PROBLEME INVERSE MEG/EEG1

Sylvain Baillet

Groupe Signal & Image

Laboratoire de Neurosciences Cognitives & Imagerie C�r�brale

CNRS UPR 640 Ð LENA

H�pital deÊla Salp�tri�re, Paris.

Sylvain.Baillet@chups.jussieu.fr

HYPERLINK

1 Ce document contient de larges extraits issus de :Baillet, S., Mosher, J.C. & Leahy, R.M., ÇÊMapping Humain Brain Functions using Intrinsic Electromagnetic SignalsÊÈ,article invit� dans IEEE Signal Processing Magazine, en cours de r�vision (mai 2001).

1. The Physics of MEG and EEG : Source and Head Models2

MEG and EEG are electrophysiological measurements made non-invasively outside the body. Given a set of MEG

or EEG signals from an array of external sensors, the inverse problem is the estimation of current sources within the

brain that produced these signals. Before we can make such an estimate (discussed in the next section), we must first

understand the forward problem, which models the scalp potentials and external fields given an internal current

distribution. In this section we review basic source and head modeling approaches, as well as the definitions of

ÒleadÓ and ÒforwardÓ fields.

1.1 Quasi-static Approximation of Maxwell Equations

The relevant frequency spectrum for electrophysiological signals in MEG and EEG is typically below 1 kHz, and

most studies deal with frequencies between 0.1 and 100 Hz. Consequently, the physics of MEG and EEG can be

described by the quasi-static approximation of MaxwellÕs equations. The quasi-static current flow ( )'rJ at location

r_ is therefore divergence free and can be related rather simply to the magnetic field at location r through the well-

known Biot-Savart law,

( ) ''

')'(

4 30 dv∫ −

−×=rr

rrrJrB

πµ

(1)

We may further partition the total current flow into a primary current flow ( )'rJ P and a volume current flow ( )'rJV

.

We define the volume current flow ( )'rJV as ÒpassiveÓ current that results from the effect of the electric field on

extracellular charge carriers ( ) ( ) ( )''' rErrJ σ=V. From the quasi-static assumption, the electric field is simply the

negative gradient of a potential, V−∇=E , and ( )'rσ is the conductivity, which we will here assume to be

isotropic. The primary current flow ( )'rJ P is the ÒdriverÓ in the current flow and can be defined at the

macroscopic level as simply that portion that is not volume current,

( ) ( ) ( ) ( ) ( ) ( )'''''' rrrJrJrJrJ VPVP ∇−=+= σ (2)

The total current density may not have any volume currents (for instance, a closed loop of current), but every current

density must have a primary current that generates the total current distribution.

In MEG and EEG, we are most interested in the locations of the primary currents, since they represent the

regions of active cell assemblies. The volume currents are necessary to Òclose the loopÓ to create divergence-free

total current densities, but volume currents are otherwise uninteresting. We therefore adjust our forward models to

emphasize the primary current distribution, but of course every model must still account for the effects of the

volume currents.

If we assume that the head consists of a set of contiguous regions each of constant isotropic conductivity,

we can alter the Biot-Savart law above both to emphasize the primary current and to emphasize the boundary

regions of the head. Using standard vector identities, the Biot-Savart law becomes (cf. [1SEQMERGEFORMAT]),

( ) ( ) ( ) ( )∫∑ ×−−−+=

ijSij

ijji V '

'30

0'

''

4dS

rr

rrrrBrB σσ

πµ

, (3)

where the summation is over all boundaries, and ( )rB0 is the magnetic field due to only the primary current. Typical

macroscopic boundaries are the inner skull surface, outer skull surface, and scalp.

This general equation states that the magnetic field can be calculated if we know the primary current

distribution and the potential ( )'rV on all surfaces. We can create a similar equation for the potential itself, although

the derivation is somewhat tedious ([2], cf. [SEQMERGEFORMAT1]), yielding

2 Voir �galement dans ce recueil ÇÊBases Physiologiques & Physiques de la MEGÊÈ par L. Garnero.

( ) ( ) ( ) ( ) ( )∑ ∫ ⋅−−−+=+

ij S

ijjiji

ij

VVV ''300

'

''

2

12 dS

rr

rrrrr σσ

πσσσ , (4)

for the potential on surface S ij where ( )r0V is the potential at r due to the primary current distribution.

These two equations therefore represent the integral solutions to the forward problem. We specify a

primary current distribution ( )'rJ P , for which we can then calculate a primary potential and a primary magnetic

field,

( ) ''

')'(

4

13

00 r

rr

rrrJr dV P

−−⋅= ∫πσ

, ( ) ''

')'(

4 30

0 rrr

rrrJrB dP

−−×= ∫π

µ(5)

The primary potential ( )r0V is then used to solve (4) for the potential on all surfaces, and therefore solves the

forward problem for EEG, which measures potential differences on the scalp. These surface potentials ( )'rV and

the primary magnetic field ( )rB0 are then used to solve (3) for the external magnetic fields. Unfortunately, the

solution of (4), a Fredholm integral of the second kind, has analytic solutions only for special shapes and must

otherwise be solved numerically. We return to specific solutions of the forward problem below but first we discuss

the types of Òsource modelsÓ used to describe the primary current distributions.

1.2 Source models

1.2.1 Biological sources

The most likely source of neural primary currents are the pyramidal cells in the thin cortical layer. As shown in

Fig.SEQMERGEFORMAT1, these cells are characterized by long apical dendrites. Extraordinarily small currents

flow in these dendritic branches as a function of excitatory postsynaptic potentials (EPSPs). Calculations such as

those shown in [1] suggest each synapse along a dendrite may contribute as little as a 20 fA-m current source,

probably too small to measure in E/MEG. Empirical observations instead suggest we are seeing sources on the

order of 10 nA-m, in other words the cumulative summation of millions of synaptic junctions in a relatively small

region. Thus macrocellular models ignore the microcellular details and focus instead on the cumulative effects of these

cells over a region of cortex. Nominal calculations of neuronal density and cortical thickness suggest that the cortex

has a macrocellular current density on the order of 100 nA/mm2 [1]. If we assume the cortex is about 4 mm thick,

then a small patch 5 mm x 5 mm would yield a net current of 10 nA-m, consistent with empirical observations and

invasive studies.

Figure 1: Basics of MEG/EEG generators

1.2.2 Current dipoles

Let us assume a small patch of activated cortex is centered at location rq and that our observation point r is some

distance from this patch. If we insert this current source into the primary magnetic field model (5), the field can be

well approximated as

( ) '4 3

00 r

rr

rrqrB d

q

q

−

−×≅

πµ

(6)

where ∫≡ ')'( rrJq dP is defined as the equivalent current dipole. The current dipole is a straightforward extension of

the better-known model of the paired-charges dipole in electrostatics. Though visually close to some simplistic

geometrical representation of pyramidal cell assemblies, it is essential to bear in mind that we are dealing with

different scales here. Neural macrocolumns typically extend at the millimeter scale, whereas the focal current dipole

is a concept, or a metaphor, equivalent to the unidirectional activation of up to several square centimeters of gray

matter.

The current dipole model is the workhorse of E/MEG processing, since any arbitrary primary current

density can always be broken down into small patches, each patch represented by an equivalent current dipole. A

problem arises, however, when too many of these patches are required to represent a single large patch. These larger

patches may be more simply represented by a multipolar model, rather than many individual dipoles.

1.2.3 Higher-order parametric source models: the multipolar approach

We now consider a slightly larger patch, such that (6) is no longer a good approximation. Rather than break the

patch into smaller dipolar patches, we may instead perform a Taylor series expansion of the denominator in (5).

Rearrangement of the terms yields analogies to the current dipole in the form of the magnetic dipole and the magnetic

quadrupole,

∫ ×≡ ')'(' rrJrm dP , ∫ ×≡ ''))'('( rrrJrM dP (7)

Other definitions are possible, such as a current quadrupole, for both the primary magnetic and primary potential

models, which we have omitted for simplicity in this discussion. A large patch, therefore, may be represented by its

source location and its moment parameters. By contrast, if we restricted ourselves to only current dipole models,

then this same large patch would require several location parameters and several moment parameters. As we will see

below, location parameters complicate the inverse procedures so that in many cases multipolar model may be

attractive.

The first definitions for multipolar sources were established in magnetocardiography [3] and were not

extended to MEG until recently. The truncated Taylor series expansion of the magnetic fields in MEG naturally

gives rise to multipolar components (dipole, quadrupole, octupole, É). For distributed sources confined to a patch

of cortex some distance from the sensor, the contributions to the magnetic field from octupolar and higher order

terms drop off rapidly with distance, so that restricting sources to dipolar and dipole + quadrupole is probably

sufficient to represent most plausible cortical sources. We may discriminate between the following types of sources:

(i) point sources that are exactly represented as point current dipoles, (ii) highly focal sources that can be represented by

a magnetic dipole moment model, and (iii) locally distributed sources that can be represented by a first-order multipole

(dipole + quadrupole) model [4]. An alternative approach to multipolar models of brain sources can be found in [5].

1.3 Head models

1.3.1 Spherical volume conductor models

Given a specified primary current density, the general solution of the MEG forward model first requires solution of

the EEG forward model, which in turn requires the solution of a non-trivial integral equation. If the head can be

modeled as a single homogeneous sphere, then analytic solutions of (4) are well-known. If the head is a set of

concentric homogeneous shells (e.g. brain, skull, and scalp), then the solution involves an infinite series expansion,

but simpler approximations have recently been developed. In [6], we review the development and approximations of

these equations for both the EEG and MEG cases (see also [7], [8]).

Consider the case of a current dipole in a multishell spherical head model. If we examine the radial

component of the magnetic field, it is relatively straightforward to show that the contributions of the surface

potentials in (3) vanish, and we are left with the remarkably simple formula,

( )3

00

4)()(

r'r

qr

rrB

rrB

rr

−⋅×=⋅=⋅≡ qr

rrrB

πµ

(8)

Thus these radial MEG measurements are a function of the primary current only, such that the volume currents do

not contribute to this measurement. The conductivity profiles of the shells have become unimportant in the

modeling due to their axial symmetry. However, non-radial measurements must include the effects of volume

currents. Surprisingly, the effects of the volume currents are independent of conductivity provided the sphere

conductivity profile is axially symmetric [9], [10], [11]. Because of the simplicity of the radial model (8) and the

absence of the conductivity profile in the full field model, a mistaken assumption is sometimes made that MEG is

insensitive to volume currents (Fig. 2).

A further interesting note is that if q is radially oriented in the above equation, then )(rrB is zero, and

therefore so is the total magnetic field outside the sphere. In other words, a radially-oriented dipole inside a spherical

conductor creates volume currents that identically cancel the primary field everywhere outside the sphere. As shown

in [11], however, this cancellation is extremely sensitive to the sphere assumption, and even a slight deviation from

spherical allows a radial dipole to generate an external field.

1.3.2 Realistic volume conductor models

High-resolution MR imaging of the head is now readily performed and routinely included as part of most MEG data

acquisition protocols. These images are primarily used to relate source locations found using MEG to the underlying

neuroanatomical structures visualized with MR. These images can also be used to extract morphological information

about individual head shape (see Fig 3) that can be used to improve the accuracy of the forward model. As

mentioned above, however, the forward problem must in general be solved numerically for arbitrary head shapes.

Although the general field of computational electromagnetics is very broad and often complex, EEG/MEG falls into the

relatively simple category of Òpotential problems,Ó since the quasistatic Maxwell equations are mathematically similar

to problems involving the Newtonian potential.

Although often presented as quite different, the finite element method (FEM) and the boundary element

method (BEM) for numerically solving the forward problem are actually very similar. Recall from above that the

0current is divergence free, and that we have partitioned the current into primary and volume currents. The

differential equation relating primary to volume current is therefore

Figure 2: Schematic representation of a 4-shellspherical head model overlaid on a saggittal image MRimage through the head. The four concentric shellsreflect approximately the shape and conductivity profile

( 51 toσ ) of brain, cerebrospinal fluid, skull and scalp.

Also illustrated in the figure is a current dipole (yellowarrow) with associated volume currents (green).

))()(()( rrrJ VP ∇⋅∇=⋅∇ σ . (9)

We may partition the head volume into millions of tiny tetrahedra, assign the appropriate primary currents at the

desired locations, assign the appropriate conductivity to each tiny volume, then numerically solve a consistent series

of equations for the potentials. The boundary conditions at each tetrahedron are that the current normal through the

surface, and the potentials, be conserved. Because each tetrahedron interacts only directly with its immediate

neighbors, the system is quite sparse, and optimized sparse algorithms can be used. Once the surface potentials are

found, solution for the magnetic field is relatively straightforward [6, 12].

Typically, we assume the head consists of a small set of relatively large isotropic homogeneous regions:

brain, skull, scalp. Rather than create tiny tetrahedra, each volume region can instead be tessellated at its boundaries

by a dense set of triangles, and the same boundary conditions applied as in FEM. Although the number of triangles

may be substantially less than the number of tetrahedra, the matrix system of equations is now denser, yielding a

BEM approach of about the same computational complexity as its FEM equivalent. Rather than either of the above

approaches, the most common form of BEM used in EEG and MEG is actually a third variation on this theme,

which involves the solution of the Fredholm integral of the second kind (3). This boundary integral equation, derived

in 1967 by Gesolowitz [13], allows an ÒindirectÓ formulation of the BEM to be generated [14]. In more complicated

electromagnetic situations, such an integral may not be available or easily derived, and therefore the use of indirect

BEM approaches seems somewhat peculiar to EEG and MEG modeling.

Regardless of the method used, the other major design criteria in FEM and BEM are the element

approximation and the error method used to control the solution of the potentials. The potentials on each

tetrahedron or triangle are usually assumed to be either constant or linear across their faces. The error in these

approximations may be controlled either at discrete points or as integrals over the elements. In [6] we review the

literature on these techniques and investigate their performance.

BEM and FEM approaches for realistic head modeling are becoming increasingly practical because of faster

processors and the use of computationally efficient methods. Most of the BEM and FEM techniques can have

extensive Òone-timeÓ computations carried out using only the anatomy of the subject and no consideration of the

Figure 3: examples of surfaces extracted from high-resolution MR images. The following surfaces were all extracted using an automatedmethod described in [58]: (a) high-resolution brain surface, (b) smoothed brain surface, (c) skull surface, (d) scalp surface. The surfacesin b,c, and d are used as input to a boundary element code for computing forward MEG and EEG fields. The remaining figures arehigh-resolution cortical surfaces extracted from the same MR image and corrected to have the topology of a sphere. These can be used forcortically-constrained MEG or EEG imaging: (e) high resolution cortical surface, (f) smoothed representation obtained using relaxationmethods similar to that described by [39] to allow improved visualization of deep sulcal features; (g) high resolution and (h) smoothedrepresentations of the cortex with approximate measures of curvature overlaid.

actual primary currents. Once these computations are completed, the results are captured into Òtransfer matricesÓ (cf.

[6, 15]) that allow for rapid run-time computation of the EEG or MEG solution given an arbitrary primary

distribution. Run-time costs can be reduced further by precomputing the forward solution at a grid of possible

source locations and then simply interpolating these values at run-time to obtain solutions for arbitrary source

locations [15].

1.3.3 Open Issues for Accurate Head Modeling

Although realistic head modeling represents an important advance over the spherical model, both fundamental and

practical issues still need to be tackled to further improve BEM and FEM accuracy. Our experience with realistic

head phantom studies using a human skull was that a simple three layer (brain, skull, scalp) model was insufficient to

achieve high accuracy [16]. These models ignore features such as fat layers in the scalp, bone marrow in the skull, and

the cerebral spinal fluid surrounding the brain. The air-filled sinuses, which are often indistinguishable from bone,

also can confound these models, since both have very low intensity in standard T1-weighted anatomical MR images.

More sophisticated segmentation methods, in conjunction with more complex BEM or FEM models that better

describe the different parts of the head, would lead to improved forward model accuracy.

Knowledge of the conductivity profile of head tissues is also of critical importance to realistic head

modeling. Most of the head models used by the bioelectromagnetism community consider typical values for the

conductivity of the brain, skull and skin. Skull is typically assumed to be 40 to 90 times more resistive than brain and

scalp, which are assumed to have similar conductive properties. These values are measured in vitro from postmortem

tissue, where conductivity can be significantly altered compared to in vivo values [17]. Consequently, recent research

efforts have focused on in vivo measures.

Two approaches are currently being considered. The first is Electrical Impedance Tomography (EIT) which

proceeds by injecting a small current (1-10 microA) between pairs of EEG electrodes and measuring the resulting

potentials at all electrodes. Given a model for the head geometry, EIT solves an inverse problem by minimizing the

error between the measured potentials on the rest of the EEG leads and the model-based computed potentials, in

terms of parameters of the conductivity profile. Recent simulation results with three or four-shell spherical head

models have demonstrated the feasibility of this approach [18, 19].

SEQARABICThe second approach is more tentative and uses two properties of the magnetic resonance signal. The

first technique uses the shielding effects of induced eddy currents on spin precession and could in principle help

determine the conductivity profile at any frequency [20]. The first experiment with a phantom was realized with a 7T

MR system and there is no indication whether relevant measurements could be obtained on a standard 1.5 T magnet.

The second technique uses diffusion-tensor imaging with MRI (DT-MRI) that probes the microscopic diffusion

properties of water molecules within the tissues of the brain [21]. The diffusion values can then be tentatively related

to the conductivity of these tissues [22]. Both of these MR-based techniques are still under investigation, but given

the poor signal-to-noise ratio (SNR) of the MR in bone regions, which is of critical importance for the forward EEG

problem, the potential for fully 3D impedance tomography with MR remains speculative.

Studies have shown that anisotropic conductivities play a major role on the calculation of electric potentials

[23] and consequently on EEG source localization [24, 25], but have a lesser effect on MEG [26]. Because some

head tissues have strong anisotropic conduction properties (e.g., the skull and white matter), FEM would be a natural

approach for approximating each small region of varying conductivity; however, use of these more complex models

must be preceded by a technique for reliably imaging the anistropic conductivity of the subjectÕs head.

1.4 Lead Fields and Forward Fields

With the introduction of the source and head models for solution of the forward problem, we can now provide a

few key definitions and linear algebra models that will clarify the different approaches taken in the inverse methods

in the next section. See [27] for a more detailed review of the design of these matrices.

1.4.1 Discrete Fields

Regardless of the specifics of the head model, by electromagnetic superposition the forward model is linear in the

moment, and we may write the relationship between the moment for a dipole at rq and the measurement at sensor

location r as the inner product of a lead field vector g(r,rq) and the dipole moment q. We assume dipole moments of

the 3x1 Cartesian form [ ]Tzyx qqq ,,=q , where the individual components [ ]T

xq 0,0, , [ ]Tyq 0,,0 and

[ ]Tzq,0,0 represent Òelemental dipoles.Ó By analogy, the extension of this model for the three magnetic dipole

moments or nine magnetic quadrupole moments is straightforward, but for simplicity we will restrict the discussion

to current dipoles.

The canonical elemental dipoles (i.e., with qx=qy=qz=1) form a basis for any dipole located at rq with any

orientation and amplitude. This orthogonal dipole triplet is sometimes referred to as a regional source or rotating dipole

[27, 28] which serves as a basic focal model for cortical activation within cortical folds in the neighborhood of rq.

The three components of the lead field vector g(r,rq) are formed as the solution to either the magnetic or electric

forward problem for each of these elemental dipoles. Explicit forms of g(r,rq) are summarized in [6].

For many dipoles (each representing a patch of cortex), we simply sum the individual contributions. An

EEG or MEG measurement may be therefore be represented as ( ) ∑ ⋅= iqim qrrgr ),( . The measurements by an

EEG or MEG array are made at N sensors, and we can concatenate these measurements into a matrix linear algebra

form,

( )

( )

( ) ( )

( ) ( )qG

q

q

r,rgr,rg

r,rgr,rg

r

r

m

NN

11

=

=

=

pqpT

qT

qpT

qT

Nm

m

M

L

MOM

L

M1

1

11

, (10)

where G is the Ògain matrixÓ relating the set of p dipoles to the set of discrete sensor locations, and m is a generic set

of N MEG or EEG measurements. Each column of G relates an elemental dipole to the array of sensor

measurements and may be called the forward field, gain vector, or scalp topography, of the current dipole source. This

forward field is effectively sampled by the discrete locations of the sensors. Alternatively, we may look along a row of

G and observe that we are calculating the lead field vector for the same sensor location and many different dipole

locations. For reasons of reciprocity, these discrete dipole locations are effectively samples of the lead field that would be

generated if the sensors were used to generate currents in the brain.

1.4.2 Spatio-temporal Models

The above model is easily extended to multiple time samples. The most common model is to concatenate each

successive vector into a spatio-temporal data matrix. The observed set of measurements over an N-sensor array for p

sources can be expressed as a linear forward spatio-temporal model of the form GQM = , where the observed

forward field M (N-sensors x T-time samples) can be expressed in terms of the gain matrix G (N-sensors x

3pÐelemental dipoles) and a set of dipole moments Q (3p x T).

In this model, each row of the matrix Q represents the time series for an elemental dipole (e.g. x, y, or z-

directed). The three elemental dipoles at a fixed location represent a dipole of arbitrary orientation, but if the three

time series associated with the elemental dipoles are allowed to have arbitrary time series, then the dipole appears to

ÒrotateÓ in its position as a function of time. A common practice, therefore, in spatio-temporal modeling is to

constrain each dipole to a ÒfixedÓ orientation by factoring out a unit orientation vector u, such that time series for

each dipole may be represented as

[ ] T

z

y

x

zz

yy

xx

Tss

u

u

u

Tqq

Tqq

Tqq

us=

=

)()1(

)()1(

)()1(

)()1(

L

L

L

L

(11)

where u is constrained to have unity norm. For p sources and T time samples, all with fixed orientation, the spatio-

temporal model can therefore be modified as

( )

( )

( )

( )

( ) ( )

( ) ( )T

iiTp

T

pT

qpNT

qN

pT

qpT

q

NN Tm

Tm

m

m

SurA

s

s

ur,rgur,rg

ur,rgur,rg

r

r

r

r

M11

}),({

,

,

1,

1, 1

11

1111

=

=

= M

L

MOM

L

L

MO

L

M (12)

The source parameters that define our forward model are the set of dipole locations and orientations },{ iqi ur for p

sources, and the corresponding time series for each dipole, which are the columns of the time series matrix S . Each

column of A now corresponds to a single dipole at location qir with orientation iu . Embedded in the lead field

vector ( )qji r,rg are all of the important head model parameters that convert these dipolar sources into

measurements.

2. Imaging Electrical Activity in the Brain: the Inverse Problem

We have seen in the previous section that the sources of the primary currents giving rise to EEG and MEG signals

are cortical cell assemblies that are typically represented by equivalent current dipoles. Parametric and imaging methods

are two general approaches to the estimation of these sources. The parametric methods assume that the sources can

be represented by a few equivalent current dipoles. They solve a nonlinear inverse problem to find the location,

orientation and dynamic behavior of these sources. By extension, parametric multipole models can equivalently be

used for cases where the cell assemblies extend over larger areas of cortex that can be modeled with a single dipole.

The imaging methods allow a current dipole in each of many tens of thousands of tessellation elements on a cortical

surface, where the orientation is typically constrained to be the local surface normal. The inverse problem in this case

is linear, since the only unknowns are the amplitudes of the dipoles in each tessellation element. In this case the

problem is hugely underdetermined and regularization methods are required to restrict the range of allowable

solutions. In this section we will describe parametric and imaging approaches, contrasting the underlying

assumptions and the limitations inherent in each.

2.1 Least-squares Source Estimation

2.1.1 Principles

In the presence of measurement errors, the forward model may be represented as NSurAM += Tiqi }),({ , where

N is a spatio-temporal noise matrix. Our goal is to determine the set },{ iqi ur and the time series S that best describe

our data, given the measurements. The earliest and most straightforward strategy is to fix the number of candidate

sources p and use a nonlinear estimation algorithm to minimize the squared error between the data and the fields

computed from the estimated sources using the chosen forward model. Each dipole comprises three nonlinear

location parameters denoted qir , three quasi-linear parameters iu (constrained to have unity norm), and T linear

amplitude values. The terms ÒnonlinearÓ, Òquasi-linear,Ó and ÒlinearÓ will become apparent below.

For p dipole candidates, we define the measure of fit in the least-square sense as the square of the Frobenius

norm,

( ) 2}),({},{

F

TiqiiqiLSJ SurAMS,ur −= (13)

A brute force approach is to let a nonlinear search program attempt to minimize LSJ for all of the parameters;

however, a simple optimal modification greatly reduces the computational burden. For any selection of },{ iqi ur , the

matrix S that will minimize LSJ is

MAS #= , (14)

where #A is the pseudoinverse of }),({ iqi urAA = . If A is of full column rank, then the pseudoinverse may be

explicitly written as TT AAAA 1# )( −= [29]. We can then solve (13) in },{ iqi ur by minimizing the adjusted cost

function:

( ) ( ) 22#2# )(},{FFF

iqiLSJ MPMAAIMAAMur A⊥=−=−= , (15)

where ⊥AP is the orthogonal projection matrix on the left null space of A. Thus the LS problem can be optimally solved in

the limited set of nonlinear parameters },{ iqi ur with an iterative minimization procedure such as a Nelder-Meade

downhill simplex method. The linear parameters in S are then optimally estimated from (14). In [27], we discuss

additional simplifications of (15) using singular value decompositions (SVDs).

This least-squares model can be applied to a block of time samples, or to each time sample

individually. When applied sequentially to a set of individual time slices, the result is called a Òmoving

dipoleÓ model, since the location is not constrained. Alternatively, by using the entire block of data in

the least-squares fit, the dipole locations are fixed over the entire interval. This fixed dipole model

approach has proven to be quite successful in both EEG and MEG and still remains the most widely

used approach to processing experimental and clinical data.

2.1.2 Limitations

In implementing the least-squares approach, the very first issue the researcher encounters is how many sources to

use. This serious issue is still resolved in practice through the experience of expert data analysts. Often several model

orders are tested and the result that seems most physiological plausible is selected. Caution is obviously required

since a sufficiently large number of sources can be made to fit any data set, regardless of its quality. Furthermore, as

the number of sources increases, the nonconvexity of the cost function results in increased chance of trapping in

undesirable local minima. This latter problem can be dealt with using stochastic search or multistart optimization

algorithms.

2.2 Scanning methods

Alternatives to least-squares avoid the non-convexity issue by scanning a region of interest that can range from a

single location to the whole brain volume for possible sources. An estimator of the contribution of each putative

source location to the data can be derived either via spatial filtering techniques or signal classification indices. An

attractive feature of these methods is that they do not require any prior on the number of underlying sources.

2.2.1 Spatial Filtering and Beamforming Approaches

A beamformer operates by linearly weighting a measurement vector m(t) to achieve an output response

)()( tty T mw= . For E/MEG applications we design the weighting vector w to respond to signals originating at a

particular location qr in the brain, i.e. )( qrww = . Ideally 0=y when qr is not a true source location. In scanning

mode, the beamformer operates by scanning through a list of locations, with a different weight for each location, and

monitoring the output variable y for indications of a source at each location.

The principal differences between MEG/EEG and other beamforming applications are the lack of a phase

delay and the spatial diversity of the signal across the array. In most SONAR and RADAR work, the signal arrives as

a narrow frequency plane wave, and the algorithms exploit the phasing difference between sensors. Because the

frequencies are so low in MEG/EEG, the signal arrives virtually instantaneously at all sensors; however, these neural

signals are detected in their near-field and the amplitude diversity across the array can be exploited to provide

localizing power to the beamformer. Excellent reviews of standard and high-resolution beamforming methods can

be found in [30] and [31].

A single dipole scan is the simplest form of E/MEG beamformer. For each dipole location and orientation

of interest we form the weight vector 2

),(),( iqiq urauraw = . This produces unity output for a unit strength

source at location qr , but has uncontrolled Òside lobesÓ and has poor selectivity in identifying dipoles.

A second category of beamformer uses a statistically optimum approach to select the weight vector. The

general idea is to attempt not only to identify a source at a particular location, but also create weights that block

signals from other source locations. Adaptive methods can be used to track time dependent changes in the signals,

but we will restrict discussion here to fixed weight methods. As we first presented in [32], linearly constrained minimum

variance beamforming (LCMV) [30, 33] can be used in E/MEG processing where the response of the beamformer is

designed to minimize the output power subject to the constraint that signals from the location of interest are passed

with a specified (typically unity) gain:

wRww

mTmin , subject to fwC =T

(16)

where })()({ Tm ttE mmR ≡ , and C is a matrix of constraints, with the desired gain responses in f . The solution

for the optimal w is [30]

fCRCCRw 111 ][ −−−= mT

m (17)

In [32], we presented studies where C and f constrain the gain of three elemental dipoles at a single location to

unity. The beamformer minimizes its output power by adjusting its weights to null interfering sources, but the

constraints allow sources from the desired location to pass through this spatial filter. More elaborate constraints may

be designed using eigenvectors spanning a desired region of sources, but at the expense of reduced degrees of

freedom in fitting the weights and hence nulling of interferers.

Although the presence of uncorrelated noise will generally ensure that mR is invertible, an additional

diagonal regularizing parameter îµ can be added to the matrix, effectively stabilizing the inverse. If we constrain the

dipole orientation to be normal to the cortex to form our fixed dipole unity-gain constraint 1),( == wurawC iqiTT ,

then we have the analogous weighting vector for Synthetic Aperture Magnetoencephalography (SAM) [34]

[ ][ ] aîRa

aîRw

1

1

]

]−

−

+

+=µ

µ

mT

m . (18)

SAM does not plot the output power of the beamformer directly, choosing instead to scale the power by an

estimated noise-only power, yielding a Statistical Parametric Map (SPM) of neural activity; see [34] for further details.

2.2.2 Limitations of Beamforming Methods

In theory, LCMV has the appeal of being a Òvirtual depth electrode,Ó since we could in theory design LCMV weights

for every location within the brain and monitor the output power of the beamformer for each location. As discussed

in [30, 33], however, the principal limitation of LCMV techniques is signal cancellation, which occurs either due to

imprecise model specification or partial correlation with other sources. Modeling errors result in inaccurate

constraints, and the beamformer consequently attenuates desired neural signals as if they were interference.

Furthermore, in event related studies, neural activity at one location is often partially correlated with activity at other

locations. Power minimization in the LCMV will use the partial correlations of these other sources to partially

attenuate the desired signal. In other words, the desired neural activity must be orthogonal to all other activity in

order to be processed correctly by the LCMV beamformer. In [32], we present an example of monitoring for an

isolated EEG transient event, which meets some of these orthogonality requirements.

The class of high-resolution beamforming methods can overcome the limitations of the LCMV beamformer using

subspace scanning and data fitting techniques. The MUSIC algorithm is one such method and we now describe its

application to E/MEG

2.2.3 From classical MUSIC to RAP-MUSIC

The Multiple Signal Classification Approach (MUSIC) was developed in the array signal processing community [35]

before being adapted to M/EEG source localization [27]. We will restrict our brief description of the MUSIC

approach here to dipole sources with fixed orientation, although it can be extended to regional sources and even

multipoles. As before, let M be a N × T spatio-temporal matrix containing the data set under consideration for

analysis, and let the data be a mixture of S sources. Our model remains NSurAM += Tiqi }),({ , but we represent

the individual sources more explicitly as

NsuraM += ∑=

S

i

Tiiqi

1

),( , (19)

where ),( iqi ura is a single column from the gain matrix A , representing a single fixed dipole. Let TVUîM = be the

singular value decomposition (SVD) of M [29]. The set of left singular vectors is an orthonormal basis for the

subspace spanned by the data. Provided that N > S, the SNR is sufficiently large, and noise is i.i.d. at the sensors,

one can define a basis for the signal and noise subspaces from the column vectors of U. The signal subspace is

spanned by the S first left singular vectors in U, denoted US, while the noise subspace is spanned by the remaining

left singular vectors. The best rank S approximation of M is given by ( )MUUM TSSS = and ( )T

SSS UUIP −=⊥ is

the orthogonal projector onto the noise subspace.

As introduced in [27], we can use this projection operator to create a new cost function,

( )2

2

2

2

),(

),(,

iqi

iqiS

iqiJura

uraPur

⊥

= , (20)

which is zero at each of the true source locations and orientations iqi ur , . The advantage over least-squares is that

each source can be found by scanning through the possible set of locations and orientations for a single source,

rather than searching simultaneously for the set of all sources. An additional strong feature is that the orientations

can be factored out of the search as well, much like their linear counterparts in the least-squares equations, yielding

the cost function

( )2

2

)(

)(

Fqi

FqiS

qiJrG

rGPr

⊥

= (21)

where )( qirG is the three column gain matrix corresponding to a single dipole (i.e., the three gains for the three

elemental dipoles). At every valid location rqi, this cost function is theoretically zero. The optimal orientation is found

using a simple eigenvector analysis (hence its Òquasi-linearÓ status) [27],[36], and the optimal time series are found as

in the least-squares approach.

By evaluating J(rqi) on a predefined set of grid points and then taking the inverse of J(r), a ÒMUSICÓ map is

readily obtained with S peaks being representative of locations close to the original S generators. Practical limitations

often make this Òpeak-pickingÓ step rather tedious as limited precision of the source and head modeling never allow

J(rqi) to reach zero. In the presence of multiple sources, the MUSIC map often appears as a set of smooth three-

dimensional peaks of various heights. Recent improvements to the original MUSIC scanning method include the

recursive estimation of multiple sources, which solves this Òpeak-pickingÓ issue (Recursively Applied and Projected

MUSIC, RAP-MUSIC, [37]). Extensions of the MUSIC approach based on the estimation of the signal subspace

from time-frequency analysis of MEG signals have been applied to the localization of characteristic brain rhythmic

activity [38].

2.2.4 Limitations

Standard MUSIC approaches rely on each source having a time series independent of any other combination of

sources; hence, MUSIC will fail when two sources with strongly correlated time-series are active. Indeed,

synchronous activation of an arbitrary number of sources with unit time series correlation will produce a rank one

signal. This synchronous problem can be corrected by adjusting the concept of single dipole models. In [36], we

introduce the concept of p-dipole spatial topographies that allow the same MUSIC approaches to be applied to

synchronous sets of dipoles.

Stationarity of the underlying signals is also critical here for the proper estimation of the subspaces.

Because SNR is low in unaveraged recordings, it is likely that no reliable signal subspace identification can be done

without averaging. Working with averaged data will improve SNR and increase noise whiteness. Except for sources

from components of higher-frequency and smaller amplitudes, for which the relative SNR might be too low, running

MUSIC scans on averaged data is optimal. As rank determination is optimal when noise is white, it is also necessary

to apply whitening techniques based on the baseline of the recordings prior to any evoked brain activity [27].

2.3 Imaging approaches

2.3.1 Cortically-distributed source models

Imaging approaches to the M/EEG inverse problem consist of methods for estimating the amplitudes of a dense set

of dipoles distributed at fixed locations within the head volume. In this case, since the locations are fixed, only the

linear parameters need be estimated and the inverse problem reduces to a linear one with strong similarities to those

encountered in image restoration and reconstruction, i.e., the imaging problem involves solution of the linear systemTASM = for the dipole amplitudes, S .

The most basic approach consists of distributing regional dipoles over a predefined volumetric grid similar

to the ones used in the scanning approaches. However, since primary sources are widely believed to be restricted to

cortex, the image can be plausibly constrained to sources lying on the cortical surface which has been extracted from

an anatomical MR image of the subject [39]. Following segmentation of the MR volume, dipolar sources are placed

at each node of a triangular tessellation of the surface of the cortical mantle. Since the apical dendrites that produce

the measured fields are oriented normal to the surface, we can further constrain each of these elemental dipolar

sources to be normal to the surface. The highly convoluted nature of the human cortex requires that a high-

resolution representation contains on the order of ten to one hundred thousand dipole ÒpixelsÓ. The inverse

problem is therefore hugely underdetermined and imaging requires the use of either explicit or implicit constraints

on the allowed current source distributions. Typically this has been accomplished through the use of regularization

or Bayesian image estimation methods.

2.3.2 Bayesian Formulation of the Inverse Problem.

For purposes of exposition, we will describe imaging methods from a Bayesian perspective. Consider the problem of

estimating the matrix S of dipole amplitudes at each tessellation element from the spatio temporal data matrix M ,

which are related in the noiseless case by TASM = . The ith row of S contains the amplitude image across the

cortex at time i. From Bayes theorem, the posterior probability for the amplitude matrix S conditioned on the data

M is given by

)(

)()()(

M

SM/SS/M

p

ppp = (22)

where )( M/Sp is the conditional probability for the data given the image and )(Sp is a prior distribution reflecting

our knowledge of the statistical properties of the unknown image. While Bayesian inference offers the potential for a

full statistical characterization of the sources through the posterior probability, in practice images are typically

estimated by maximization of the posterior or log-posterior probability:

( ) ( ) ( ) ( )SSMSSMSSS

pppp lnlnmaxargmaxargö +≡= (23)

The term )( M/Sp is the log likelihood for the data that depends on the forward model and the true source

distribution. Typically, MEG and EEG data are assumed to be corrupted with additive Gaussian noise which we

assume here is spatially and temporally white (generalizations for colored noise are straightforward). The log

likelihood is then simply given by, within a constant,

( ) 2

22

1ln

F

Tp ASMSM −−=σ

. (24)

The prior is a probabilistic model that describes our expectations concerning the statistical properties of the source

for which we will assume an exponential density

)}(exp{1

)( SS fz

p β−= (25)

where β and z are scalar constants and )(Sf is a function of the image S . This form encompasses both multivariate

Gaussian models and the powerful class of Gibbs distributions or Markov random field models. Combining the log

likelihood and log prior gives the general form of the log posterior whose minimization yields the maximum a

posteriori or MAP estimate:

( ) ( )SASMS fU T �2

+−= , (26)

where 22σβλ = . We can now give a brief overview of the imaging methods as special cases of minimization of

the energy function in (26).

2.3.3 Linear Imaging Methods

In the case of a Gaussian image, the log prior has the form:

( ) { }TStrf SSCS 1−= , (27)

where 1−SC is the inverse spatial covariance of the image; this model assumes that the image is independent from

one time sample to the next. The corresponding energy function U(S) is quadratic in S and the minimum is given by

( ) MFMIAAWWAWWS λ=+=−1

�ö TTTTT . (28)

where we have factored T

S WWC =−1. We note that for this case the posterior is Gaussian and the MAP estimator is

equivalent to the minimum mean squared error estimator and or Wiener solution.

We can also interpret (26) as a Tikhonov regularized form of the inverse problem [40] [41], where the first

term measures the fit to the data and the last is a regularizing function that measures smoothness of the image. The

scalar λ is the regularization parameter that can be chosen using cross-validation methods or the L-curve. Within this

regularized interpretation of (26), several forms of W have been proposed for M/EEG imaging applications: (i) the

identity matrix which produces a regularized minimum norm solution [42]; (ii) the column normalized minimum

norm in which W is a diagonal matrix with elements equal to the norm of the corresponding column of A [43]; (iii)

W computes a spatial derivative of the image of first order [44] or Laplacian form [45]; (iv) W is diagonal with

elements computed from the output of a beamformer or MUSIC scan evaluated for each dipole pixel in turn [39, 46,

47], where these values are used as estimates of the signal power originating at each pixel.

The underdetermined nature of the inverse problem in M/EEG is such that these linear methods produce

very low resolution solutions. Focal cortical sources tend to spread over multiple cortical sulci and gyri. In some

applications this may be sufficient to draw useful inferences from the resulting images. However, the images formed

do not reflect the generally sparse focal nature of event-related cortical activation that is visualized using the other

functional imaging modalities of PET and fMRI. In an attempt to produce more focal sources, the FOCUSS method

[48] uses an iterative re-weighting scheme in which the diagonal weight matrix W is updated at each iteration to equal

the magnitude of the current image estimate. This approach does indeed produce sparse sources, but can be highly

unstable with noisy data [25].

An interesting approach to interpretation of minimum norm images formed using (28) was proposed by Dale et al

[47], in which an image of signal to noise ratio is computed by normalizing each pixel value computed using (28)

with an estimate of the noise sensitivity of that pixel, i.e. for the case of white Gaussian noise, each value in TSö is

normalized by the noise sensitivity given by the corresponding diagonal elements of TFF [47]. This has the

interesting property of generally reducing the amount by which activity spreads across multiple sulci and gyri when

compared to the standard minimum norm image; these images can also be used to make statistical inferences about

the probability of a source being present at each location.

Comment aller plus loinÊ?Ce qui reste marquant � la lecture de ces �tudes est le caract�re global des a priori ou des contraintes introduites.Comment tenir compte des connaissances physiologiques que nous poss�dons, pour aller au-del� de la simple contrainte dedip�les perpendiculaires � la surface du cortexÊ?En lÕ�tat actuel de nos connnaissances, la cytoarchitecture du cortex ne r�v�le pas de zones � fortes sp�cificit�sfonctionnelles qui puissent �tre identifi�es chez tous les sujets. Cependant, lÕexploitation de la morphologie particuli�re ducortex et son int�gration dans des mod�les plus complets nÕa pas encore �t� v�ritablement exploit�e.

Ainsi, sans m�me faire correspondre structure et fonction, il semble envisageable de mettre en �vidence une organisationparticuli�re dans les circonvolutions m�mes du cortex. Les �tudes de Welker montrent par exemple des interconnexionspr�f�rentielles de gyrus � gyrus, plut�t que dÕun gyrus au sillon adjacent. M�me les connexions interh�misph�riques sontpr�dominantes de gyrus � gyrus. LÕarborisation dendritique, la vascularisation, la lamination corticale sont bien plusd�velopp�es sur les convexit�s gyrales que sur les bords ou dans le fond dÕun sillon (Welker 1990).Cette architecture sp�cifique se trouve �galement confirm�e par des organisations fonctionnelles. Ainsi, il appara�t parexemple que dans lÕaire sensorielle primaire de la main, les gyrus seraient associ�s � la face ventrale, et les sillons � la facedorsale de la main (dont le r�le fonctionnel est moins important) (Wong 1998).

Bien �videmment, cette conception est sans doute tr�s simpliste et est susceptible dÕ�tre aussi contestable que les sch�masfonctionnels classiques associ�s � la cytoarchitecture. Mais elle pose la question de la possibilit� dÕintroduire desinformations plus locales dans le mod�le de sources associ� � lÕestimateur.

CÕest cette id�e que nous allons maintenant formaliser.

Welker, W., "Why does cerebral cortex fissure and fold ?", Cerebral Cortex, Vol. 8B pt. 2, pp. 3-136, 1990.

Wong, P., "Potential fields, EEG maps, and cortical spike generators", Electroenceph. Clin. Neurophysiol.,Vol. 106, pp. 138-141, 1998.

2.3.4 Non-Gaussian Priors

In an attempt to produce more physiologically plausible images than can be obtained using linear methods, a large

number of researchers have investigated alternative methods that can collectively be viewed as selecting alternative

(non-quadratic) energy functions )(Sf in (26). From a regularization perspective, these have included entropy

metrics and Lp norms with values of p<2, i.e. pSf =)(S [49]. For the latter case, solutions will become increasingly

sparse as p is reduced. For the special case of p=1, the problem can be modified slightly to be recast as a linear

program [49]. This is achieved by replacing the quadratic log-likelihood term with a set of underdetermined linear

inequality constraints where the inequalities reflect expected mismatches in the fit to the data due to noise. The L1

cost can then be minimized over these constraints using a linear simplex algorithm. The attraction of this approach is

that the properties of linear programming problems guarantee that there exists an optimal solution for which the

number of non-zero pixels does not exceed the number of constraints, or equivalently the number of measurements.

Since the number of pixels far outweigh the number of measurements, the solutions are therefore guaranteed to be

sparse. This idea can be taken even further by using the Lp quasi-norm for values of p<1. In this case, it is possible to

show that there exists a value 0<p<1for which the resulting solution is maximally sparse [43].

An alternative to the use of simple algebraic forms for the energy function )(Sf is to explicitly define a

prior distribution that captures the desired statistical properties of the images. This can be done using the class of

Markov Random Field (MRF) models. MRFs are a powerful framework which have been extensively investigated in

image restoration and reconstruction for statistical modeling of a range of image properties [50]. A key property of

MRFs is that their joint statistical distribution can be constructed from a set of potential functions defined on a local

neighborhood system. Thus the energy function ( )Sf for the prior can be expressed as

( ) ( )∑=

Φ=J

jjf

1

SS , (29)

where ( )SjΦ is a function of a set of dipole pixel sites on the cortex that are all mutual neighbors. In this way, the

MRF model can capture local interaction properties between image pixels and their neighbors. The total number, J,

of these functions depends on the number of pixels and the number of different ways in which they are allowed to

interact with their neighbors. Among the simplest MRF image models are those in which each of the potential

functions involves a pair of neighboring pixel values. To model smoothness in images an appropriate choice of

potential function might be the squared difference between these neighboring pixels.

In the case of E/MEG, the model should reflect the observation that cortical activation appears to exhibit a

sparse focal structure, i.e. during an event related E/MEG study, most of the cortex is not involved in the response,

and those areas that are, correspond to focal regions of active cell assemblies. To capture these properties, a highly

non-convex potential function defined on the difference between each pair of neighboring pixel values was used in

[51]. This prior has the effect of favoring the formation of small discrete regions of active cortex surrounded by

regions of near-zero activity. An alternative model was proposed in [52] where a binary random field, x, was used to

indicate whether each dipole pixel was either active (x=1) or inactive (x=0). A MRF was defined on this binary field

to capture the two desired properties of sparseness and spatial clustering of active pixels; the parameters of this prior

could then be adjusted to achieve differing degrees of sparseness and clustering [52].

The MRF-based image priors lead to non-convex [51] and integer [52] programming problems in

computing the MAP estimate. Computational costs can be very high for these methods since although the priors

have computationally attractive neighborhood structures, the posteriors become fully coupled through the likelihood

term. Furthermore, to deal with non-convexity and integer programming issues some form of deterministic or

stochastic anealing algorithms must be used [53].

Limitations pratiques du recuit simul� avec �chantillonneur de GibbsÊ:Sans entrer ici dans les d�tails de la d�termination du MAP, citons simplement la m�thode de recuit simul� associ� �lÕ�chantillonneur de Gibbs qui permet dÕatteindre lÕ�quilibre thermodynamique correspondant au MAP, malgr� lapr�sence possible de nombreux minima locaux de la fonction dÕ�nergie globale (Geman & Geman 1984). Cetalgorithme stochastique justifie dÕailleurs lÕadoption dÕun cadre probabiliste pour la r�gularisation.Cette approche n�cessite un balayage de lÕensemble des pixels (donc des sources dipolaires corticales en M/EEG) avecperturbation locale du champ dÕintensit�. Gr�ce � la d�finition du syst�me de voisinage, cet algorithme se pr�te bien � laparall�lisation.Cependant, en pratique, la visite r�p�t�e de chaque site avec rafra�chissement est dÕautant plus rapide que les intensit�sdes sources sont cod�es sur peu de bits. De plus, lÕ�chantillonnage de la densit� de probabilit� a posteriori est effectu� enaccord avec le voisinage a posteriori du pixel courant. Dans le cadre de la restauration dÕimage, le voisinage a posterioriest presque identique � celui a priori car lÕop�rateur d�formant A est � support tr�s limit� (chaque pixel de lÕimageobserv�e est un m�lange de peu de pixels voisins).Mais en reconstruction dÕimage, lÕop�rateur tend � m�langer en tout point dÕobservation les intensit�s dÕune grande partiede lÕimage initiale. Dans le cas dÕune transform�e de Fourier, ou pour le mod�le de production des donn�es M/EEG,lÕop�rateur prend lÕobjet dans son ensemble comme support (Nikolova 1998).Au final, pour lÕapplication qui nous int�resse ici, lÕalgorithme de recuit simul� avec �chantillonneur de Gibbs reste peupratique.La souplesse offerte par les fonctions de potentiels en termes du champ dÕintentsit� des sources (et introduisant alors unprocessus de contours/lignes implicites), a permis de d�velopper un grand nombre dÕalgorithmes d�terministes, sous-optimaux quand la fonction de potentielle choisie est non convexe.Ces algorithmes n�cessitent une optimisation non lin�aire sur les intensit�s des champs de pixels. Des m�thodes degradient et de gradient modifi� ont �t� largement employ�es, ainsi que dÕautres techniques qui sont des raccourcis delÕalgorithme de Gibbs original. Nous citerons par exemple lÕalgorithme de modes conditionnels it�r�s (IteratedConditional Modes, ICM Besag 1986) et de recuit � champ moyen (Mean Field Annealing, MFA Geiger &Girosi 1991).Citons aussi la m�thode dÕintroduction graduelle de non-convexit�s (Graduated NonÐConvexity, GNC) quipr�sente un compromis int�ressant entre lÕaspect bien pos� de la r�gularisation tout en introduisant petit � petit despropri�t�s de non-convexit� pouvant assurer la d�tection de sauts dÕintensit� (Blake & Zisserman 1987), (Nikolova1998)

Geman, S., Geman, D., "Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images",IEEE Trans. on Pattern Anal., Vol. PAMI-6, pp. 721-741, 1984.Nikolova, M., Idier, J., Mohammad-Djafari, A.,Inversion of large-support ill-posed linear operators usinga piecewise Gaussian MRF, IEEE Trans. on Image Processing, 7,pp. 571-85, 1998.Besag, J., "On the statistical analysis of dirty pictures", J. Roy. Statist. Soc., N¡3, pp. 259-302, 1986.

Geiger, D., Girosi, F., "Parallel and deterministic algorithms from MRF's: surface reconstruction", IEEETrans. on Pattern Anal. Mach. Intell., Vol. 13, N¡5, pp. 401-412, 1991.Blake, A., Zisserman A., "Visual Reconstruction", MIT Press, Cambridge: MA, 1987.

2.3.5 Limitations of Imaging Approaches and Hybrid Alternatives

The imaging approaches are fundamentally limited by the huge imbalance between the numbers of spatial

measurements and dipole-pixels. As we have seen, methods to overcome the resultant ambiguity range from

standard minimum-norm based regularization to the use of physiologically based statistical priors. Nonetheless, we

should emphasize that the class of images that provide reasonable fits to the data is very broad, and selection of the

ÒbestÓ image within the class is effectively done without regard to the data. In contrast, the dipolar and multipolar

methods control this ambiguity through a more explicit specification of the source model. This may lead to

improved confidence in the estimated sources, but at the potential cost of missing sources that do not conform to

the chosen model, and to the added complexity of interpreting the resulting solutions.

Figure 4 - MEG modeling and imaging. The data described here were acquired on a 122-MEG sensor array (a). The task for thesubject consisted in making self-paced movements of the forefinger of the right hand. Insert (b) displays some typical magnetic fieldmaps recorded outside the head from about Ð9.8ms to 47.9ms about the movement onset. The increase in magnetic field clearly occursabove the contralateral (left) side of the head (i.e. where the primary sensori-motor response occurs). Head models are built from theindividual MRI. (c) displays the best-fitting 3-shell approximation and (d) the BEM surface tessellation for realistic head modeling.The LS fit model shows a single dipole in the contralateral central sulcus. Location is satisfactory, but there is no information aboutthe source extension. MN imaging is spread over the central sulcus region and mostly on the corresponding gyral crowns with minorartifacts at some distant regions. The hybrid RAP-MUSIC/cortical remapping approach also finds activation about the centralsulcus. Detailed analysis of the localization in the MR volume indicates the source region covers the omega-shaped region of the handarea in both the primary sensory and motor regions. Further analysis in the [-400, 100] ms time window revealed multipolar sourceactivity in the supplementary motor area and the ispsilateral somato-sensory region, as an indication of significant brain activityrelated to movement preparation (see corresponding time series).

Recently we have been exploring the idea of remapping estimated dipolar and multipolar solutions onto cortex as a

hybrid combination of the parametric and imaging approaches [54]. In this way we can rapidly find a solution to the

inverse problem using, for example, the MUSIC scanning method. We then fit each source in turn to the cortex by

solving a local imaging problem to compute an equivalent patch of activated cortex whose magnetic fields or scalp

potentials match those of the estimated dipole or multipole. Examples of solutions found using this method are

shown in Fig 4.

A second hybrid approach to source estimation draws elements from the imaging and parametric approaches by

specifying a prior distribution consisting of a set of activated cortical regions of unknown location, size and

orientation. By constructing and sampling from a posterior distribution using Markov Chain Monte Carlo methods,

Schmidt et al [55] are able to investigate the parameter space for this model and provide estimates, together with

confidence values, of the true source distribution. As with the other physiologically-based Bayesian models, this

approach has high computational costs.

3. Conclusion and perspectives

As we have attempted to show, M/EEG source imaging encompasses a great variety of signal modeling and

processing methods. We hope that this paper serves as an introduction that will help to attract signal processing

researchers to explore this fascinating topic in more depth. We should emphasize that this paper is not intended to

be a comprehensive review, and for the purposes of providing a coherent introduction, we have chosen to present

the field from the perspective of the work that we have done over the last several years.

The excellent time resolution of E/MEG gives us a unique window on the dynamics of human brain functions.

Though spatial resolution is the AchillesÕ heel of this modality, future progress in modeling and applying modern

signal processing methods may prove to make M/EEG a dependable functional imaging modality. Potential

advances in forward modeling include better characterization of the skull, scalp and brain tissues from MRI and in

vivo estimation of the inhomogeneous and anisotropic conductivity of the head. Progress in inverse methods will

include methods for combining E/MEG with other functional modalities3 and exploiting signal analysis

methodologies, such as Independent Component Analysis4, to better localize and separate the various components

of the brainÕs electrical responses. Of particular importance are methods for understanding the complex interactions

between brain regions using single-trial signals to investigate transient phase synchronization between sensors [56]

SEQMERGEFORMATor directly within the M/EEG source map [57].

Acknowledgments

The authors are grateful to Marie Chupin and David W. Shattuck for their help in preparing the illustrations.

3 Voir dans ce recueilÊ: ÇÊFusion de donn�esÊÈ, par D. Schwartz.4 Voir les copies de transparents deÊ:Ê ÇÊPr�traitement des donn�esÊÈ, par S. Baillet.

Annexe - Quelques �l�ments de compariason entre une s�lection de modalit�s dÕimagerie

MEG EEG fMRI PET

PhysiologicalOrigin

Coherent spatio-temporal organization ofintra & extra-cellular currents triggered byexcitatory postsynaptic potentials at the apicaldendritic trees of neural cell assemblies

Regional alterationsin blood oxygenation

Regional alterationsin metabolism andblood flow

Spatial Resolution

Nominally 5mm butstrongly model-dependent

Ú

Nominally 5mm butstrongly model-dependent.

Ø

1-3mm 2-10mm

5s-8s 20s-1minTemporalResolution

0.4ms-1msVirtually limited by the ADC sampling rate Limited by the intrinsic hemodynamic /

metabolic phenomena

Invasiveness Non invasive

Non invasive buthigh-magnetic fieldenvironment, from0.5T to 7T (typ. 1.5T)

Intravenous injectionof radioactive isotope

Cost Ø Ú Ø Ø

Robustnesstowards headmodelapproximations

Ú Ø

Sensitivity todeeper brainstructures

Ø Ú

Ú and Ø are relative positive or negativefeatures within a row respectively.

R�f�rences

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