce the n n matrixes W such that Wij = wij/i, and the function S,R

n Rn such that S(x) is the vector of coordinates Si(xi), if x = (x1, , xn). We rewrite (3)in vector form and obtain the following system ofn ordinary differential equations

V = LV+WS(V) + Iext, (4)

where L is the diagonal matrix L = diag(1/i).In terms of units, the left and righthand sides of this equations are in units of, say mV ms1.

Therefore Iext, despite its name, is not a current. Note that since S(V) is an activity, its unit is ms1

and hence W is in mV.

2.1.2 The activity-based model

The assumption is that the shape of a PSP depends only on the nature of the presynaptic cell, that is

P SPij(t) = wijP SPj(t).

As above we suppose that P SPi(t) satisfies the differential equation (2) and define the time-averagedfiring rate to be

Aj(t) =

t0

P SPj(t s)j(s) ds.

A similar derivation yields the following set ofn ordinary differential equations

idAi(t)

dt+ Ai(t) = Si

j

wijAj(t) + Iiext(t)

i = 1, , n.

We include the is in the sigmoids Si and rewrite this in vector form

A = LA+ S(WA+ Iext), (5)

The units are ms2 for both sides of the equation. W is expressed in mV ms and Iext is in mV.

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2.2 The continuum models

We now combine these local models to form a continuum of neural masses, e.g., in the case of amodel of a significant part of the cortex. We consider a subset ofRq, q = 1, 2, 3 which weassume to be connected and compact, i.e. closed and bounded. This encompasses several cases ofinterest.

When q = 1 we deal with one-dimensional sets of neural masses. Even though this appears tobe of limited biological interest, this is one of the most widely studied cases because of its relativemathematical simplicity and because of the insights one can gain of the more realistic situations.

When q = 2 we discuss properties of two-dimensional sets of neural masses. This is perhapsmore interesting from a biological point of view since can be viewed as a piece of cortex wherethe third dimension, its thickness, is neglected. This case has received by far less attention than theprevious one, probably because of the increased mathematical difficulty. Note that we could alsotake into account the curvature of the cortical sheet at the cost of an increase in the mathematicaldifficulty. This is outside the scope of this paper.

Finally q = 3 allows us to discuss properties of volumes of neural masses, e.g. cortical sheetswhere their thickness is taken into account [33, 6].

The theoretical results that are presented in this paper are independent of the value of q.We note V(r, t) (respectivelyA(r, t)) the n-dimensional state vector at the point r of the con-

tinuum and at time t. We introduce the n n matrix function W(r, r) which describes how theneural mass at point r influences that at point r at time t. We call W the connectivity matrixfunction. In particular, W(r, r) = W, the matrix that appears in equations (4) and (5). More pre-cisely, Wij(r, r) describes how population j at point r influences population i at point r at time t.Equation (4) can now be extended to

Vt(r, t) = LV(r, t) +W(r, r)S(V(r, t)) dr + Iext(r, t), (6)

and equation (5) to

At(r, t) = LA(r, t) + S

W(r, r)A(r, t)) dr + Iext(r, t)

. (7)

It is important to discuss again the units of the quantities involved in these equations. For equation(6), as for equation (3) the unit is mV ms1 for both sides. Because of the spatial integration,W is in mV ms1 mmq , q is the dimension of the continuum. To obtain a dimensionlessequation we normalize, i.e. divide both sides of the equation, by the Frobenius norm WF of theconnectivity matrix function W (see appendix A.1 for a definition). Equivalently, we assume thatWF = 1.

We have given elsewhere [17], but see proposition 3.2 below for completeness, sufficient con-ditions on W and Iext for equations (6) and (7) to be well-defined and studied the existence andstability of their solutions for general and homogeneous (i.e. independent of the space variable)external currents. In this article we analyze in detail the case of stationary external currents, i.e.independent of the time variable, and investigate the existence and stability of the correspondingstationary solutions of (6) and (7).

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A significant amount of work has been devoted to this or closely related problems, startingperhaps with the pioneering work of Wilson and Cowan [56]. A fairly recent review of this work, andmuch more, can be found in a paper by Coombes [8]. Amari [1] investigated the problem in the casen = q = 1 when the sigmoid function is approximated by a Heaviside function and the connectivityfunction has a Mexican-hat shape. He proved the existence of stable bumps in this case. His workhas been extended to different firing-rate and connectivity functions [24, 35, 36, 47, 23, 22].

The case n = 1, q = 2 has been considered by several authors including [44, 45] for generalfiring-rate functions and Gaussian-like connectivity functions, and [3] when the firing-rate functionsare approximated by Heaviside functions.

Extending these analysis to two- or three-dimensional continuum is difficult because of the in-crease in the degrees of freedom in the choice of the connectivity function. The case n = 2, q = 1has been studied in [55, 4] when the firing-rate functions are approximated by Heaviside functions

and the connectivity function is circularly symmetric while the case n = 2, q = 2 is mentioned asdifficult in [14].In all these contributions, the proof of the existence of a bump solution is based upon the original

Amaris argument [1] which works only when q = 1 and the firing rate function is approximated bya Heaviside function. Solutions are usually constructed using a variant of the method of the singularperturbation construction, e.g., [45] which is usually fairly heavy. Sufficient conditions for theirstability are obtained by a linear stability analysis which in general requires the use of Heavisidefunctions instead of sigmoids.

The approach that we describe in this paper is a significant departure from the previous ones. Byusing simple ideas of functional analysis we are able to

1. Prove the existence and uniqueness of a stationary solution to equations (6) and (7) for anydimensions n and q, arbitrary connectivity functions and general firing-rate functions.

2. Obtain very simple conditions for the absolute stability of the solution in terms of the spectrumof the differential of the nonlinear operator that appears in the righthand side of equations (6)and (7).

3. Construct a numerical approximation as accurate as needed of the solution, when it exists, forany stationary input.

4. Characterize the sensitivity of the solutions to variations of the parameters, including the shapeof the domain .

To be complete, let us point out that equations of the type of ( 6) and (7) have been studied in puremathematics, see e.g.[28]. They are of the Hammerstein type [27, 51]. This type of equations hasreceived some recent attention, see [2], and progress have been made toward a better understand-ing of their solutions. Our contributions are the articulation of the models of networks of neuralmasses with this type of equation, the characterization of persistent activity in these networks asfixed points of Hammerstein equations, the proof of the existence of solutions, the characterizationof their stability and the analysis of their sensitivity to variations of the parameters involved in theequations.

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3 Existence of stationary solutions

In this section we deal with the problem of the existence of stationary solutions to (6) and (7) for agiven stationary external current Iext.

As indicated in the previous section, we use functional analysis to solve this problem. Let F bethe set L2n() of square integrable functions from to R

n. This is a Hilbert, hence a Banach, spacefor the usual inner product

V1, V2 =

V1(r)TV2(r) dr,

where V is the complex conjuguate of the vector V. This inner product induces the norm V2F =

i=1, ,n

|Vi(r)|2 dr, see appendix A.1. F is the state space. Another important space is

L2nn( )), the space of square integrable n n matrices, see appendix A.1 for a precisedefinition. We assume that the connectivity matrix functions W(, ) are in this space, see proposi-tions 3.1 and 3.2 below.

We also identify L2nn( )) with L (F) (the space of continuous linear operators on F) asfollows. IfW L2nn( )) it defines a linear mapping

W : F F such thatX W X =

W(., r)X(r)dr

For example this allows us to write (6) and (7)

Vt = LV+W S (V) + Iext

At = LA+ S(W A+ Iext)

We first recall some results on the existence of a solution to (6) and (7) that will be used in thesequel.

We denote by J a closed interval of the real line containing 0. A state vectorX(r, t) is a mappingX : J F and equations (6) and (7) are formally recast as an initial value problem:

X(t) = f(t,X(t))X(0) = X0

(8)

where X0 is an element ofF and the function f from J F is defined by the righthand side of (6),in which case we call it fv , or (7), in which case we call it fa. In other words, equations (6) and (7)become differential equations defined on the Hilbert space F.

We need the following two propositions that we quote without proof [17].

Proposition 3.1 If the following two hypotheses are satisfied.

1. The connectivity functionW is in L2nn( )) (see appendix A.1),

2. At each time instantt J the external currentI is in C(J; F), the set of continuous functionsfrom J to F,

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then the mappings fv andfa are from J F to F, continuous, and Lipschitz continuous with respectto their second argument, uniformly with respect to the first.

Proposition 3.2 If the following two hypotheses are satisfied

1. The connectivity functionW is in L2nn( )),

2. the external currentIext is in C(J; F), the set of continuous functions from J to F,

then for any function X0 in F there is a unique solution X, defined on R (and not only on J) andcontinuously differentiable, of the abstract initial value problem (8) forf = fv andf = fa.

This proposition says that, given the two hypotheses and the initial condition, there exists a uniquesolution to (6) or (7) and that this solution is in C1(R; F), the set of continuously differentiablefunctions from R to F.

We now turn our attention to a special type of solutions of (6) and (7), corresponding to stationaryexternal currents. We call these solutions, when they exist, stationary solutions. The currents Iextare simply in F.

A stationary solution of (6) or (7) is defined by

X = fL(X), (9)

where the function fL, F F, is equal to fLv defined by

fLv (V)(r) =

WL(r, r)S(V(r)) dr + ILext(r), (10)

or to fLa defined by

fLa (A)(r) = SL

W(r, r)A(r) dr + Iext(r)

, (11)

whereWL = L1W, SL = L1S and ILext = L1Iext.

We now recall the

Definition 3.3 A continuous mapping M : F F (linear or nonlinear) is called compact providedthat for each bounded subsetB ofF, the setM(B) is relatively compact, i.e. its closure is compact.

We then consider the nonlinear mapping gLv : F F

gLv (V)(r) =

WL(r, r)S(V(r)) dr (12)

and the linear mappings ga and gLa

ga(A)(r) =

W(r, r)A(r) dr, (13)

gLa (A)(r) =

WL(r, r)A(r) dr. (14)

We have the following

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Proposition 3.4 IfW L2nn( ), gLv andgLa are compact operators ofF.

Proof. We know from proposition 3.1 that gLv is continuous and prove that for each sequence{Vn}

n=1 ofF there exists a subsequence {Vnj}

j=1 such that g

Lv (Vnj ) is convergent in F.

Because of the definition 2.1 of S the sequence {An = S(Vn)}n=1 is bounded in F byC = Sm

n|| > 0. We prove that there exists a subsequence {Anj}

j=1 such that {g

La (Anj ) =

gLv (Vnj )}j=1 converges in F.

Since Since F is separable, its unit ball is weakly compact and because {An}n=1 is boundedthere exists a a subsequence {Anj}

j=1 of{An}

n=1 that converges weakly in F towardA. Because

of Fubinis theorem, for almost all r (noted a.s.) the function r W(r, r) is in F. Therefore,a.s., Bnj = g

La (Anj ) B.

Since AF liminfj AnjF C,A is also bounded by C in F. It is easy to show that

Bnj B2

F 2CWF and we can apply Lebesgues Dominated Convergence Theorem to thesequenceBnj (r) B(r) and conclude that Bnj BF 0, i.e., gLv (Vnj ) is convergent in F.

A small variation of the proof shows that gLa is compact.From proposition 3.4 follows the

Proposition 3.5 Under the hypotheses of proposition 3.4, if Iext F, fLv and fLa , are compact

operators ofF.

Proof. The operators X ILext and X Iext are clearly compact under the hypothesis Iext F, therefore fLv is the sum of two compact operators, hence compact. For the same reason ga + Iextis also compact and so is fLa = S

L(ga + Iext) because SL is smooth and bounded.We can now prove the

Theorem 3.6 IfW L2nn( ) andIext F, there exists a stationary solution of(6) and (7).

Proof. A stationary solution of (6) (respectively of (7)) is a fixed point of fLv (respectively offLa ).

Define the set Cv = {V F|V = fLv (V) for some 0 1}. Because of lemma A.2for allV Cv we have

VF (gLv (V)F + I

LextF) (Sm

n||WLF + I

LextF),

hence Cv is bounded.Similarly define the set Ca = {A F|A = fLa (A) for some 0 1}. Because of

lemma A.2 for all A Ca we have AF Sm

n||, hence Ca is bounded.The conclusion follows from Schaefers fixed point theorem [16].

4 Stability of the stationary solutions

In this section we give a sufficient condition on the connectivity matrix W to guarantee the stabilityof the stationary solutions to (6) and (7).

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4.1 The voltage-based model

We define the corrected maximal connectivity function Wcm(r, r) by Wcm = WDSm, whereDSm is defined in definition 2.1. We also define the corresponding linear operator hm : F F

hm(V)(r) =

Wcm(r, r)V(r) dr

which is compact according to proposition 3.4. Its adjoint, noted hm is defined1 by

hm(V)(r) =

WTcm(r, r)V(r) dr,

and is also compact. Hence the symmetric part hsm =12

(hm + hm), the sum of two compact

operators, is also compact. Furthermore we have V, hm

(V) = V, hsm

(V), as can be easilyverified. It is also self-adjoint since, clearly, hsm = h

sm .

We recall the following property of the spectrum of a compact self-adjoint operator in a Hilbertspace (see, e.g., [13]).

Proposition 4.1 The spectrum of a compact, self-adjoint operator of a Hilbert space is countableand real. Each nonzero spectral value is an eigenvalue and the dimension of the correspondingeigenspace is finite.


Theorem 4.2 A sufficient condition for the stability of a stationary solution to (6) is that all theeigenvalues of the linear compact, self-adjoint, operatorhL, sm be less than 1, where h

L, sm is defined

by

hL, s

m (x)(r) =

1

2L

1/2

(WT

cm(r

, r) + Wcm(r, r

))L1/2

x(r

) dr

x F,

where hL, sm is the symmetric part of the linear compact operatorhLm : F F:

hLm(x)(r) =

L1/2Wcm(r, r)L1/2 x(r) dr

Proof. The proof of this theorem is a generalization to the continuum case of a result obtainedby Matsuoka [40].

Let us note S the function (DSm)1S and rewrite equation (6) for an homogeneous input Iextas follows

Vt(r, t) = LV(r, t) +

Wcm(r, r)S(V(r, t) dr + Iext(r).

Let U be a stationary solution of (6). Let also V be the unique solution of the same equation

with initial some condition V(0) = V0 F, see proposition 3.2. We introduce the new functionX = V U which satisfies

Xt(r, t) = LX(r, t) +

Wcm(r, r)(X(r, t)) dr = LX(r, t) + hm((X))(r, t)

1By definition, V1, hm(V2) = hm(V1),V2, for all elements V1,V2 ofF.

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where the vector (X) is given by (X(r, t)) = S(V(r, t)) S(U(r)) = S(X(r, t) + U(r)) S(U(r)). Consider now the functional

(X) =

n

i=1

Xi(r,t)0

i(z) dz

dr.

We note that

z i(z) < 0 for z < 0 and 0 < i(z) z for z > 0, i(0) = 0, i = 1, , n.

This is because (Taylor expansion with integral remainder):

i(z) = Si(z + Ui) Si(Ui) = z10

Si(Ui + z) d,

and 0 < Si 1 by construction of the vector S. This implies that the functional (X) is strictlypositive for all X F = 0 and (0) = 0. It also implies, and this is used in the sequel, thatzi(z) i(z)2.The time derivative of is readily obtained:

d(X)

dt=

T(X(r, t))Xt(r, t)) dr = (X), Xt

We replace Xt(r, t)) by its value in this expression to obtain

d(X)

dt = (X), LX + (X), hm((X))

Because of a previous remark we have

XT(r, t))L(X(r, t)) T(X(r, t))L(X(r, t)),

and this provides up with an upper bound for d(X)dt

:

d(X)

dt (X), (L+ hsm).(X) =

L1/2(X), (Id + hL, sm )L

1/2(X)

,

and the conclusion follows.Note that we have the following

Corollary 4.3 If the condition of theorem 4.2 is satisfied, the homogeneous solution of (6) is unique.

Proof. Indeed, the result of theorem 4.2 is independent of the particular stationary solution Uthat is chosen in the proof.

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4.2 The activity-based model

We now give a sufficient condition for the stability of a solution to (7). We define the maximalcorrected connectivity matrix function Wmc = DSmW and the linear compact operator km fromF to F

km(x)(r) =

Wmc(r, r)x(r) dr.

Theorem 4.4 A sufficient condition for the stability of a solution to (7) is that all the eigenvalues ofthe linear compact operatorkLm be of magnitude less than 1, where k

Lm is defined by

kLm(x)(r) =

L1/2Wmc(r, r)L1/2x(r) dr x F

Proof. Let U be a stationary solution of (7) for a stationary external current Iext(r). As in theproof of theorem 4.2 we introduce the new functionX = VU, where V is the unique solution ofthe same equation with initial conditions V(0) = V0 F, an element ofC(J, F). We have

Xt(r, t) = LX(r, t)+

S

W(r, r)V(r, t) dr + Iext(r)

S

W(r, r)U(r) dr + Iext(r)

Using a first-order Taylor expansion with integral remainder this equation can be rewritten as

Xt(r, t) = LX(r, t)+10

DS

W(r, r)U(r) dr + Iext(r) +

W(r, r)X(r, t) dr

d

W(r, r)X(r, t) dr

Consider now the functional (X) = 12

X2F. Its time derivative is:

d(X)

dt= X, Xt

We replace Xt(r, t)) by its value in this expression to obtain

d(X)dt

= X, LX + X, m(X)km(X) ,

where the nonlinear operator m is defined by

m(X)(r, t) =

10

DS

W(r, r)U(r) dr +

W(r, r)X(r, t) dr

DS1m d,

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a diagonal matrix whose diagonal elements are between 0 and 1. We rewrite d(X)dt in a slightlydifferent manner, introducing the operator kLm

d(X)

dt=

L1/2X, L1/2X

+

m(X)L

1/2X, kLm(L1/2X)

,

From the Cauchy-Schwarz inequality and the property ofm(X) we obtainm(X)Y, kLm(Y) m(X)YFkLm(Y)F YFkLm(Y)F Y = L1/2XA sufficient condition for d(X)dt to be negative is therefore that k

Lm(Y)F < YF for all Y.

5 Numerical experimentsIn this section and the next we investigate the question of effectively, i.e. numerically, computingstationary solutions of (6) which is equivalent to computing solutions of (9). Similar results areobtained for (7).

In all our numerical experiments, we assume the sigmoidal functions Si, i = 1, , n intro-duced in the definition 2.1 to be of the form

Si(v) =1

1 + esi(vi). (15)

This function is symmetric with respect to the "threshold" potential i, see section 6.1.3, and variesbetween 0 and 1. The positive parameter si controls the slope of the ith sigmoid at v = i, seesection 6.1.4 and figure 2.

5.1 Algorithm

We now explain how to compute a fixed point Vf of equation (10) in which we drop for simplicitythe upper index L and the lower index ext:

Vf = W S(Vf) + I (16)

The method is iterative and based on Banachs fixed point theorem [16]:

Theorem 5.1 Let X be Banach space andM : X X a nonlinear mapping such that

x, y X, M(x) M(y) q x y , 0 < q < 1

Such a mapping is said contracting. M has a unique fixed point xf and for all x0 X andxp+1 = M(xp) then (xp) converges geometrically to xf.

Note that this method only allows to compute the solution of ( 9) when it admits a unique solutionand f is contracting. However it could admit more than one solution (recall it always has a solution,see theorem 3.6) or f could be non-contracting. Another method has to be found in these cases.

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In our case : X = F = L2n() where is an open bounded set ofRn and M = fv . Accordingto lemmas A.2 and A.1, ifDSm WF < 1, fv is contracting.

Each element of the sequence Vp, p 0 is approximated by a piecewise constant functionVp, h, where h is the step of the approximation, defined by a finite number of points rh,j , 1 j 1h . In order to avoid difficulties because Vp, h L

2n(), hence defined almost everywhere,

we assume that W and I are smooth. It is not a restriction because every function ofL2n() can beapproximated by a smooth function. As the bump solution is smooth as soon as W, I are smooth, wecan use the multidimensional Gaussian quadrature formula [46, 49] with N points (in the examplesbelow, usually N = 20) on each axis. In order to interpolate the values of the bump from a finite(and small) number of its values Vn(rh,j,Gauss), we use Nystrms method [28, Section: Fredholmequation, numerical methods] stated as follows:

Vp(r) =j

gjWp(r, rp,j,Gauss)S(Vp(rp,j,Gauss)) + I(r)

where the gjs are the weights of the Gaussian quadrature method and the points rp,j,Gauss are chosenaccording to the Gauss quadrature formula. It is to be noted that the choice of a particular quadratureformula can make a huge difference in accuracy and computing time, see appendix A.2.

Having chosen the type of quadrature we solve with Banachs theorem:

Vfh = Wh S(V

fh) + Ih, (17)

i.e., we compute the fixed point at the level of approximation defined by h.The following theorem ensures that limh0V

fh = V

f:

Theorem 5.2 Assume that limh0Wh = W in L2nn

( ) , then max1j

1

h

|Vh(rh,j)

Vf(rh,j)| = O(ah)h

0 with ah = W WhF

Proof. The proof is an adaptation of [34, Theorem 19.5].

5.2 Examples of bumps

We show four examples of the application of the previous numerical method to the computation ofbumps for various values ofn and q.

There are n populations (V = [V1, , Vn]T, W L2nn( )), some excitatory and someinhibitory. = [1, 1]q . We characterize in section 6.2 how the shape of influences that of thebumps. The connectivity matrix is of the form

Wij(r, r) = ij exp

12

r r,Tij (r r)

, (18)

with Tij Mqq is a q q symmetric positive definite matrix. The weights ij , i, j = 1, , n

form an element ofMnn and T =

T11 T12T21 T22

is an element ofMnqnq. The weights

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are chosen so that DSmWF < 1. The sign ofij , i = j, indicates whether population j excitesor inhibits population i. The bumps are computed using the algorithm described in the previoussection.

First example: n = 2, q = 2, constant currentFigure 3 shows an example of a bump for the following values of the parameters:

=

0.2 0.10.1 0.2

I = [0.3, 0]T T =

40 0 12 00 40 0 128 0 20 00 8 0 20

There is one excitatory and one inhibitory population of neural masses.

1

0.5

0

0.5

1

1

0.5

0

0.5

1

0.314

0.312

0.31

0.308

0.306

0.304

0.302

Shape of V1

1

0.5

0

0.5

1

1

0.5

0

0.5

1

5

4

3

2

1

0

1

2

x 103

Shape of V2

Figure 3: Example of a two-population, two-dimensional bump with constant external currents (seetext).

Second example: n = 2, q = 2, non constant currentFigure 4 shows a different example where the external current I is still equal to 0 for its secondcoordinate and is not constant but equal to its previous value, 0.3 to which we have added acircularly symmetric 2D Gaussian centered at the point of coordinates (0.5, 0, 5) of the square with standard deviation 0.18 and maximum value 0.2. It is interesting to see how the shape

of the previous bump is perturbed. The matrix is the same as in the first example. ThematrixT is equal to

T =

5 0 1 00 5 0 1

16 0 40 00 16 0 40

,

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corresponding to a spatially broader interaction for the first population and narrower for thesecond.

1

0.5

0

0.5

1

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

Shape of V1

1

0.5

0

0.5

1

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

3

2.5

2

1.5

1

0.5

0

0.5

1

1.5

x 103

Shape of V2

Figure 4: Example of a two-population, two-dimensional bump with Gaussian shaped external cur-rent (see text).

Third example: n = 3, q = 2, constant currentFigure 5 shows an example of a bump for three neural populations, two excitatory and oneinhibitory, in two dimensions. We use the following values of the parameters:

=

.442 1.12 0.8750 0.1870 0.0850

0.128 0.703 0.7750

T

I = [0, 0]T T =

40 0 12 0 12 00 40 0 12 0 128 0 20 0 9 00 8 0 20 0 9

40 0 12 0 12 00 40 0 12 0 12

Fourth example: n = 2, q = 3, constant current We show an example of a 3-dimensional bumpfor two populations of neural masses. The parameters are:

= 0.2 0.1

0.1 0.2

I = [0, 0]T

T = 40Id3 12Id3

8 Id3 20Id3

,

where Id3 is the 3 3 identity matrix.

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1

0.5

0

0.5

1

1

0.5

0

0.5

1

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

Shape of V1

1

0.5

0

0.5

1

1

0.5

0

0.5

1

1

0

1

2

3

4

5

x 103

Shape of V2

1

0.5

0

0.5

1

1

0.5

0

0.5

1

8

7

6

5

4

3

2

1

x 103

Shape of V3

Figure 5: Example of a three-population, two-dimensional bump (see text).

Figure 6: Example of a two-population, three-dimensional bump, isosurfaces are shown. Trans-parencies increases linearly from red to blue.

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6 Sensitivity of the bump to variations of the parameters

In this section we characterize how the solutions of (9) vary with the parameters that appear in theequation. These parameters are of two types: first we have a finite number of real parameters such asthe external currents, the weights in the connectivity matrix W, or the parameters of the sigmoids,and, second, the shape of the domain , a potentially infinite-dimensional parameter.

We focus on the voltage-based model, the analysis in the activity-based case is very similar. Westart with a set of general considerations in the finite dimensional case which we then apply to thevarious cases. We then tackle the more difficult case of the dependency with respect to the shape of.

As fv is a smooth function of the parameters (I,, S. . . ), one can show (by extending Banachstheorem) that the fixed point Vf inherits the smoothness offv.

6.1 The finite dimensional parameters

We introduce the linear operator2 W DS(Vf) : F F such that

W DS(Vf) V(r) =

W(r, r)DS(Vf(r))V(r) dr V F


Proposition 6.1 The derivative Vf of the fixed pointVf with respect to the generic parametersatisfies the equation

(Id W DS(Vf)) Vf = b(,Vf), (19)

where b(,Vf

) = (W) S(Vf

) + W (S(Vf

)) + I.

Proof. Taking the derivative of both sides of (16) with respect to , we have:

Vf = W DS(Vf) V

f + (W) S(Vf) + W (S(V

f)) + I,

hence we obtain equation (19).Note that S(Vf) is the partial derivative of the vector S with respect to the scalar parameter evaluated at V = Vf.

Because of the assumption DSmWF < 1 the linear operator J = Id W DS(Vf) isinvertible with

J1 =

p=0 W DS(Vf)

p

,

and the series is convergent.V

f is thus obtained from the following formula

Vf = J1b(,Vf),

2W DS(Vf) is the Frechet derivative of the operator fv at the pointVf ofF.

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the righthand side being computed byx0 = b(,V

f)xp+1 = x0 +W DS(V

f) xp p 0

We now apply proposition 6.1 to the study of the sensitivity of the bump to the variations of theparameters.

6.1.1 Sensitivity of the bump to the exterior current

When = I1, we find:

I1Vf = J1

10

00

This inequality is to be understood component by component. It predicts the influence ofI1 onVf.For example, with the parameters and T used in figure 3 but with an external current equal to 0,we obtain the bump shown in figure 7 (top) with the derivatives shown at the bottom of the samefigure. We also show in figure 8 ofV1 and V2 along the diagonal and the x-axis for different valuesofI1 close to 0. The reader can verify that the values increase with I1, as predicted.

6.1.2 Sensitivity of the bump to the weights

For = ij , one finds:J ijV

f = ijW DS(Vf)

We then have

= a : We find

aVf(r) = J1

exp

1

2r , T11 (r )

0

0 0

DS(Vf) =

J1

exp

1

2r ,T11 (r ) S

1(Vf1 ())

0

00

The fixed point is an increasing function of the excitatory parameter a.

= b : We find

bVf(r) = J1

0 exp

1

2r ,T12 (r )

0 0 DS(Vf) =

J1

exp

1

2r ,T12 (r )

S

2(Vf2 ())

0

00

The fixed point is a decreasing function of the inhibitory parameter b, see figure 9.

The other cases are similar.

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Figure 7: A bump corresponding to the following parameters and T are the same as in figure 3,I = [0 0]T (top). Derivative of the bump with respect to the first coordinate, I1, of the exteriorcurrent (bottom). We verify that it is positive (see text).

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-1 -0.5 0 0.5 1-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

diagonal

abscissa

-1 -0.5 0 0.5 1-2

0

2

4

6

8

10x 10

-3

diagonal

-1 -0.5 0 0.5 11

2

3

4

5

6x 10

-3

abscissa

Figure 8: Cross sections of V1 (left) and V2 (right) for I1 = 0.001 (green), I1 = 0 (black) andI1 = 0.001 (blue). I2 = 0 in all three cases. To increase the readibility of the results we haveapplied an offset of0.001 and 0.002 to the black and blue curves on the righthand side of the figure,respectively.

-1 -0.5 0 0.5 1-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

diagonal

abscissa -1 -0.5 0 0.5 1-2

0

2

4

6

8

10x 10-3

diagonal

-1 -0.5 0 0.5 11

2

3

4

5

6x 10

-3

abscissa

Figure 9: Cross sections ofV1 (left) and V2 (right) for b = 0.101 (green), b = 0.1 (black) andb = 0.099 (blue). To increase the readibility of the results we have applied an offset of 0.001 and+0.002 to all black and blue curves, respectively.

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6.1.3 Sensitivity of the bump to the thresholds

When = i, i = 1, 2 we have from the definition (15) ofS and with the notations of proposition6.1:

b(, Vf) = W DS(Vf) ei, i = 1, 2,

where e1 = [1, 0]T, e2 = [0, 1]T. We show in figure 10 some cross sections of the bump Vf

obtained for the same values of the parameters as in figure 3 and three values of the threshold vector.

-1 -0.5 0 0.5 1-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

diagonal

abscissa -1 -0.5 0 0.5 1-2

0

2

4

6

8

10x 10

-3

diagonal

-1 -0.5 0 0.5 11

2

3

4

5

6x 10

-3

abscissa

Figure 10: Cross sections ofV1 (left) and V2 (right) for = 0.101[1, 1]T (green), = 0 (black)and = 0.1[1, 1]T (blue). To increase the readibility of the results we have applied an offset of0.001 and +0.002 to all black and blue curves, respectively.

6.1.4 Sensitivity of the bump to the slope of the sigmoid

When = si, i = 1, 2 we have from the definition (15) ofS and with the notations of proposition6.1:

b(, Vf) = W DS(Vf) sVf

,

where the matrix s is given by

s =

1s1

0

0 1s2

Figure 11 shows the two coordinates sVf1 and sV

f2 for s1 = s2 = s of the derivative of the bump

Vf at s = 1 obtained for the same values of the other parameters as in figure 3, except the intensitywhich is equal to 0.

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1

0.5

0

0.5

1

1

0.5

0

0.5

1

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

x 104

Shape of sV

1

1

0.5

0

0.5

1

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

3

2.5

2

1.5

1

0.5

0

x 104

Shape of sV

2

Figure 11: Plot of the derivative with respect to the slope of the sigmoids of the the bump obtainedwith the same parameters , I andT as in figure 3.

6.2 Sensitivity of the bump to variations of the shape of the domain

We expect the bump to be somewhat dependent on the shape of. It would nonetheless be desirablethat this dependency would not be too strong for the modeling described in this paper to have somebiological relevance. Indeed, if the bumps are metaphores of persistent states, we expect them to berelatively robust to the actual shape of the cortical part being modeled. For example if we take to

be a representation of the primary visual cortex V1 whose shape varies from individual to individualit would come as a surprise if the shape of a bump induced by the same spatially constant stimuluswould be drastically different.

Technically, in order to study the dependence ofVf with respect to we need to assume that is smooth, i.e. its border is a smooth curve (q = 2) or surface (q = 3) unlike the previousexamples where was the square [1, 1]2. But even a difficulty arises from the fact that the set ofregular domains is not a vector space, hence the derivative of a function (the bump) with respect toa domain has to be defined with some care. The necessary background is found in appendix A.3.

We make explicit the fact that the connectivity function W has been normalized to satisfyWF = 1 by writingW(r, r, ) where, with some abuse of notation

W(r, r, ) = W(r, r)/J() with J() =

W(r, r)2F dr dr

Theorem 6.2 Let us assume that is a smooth bounded domain ofRq. IfW is in W1,2nn( ),Iext is in W1

,2n () (see appendix A.1 for a definition) the material derivative (see appendix A.3 for

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a definition) Vfm(r, ) of the bump Vf satisfies the following equation:

Vfm(r, ,X) =

W(r, r, )DSVf(r, )

Vfm(r

, ,X) dr (20)

+

W(r, r, )SVf(r, )

divX(r)dr (21)

+

D1W(r, r, )X(r)S

Vf(r, )

dr (22)

+

D2W(r, r, )X(r)S

Vf(r, )

dr (23)

J(), X

J() Vf(r, ) Iext(r) + DIext(r) X(r) (24)

where Di, i = 1, 2 indicates the derivative with respect to the ith variable and J(), X is theGteaux derivative ofJ() with respect to the vector fieldX:

J(), X = lim0

J(()) J()

,

where X is defined in the proof below. We have

J(), X =1

2J()

W(r, r, )2F X(r), N(r) dr da(r)+

W(r, r, )2F X(r), N(r) da(r) dr

,

where da is the surface element on the smooth boundary of , andN its unit inward normalvector.

Proof. The proof uses ideas that are developed in [12, 48], see also the appendix A.3. We wantto compute :

Vfm(r, , X) = lim0

Vf(r(), ()) Vf(r, )

from equation (9). As far as the computation of the derivative is concerned only small deformationsare relevant and we consider the first order Taylor expansion of the transformation T:

T(, r) = T(0, r) + T

(0, r) = r+ X(r)

We define:

1

()W(r(), r

, ())S(Vf

(r

, ()))dr

W(r, r, )S(Vf(r, )) dr

+ Iext(r+ X(r)) Iext(r)

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In the first integral, we make the change of variable r r + X and obtain:

1

W(r+ X(r), r + X(r), + X)S(Vf(r + X(r), + X))|detJ(r)|dr

We have :detJ(r

) = 1 + div(X(r)) + o().

Hence for sufficiently small detJ > 0. Moreover:

lim0

detJ = 1 lim0

detJ(r) 1

= div(X(r)),

and

W(r+ X(r), r + X(r), + X) =

W(r, r) + D1W(r, r)X(r) + D2W(r, r

)X(r)

J(), X

J()W(r, r, ) + o(),

where Di, i = 1, 2 indicates the derivative with respect to the ith argument. Thus we have:

=

W(r, r, )S(Vf(r + X(r), + X)) S(Vf(r, ))

detJ(r

)dr

+ W(r, r, )S(Vf(r, ))detJ(r

) 1dr

+

D1W(r, r, )X(r)S(Vf(r + X(r), + X))detJ(r

)dr

+

D2W(r, r, )X(r)S(Vf(r + X(r), + X))detJ(r

)dr

J(), X

J()

W(r, r, )S(Vf(r + X(r), + X))detJ(r)dr

+ DIext(r) X(r)

Because

lim0

S(Vf(r + X(r), + X)) S(Vf(r, ))

= DS(Vf(r, ))Vfm(r, ,X),

andW(r, r, )S

Vf(r, )

dr = Vf(r, ) Iext(r), we obtain equation (20). The value

ofJ(), X is obtained from corollary A.8.

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Equation (20) is of the same form as before:

(J Vfm)(r, ,X) =

W(r, r, )SVf(r, )

divX(r)dr

+

D1W(r, r, )X(r)S

Vf(r, )

dr

+

D2W(r, r, )X(r)S

Vf(r, )

dr

J(), X

J()(Vf(r, ) Iext(r))

This result tells us that the shape of the bump varies smoothly with respect to the shape of the domain

.

6.2.1 Numerical application for the domain derivatives

We show in figure 12 the bump Vf for equal to the unit disc D(0, 1) and in figure 13 the one

for equal to the ellipse3 Ellipse(1.2, 1) of equation r2

1

a2 + r22 1 = 0. The values of the weight

parameters are the same as in figure 3 and I = [0, 0]T. The matrix T is equal to

T =

40 0 10 00 10 0 12

12 0 40 00 40 0 40

Note that because the diagonal elements are not equal for T11, T12 and T13, W is not circularlysymmetric and so is the bump in figure 12 despite the fact that is circularly symmetric.Finally we show in figure 14 the two coordinates of the shape (material) derivative of the first

bump in the direction of the field X corresponding to the transformation

T(, r) = r+

(a 1)r1

0

T(1) transforms the disc D(0, 1) into the ellipse Ellipse(a, 1), X(r) = [(a 1)r1, 0]T.

Thus divX = (a 1) and, because of (18):

(J Vfm)(r, ,X) =

a 1

J(), X

J()V

f

I

+

D1W(r, r, )

X(r) X(r)

SVf(r, )

dr,

3Ellipse(a, b) represents the ellipse lying along the first axis of coordinates with semimajor axis a and semiminor axisb.

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and

J(), X =

W(r, r, )2F X(r), N(r) dr da(r)

J()

As the Gaussian quadrature formula holds for a rectangular domain, we use polar coordinates tomap the disk (or the ellipse) to a square. For our numerical study we can simplify these expressions(the matrixes Tij are symmetric) :

D1W(r, r, )

X(r) X(r)

SVf(r, )

dr

i

=

j

r r

TTij

X(r) X(r)

Wij(r, r

, )Sj

Vfj (r

, )

dr i = 1, , n

Thus we can use a simple modification of the algorithm that computes W S(V) to obtain theprevious expression.

J() and J(), X are computed with a Gauss quadrature formula. For a circle in polarcoordinatesN(r) = r.

Let us be a bit more precise. In the case showed in figure 12, we choose I = 0. Using Banachstheorem we compute VfGauss for N = 30 and use Nystrms interpolation to compute V

fNys for

n = 100 (for example) points on each axis.Then, using VfGauss, we compute V

fm,Gauss for N points. But the equation for V

fm reads:

Vfm = W.DS(Vf).Vfm + J

(), X

Having computed a Nystrm interpolation ofn points for Vfm = W.DS(Vf).Vfm + J

(), X,

we again use a Nystrm interpolation with the last equation to compute Vfm,Nystrom for n points oneach axis.

We used this numerical method in every previous example related to the computation of deriva-tives.

7 Conclusion and perspectives

We have studied two classes (voltage- and activity-based) of neural continuum networks in the con-text of modeling macroscopic parts of the cortex. In both cases we have assumed an arbitrary num-ber of interacting neuron populations, either excitatory or inhibitory. These populations are spatiallyrelated by non-symmetric connectivity functions representing cortico-cortical, local, connections.External inputs are also present in our models to represent non-local connections, e.g., with othercortical areas. The relationship between (average) membrane potential and activity is describedby nondegenerate sigmoidal nonlinearities, i.e., not by Heaviside functions which have often beenconsidered instead in the literature because of their (apparent) simplicity.

The resulting nonlinear integro-differential equations are of the Hammerstein type [27] and gen-eralise those proposed by Wilson and Cowan [56].

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Figure 12: The unit disk and its bump Vf.

Figure 13: Bump associated to the ellipse with major axis along the r1 coordinate and the minor axisalong the r2 coordinate. The ratio of the axes length is a = 1.2, see text.

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Figure 14: The shape derivative Vfm for a = 1.2.

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Departing from most of the previous work in this area we relax the usual assumption that the do-main of definition where we study these networks is infinite, i.e. equal to R or R2 and we explicitelyconsider the biologically much more relevant case of a bounded subset ofRq, q = 1, 2, 3, obvi-ously a better model of a piece of cortex.

Using methods of functional analysis, we have studied the existence and uniqueness of a sta-tionary, i.e., time-independent, solution of these equations in the case of a stationary input. Thesesolutions are often referred to as persistent states, or bumps, in the literature.

We have proved that, under very mild assumptions on the connectivity functions, such solutionsalways exist (this is due in part to the fact that we do not use Heaviside functions).

We have provided sufficient conditions on the connectivity functions for the solution to be ab-solutely stable, that is to say independent of the initial state of the network. These conditions canbe expressed in terms of the spectra of some functional operators, that we prove to be compact, that

arise very naturally from the equations describing the network activity.We have also studied the sensitivity of the solution(s) to variations of such parameters as theconnectivity functions, the sigmoids, the external inputs, and the shape of the domain of definitionof the neural continuum networks. This last analysis is more involved than the others because of theinfinite dimensional nature of the shape parameter. An analysis of the bifurcations of the solutionswhen the parameters vary over large ranges requires the use of techniques of bifurcation analysis forinfinite dimensional systems and is out of the scope of this paper.

We believe and we hope by now to have convinced the reader that the functional analysis frame-work that we have used in this paper is the right one to try and answer some of the mathematicalquestions that are raised by models of connected networks of nonlinear neurons. We also believethat some of these also begin to answer biological questions since these networks models, despitethe admitedly immense simplifications they are built from, are nonetheless metaphores of real neuralassemblies.

A Notations and background material

A.1 Matrix norms and spaces of functions

We note Mnn the set ofn n real matrixes. We consider the Frobenius norm on Mnn

MF =

ni,j=1

M2ij ,

and consider the space L2nn( ) of the functions from to Mnn whose Frobenius norm

is in L2( ). IfW L2nn( ) we note W2F = W(r, r)2F dr dr. Note that

this implies that each element Wij , i, j = 1, , n in in L2( ). We note F the set L2n() ofsquare-integrable mappings from toRn and xF the corresponding norm. We have the following

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Lemma A.1 Given x L2n() andW L2nn( ), we define y(r) =W(r, r

)x(r) dr.This integral is well defined for almost all r, y is in L2n() and we have

yF WF xF.

Proof. Since each Wij is in L2( ), Wij(r, .) is in L2() for almost all r, thanks to Fubinistheorem. So Wij(r, .)xj(.) is integrable for almost all r from what we deduce that y is well-definedfor almost all r. Next we have

|yi(r)| j

Wij(r, r) xj(r

) dr

and (Cauchy-Schwarz):

|yi(r)| j

W2ij(r, r) dr

1/2xj2,

from where it follows that (Cauchy-Schwarz again, discrete version):

|yi(r)|

j

xj22

1/2

j

W2ij(r, r) dr

1/2

= xF

j

W2ij(r, r) dr

1/2

,

from what it follows that y is in L2n() (thanks again to Fubinis theorem) and

y2F x2F

i,j

W2ij(r, r) dr dr = x2FW

2F.

We also use the following

Lemma A.2 For each V ofF, S(V) is in F and we have

S(V)F Sm

n||

For all V1 andV2 in F we have

S(V1) S(V2)F DSmV1 V2F,

where DSm is defined in Definition 2.1.

Proof. We have S(V)2F =n

i=1

(Si(Vi(r)))2 dr S2mn||, where || is the Lebesguemeasure of (its area). Similarly, S(V1) S(V2)2F =

ni=1 (Si(V

1i (r)) Si(V

2i (r)))

2 dr

(DSm)2n

i=1(V

1

i (r)V2

i (r))2

dr = (DSm)2

V1V22

F In theorem 6.2 we use the Sobolevspaces W1,2n () and W1,2nn( ). W

1,2n () is the set of functions X : R

n such thateach component Xi, i = 1, , n is in W1,2(), the set of functions of L2() whose first orderderivatives exist in the weak sense and are also in L2() (see [16]). Similarly W1,2nn( ) isthe set of functions X : Mnn such that each component Xij , i, j = 1, , n is in is inW1,2().

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A.2 Choice of the quadrature method

We emphasize the importance of the choice of a specific quadrature formula using the following

example :1

1

etdt = e 1/e where we compare a 0th-order finite elements methods with and

Gauss method (the parameters of the Gauss quadrature formula are computed with a precision of1016 using Newtons method).

Method ValueExact 2.350 402 387 287 603...0th-order (N=1000) 2.351 945...finite elementGauss (N=5) 2.350 402 386 46...

The Gauss method is far more powerful and allows us to compute bumps in 3D for an arbitrarynumber of populations.

A.3 Shape derivatives

As it has already been pointed out, the computation of the variation of the bump with respect to theshape of the region is difficult since the set U of regular domains (regular open bounded sets) ofR

q does not have the structure of a vector space. Variations of a domain must then defined in someway. Let us consider a reference domain U and the set A of aplications T : Rq which areat least as regular as homeomorphisms, i.e. one to one with T and T1 one to one. In detail

A = Tone to one, T, T1 W1,(,Rq) ,where the functional space W1,(,Rq) is the set of mappings such that they and their first orderderivatives are in L(, Rq). In detail

W1,(,Rq) = {T : Rq such that T L(, Rq) and iT L(, Rq), i = 1, , q}

Given a shape function F : U Rq, for T A, let us define F(T) = F(T()). The key point isthat since W1,(,Rq) is a Banach space we can define the notion of a derivative with respect tothe domain as

Definition A.3 F is Gteaux differentiable with respect to if and only ifF is Gteaux differen-tiable with respect to T.

In order to compute Gteaux derivatives with respect to T we introduce a family of deformations

(T())0 such that T() A for 0, T(0) = Id, and T() C1

([0, A]; W1,

(,R

q

)), A >0. From a practical point of view, there are many ways to construct such a family, the most famousone being the Hadamard deformation [25] which goes as follows.

For a point r we note

r() = T(, r) with T(0, r) = r() = T(, ) with T(0, ) = )

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Let us now define the velocity vector field X corresponding to T() as

X(r) =T

(0, r) r

From definition A.3 follows the

Definition A.4 The Gteaux derivative of a shape function F() in the direction ofX, denotedF(), X, is equal to

F(), X = lim0

F(()) F()

We also introduce the

Definition A.5 The material derivative of a function f(r, ), notedfm(r, , X) is defined by

fm(r, , X) = lim0

V(r(), ()) V(r, )

,

and

Definition A.6 The shape derivative of a function f(r, ), notedfs(r, , X) is defined by

fs(r, , X) = lim0

f(r, ()) fr, )

,

The following theorem whose proof can be found, e.g., in [12, 48] relates the Gteaux derivativeand the shape derivative

Theorem A.7 The Gteaux derivative of the functional F() =

f(r, ) dr in the direction ofX is given by

F(), X =

fs(r, , X) dr

f(r, ) X(r), N(r) da(r),

where N is the unit inward normal to andda its area element.

The following corollary is used in the proof of theorem 6.2.

Corollary A.8 The Gteaux derivative of the functional F() =

f(r) dr in the direction ofXis given by

F(), X

=

f(r)X(r), N(r)

da(r),

where N is the unit inward normal to andda its area element.

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References

[1] S.-I. Amari. Dynamics of pattern formation in lateral-inhibition type neural fields. BiologicalCybernetics, 27(2):7787, jun 1977.

[2] J. Appell and C.-J Chen. How to solve Hammerstein equations. Journal of integral equationsand applications, 18(3):287296, 2006.

[3] P. Blomquist, J. Wyller, and G.T. Einevoll. Localized activity patterns in two-population neu-ronal networks. Physica D, 206:180212, 2005.

[4] P. Bressloff. Spontaneous symmetry breaking in selforganizing neural fields. BiologicalCybernetics, 93(4):256274, oct 2005.

[5] D.P. Buxhoeveden and M.F. Casanova. The minicolumn hypothesis in neuroscience. Brain,125:935951, 2002.

[6] L. M. Chalupa and J.S. Werner, editors. The visual neurosciences. MIT Press, 2004.

[7] C.L. Colby, J.R. Duhamel, and M.E. Goldberg. Oculocentric spatial representation in parietalcortex. Cereb. Cortex, 5:470481, 1995.

[8] Stephen Coombes. Waves, bumps, and patterns in neural fields theories. Biological Cybernet-ics, 93(2):91108, 2005.

[9] Olivier David, Diego Cosmelli, and Karl J. Friston. Evaluation of different measures of func-tional connectivity using a neural mass model. NeuroImage, 21:659673, 2004.

[10] Olivier David and Karl J. Friston. A neural mass model for meg/eeg: coupling and neuronaldynamics. NeuroImage, 20:17431755, 2003.

[11] P. Dayan and L. F. Abbott. Theoretical Neuroscience : Computational and MathematicalModeling of Neural Systems. MIT Press, 2001.

[12] M.C. Delfour and J.-P. Zolsio. Shapes and geometries. Advances in Design and Control.Siam, 2001.

[13] Jean Dieudonn. Foundations of Modern Analysis. Academic Press, 1960.

[14] K. Doubrovinski. Dynamics, stability and bifurcation phenomena in the nonlocal model of cor-tical activity. U.u.d.m. project report 2005:8, Uppsala University, Department of Mathematics,

jun 2005.

[15] Bard Ermentrout. Neural networks as spatio-temporal pattern-forming systems. Reports onProgress in Physics, 61:353430, 1998.

[16] L.C. Evans. Partial Differential Equations, volume 19 of Graduate Studies in Mathematics.Proceedings of the American Mathematical Society, 1998.

INRIA


41/44


[17] Olivier Faugeras, Franois Grimbert, and Jean-Jacques Slotine. Stability and synchronizationin neural fields. Technical Report RR-6212, INRIA, 2007.

[18] W.J. Freeman. Mass action in the nervous system. Academic Press, New York, 1975.

[19] S. Funahashi, C.J. Bruce, and P.S. Goldman-Rakic. Mnemonic coding of visual space in themonkeys dorsolateral prefrontal cortex. J. Neurophysiol., 61:331349, 1989.

[20] W. Gerstner and W. M. Kistler. Mathematical formulations of hebbian learning. BiologicalCybernetics, 87:404415, 2002.

[21] F. Grimbert and O. Faugeras. Bifurcation analysis of Jansens neural mass model. NeuralComputation, 18(12):30523068, December 2006.

[22] Y. Guo and C.C. Chow. Existence and stability of standing pulses in neural networks: Iistability. SIAM Journal on Applied Dynamical Systems, 4:249281, 2005.

[23] Yixin Guo and Carson C. Chow. Existence and stability of standing pulses in neural networks:I. existence. SIAM Journal on Applied Dynamical Systems, 4(2):217248, 2005.

[24] B.S. Gutkin, G.B. Ermentrout, and J. OSullivan. Layer 3 patchy recurrent excitatory con-nections may determine the spatial organization of sustained activity in the primate prefrontalcortex. Neurocomputing, 32-33:391400, 2000.

[25] J. Hadamard. Mmoire sur un problme danalyse relatif lquilibre des plaques lastiquesencastres. Mmoire des savants trangers, 1968. CNRS, Paris.

[26] Stefan Haeusler and Wolfgang Maass. A statistical analysis of information-processing proper-

ties of lamina-specific cortical microcircuits models. Cerebral Cortex, 17:149162, jan 2007.[27] A. Hammerstein. Nichtlineare integralgleichungen nebst anwendungen. Acta Math., 54:117

176, 1930.

[28] Michiel Hazewinkel, editor. Encyclopaedia of Mathematics. Springer, 2001.

[29] J. J. Hopfield. Neurons with graded response have collective computational propertieslike those of two-state neurons. Proceedings of the National Academy of Sciences, USA,81(10):30883092, 1984.

[30] F.C. Hoppenstaedt and E.M. Izhikevich. Weakly Connected Neural Networks. Springer-Verlag,New York, 1997.

[31] Ben H. Jansen and Vincent G. Rit. Electroencephalogram and visual evoked potential genera-

tion in a mathematical model of coupled cortical columns. Biological Cybernetics, 73:357366,1995.

[32] Ben H. Jansen, George Zouridakis, and Michael E. Brandt. A neurophysiologically-basedmathematical model of flash visual evoked potentials. Biological Cybernetics, 68:275283,1993.

RR n 0123456789


42/44


[33] E.R. Kandel, J.H. Schwartz, and T.M. Jessel. Principles of Neural Science. McGraw-Hill, 4thedition, 2000.

[34] M.A. Krasnoselskii, G.M. Vainikko, P.P. Zabreiko, and V.Ya. Stetsenko. Approximate solu-tions of operator equations. Wolteers-Noordhoff, 1972. Translated by D. Louvish.

[35] C.L. Laing, W.C. Troy, B. Gutkin, and G.B. Ermentrout. Multiple bumps in a neuronal modelof working memory. SIAM J. Appl. Math., 63(1):6297, 2002.

[36] C.R. Laing and W.C. Troy. Two-bump solutions of amari-type models of neuronal patternformation. Physica D, 178(3):190218, apr 2003.

[37] F.H. Lopes da Silva, W Blanes, S.N. Kalitzin, J. Parra, P. Suffczynski, and D.N. Velis. Dy-namical diseases of brain systems: different routes to epileptic seizures. IEEE transactions inbiomedical ingeneering, 50(5):540548, 2003.

[38] F.H. Lopes da Silva, A. Hoeks, and L.H. Zetterberg. Model of brain rhythmic activity. Kyber-netik, 15:2737, 1974.

[39] F.H. Lopes da Silva, A. van Rotterdam, P. Barts, E. van Heusden, and W. Burr. Model ofneuronal populations. the basic mechanism of rhythmicity. M.A. Corner, D.F. Swaab (eds)Progress in brain research, Elsevier, Amsterdam, 45:281308, 1976.

[40] Kiyotoshi Matsuoka. Stability conditions for nonlinear continuous neural networks with asym-metric connection weights. Neural Networks, 5:495500, 1992.

[41] E.K. Miller, C.A. Erickson, and R. Desimone. Neural mechanisms of visual working memory

in prefrontal cortex of the Macaque. J. Neurosci., 16:51545167, 1996.[42] V.B. Mountcastle. Modality and topographic properties of single neurons of cats somatosen-

sory cortex. Journal of Neurophysiology, 20:408434, 1957.

[43] V.B. Mountcastle. The columnar organization of the neocortex. Brain, 120:701722, 1997.

[44] D.J. Pinto and G.B. Ermentrout. Spatially structured activity in synaptically coupled neuronalnetworks: 1. traveling fronts and pulses. SIAM J. of Appl. Math., 62:206225, 2001.

[45] D.J. Pinto and G.B. Ermentrout. Spatially structured activity in synaptically coupled neuronalnetworks: 2. standing pulses. SIAM J. of Appl. Math., 62:226243, 2001.

[46] William H. Press, Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling. NumericalRecipes in C. Cambridge University Press, 1988.

[47] J.E. Rubin and W.C. Troy. Sustained spatial patterns of activity in neuronal populations withoutrecurrent excitation. SIAM journal on applied mathematics, 64(5):16091635, 2004.

[48] J. Sokolowski and J.-P. Zolsio. Introduction to shape optimization. Shape sensitivity analysis.,volume 16 ofSpringer Ser. Comput. Math. Springer-Verlag, Berlin, 1992.

INRIA


43/44


[49] J. Stoer and R. Bulirsch. Introduction to Numerical Analysis. Springer-Verlag, 1972.

[50] Alex M. Thomson and A. Peter Bannister. Interlaminar connections in the neocortex. CerebralCortex, 13:514, January 2003.

[51] F.G. Tricomi. Integral Equations. Dover, 1985. Reprint.

[52] A. van Rotterdam, F.H. Lopes da Silva, J. van den Ende, M.A. Viergever, and A.J. Hermans.A model of the spatial-temporal characteristics of the alpha rhythm. Bulletin of Mathematical

Biology, 44(2):283305, 1982.

[53] F. Wendling, F. Bartolomei, JJ. Bellanger, and P. Chauvel. Interpretation of interdependenciesin epileptic signals using a macroscopic physiological model of the eeg. Clinical Neurophysi-ology, 112(7):12011218, 2001.

[54] F. Wendling, J.J. Bellanger, F. Bartolomei, and P. Chauvel. Relevance of nonlinear lumped-parameter models in the analysis of depth-eeg epileptic signals. Biological Cybernetics,83:367378, 2000.

[55] Herrad Werner and Tim Richter. Circular stationary solutions in two-dimensional neural fields.Biological Cybernetics, 85(3):211217, sep 2001.

[56] H.R. Wilson and J.D. Cowan. A mathematical theory of the functional dynamics of corticaland thalamic nervous tissue. Biological Cybernetics, 13(2):5580, sep 1973.

RR n 0123456789


44/44

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