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Habilitation `a diriger des recherches Ecole Doctorale Sciences Fondamentales et Applications, Universit´ e de Nice Sophia Antipolis par Maureen CLERC n´ ee GALLAGHER Brain Functional Imaging: simulation, calibration and estimation Soutenue le 27 novembre 2007 devant le jury compos´ e de M. Laurent BARATCHART examinateur M. Jacques BLUM rapporteur M. Patrick CHAUVEL examinateur M. Olivier FAUGERAS directeur M. Jan DE MUNCK rapporteur Mme Line GARNERO rapporteur M. Renaud KERIVEN examinateur
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Page 1: Brain Functional Imaging: simulation, calibration and ...

Habilitation a diriger des recherches

Ecole Doctorale Sciences Fondamentales et Applications,Universite de Nice Sophia Antipolis

par

Maureen CLERC nee GALLAGHER

Brain Functional Imaging: simulation, calibration andestimation

Soutenue le 27 novembre 2007 devant le jury compose de

M. Laurent BARATCHART examinateurM. Jacques BLUM rapporteurM. Patrick CHAUVEL examinateurM. Olivier FAUGERAS directeurM. Jan DE MUNCK rapporteurMme Line GARNERO rapporteurM. Renaud KERIVEN examinateur

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2

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Contents

Introduction 1

1 Computational electromagnetics with Boundary Elements 51.1 A common formalism for Boundary Element Methods . . . . 6

1.1.1 The double-layer BEM . . . . . . . . . . . . . . . . . . 81.1.2 The symmetric BEM . . . . . . . . . . . . . . . . . . . 8

1.2 Non-nested geometries . . . . . . . . . . . . . . . . . . . . . . 101.3 The Fast Multipole Method for large problems . . . . . . . . . 121.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Conductivity model calibration 172.1 Electrical Impedance Tomography . . . . . . . . . . . . . . . . 192.2 Simultaneous source and conductivity estimation . . . . . . . 202.3 Anisotropic models . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.1 Skull and white matter anisotropy . . . . . . . . . . . . 212.3.2 Nerve model for Functional Electrical Stimulation (FES) 22

2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Cortical activity estimation 253.1 Cortical mapping . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Source estimation with distributed source models . . . . . . . 303.3 Single trial MEG/EEG analysis . . . . . . . . . . . . . . . . . 323.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Conclusion 37

Publications 42

References 51

i

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ii CONTENTS

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Introduction

The study of human bioelectricity was initiated with the discovery of elec-trocardiography at the turn of the twentieth century, followed by electroen-cephalography (EEG) in the 1920’s, magnetocardiography in the 1960’s, andmagnetoencephalography (MEG) in the 1970’s. Biomagnetic and bioelectricfields have the same biological origin: the displacement of charges withinactive cells called neurons [59].

There are several scales at which bioelectricity can be described and mea-sured: the microscopic scale, with microelectrodes placed inside or in a veryclose vicinity to neurons, and the mesoscopic scale, with intracortical record-ing of local field potentials (i.e. the electric potential within the cortex),below a square millimeter. Non-invasive measurements of the electric po-tential via EEG or the magnetic field via MEG are taken on the scalp, andthe spatial extent of brain activity to which these measurements can be re-lated has not yet been elucidated, but lies between a square millimeter anda square centimeter. Given the size of the head, and the time scale of in-terest - the millisecond - the quasistatic approximation can be applied tothe Maxwell equations [46]. The electromagnetic field is thus related to theelectric sources by two linear equations: the Poisson equation for the electricpotential, and the Biot-Savart equation for the magnetic field.

Reconstructing the cortical sources from the electromagnetic field mea-sured on EEG and MEG sensors requires that the inverse problem of MEGand EEG (denoted collectively as MEEG for short) be solved. It is a difficultproblem because it is ill-posed, and this has led to a large body of literature,concerning both theoretical [10, 23] and computational aspects [8, 28, 35].

This habilitation deals with models and computations related to the in-verse MEEG problem. Before explaining in more detail the purpose of myresearch in this domain, I briefly sketch the type of research and of applica-tions which this work is related to.

Clinical research in neurophysiology aims at understanding the mechanismsleading to disorders of the brain and the central nervous system, in order

1

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2 INTRODUCTION

to improve diagnosis and eventually propose new therapies. The clinical do-mains in which EEG and MEG are most routinely used include epilepsy,schizophrenia, depression, attention deficit disorders. Clinicians are espe-cially interested in the time courses of the measured signals: their experiencein the visual analysis of superimposed sensor time courses allows them todetect abnormal patterns. The source localization performed by clinicians isgenerally limited to simple dipole scanning methods.

EEG and MEG rely on passive measurements, with no applied electro-magnetic field. In contrast, active techniques using bioelectric stimulation,are currently being developed and tested on patients, to treat disorders suchas Parkinson’s disease, chronic pain, dystonia and depression. Implantedintracortical or extradural electrodes can deliver a electrical stimulation tospecific brain areas (Deep Brain Stimulation or DBS). Less invasive, Tran-scranial Magnetic Stimulation (TMS), uses a time-varying magnetic field toinduce a current within the cortex. All of these techniques can be studiedwith the same equations, models, and numerical tools as the ones used forMEEG. Moreover, in order to understand the physiological mechanisms trig-gered by these stimulations, simultaneous TMS/EEG and DBS/MEEG canbe performed and analyzed [74].

Devices such as neural or cochlear implants, and Brain Machine/ComputerInterfaces are examples of neuroengineering, whose developments are re-ported for instance in the IEEE Trans. on Neural Systems and RehabilitationEngineering or the new online Journal of Neuroengineering and Rehabilita-tion. This multidisciplinary and rapidly growing domain of research aims torestore or supplement a bodily function by a numerical interface, or some-times a bypass, based on bioelectricity. In the case of an interface, the systemmust be able either to measure and interpret a bioelectric signal (such as theEEG) in terms of an action to be performed by a computer or machine, orto interpret an external signal (for instance an auditory stimulus) and trans-form it into a bioelectric signal which the body (brain or nerve bundle) canprocess. In the case of a bypass, the system should replace a missing link inthe neural system (such as in the case of a spinal chord injury), by analyzinga bioelectric signal and transmitting it to another component in the chain.The huge difficulty to be overcome is the bidirectionality of the informationtransmission in the biological chain.

In cognitive neuroscience, much of our knowledge of the brain has beenacquired from intracortical recordings in cat and monkey brains, as well asperoperative recordings on human brains. Although the advent of functionalMagnetic Resonance Imaging (fMRI) in the 1980’s has opened a unique per-spective on the localization of human brain cognitive function, timing issuesremain difficult to resolve. Because of their high time resolution, EEG and

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INTRODUCTION 3

MEG are very useful for analyzing oscillatory activity, and the timings ofactivations between different brain regions. And because of its strictly non-invasive nature, MEEG is also well-suited to the study of the developmentof human brain function, from infancy to adulthood.

Lastly, an area in which bioelectrical computational models are beingapplied concerns the safety issues related to the exposure of the body toelectromagnetic fields, such as those generated by mobile phones and anten-nas, and, in the clinical domain, by Transcranial Magnetic Stimulation andMRI scanners.

This habilitation thesis presents the research which I conducted after joiningthe Odyssee team in 2001. My PhD thesis and post-doc were in quite a differ-ent field, the statistical analysis of signals and images using wavelets [Cle99,CM03]. Estimating the statistics of certain warped non-stationary pro-cesses led me to study the Shape from Texture problem in Computer Vi-sion [CM02, Cle02]. Joining Odyssee when the project was reorienting itselftoward the study the Computational Neuroscience of Vision, I had the op-portunity to focus on EEG and MEG, particularly on problems related toelectromagnetic modeling, to which I had been acquainted during my Mas-ters in Numerical Analysis. My expertise in wavelets is also proving usefulfor the interpretation of EEG/MEG signals.

The document is divided in three chapters, covering three distinct aspectsof Brain Functional Imaging. I describe them here, along with the mainpublication relative to each topic.

The first chapter focuses on new computational methods, developed withthe aim of improving the solution of the forward problem of MEEG, in termsof

• accuracy: the symmetric BEM [KCA+05], which was developped in thepost-doctoral assignment of Jan Kybic, and the PhD thesis of GeoffrayAdde, both of which I co-directed;

• speed and memory requirements: the Fast Multipole Method [KCF+05],developed during and after Jan Kybic’s post-doc;

• general head models: the generalized geometries [KCF+06], also devel-oped during and after Jan Kybic’s post-doc.

The second chapter deals with the modeling and calibration of conduc-tivity, describing two types of methods which we are developing for their invivo estimation:

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4 INTRODUCTION

• Electrical Impedance Tomography [CBA+05], [CAK+05];

• simultaneous source/conductivity estimation [VBC06], [VCB07], in Syl-vain Vallaghe’s PhD thesis, which I am co-directing,

as well as a new model for Functional Electrical Simulation being developedwith Sabir Jacquir in the course of his post-doc which I am co-directing withDavid Guiraud (Demar, INRIA and LIRMM).

The third chapter is devoted to the estimation of cortical activity fromMEG or EEG measurements. In this domain, we propose

• a new Cortical Mapping method to transport data from the sensors tothe cortical surface [CK07];

• a new algorithm to regularize distributed source methods [ACF+03,ACK05], in Geoffray Adde’s thesis, which I co-directed;

• a template-based time-frequency-topography approach for the analysisof single-event MEG/EEG data [BPC06a, BPC06b, BCP07], devel-opped in Christian Benar’s post-doctoral assignment, which I directed.

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Chapter 1

Computationalelectromagnetics withBoundary Elements

Boundary Element Methods (BEM), which arise when boundary integralequations are discretized, are very useful in solving certain types of PartialDifferential Equations, such as those involving the Laplacian or the bilapla-cian, in piecewise-homogeneous media [21]. They have been introduced fortyyears ago in electrocardiography [12, 13]

The use of BEM for EEG and MEG was pioneered by Geselowitz [37]in 1967, and further developed by Sarvas [70] in 1987 and de Munck [30]in 1992 who proposed a piecewise linear (P1) formulation. In the sequel werefer to the Geselowitz method with P1 discretization as the “classical”, or“double-layer” BEM.

The forward problem in MEG/EEG computes an electromagnetic fieldwhich is due to electrical activity within the brain. In the quasistatic ap-proximation, valid at frequencies below 1 kHz, the electric component isdecoupled from the magnetic component and can be computed separately.The electric potential V is the solution of an electrostatics problem, via thePoisson equation ∇ · (σ∇V ) = f , in which σ represents the (spatially vary-ing) conductivity of the head tissues, and f is related to the brain electricalactivity Jp by f = ∇ · Jp.

Solving the forward problem requires a conductivity model for the compu-tational domain. For a simple model consisting of concentric spheres withdifferent conductivities, with spherical harmonics, one can derive analyticalexpressions for the electric potential and the magnetic field, and accelerationmethods have been proposed for the numerical implementation of the se-

5

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6CHAPTER 1. COMPUTATIONAL ELECTROMAGNETICS WITH BOUNDARY ELEMENTS

ries [29, 18]. The three-sphere model has been widely used for EEG becauseof its simplicity: it does not require any mesh generation, and the compu-tations are very fast [71]. When a MRI of the subject’s head is available,the main tissues of the head can be localized and meshes of the interfacescan be generated. Using a piecewise-constant conductivity model, one cantransform the Poisson equation into a set of Laplace equations, connected bycontinuity conditions on the interfaces. Then the Green representation the-orem leads to a set of Boundary Integral Equations, and after discretization,to the BEM.

We have introduced a general formalism of boundary integral representationsfor the forward problem of EEG, which has allowed us to derive different typesof BEM, and to apply these methods to non-nested geometries. We have alsoproposed an acceleration method, called the Fast Multipole Method, in orderto handle the large dense matrix-vector multiplications which arise from theBEM.

1.1 A common formalism for Boundary Ele-

ment Methods

Our research in Boundary Elements was initiated by comparing the classicalBEM and the FEM. At the time, the comparison of the precision of bothapproaches was unfavorable to the BEM [CDF+02], even when using theIsolated Problem Approach, which is intended to improve the precision [47],as is shown in Figure 1.1. Our quest for a better precision of the BEMled us to explore, with T. Abboud, more advanced representation theorems,which had initially been developed for computational electromagnetics in thegroup of J-C Nedelec [61]. The Green Representation Theorem expresses apiecewise harmonic function as a combination of boundary integrals of itsjumps and the jumps of its normal derivative across interfaces1.

1Notation : ∂nV = n · ∇V denotes the partial derivative of V in the direction of aunit vector n. The restriction of a function f to a surface Sj will be denoted fSj

. Wedefine the jump of a function f : R

3 → R across Sj as

[f ]Sj= f−

Sj− f+

Sj,

the functions f− and f+ on Sj being respectively the interior and exterior limits of f :

for r ∈ Sj , f±

Sj(r) = lim

α→0±f(r + αn).

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1.1. A COMMON FORMALISM FOR BOUNDARY ELEMENT METHODS7

0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

dipole excentricity

RDM, Mesh size 0.15

IPA

FEM

Sym BEM

Figure 1.1: In a three-sphere model with a dipolar source, the relative errorbetween the analytical potential and the potential computed with the double-layer BEM (IPA), the FEM and the symmetric BEM, as the dipole positioneccentricity increases along the horizontal axis.

Let us consider an open region Ω and a function u such that ∆u = 0 in Ωand in R

3\Ω. Let G(r) = 14π‖r‖

be the fundamental solution of the Laplaciansuch that −∆G = δ0. The Green Representation Theorem states that, for apoint r belonging to ∂Ω,

u−(r)+u+(r)2

= −

∂Ω

[u] ∂n′G(r − r′)ds(r′) +

∂Ω

[∂n′u]G(r − r′)ds(r′) .

As shown in [KCA+05], this representation also holds when Ω is the unionof disjoint open sets: Ω = Ω1 ∪ Ω2 ∪ . . .ΩN , with ∂Ω = S1 ∪ S2 ∪ . . . SN , asin Figure 1.2 (left). In this case, for r ∈ Si,

u−(r)+u+(r)2

= −N∑

j=1

Sj

[u]Sj∂n′G(r − r′)ds(r′)

+

Sj

[∂n′u]SjG(r − r′)ds(r′) (1.1)

The notation is simplified by introducing two integral operators, called the“double-layer” and “single-layer” operators, which map a scalar function fon ∂Ω to another scalar function on ∂Ω:

(Df

)(r) =

∫∂Ω

∂n′G(r−r′)f(r′) ds(r′)

and(Sf

)(r) =

∫∂Ω

G(r−r′)f(r′) ds(r′). For a given operator D, its restriction

which maps a function of Sj to a function of Si is denoted Dij.

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8CHAPTER 1. COMPUTATIONAL ELECTROMAGNETICS WITH BOUNDARY ELEMENTS

1.1.1 The double-layer BEM

To apply the representation theorem to the forward problem of EEG, a har-monic function must be produced, which relates the potential and the sources.Decomposing the source term as f =

∑i fi where the support of each fi lies

inside homogeneous region Ωi, consider vΩisuch that ∆vΩi

= fi holds inall R

3. The function vd =∑N

i=1 vΩisatisfies ∆vd = f and is continuous

across each surface Si, as well as its normal derivative ∂nvd. The functionu = σ V − vd is a harmonic function in Ω, to which (1.1) can be applied.Since [u]Si

= (σi − σi+1)Vj and [∂nu] = 0, we obtain, on each surface Si,

σi + σi+1

2Vj +

N∑

j=1

(σj − σj+1) Dij Vj = vd , (1.2)

i.e. the formula established by Geselowitz [37]. The classical BEM corre-sponds to a double-layer potential formulation because it involves the double-layer operator D.

An extension of the Green Representation Theorem represents the di-rectional derivative of a harmonic function as a combination of boundaryintegrals of higher order. This requires two more integral operators: theadjoint D∗ of the double-layer operator, and a hyper-singular operator N

defined by(Nf

)(r) =

∫∂Ω

∂n,n′G(r − r′)f(r′) ds(r′). If r is a point of Si,

−∂nu

−(r) + ∂nu+(r)

2= +N[u] − D

∗[∂nu] (1.3)

The Geselowitz formula exploited the first boundary integral representationequation (1.1), whereas [KCA+05] exploits both (1.1) and (1.3). This allowsus to derive two new Boundary Element Methods within a unified setting: adual, single-layer potential formulation, and a formulation combining single-and double-layer potentials. In the next section, we focus on the latter,called the symmetric Boundary Element Method, which proves to be a verypromising alternative to the classical, double-layer BEM.

1.1.2 The symmetric BEM

The originality of the symmetric Boundary Element Method is to considerone piecewise harmonic function per domain: the function uΩi

equal to V−vΩi

σi

within Ωi and to −vΩi

σioutside Ωi. This function uΩi

is indeed harmonic in

R3\∂Ωi, and the representation equations (1.1) and (1.3) can be applied,

leading to a system of integral equations involving two types of unknowns:the potential Vi and the normal current (σ∂nV )i on each interface.

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1.1. A COMMON FORMALISM FOR BOUNDARY ELEMENT METHODS9

The surfaces are represented by triangular meshes. To fix ideas, we con-sider a three-layer geometrical model for the head. Conductivities of eachdomain are respectively denoted σ1, σ2 and σ3. The surfaces enclosing thesehomogeneous conductivity regions are denoted S1 (inner skull boundary),

S2 (skull-scalp interface) and S3 (scalp-air interface). Denoting ψ(k)i the P0

function associated to triangle i on surface Sk, and φ(l)j the P1 function asso-

ciated to node j on surface Sl, the potential V on surface Sk is approximatedas VSk

(r) =∑

i x(k)i φ

(k)i (r), while p = σ∂nV on surface Sk is approximated

by pSk(r) =

∑i y

(k)i ψ

(k)i (r).

As an illustration, considering the source term to be resricted to the braincompartment Ω1, the variables

(xk

)i= x

(k)i and

(yk

)i= y

(k)i satisfy the linear

system:

(σ1+σ2)N11 −2D∗11 −σ2N12 D∗

12 0−2D11 (σ−1

1+σ−1

2)S11 D12 −σ−1

2S12 0

−σ2N21 D∗21 (σ2+σ3)N22 −2D∗

22 −σ3N23

D21 −σ−12

S21 −2D22 (σ−12

+σ−13

)S22 D23

0 0 −σ3N32 D∗32 σ3N33

x1

y1

x2

y2

x3

=

b1

c1

000

(1.4)

where b1 and c1 are the coefficients of the P0 (resp. P1) boundary elementdecomposition of the source term ∂nvΩ1

(resp. −σ−11 vΩ1

).The blocks Nij and Dij map a potential Vj on Sj to a quantity defined on

Si. The blocks Sij map a normal current pj on Sj to a quantity defined onSi. The resulting matrix is block-diagonal, and symmetric, whence the name“symmetric BEM”.

The symmetric BEM introduces an additional unknown into the problem:the normal current, and uses an additional set of representation equationslinking the normal current and the potential. It demonstrates significantlyhigher accuracy than the double-layer BEM (see Figure 1.1). It is also moreaccurate than the Finite Element Method, although this comparison maybe unfair with regard to the implementation of the FEM used, which doesnot represent the dipolar source as precisely as the BEM or the analyticalsolution.

Our common formalism for the integral formulations of the forward EEGproblem, besides providing a better understanding of the Boundary ElementMethods used in EEG and MEG, also leads to a more precise forward problem

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10CHAPTER 1. COMPUTATIONAL ELECTROMAGNETICS WITH BOUNDARY ELEMENTS

for EEG with no additional meshing (but larger system matrices).

For the forward MEG problem, the magnetic field is computed from theelectric field and the primary source distribution using the Biot and Savartequation, as proposed by Ferguson, Zhang and Stroink [36].

This new methodology was implemented by G. Adde in a C++ codecalled “OpenMEEG’: an Opensource software package for solving the for-ward and inverse problems of MEG and EEG2. In order to benefit to alarger MEG/EEG community, OpenMEEG has been integrated to Brain-VISA, a human brain mapping library distributed freely by CEA-SHFJ3.Within BrainVISA, OpenMEEG can be used in conjunction with Brain-Storm: for instance, the symmetric BEM forward model of OpenMEEG canbe selected as a lead field for the MUSIC methods of BrainStorm.

The symmetric BEM is also being integrated within the C++ packageSimbio4, in collaboration with A. Anwander and T. Knoesche of the MaxPlanck Institute, Leipzig. This could lead to an integration into the com-mercial software Advanced Source Analysis (ASA) by Advanced Neuro Tech-nology (ANT), of which Simbio represents the computational core.

1.2 Non-nested geometries

Boundary Element Methods (BEM) require a geometrical model describingthe interfaces between tissues with different conductivities. The head mod-els classically used for solving the forward and inverse problems of magneto-and electro-encephalography (MEG/EEG) consist of nested volumes. How-ever, in many cases, more accurate geometrical models of the head are neces-sary [17, 64]. The electric potential on the scalp is affected by inhomogeneitiesof the skull, such as the fontanellas for infants, or the occurrence of holes orflaps, even after they have been filled in after an operation. Such skull par-ticularities should be incorporated in the head model. The presence of largeventricles filled with CerebroSpinal Fluid is also difficult to handle with anested volume model. The boundary element method is not restricted tonested geometries [46]. However, apart from a few exceptions [48, 2], mostimplementations of the BEM only consider nested volumes, making it com-plicated to accommodate more general geometries [17, 64]. In order to usethe symmetric Boundary Element Method with a natural, volume-wise de-scription of the geometry, we have implemented it for such generalized head

2https://gforge.inria.fr/projects/openmeeg/3http://www.brainvisa.info4http://www.simbio.de/

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1.2. NON-NESTED GEOMETRIES 11

ΩN+1 σN+1

SN

σN

ΩN

σ1

Ω1

Ω2

S1

S2

σ2

ΩN+1

Ωα

Ωβ

Sαβ

Figure 1.2: Traditionally, the head is modeled as a set of nested regions(left), while a real head geometry is much more complex. The model pro-posed in [KCF+06] assumes piecewise constant conductivity, with arbitrarypartitioning (right).

σ = 10−4 σ = 10−2 σ = 10−1 σ = 1

Figure 1.3: The surface potential for the realistic head models for differentvalues of the hole compartment conductivity. The same colourmap was usedfor all images (red is positive, blue is negative). After mean subtraction.

models. In [KCF+06], we present our implementation and show the numeri-cal incidence of such detailed models on the forward EEG.

A residual difficulty certainly lies in the mesh generation: local mesh re-finement is needed at intersections between surfaces which meet at a sharp an-gle. Resistor-mesh models can incorporate skull defects such as holes rathereasily as far as meshing is concerned [24]. The mesh generation may howevernot remain an issue much longer, as more interactive and practical mesh-generation methods become available [20].

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12CHAPTER 1. COMPUTATIONAL ELECTROMAGNETICS WITH BOUNDARY ELEMENTS

1.3 The Fast Multipole Method for large prob-

lems

The solution of the EEG forward problem, which computes the electric poten-tial due to a known distribution of cortical sources, requires the inversion ofa linear system (1.4) of the form Au = c. In the Boundary Element Method,the matrix, whose elements can be seen as the interaction of boundary el-ements through singular integral operators, is dense. When sophisticatedhead models such as those described in Section 1.2 are employed and thenumber of degrees of freedom of the system grows large (over 10000), solvingthe linear system by direct inversion becomes impossible. Instead, iterativemethods, such as MINRES, which only access the matrix through matrix-vector products, are used.

With the Fast Multipole Method (FMM), the product Au can be com-puted without the matrix being formed explicitly and the overall complexityof calculating the product Au for a u of size N is decreased from O(N2)to O(N). The two main advantages of the FMM are to reduce the mem-ory consumption, and to accelerate the computations. There is extensiveliterature dealing with the FMM [25, 15, 34, 32, 67] for gravitational or elec-tromagnetic scattering calculations. In parallel to our own developments,the FMM has been applied to the electrostatic Maxwell problem and thesymmetric BEM, but with only one interface [63].

The matrix-vector products in the BEM consist in computing the inter-action between elements in two sets A and B, under the form

cj =∑

i∈A

Aijui, for all j ∈ B .

The elements from A and B correspond to the support of the basis functions,i.e. either to triangles (for piecewise constant P0 elements), or to sets oftriangles with a common vertex (for piecewise linear P1 elements).

In the boundary element methods introduced for the forward EEG prob-lem, each of the boundary integral operators has a kernel which involves thefunction 1/‖r − r′‖. The gist of the Fast Multipole Method is to decouplethe two variables r and r′, when they are sufficiently spatially separated withrespect to a center of expansion C

‖r′ − C‖ > λ‖r − C‖︸ ︷︷ ︸well-separateness

=⇒1

‖r − r′‖=

L∑

n=0

n∑

m=−n

I−mn (C−r)Om

n (r′−C)+error

where Imn resp. Om

n are inner resp. outer spherical harmonics [34]. The

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1.3. THE FAST MULTIPOLE METHOD FOR LARGE PROBLEMS 13

generalized product (similar to matrix multiplication) of the above equationis denoted by ⊙.

Using the type of spherical harmonic decomposition described above, foreach pair of elements i in A, resp. j in B sufficiently well-separated withrespect to a center of expansion C, and the matrix term Aij can be approx-imated with an a priori precision ε:

| Aij − φj(C) ⊙ φi(C) |≤ ε, (1.5)

where φj(C), and φi(C) are called an outer (far, or multipole) resp. inner(near, or local) expansion.

Different algorithms can then be proposed according to different choicesof centers of expansions. The simplest FMM algorithm is the ‘grouping’ or‘middle-man’ algorithm. It is based on dividing the elements in A and B intospatially constrained cells, typically by partitioning the space into rectangularcells of identical size [15]. To each cell Ak (resp. Bl) is associated a center ofexpansion Ck (resp. Cl). The interaction between well-separated cells Ak,Bl is then carried out using an approximation

i∈Ak

Aijui ≈ φj(Cl) ⊙(∑

i∈Ak

φi(Cl)ui

)

︸ ︷︷ ︸eΦ(Ak,Cl)

= φj(Cl) ⊙ Φ(Ak,Cl)

derived from (1.5). This algorithm has complexity of O(N3/2), where N is thenumber of elements ‖A‖ ≈ ‖B‖. One may introduce a translation operator Tto convert an outer expansion at point Ck into an inner one around point Cl

φi(Cl) = TCkClφi(Ck) .

In this way, the middle-man algorithm can be improved so that the innerfields Φ(Ak,Cl) can be calculated more efficiently. Instead of computing

Φ(Ak,Cl) for each Cl, the Φ(Ak,Ck) have to be calculated only once, andare translated to the centers of all other cells Bl . The improved algorithmis called a single-level FMM algorithm and has an asymptotic complexity ofO(N4/3). We applied it to the forward problem of EEG in [CKF+02].

In order to improve the single-level FMM, we build a hierarchy of cellsof different sizes, so that an optimal cell-size can be chosen depending onthe interaction distance. This leads to a multi-level FMM algorithm, oftensimply called FMM [15]. We create trees A resp. B from the input set A resp.output set B. Children of each non-leaf cell (tree node) X are themselvescells contained inX. Two translation operators, an outer-to-outer translation

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14CHAPTER 1. COMPUTATIONAL ELECTROMAGNETICS WITH BOUNDARY ELEMENTS

a a a a a a

a

a a a a

a

a

a a

localinteraction yi+

S

Tree A

a

Tree B

inner-innerS

outer-inner TRouter-outer

R

a a

R

ia

ja

xj

Figure 1.4: The principle of the FMM (from [KCF+06]): The outer fields arepropagated up the tree A during the up-sweep phase using operator R. Theyare transfered to tree B and converted to inner fields using operator T. Theinner fields are propagated down tree B using operator S. At leaf cells, themultipole interactions calculated from the inner-field coefficients are summedwith local interactions.

operator R and an inner-to-inner translation operator S are introduced :

φj(Cl) = RCkClφj(Ck)

φj(Cl) = SCkClφj(Ck)

The operator R is used to calculate the outer field for each non-leaf cell X inthe tree A by summing the outer fields of all its children Y — an up-sweep.Similarly, during the down-sweep, operator S translates the inner-field fromnon-leaf cells in tree B to their children (Figure 1.4).

To minimize the number of local interactions (i.e. explicit computationand summation of Aij ui), we use local interactions exclusively on pairs ofleaf cells which are not well-separated. A classical approach is to descendsimultaneously from the root to the leaves in both trees A, B (which mustbe identical), and, at each level, to include all valid interactions that havenot been already treated at higher levels, allowing only interactions between

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1.4. DISCUSSION 15

cells at the same level [15, 25]. Our method is more general as it does notrequire the trees A, B to be identical [KCF+05].

The FMM version of the linear system inversion is shown to be as accu-rate as the direct method, yet for large problems it is both faster and moreeconomical in terms of memory requirements. At the time of publication, thiswas the only implementation capable of accurately solving the MEG/EEGforward problem for realistic head models described by meshes with over30,000 points (70,000 unknowns) on a single personal computer [KCF+05].

1.4 Discussion

Our efforts on the forward problem have led us to develop at Odyssee a newformulation for the Boundary Integral Equations of EEG and MEG, whosediscretization, the Symmetric BEM, provides solutions with much better ac-curacy than the pre-existing Boundary Element solutions. The integration ofthis code into BrainVISA makes this method accessible to the neuroimagingcommunity at large. The Fast Multipole Method has been proposed to ac-celerate the solution of the forward problem, when the system to be invertedbecomes very large. Other acceleration techniques, for instance relying onadjoint problems [2], could further improve the speed.

The forward problem is the fundamental link between the unknown sourcedistribution and the measured data. The development of high-quality nu-merical methods for its solution is a necessity in order to obtain high-qualityinverse solutions. The next chapters will show how, in addition to the elec-tromagnetic source imaging problem, the symmetric BEM can be used tosolve a conductivity estimation problem, to model electrode-nerve interfacefor Functional Electrical Stimulation, and to solve a continuation problemcalled “cortical mapping”.

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16CHAPTER 1. COMPUTATIONAL ELECTROMAGNETICS WITH BOUNDARY ELEMENTS

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Chapter 2

Conductivity model calibration

The importance of correct realistic geometry for electromagnetic brain imag-ing has been recongnized for over a decade [27, 76]. In EEG the geometricalhead model has a considerable influence on results [53, 80]. In MEG, theinfluence of the head model is less critical than in EEG. In the extreme caseof a spherical conductivity geometry (such as in nested spheres), the mag-netic field outside the outer sphere is in fact independent of the conductivityprofile [70]. On this account it is often assumed that the influence of conduc-tivity on MEG is negligible, although for realistic head models, conductivitydoes have an influence [77].

Starting from a subject’s Magnetic Resonance Image (MRI), many com-putational tools now make it possible to segment the different tissues presentin the image, and mesh their interfaces, yielding subject-dependent, realistichead models. The forward MEG/EEG problems can then be solved using ei-ther Finite Difference, Resistor Mesh, Boundary Element or Finite ElementMethods. The Boundary Element Method assumes that the conductivityis piecewise constant, and uses discretizations based on the meshes of theinterfaces between tissues (see Chapter 1).

The use of sophisticated subject-dependent geometry in computationalbioelectromagnetism also requires precise conductivity values, adapted to thegeometrical model. This is all the more crucial as tissue conductivities varyfrom subject to subject [3], and even vary over time, for a given subject [52].These conductivity values should therefore ideally be calibrated at the timeof each experiment.

For Boundary Element Models, one of the parameters which has a greatinfluence on the source localization is the the skull/scalp conductivity ratio.The three pictures of Figure 2.1 show the variability of dipole localizationfrom EEG data, with three different assumptions on the conductivity ratio.

17

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18 CHAPTER 2. CONDUCTIVITY MODEL CALIBRATION

σscalp/σskull = 80 σscalp/σskull = 40 σscalp/σskull = 20

Figure 2.1: Single dipole, estimated by MUSIC method, from an EEG inter-ictal spike, for three different scalp/skull conductivity ratios (courtesy of J.M.Badier).

In this chapter, we focus on two methods for conductivity calibration,both of which impose a current source and measure the corresponding electricpotential on the scalp by EEG. The two methods vary in the way the currentsource is imposed.

The principle of Electrical Impedance Tomography (EIT) is to imposea (very small) source of current on the scalp, and to measure the resultingelectrical potential on the scalp by EEG. The conductivity values are es-timated by minimizing the difference between the measured potential andthe potential predicted by a forward EIT simulation. EIT has recently beenshown appropriate for estimating the values of tissue conductivities insidethe head, particularly the skull-to-scalp conductivity ratio [40, 14, 45]. Oursolution for EIT, which we present in Section 2.1, is based on the symmetricBoundary Element Method.

The other conductivity calibration method presented uses a sensory stim-ulation to evoke a cortical current source. The source localization can beperformed by using MEG data, reputedly less sensitive to conductivity thanEEG. Then, the conductivity values can be estimated, as in EIT, by minimiz-ing the difference between measured and simulated electric potentials [40].Instead of using concurrent MEG data, we propose a method which reliesonly on EEG data, and simultaneously estimates the source localization andthe tissue conductivities. This method, being developed by Sylvain Vallaghein his PhD thesis, is presented in Section 2.2.

Realistic head models with anisotropic conductivity require the use ofFinite Element or Finite Difference Methods [51]. However, anisotropy with

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2.1. ELECTRICAL IMPEDANCE TOMOGRAPHY 19

a very regular structure can be handled with Boundary Element Methods.This is the case of the cylindrically symmetric nerve model, which is currentlybeing used to model electrode-nerve interaction for Functional Electric Stim-ulation (section 2.3).

2.1 Electrical Impedance Tomography

In Boundary Element Methods, the conductivity values, assumed region-wiseconstant, should be viewed as “effective”, i.e. indissociable from the geomet-rical interfaces. Indeed, adapting the conductivity values can compensate forinaccuracies in geometry [39].

A BEM solution for EIT has previously been proposed, by Goncalvesand de Munck [40]. It is not completely satisfactory, because the matrix isdifficult to invert, the current injection difficult to model, and the resultingmethod is slow, not allowing it to be applied routinely.

In contrast, current injection is very easy to model with the symmetricBEM, since the normal current is an explicit unknown [KCA+05]. Let usconsider the same three-layer head model as in Section 1.1.2. The scalp cur-rent is discretized with P0 elements on surface S3: j =

∑jiψ

(3)i (r). Denoting

ψ(k)i the P0 function associated to triangle i on surface Sk, and φ

(k)i the P1

function associated to node i on surface Sk, the potential V on surface Sk isapproximated by VSk

(r) =∑

i x(k)i φ

(k)i (r), while p = σ∂nV on surface Sk is

approximated by pSk(r) =

∑i y

(k)i ψ

(k)i (r).

The variables(xk

)i= x

(k)i ,

(yk

)i= y

(k)i , and (j)i = ji are related by the

linear system [CBA+05]:

(σ1+σ2)N11 −2D∗11 −σ2N12 D∗

12 0−2D11 (σ−1

1+σ−1

2)S11 D12 −σ−1

2S12 0

−σ2N21 D∗21 (σ2+σ3)N22 −2D∗

22 −σ3N23

D21 −σ−12

S21 −2D22 (σ−12

+σ−13

)S22 D23

0 0 −σ3N32 D∗32 σ3N33

x1

y1

x2

y2

x3

=

00

−D∗23j

σ−13

D23j(−1

2I33 + D∗

33)j

(2.1)

The blocks Nij,Sij and Dij only depend on the geometric structure of themeshes, and not on the conductivities.

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20 CHAPTER 2. CONDUCTIVITY MODEL CALIBRATION

The EEG forward problem (1.4) and the EIT forward problem (2.1) havethe same system matrices, therefore the only additional assembly requiredfor EIT concerns the right-hand side “source” vector.

For a given scalp current injection profile jm, the scalp potential vm

is measured, and for a conductivity profile σ, a simulated scalp potentialV (σ, jm) is computed with the forward EIT problem. The inverse EIT prob-lem consists in finding the conductivity values which minimize the cost func-tion

E(σ) =1

2

M∑

m=1

‖V (σ, jm) − vm‖2meas . (2.2)

Given the very simple dependence of the forward EEG system matrix on theconductivity parameters, a gradient of E(σ) is readily computed, and thecost can be minimized with a gradient descent algorithm [CAK+05].

When acquiring experimental data at the EEG laboratory at La Tim-one, with the assistance of P. Marquis, we faced a practical difficulty: theavailable stimulator was impossible to control in terms of delivered currentamplitude. For this reason, we propose an alternative inverse EIT methodin which the current intensities are also estimated [CAK+05]. The estimatedcurrent intensity was found to vary during a given experiment. The practicaldifficulties encountered with the electrical stimulator led us to acquire a spe-cific stimulator dedicated to this application, whose intensity is controllableto a precision of 0.25 µA. This stimulator was designed by the Demar team1

and manufactured by the MXM laboratories2 [43].Our first experiments demonstrate the ability of the method to recover

the skull-scalp conductivity ratio, on simulations and on measured data onone subject. The method is in course of validation on more subjects.

2.2 Simultaneous source and conductivity es-

timation

As explained above, regardless of the care taken in designing realistic headgeometries and the forward model, one cannot expect good precision in sourcereconstruction unless the conductivity model is properly calibrated.

To turn a curse into a remedy, the high dependency of EEG to the scalp-to-skull conductivity ratio can be used to estimate this ratio itself, as pro-posed by Goncalves and de Munck in their paper on conductivity estimation

1http://www-sop.inria.fr/demar/2http://www.mxmlab.com

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2.3. ANISOTROPIC MODELS 21

by Somatosensory Evoked Field-constrained Somatosensory Evoked Poten-tials (SEF-constrained SEP) [41]. The MEG, less sensitive to conductivitydifferences than EEG, can be used to localize a focal source, representingthe primary response of a somatosensory median nerve stimulation. For thesame stimulation but with EEG measurements, the source previously esti-mated from MEG measurements can then be incorporated in the forwardmodel, and only the conductivities of the tissues remain to be estimated.

In the ongoing PhD thesis of Sylvain Vallaghe, we extend the reasoningeven further in order to develop a conductivity estimation method whichdoes not rely on MEG measurements. This will lead to greater practicality,since all EEG laboratories do not have easy access to a MEG center. Forfocal sources, such as the primary response of SEP, our results based on realmeasurements show that it is possible to estimate the source localization andthe tissue conductivities simultaneously [VBC06].

The cost function shows good sensitivity to the scalp / skull ratio, andits robustness is improved by the use of a cortically constrained dipolar sourcemodel [VCB07]. The EIT approach and the simultaneous source/conductivityestimation will be compared on the same set of four or five subjects in orderto confirm our first results.

In practical terms, both of our methods – the EIT as well as the si-multaneous source/conductivity estimation using SEP – require an electricalstimulator, but the SEP does not need the injected current to be preciselycontrolled, and therefore should be easier to deploy as it requires apparatusreadily available in EEG laboratories.

2.3 Anisotropic models

2.3.1 Skull and white matter anisotropy

Some tissues have conductivities which are better modeled as anisotropictensors. The skull is a composite medium, made of different types of softand hard bone, and the resulting effective conductivity should be modeledas anisotropic, with the radial conductivity about 10 times lower than thetangential [60]. Incorporating anisotropy for the skull conductivity is geomet-rically relatively simple. In a nested sphere geometry, it can be computedanalytically [85]. For more realistic head models, unfortunately, the BEMcannot be used, because there is no translation-invariant Green function asso-ciated with the modified anisotropic Laplacian, making the BEM ill-adaptedto handling anisotropic conductivities other than in very simple geometries(planar, or cylindrical). The Finite Difference and the Finite Element Meth-

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22 CHAPTER 2. CONDUCTIVITY MODEL CALIBRATION

ods, based on volume meshes, can assume conductivity distributions withmore intricate variations, including anisotropic, tensor-valued conductivityfields. Several groups have proposed such anisotropic FEM implementations,whether for the skull [60] or the white matter [49, 44, 83]. A Resistor MeshMethod has also been proposed to handle the anisotropic conductivity of theskull [24]. The implicit mesh finite element discretization developed by T.Papadopoulo also handles anisotropic conductivities [PVC06].

The white matter conductivity has an influence on the EEG and MEG,although to a lesser extent than that of the skull. Tuch and Wedeen [75]have derived a linear relationship between the diffusion and the conductiv-ity tensors, which has since been applied to derive realistic white-matterconductivity models for MEG and EEG [49, 44, 83]. However, the linearrelationship has never been properly assessed by conductivity measurementson tissue samples for instance, and it could be interesting to compare DT-MRI measurements and electrical conductivity measurements with EIT onthe same tissue samples.

2.3.2 Nerve model for Functional Electrical Stimula-tion (FES)

Functional Electrical Stimulation (FES) consists in applying an electrical cur-rent with the purpose of restoring, or improving a bodily function [42]. As anexample, cochlear stimulation can partially restore the sense of audition tohearing impaired people3. The goal of the STIC-Sante GENESYS project isto study Functional Electrical Stimulation for the restoration of bladder con-trol. The project combines theoretical modeling issues, experimental in vivovalidation on small animals, hardware implementation, and peroperatory val-idation on humans. Odyssee is concerned with the modelization phase, moreprecisely, to control the propagation of the applied current within the bio-logical tissues. The electrodes used in this application are cuff-electrodes,placed around the nerve sheath. The modeling problem is closely related tothe forward problem of Electrical Impedance Tomography. Geometrically,the nerve has two particularities: its anisotropy, and its cylindrical shape.The cylindrical shape allows analytical solutions to be derived, using Besselfunctions. The drawback of analytical solutions is that the injected currentmust also have a cylindrical geometry, which is overly restrictive. Analyticalsolutions are however useful for validation purposes.

In the case of the nerve, the longitudinal direction is more conductive thanthe transverse direction [82]. Boundary Element Methods are in general ill-

3see for instance http://www.neurelec.com

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2.4. DISCUSSION 23

adapted to anisotropic conductivity. However, when the anisotropy has auniform structure, the problem admits a translation-invariant Green func-tion and, by locally defining a coordinate system adapted to the new Greenfunction, the Boundary Element Method can be applied. We are currentlyadapting the symmetric Boundary Element Method to simulate the currentpropagation involved in electrode-nerve interaction. Thanks to an accurate3D computation of the electrode-nerve interaction, it will become possibleto design the electrode geometry, in terms of the positions of the electrodecontacts, and to optimize the current injection profile, both in space and intime. Sabir Jacquir is working on this problem as a post-doc, jointly betweenDemar and Odyssee.

2.4 Discussion

The long-term goal of our work on conductivity models is

• to provide pratical procedures to acquire measurements which can beused for offline conductivity model calibration,

• to better handle the anisotropies that are the most relevant for EEG,MEG or FES.

We plan to explore the possibility of estimating the anisotropy of the skullconductivity using EIT or SEP methods, with a volume FEM.

In the future, new physical devices will undoubtedly provide differentways of collecting measurements. Impedance tomography using magnetic in-duction may be less prone to shielding from the skull layer than EIT [84]. Asmentioned in [3], it may be possible to use CT images to directly estimate theconductivities of different bone compartments, which could then be incorpo-rated in Finite Element models. Lastly, if the spatial conductivity profile canbe estimated with good accuracy, it can be used to estimate neural activity:conductivity variations can be linked both to changes in blood volume, andto cell-swelling [7]. This would provide low-cost, portable access to corticalactivity, similar to that of Diffuse Optical Tomography [19], though probablynot with such high time-resolution as obtained with MEG/EEG.

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24 CHAPTER 2. CONDUCTIVITY MODEL CALIBRATION

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Chapter 3

Cortical activity estimation

The MEG and EEG acquire data with a time resolution below a millisecond,comparable to the one at which the brain processes information. Using thequasistatic assumption, the relationship between sources and recorded datais a linear one. The two preceding chapters have focused on modeling andcomputing this linear relationship, through the integral equations and theconductivity values that arise in the forward computations. In this chapterwe focus on the interpretation of MEEG measurements in terms of corticalactivity.

Models for source activity belong to two basic families: focal or dis-tributed. Focal sources are modeled as dipoles or multipoles, and have afinite number of parameters (generally less than a few dozen) [54]. If onehas access to an MRI of the subject’s brain, the reconstructed sources canbe constrained to lie within the cortex [28], leading to the distributed sourcemodel: sources having their support along a surface or a volume, generallydiscretized, leading to a source space with a high dimensionality.

The purpose of cortical activity estimation is to obtain a spatio-temporalmap of brain activity correlated to a functional activity of interest, such as asensory or cognitive task or a pathological component of the signal. To brieflycompare MEG and EEG functional imaging with other imaging modalities:its main advantages are its low invasivity and high time-resolution; its maindrawback is the difficulty of recovering activity of cortical sources from datameasured at the sensors. This difficulty is in fact two-fold: it involves aspatial inverse problem, i.e. the recovery, at a given point in time, of thespatial distribution of sources giving rise to the observed measurements, andit also involves an interpretation problem: how to distinguish within themeasurements (or within the reconstructed cortical activity), the interesting,or task-related, activity from background cortical activity ? As a comparison,in functional MRI, there is no spatial inverse problem to solve, but in the

25

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26 CHAPTER 3. CORTICAL ACTIVITY ESTIMATION

case of MRI a data interpretation task is also necessary, because the signalto noise ratio is very low.

Since Helmholtz showed, in 1853, the existence of silent sources, whichproduce no electric (n)or magnetic field outside a bounded conductor, thequestion of the uniqueness of interior sources which may be recovered fromdata acquired on the surface deserves to be addressed [81]. The source mod-els used for MEG and EEG are either discrete (a finite sum of dipoles ormultipoles), or distributed (a surface or volume distribution of dipoles). Theuniqueness or well-posedness of the inverse problem naturally depends on thesource model used. El Badia and Ha-Duong have proved the uniqueness ofa linear superposition of dipolar sources [33]. Likewise, Amir has proved theuniqueness of a surface distribution of dipoles [4]. However, uniqueness isnot enough to guarantee well-posedness, and the inverse source localizationproblem is unstable.

Regardless of the source model, all estimation methods aim to recoverthe sources which best explain the measured data, by minimizing a dataattachment term of the form

C(J) = ‖M −GJ‖ ,

where J represents the source, M the measurements and G the gain (or leadfield) matrix computed after solving the forward problem. In the case of adiscrete source model, the solution space has a limited dimension, but thedata attachment term has many local minima. Depending on the initializa-tion, or the number of sources in the model, the recovered sources may havevery different spatial positions. In practice, regularization can be applied, forinstance by imposing stability of the source positions within a time-window(MUSIC). However, all methods based on a discrete source model fail tolocalize correctly several sources whose time courses are simultaneous. Inthis respect, we place great hopes in a new method developed by Baratchart,Leblond and colleagues in the APICS project team at INRIA Sophia Antipo-lis. Their method uses rational approximation theory to recover singularitieswithin planar circular slices of the cortex modeled as a sphere [10, 9, 11].It requires that the data be known on the surface of the inner sphere. Weare collaborating with this group to propose a Cortical Mapping solution,which transports the Cauchy data (potential and its normal derivative) forthe Laplacian inwards onto the cortex (Section 3.1).

There is an important distinction between the knowledge of the potentialand normal current on the surface of the cortex and the 3D localizationof sources within the cortical volume (or distributed along a surface whichfollows the cortical foldings). The terminology “surface of the cortex” denotes

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3.1. CORTICAL MAPPING 27

the smooth envelope of the cortex, similar to the dura, or the inner surfaceof the skull. Cortical Mapping consists in estimating the electrical field onthis smooth surface. Spatially, it is comparable to measurements obtainedon subdural or extradural grids in pre-surgical explorations. This electricalfield on the cortical surface, although far more focal than those measuredon the scalp, is not descriptive enough to explain the spatial distribution ofthe activity in depth, within the intricate foldings of the cortical gyri andsulci. Pre-surgical explorations in depth can be provided by intracorticalelectrodes, such as implanted with stereo-electroencephalography (sEEG) atthe epilepsy center in La Timone, Marseille.

This chapter addresses two important aspects of cortical activity esti-mation: the spatial inverse problem, and the data interpretation problem.The first two sections are devoted to two distinct spatial inverse problemswhich can be applied to the same set of measurements: the Cortical Map-

ping problem, which estimates the potential and the current on the innerskull surface, and the source estimation problem, with a distributed sourceapproach. These two inverse problems are quite distinct because of the verydifferent natures of the surfaces on which the quantities to be reconstructedare defined. The last section deals with the processing of single-event data,and presents a new approach based on topography-time-frequency templates,developed with Christian Benar in the course of his post-doctoral assignmentat Odyssee.

3.1 Cortical mapping

The field maps measured by EEG on the scalp are far less focal than thosemeasured by MEG, because of the low skull conductivity, which acts asa spatial low-pass filter in the direction tangent to the scalp surface. Apopular way to enhance the scalp EEG data is to apply high-pass filtering,in order to obtain a more focal rendering of the activity [62, 73]. For thethree-shell spherical model, the surface Laplacian operator is applied to thedata, yielding an approximation of the normal current on the inner skullsurface [38]. The extension of the surface Laplacian to more realistic headmodels is however often applied without justification.

Cortical Mapping is used as a way of obtaining more spatio-temporal resolu-tion from the data, without the need to resort to source space. The temporalnature of the resolution enhancement deserves an explanation, since Corti-cal Mapping is a purely spatial procedure. The blurring produced by the

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28 CHAPTER 3. CORTICAL ACTIVITY ESTIMATION

scal

psk

ull

cort

exsk

ull

cort

ex

(a) true solution (b) solution with 0% noise (c) solution with 5 % noise

Figure 3.1: Realistic head model and auditory source model: 3D renderingof the potential and the flux on the scalp and on the cortex. The head isviewed from above, with the nose pointing downwards [CK07].

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3.1. CORTICAL MAPPING 29

diffusion of the potential through the skull layer also operates a temporalmixture at the sensor level between the activity coming from sources havingdistinct time courses. With this point of view, a spatial deblurring also hasthe consequence of separating the time courses associated with each spatialcomponent, leading to an enhanced time resolution. A theoretical result onharmonic continuation from Cauchy boundary data (i.e. the potential and itsnormal derivative on the boundary, for a harmonic potential), the Holmgrentheorem, guarantees existence and uniqueness of the electrical potential andnormal current on the cortical surface which produce a given potential onthe scalp, provided the potential is known on a dense portion of the scalp [?]Rather than applying the surface Laplacian on the scalp, it is possible to usethe geometry and conductivities of the realistic model, and to use the struc-tural equations of the Boundary Element Methods such as implemented forEEG, to estimate the normal current and the electrical potential under theskull. Since neither the model nor the measurements are perfectly known,and the Cauchy continuation problem is ill-posed, regularization is necessary.

Such a method was proposed by He, Wang and Wu in 1999 [50], using partof the equations of the Symmetric Boundary Element Method. We proposea new solution to the Cortical Mapping problem, using the same matrixstructure as the Symmetric BEM. Note that several methods have recentlybeen proposed to solve the Cauchy continuation problem, among which anenergy-minimizing approach [5], methods using quasi-reversibility [56],[22],and methods alternating Dirichlet and Neumann problems, with regularizingproperties [58],[16],[6]. However, the methods are seldom applied to three-dimensional geometries, and never to multiple-layer domains necessary forEEG.

An originality of our Cortical Mapping method is to use a kernel-projectionto express the Boundary Integral relationships between the potential and theflux σ∂nV on the different surfaces. The discretized variables representingthe potential and the flux are grouped in a vector X, which satisfies a linearrelationship expressing the harmonic nature of the potential in each domain:

HX = 0 . (3.1)

To solve the Cortical Mapping problem, X must both satisfy (3.1), and mini-mize a cost function composed of a data attachment term and a regularizationterm:

C(X) = ‖m−MX‖2 + ‖RX‖2

where m are the measurements and M an electrode selection operator andthe regularization represents the L2 norm of the tangential gradient of the po-tential and the flux over each surface. The kernel-projection method simply

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30 CHAPTER 3. CORTICAL ACTIVITY ESTIMATION

turns the constrained minimization problem

minX∈KerHC(X)

into an unconstrained minimization problem by writing X under the formX = PKerHY and seeking the minimum-norm solution Y which minimizesC(PKerHY ) [CK07]. The kernel-projection is easily implemented with theSVD, and can be precomputed for a given head model. As mentioned in theIntroduction, our Cortical Mapping solution is intended to serve as a first stepfor a new dipole localization procedure, based on rational approximation [10],and which requires the potential and flux data to be known on the surfaceof the cortex [9, 11], [MLMB06].

In [CAL+05] we compare our method to a Cauchy continuation problembased on a Boundary Extremal Problem for nested spheres. The results arecompared in terms of the ability of the rational approximation to detectcorrect source locations from them, for simulated data with a single sourcewhose correct location was known. The Symmetric BEM Cortical Mappingmethod yields a better level of precision of the source localization, and agreater generality since it is not restricted to a spherical geometry.

3.2 Source estimation with distributed source

models

Distributed source models offer great expressivity for the source distribu-tion [28], with the additional advantage of a linear relationship betweensource and measurement space, through a lead field matrix which can beprecomputed. The high dimensionality of the unknown parameters makes itnecessary to impose constraints on the sources [8]. With Geoffray Adde, wehave studied the regularization of source estimation with Lp norms of thegradient, more precisely the L1 and L2 norms which have been introducedin the field of image processing and regularization [69]. The L2 norm of thegradient is equivalent to applying an isotropic diffusion to the image, with atendency to blur images which have sharp contours. In contrast, the L1 gra-dient norm regularization performs an anisotropic diffusion, with a strongerdiffusion along than across isovalue lines [65]. The extension from the reg-ularization of flat images to regularization along a curved surface requiresthe use of the Laplace-Beltrami operator, and the tangential gradient to thesurface. When the surface is represented by a mesh, it is convenient to ap-proximate differential operators on the surface by using extrinsic geometrycalculus [31].

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3.2. SOURCE ESTIMATION WITH DISTRIBUTED SOURCE MODELS31

Figure 3.2: Four images on the left: Upper left: A simulated patch of activity.Upper right: Reconstruction from noiseless simulated MEG measurements,with a minimum norm algorithm. Lower left: Reconstruction with the gra-dient L2 norm regularization. Lower right: Reconstruction with the gradientL1 norm regularization. Four images on the right: same as on the left, aftercorrupting the forward simulations with 10% additive Gaussian noise on thesensors. Figures reproduced from G. Adde’s PhD thesis [1].

The respective merits of the L1 and L2 regularizations have been com-pared [ACKK03, ACK05], leading to the following conclusions:

• The source gradient L2 norm regularization can be implemented witha fast, direct (non-iterative) algorithm. Reconstructed solutions havea low spatial frequency, with blurred edges.

• The source gradient L1 norm regularization must be implemented withan iterative algorithm, and the solution proposed in [ACK05] relieson Projection on Convex Sets to ensure convergence. The algorithmis more time-consuming, but the reconstructed solutions have sharperedges.

The quality of the source reconstruction was analyzed for simulated data(see Figure 3.2). Patches of homogeneous cortical activity were placed onthe cortex, and MEG and EEG measurements were simulated with the for-ward symmetric BEM. After adding Gaussian measurement noise, the inversesource localization procedures were applied, with three different regulariza-tion terms: the L2 norm of the source distribution, the source gradient L2

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32 CHAPTER 3. CORTICAL ACTIVITY ESTIMATION

norm, and the source gradient L1 norm. The source gradient L2 and L1 normsare more robust to the additive noise than the source L2 norm. As expected,the gradient L1 norm gives more focal reconstructed activity, yielding betterresult when the simulated source has a sharp spatial activation profile. Amajor difficulty with this type of comparison is the absence of ground truthconcerning the actual spatial profile of brain activity (see the Discussionsection 3.4).

3.3 Single trial MEG/EEG analysis

In order to improve the signal to noise ratio (SNR), it is customary to aver-age many EEG or MEG trials, obtained in similar conditions. The averagingprocedure assumes that the activity of interest is identical across trials. Thisassumption seldom holds, exposing the averaged data to possible misinterpre-tation. For oscillatory activity, randomness of the phase frequently causescancellation in the average. Oscillatory activity which is not phase-lockedacross trials has been called “induced activity” in contrast to “evoked activ-ity” which is not destroyed by the averaging. The SNR of induced activitycan be improved by considering the average power in the time-frequencyplane [72]. Evoked activity is also subject to significant variability in shapeand latency from one trial to the next, due to habituation effects, fluctuationsin the level of attention and arousal, or different response strategies. Suchinter-trial variations are problematic when averaging; for instance fluctua-tions in latency have the tendency to spread out the average time-courses.In some cases variability can also be a source of information.

Much effort in the detection of intertrial variations has been directed toslow-varying ERPs (in the range of 1-20Hz) [55, 66]. Far less attention hasbeen devoted to the estimation of single-trial oscillatory activity. We havedesigned a method to track fluctuations in brain electromagnetic activity, forany set of frequency bands [BPC06b, BPC06a]. We introduce an originalmethodology for defining a reference template, based on a sensor map andon a set of Gabor time-frequency atoms that is capable of modeling both lowfrequency event-related potentials and high frequency oscillations.

Indeed, to analyze MEG/EEG data, it is important to consider the fullrange of structure contained in the data, i.e. not only the time-courses, butalso the spatial topographies and the time-frequency content [57]. Usingspatial information allows us to extract more signal from the data than whena single sensor is considered, and to disentangle processes that are mixed atthe sensor level [68, 55].

The signal S(t, θ), as a function of time t and spatial position θ, is modeled

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3.3. SINGLE TRIAL MEG/EEG ANALYSIS 33

Figure 3.3: Results for the oddball data from [BCP07]. Left: histogramsof parameters fitted in fit1 (no constraint on dispersion). The red verticalbars show the mean and standard deviation of the parameters, which werenot influenced by outliers (robust estimation). Middle: raster plot originaldata (y-axis: trials corresponding to rare events), sorted by reaction time(dark line). Right: result of the fitting procedure (fit2, with constraint onthe parameters).

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34 CHAPTER 3. CORTICAL ACTIVITY ESTIMATION

as the sum of a model signal S and a noise term:

S(t, θ) = S(t, θ|par) + E(t, θ) . (3.2)

The model signal S is composed of K repetitions (or trials) of N parametrictemplates

S(t, θ|par) =N∑

n=1

K∑

k=1

βn,kTn(t− tn,k, θ − θn,k|par) . (3.3)

We further assume that a given template Tn, n ∈ 1...N, can be decom-posed as the product of a temporal pattern An and a spatial pattern (ortopography) Mn.

Tn(t, θ) = An(t)Mn(θ) .

In [BCP07] the topography is assumed stable across trials and a singletemplate is used, simplifying the model to

S(t, θ|par) =K∑

k=1

βkA(t− tk|par)M(θ) . (3.4)

In order to handle a wide range of frequencies, the temporal template ismodeled as the linear combination of P Gabor atoms,

A(t) =P∑

p=1

βk,pe− 1

2σ2k,p

(t−dk,p)2

cos(ωk,p(t− dk,p)). (3.5)

The trial-specific parameters (latencies dk,p, amplitudes βk,p, widths σk,p

and modulating frequencies ωk,p) are estimated by a nonlinear optimization.Thus the fluctuations of the actual signal are followed across trials withina sparse signal representation. The initialization of the parameters is per-formed by a specific adaptation of the Matching Pursuit method to our multi-trial setting [BCP07]. Lastly, information coming from the whole set of trialsis used in order to increase the robustness of the fit even for low SNRs. Themodel defined by (3.2), (3.4) and (3.5) is indeed fitted to the data by the min-imization of a cost function composed of two parts. The first part maximizesthe fit to the data.

C1(par) =∑

t

[S(t) − S(t|par)

]2

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3.4. DISCUSSION 35

where par represents the sought parameters; the second part avoids over-fitting the noise by minimizing the dispersion of the parameters around theirmeans:

C2(par) =∑

k,j,p

[parp

j,k − parpj

std(parpj )

]2

.

The means and standard deviations of the parameters (parpj and std(parp

j ))are estimated after a first step which solves the minimization of the first costfunction only. The first minimization step, with only C1(par) is called fit 1

and the second with both C1(par) and C2(par) is called fit 2. In [BCP07],the method was applied to real data, from an auditory oddball experiment.Figure 3.3 shows the results of parameter estimation by fit 1 and fit 2, andthe reconstructed time courses (viewed trial by trial as a raster plot image).

The topography M(θ) over the sensors (3.4) is estimated by performinga PCA in the spatial domain. If several templates co-exist (N > 1 in (3.3)),disentangling space and time is not possible with a simple PCA, and it may benecessary to go to source space (or at least use Cortical Mapping) to initializethe spatio-temporal separation of the components. Thus the incorporationof a forward problem should help the single-trial MEG/EEG analysis.

3.4 Discussion

Estimating cortical activity is a difficult task, which suffers from the absenceof ground truth.

Concerning regularization strategies, in the image processing domain,functional spaces have been proposed, to which natural images can be as-sumed to belong [78, 79]. In functional brain imaging however, there is noconsensus about the spatial regularity of the brain electric activity. Im-planted subcortical electrodes (sEEG) are of limited help in this respect,because such electrodes are sensitive to a volume around their position, andrecovering cortical activity from sEEG also requires the solving of an inverseproblem [26]. Detailed models of distributed activity of cortical columnswill hopefully contribute to fill this gap, in conjunction with other imagingmodalities such as Optical Imaging. In order to model the neural activityat a mesoscopic scale in a biologically plausible fashion, it has been pro-posed to introduce a spatial continuum of neural masses, with their spatio-temporal connectivity. Such a model is certainly too detailed to be useful forMEG/EEG source estimation, but it would be interesting to start from thismodel, and reduce it by techniques such as Mean Field Theory, in order toevaluate the scale which is relevant for MEG/EEG.

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36 CHAPTER 3. CORTICAL ACTIVITY ESTIMATION

A natural way to constrain the inverse source estimation problem is to usea priori information coming from other functional imaging modalities (fMRI,PET), while bearing in mind that these modalities are not sensitive to thesame physiological phenomena as MEG and EEG. Anatomical information,as provided by MRI, and diffusion MRI, can also be used to advantage. Inaddition to the cortical surface which can be extracted from MRI and usedto constrain the source distribuition, white matter connectivity informationcan be used to temporally constrain the delay between activities in differentregions [KFCP07].

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Conclusion

This habilitation memoir has presented several original contributions to thefield of neuroimaging: a new boundary element method which has been usedin several forward problems (EEG, MEG, EIT); new ways of tackling theconductivity estimation problem; a new method for Cortical Mapping; newregularization algorithms for the recovery of distributed sources; and a newapproach for single-event MEG / EEG analysis.

Hopefully these methods will be useful to the neuroimaging community, aswe strive to make them accessible through opensource, integrated platforms.

The work presented here naturally represents only a very tiny step onthe long road leading to a better understanding of the brain function. It isour hope that MEG and EEG will be instrumental in unveiling some of themysteries of the brain, both in its normal and in its pathological function.This will require continuing multidisciplinary efforts, as well as multimodalintegration of information coming from different measurement devices (MRI,fMRI, DMRI, Optical Imaging, MEG and EEG).

Personally, while continuing to study computational models for MEG/EEG,I would like to focus more attention on the extraction of information from themeasurements. In this domain, there are two directions of particular inter-est to me: the first one, in Cognitive Neuroscience, concerns the study withMEG and EEG of the feedbacks occuring in visual processing. The secondone is concerned with the design of Brain Computer Interfaces which rely onscalp measurements via electroencephalography, with the aim of analyzingthe data in the source space rather than in the sensor space. This will bring usto consider EEG and MEG from the eye of the experimenter, with attachedresponsibilities: when conducting research on modeling, understanding andperhaps anticipating human perception, intention and behavior, the questionof ethics must not be disregarded.

37

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38 CHAPTER 3. CORTICAL ACTIVITY ESTIMATION

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Publications

[ACF+03] G. Adde, M. Clerc, O. Faugeras, R. Keriven, J. Kybic, and T. Pa-padopoulo. Symmetric BEM formulation for the M/EEG forwardproblem. In C. Taylor and J. A. Noble, editors, Information Pro-

cessing in Medical Imaging, volume 2732 of LNCS, pages 524–535. Springer, July 2003.

[ACK05] G. Adde, M. Clerc, and R. Keriven. Imaging methods forMEG/EEG inverse problem. In Proc. Joint Meeting of 5th In-

ternational Conference on Bioelectromagnetism and 5th Inter-

national Symposium on Noninvasive Functional Source Imaging,2005.

[ACKK03] G. Adde, M. Clerc, R. Keriven, and J. Kybic. Anatomy-basedregularization for the inverse meeg problem. In 4th International

Symposium on Noninvasive Functional Source Imaging within

the human brain and heart, Chieti, September 2003.

[BCP07] C. B’enar, M. Clerc, and T. Papadopoulo. Adaptive time-frequency models for single-trial M/EEG analysis. In Karsse-meijer and Lelieveldt, editors, Information Processing in Medi-

cal Imaging, volume 4584 of Lecture Notes in Computer Science,pages 458–469. Springer, 2007.

[BPC06a] C. Benar, T. Papadopoulo, and M. Clerc. Topography-time-frequency models for single-event m/eeg analysis. In Proceedings

of Biomag, 2006.

[BPC06b] C.G. Benar, T. Papadopoulo, and M. Clerc. Time-frequency-topography adaptive templates for single-trial meg/eeg analysis.In Proceedings of HBM, 2006.

[CAK+05] Maureen Clerc, Geoffray Adde, Jan Kybic, Theo Papadopoulo,and Jean-Michel Badier. In vivo conductivity estimation with

39

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40 PUBLICATIONS

symmetric boundary elements. In Proc. Joint Meeting of 5th

International Conference on Bioelectromagnetism and 5th Inter-

national Symposium on Noninvasive Functional Source Imaging,Minneapolis, May 2005.

[CAL+05] M. Clerc, B. Atfeh, J. Leblond, L. Baratchart, J.-P. Marmorat,T. Papadopoulo, and J. Partington. The cauchy problem appliedto cortical imaging: comparison of a boundary element and abounded extremal problem. In Christoph Michel D. Brandeis,T. Koenig, editor, Brain Topography, volume 18. Springer Scienceand Business Media B.V., October 2005.

[CBA+05] M. Clerc, J.-M. Badier, G. Adde, J. Kybic, and T. Papadopoulo.Boundary element formulation for electrical impedance tomog-raphy. In ESAIM: Proceedings, volume 14, pages 63–71. EDPSciences, September 2005.

[CDF+02] M. Clerc, A. Dervieux, O. Faugeras, R. Keriven, J. Kybic, andT. Papadopoulo. Comparison of BEM and FEM methods forthe E/MEG problem. In Proceedings of BIOMAG 2002, August2002.

[CK07] M. Clerc and J. Kybic. Cortical mapping by Laplace-Cauchytransmission using a boundary element method. Technical re-port, INRIA, 2007. accepted for publication in Inverse Problems.

[CKF+02] Maureen Clerc, Renaud Keriven, Olivier Faugeras, Jan Kybic,and Theo Papadopoulo. The fast multipole method for the directE/MEG problem. In Proceedings of ISBI, Washington, D.C.,July 2002. IEEE, NIH.

[Cle99] M. Clerc. Etude des processus localement dilates, et application

au gradient de texture. PhD thesis, Ecole Polytechnique, 1999.

[Cle02] Maureen Clerc. Weak homogeneity for Shape from Texture. Inproceedings of Texture 2002, satellite conference of ECCV 2002,2002.

[CM02] M. Clerc and S. Mallat. The texture gradient equation for recov-ering shape from texture. IEEE Trans. on Pattern Analysis and

Machine Intelligence, 24(4):536–549, April 2002.

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PUBLICATIONS 41

[CM03] Maureen Clerc and Stephane Mallat. Estimating deformationsof stationary processes. Annals of Statistics, 31(6):1772–1821,December 2003.

[CTM+96] M. Clerc, P. Le Tallec, M. Mallet, M. Ravachol, and B. Stoufflet.Optimal control for the parabolized Navier-Stokes system. InComputational Fluid Dynamics ’96, pages 139–145. John Wileyand Sons, 1996.

[KCA+05] J. Kybic, M. Clerc, T. Abboud, O. Faugeras, R. Keriven, andT. Papadopoulo. A common formalism for the integral formula-tions of the forward EEG problem. IEEE Transactions on Med-

ical Imaging, 24(1):12–28, January 2005.

[KCF+05] Jan Kybic, Maureen Clerc, Olivier Faugeras, Renaud Keriven,and Theo Papadopoulo. Fast multipole acceleration of theMEG/EEG boundary element method. Physics in Medicine and

Biology, 50:4695–4710, October 2005.

[KCF+06] J. Kybic, M. Clerc, O. Faugeras, R. Keriven, and T. Pa-padopoulo. Generalized head models for MEG/EEG: bound-ary element method beyond nested volumes. Phys. Med. Biol.,51:1333–1346, 2006.

[KFCP07] J. Kybic, O. Faugeras, M. Clerc, and T. Papadopoulo. Neu-ral mass model parameter identification for MEG/EEG. InArmando Manduca and Xiaoping P. Hu, editors, Proceedings

of SPIE Medical Imaging: Physiology, Function, and Structure

from Medical Images, 2007.

[MLMB06] M.Clerc, J. Leblond, J.-P. Marmorat, and L. Baratchart. Eegsource localization by best approximation of functions. In Pro-

ceeding of HBM, 2006.

[PVC06] T. Papadopoulo, S. Vallaghe, and M. Clerc. Implicit meshes forMEG/EEG forward problem with 3D finite element method. InProceedings of the Biomag conference, August 2006.

[VBC06] S. Vallaghe, J.-M. Badier, and M. Clerc. Simultaneous estima-tion of single dipolar source and head tissue conductivities. InProceedings of HBM, 2006.

[VCB07] S. Vallaghe, M. Clerc, and J.M. Badier. In vivo conductivityestimation using somatosensory evoked potentials and cortical

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42 PUBLICATIONS

constraint on the source. In Proceedings of ISBI 2007, pages1036–1039, apr 2007.

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