MEG Source Imaging and Dynamic Characterization.

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To cite this version:

Sheraz Khan. MEG Source Imaging and Dynamic Characterization.. Signal and Image processing.

Ecole Polytechnique X, 2010. English. �pastel-00583104�

DE L'ÉCOLE POLYTECHNIQUE

Spé ialité : Mathématiques Appliquées

par

Sheraz Khan

MEG Sour e Imaging and

Dynami Chara terization

Soutenue le15 janvier2010 devant la ommission d'examen formée de :

Jury :

Habib Ammari - E ole Polyte hnique

Sylvain Baillet - Medi al College of Wis onsin

Ali Mohammad Djafari - Laboratoire des Signaux & Systèmes (Supéle )

Josselin Garnier - Université de Paris 7

Fran ois Jouve - Université de Paris 7

Roman Novikov - E ole Polyte hnique

AsIlookthroughthe lastthreeyears,therearesomanypeoplewithoutthehelp

of them thisthesis might neverbe possible, rstand foremost my PhD dire tors

SylvainBailletandHabibAmmari,whose ontinuousguidan eandsupportmake

this thesis realizable. Julien Lefevrefor his ex ellent mentorshipduring the

se -ond andthird yearsofPhD,without himHHDpartofthethesiswouldnot have

been possible.

My olleagues at Paris and Milwaukee, Manoj, Rey, Benoit, Guillaume,

So-phie,Bethandmanyothers,there onstanthelpandmotivation, makethisthesis

through,theyarealwayswithmewhenthingsarenotworkingand heermewhen

things work out.

LastlyIa knowledge myparents,mywife and mydaughter, for their

Mythesis hasaddressedtwo omplementary aspe ts of magneti sour e imaging

using Magnetoen ephalography:

1. Imaging ofneural urrent sour esfromMEG surfa ere ordings;

2. Dynami hara terization of neural urrent patterns at thesurfa e of the

ortex.

MEG Sour e Imaging

A urate estimation of the lo al spatial extent of neural urrent a tivity is

very important for the quantitative analysis of neural urrent sour es, as

esti-mated from Magnetoen ephalography (MEG) surfa e re ordings. In asso iation

withthe ex ellent time resolution oered byMEG, this wouldrepresent amajor

advan ement innon invasive, time-resolved fun tional brainimaging.

We addressed this issue through a new method  alled Multipole Corti al

Remapping (MCR)  to a urately spe ify the spatial extent of neural urrent

sour es.

InMCR,thezeroth-orderTikhonovregularized imageofthe urrent

distribu-tion onthe ortex isrstestimated fromMEGsurfa e datafor whi hwesought

for a realisti model of neural generators. Then the resulting fun tional image

is thresholded usinga simple histogram-based prin iple. This thresholded image

is thende omposed intogroups ofa tivation patternsfollowing anautomati

equivalent multipolar de omposition of ea h urrent pat h isthen obtained. By

default, themultipolar momentsarenot readily relatedto thea tualanatomi al

supportofthea tualneural urrentsdete tedusingMEG.Hen eweintrodu edan

image remapping te hniquesof themultipolar parameters ba konto theoriginal

orti al manifold, ina Bayesian framework in luding physiologi al and

anatomi- al priors.

Chara terization of MEG Sour e Dynami s

For dynami hara terization of neural urrent patternsat thesurfa e ofthe

ortex, we used a modied Helmholtz-Hodge De omposition (HHD), whi h was

appliedonve toreldsdes ribingtheowofneural urrentsour es. Thismotion

eld stems from a generalized approa h to opti al ow estimation, developed

earlier inour team.

Opti alowistheapparentmotion duetovariationsinthepatternof

bright-ness and, under spe i onditions, may mimi the velo ity eld of an obje t.

Normally, the opti al ow is obtained in a two-dimensional domain, whi h may

prevent a ess to some essential features of the obje t's motion with respe t to

the topology or geometry of the domain onto whi h it is evolving. A new

vari-ational method to represent opti al ow on non at surfa es using Riemannian

formulationwaspreviously introdu ed byour group to over omethis issue.

Webroadened this framework andintrodu ed anewformalism to dete t

fea-turesintheresultingopti alowmodelusingamodiedandextendedframework

to the HHD on 2-Riemannian manifolds, whi h we used to hara terize neural

urrent sour es.

HHD is a te hnique used to de ompose a two-dimensional (resp.

three-dimensional) ontinuous ve tor eld into thesum of 3 distin t omponents: (1)

a non-rotational element, deriving from thegradient of a s alar potential

U

;(2) a non-diverging omponent, deriving from the rotational of a s alar potential

A

(resp. ve torial potential); (3) a harmoni ve torial part, i.e., whose Lapla ian

vanishes.

WeshowedhowHHDenablesthede ompositionandtra kingoftime-resolved

neural urrent ows asobtained from MEGsour e imagingas sour esand sinks

e.g., by dete ting relative maxima of the non-rotational s alar potential. We

hen eforth suggest to extend the analysis of brain a tivity in terms of tra king

travelling obje ts onto the orti al manifold by dete ting ve tors of largest

am-plitudes inzeroLapla ian harmoni ve tor elds.

We also onsidered HHD through a series of stru tural and fun tional brain

imaging appli ations, withvery en ouragingpreliminaryresults.

Themethods dis ussedintheHHDse tionofthethesiswereimplementedin

Matlabasplug-intotheBrainstorm(MEG/EEGdatapro essingsoftware)and

an be downloaded from: http://neuroimage.us .edu/brainstorm. A short

Contents 6

List of Symbols and Abbreviations 9

List of Figures 10

Ba kground 15

Te hniquesfor the observation ofthe Human brain . . . 15

Introdu tion to MEGand EEG: . . . 19

Neural bases ofbrain ele tromagneti signature . . . 20

Forward problem . . . 22

Maxwell's equations . . . 23

Modelingprimary urrents . . . 26

Headmodeling . . . 27

Spheri al head model . . . 27

Realisti head model . . . 29

Linear formulation . . . 32 Inverse problem. . . 33 Parametri methods . . . 35 Dipoletting . . . 35 Beamforming approa hes . . . 37 Mat h lter . . . 37

Linearly onstrained minimum-varian e (LCMV) . . . 38

Imaging methods . . . 40

Bayesian formulation . . . 41

Choi e oftheregularization parameter

λ

. . . 42

Linear estimators . . . 43

Properties ofthesour e ovarian e matrix . . . 44

Nonlinear estimators ofsour e amplitudes . . . 46

Con lusion . . . 47

Multipolar Corti alRemapping 51 Introdu tion . . . 51

Multipolarexpansionsof as alar potential . . . 52

Spheri al multipolar expansions . . . 52

Multipoleexpansionsof adistributeddipole sour e. . . 54

Multipole momentsof urrent distributions . . . 57

Multipolar orti al remapping. . . 59

Compa t parametri de omposition of orti al urrents . . . 61

Sparse-fo al imagingmodel . . . 62

Results . . . 66

Simulated data . . . 67

A ura y riteria . . . 67

Single sour e ase . . . 67

Robustnessof MCRagainst hangesinthethreshold. . . 70

Two-sour e ase . . . 72

Experimental data . . . 74

Con lusion . . . 76

Helmholtz-Hodge De omposition 79 Introdu tion . . . 79

Regularization . . . 84

Variationalformulation . . . 85

HelmholtzHodgede omposition on2-Riemannian manifold . . . 87

Theory . . . 87

Denitions . . . 87

Theorem . . . 88

Dis retization. . . 89

Feature dete tion as riti al points ofpotentials . . . 92

Simulations andresults . . . 95

Con lusion . . . 99

Appli ations of HHD 101 Introdu tion . . . 101

HDDof MEGexperimental data . . . 102

Chara terizingepilepti a tivity . . . 103

Chara terizingof epilepti a tivityinECoG . . . 105

Chara terizingepilepti a tivitywithMEGsour e imaging . . . . 107

Identi ation of orti aldevelopment inthe neonate brain . . . 109

Chara terizingtumor growth patterns . . . 113

Con lusion . . . 115

Con lusion and Future A tions 117

Grid Generation 121

MEG-ECoG sour e lo alization and dynami s omparison 125

Brainstorm's HHD-Opti alow plug-in Tutorial 133

and Abbreviations

Abbreviation Des ription

EEG Ele troen ephalography

HFO High-Frequen y Os illations

HHD Helmholtz-Hodge De omposition

MCR MultipolarCorti al Remapping

MEEG Magnetoen ephalography and

Ele troen- ephalography

MEG Magnetoen ephalography

1 EEGSetup . . . 16

2 Comparisonof brainsignals . . . 17

3 MEGSetup . . . 17

4 Spatialandtemporal resolution ofdierent brainimagingmethods. . 18

5 MEEGeldpattern . . . 20

6 Cerebral Cortex . . . 21

7 Modeling of head regions . . . 25

8 Current Distribution Model . . . 27

9 Spheri alhead model, where asphere is ttedto thehead geometry. . 28

10 Realisti head Modeling . . . 31

11 DipoleFitting . . . 36

12 MUSICs an . . . 39

13 L-Curve . . . 43

14 GCV urve . . . 44

15 LCMVvs Minimnorm . . . 45

16 Spheri alharmoni multipole omponents . . . 54

17 DistributedDipoleModel . . . 56

18 Lo al urrent distribution . . . 58

19 A ura y ofMCR . . . 68

20 Subspa e orrelation estimationof MCR. . . 69

22 A ura y ofMCR . . . 71

23 Illustrationof the bootstrapestimateof onden eintervals. . . 72

24 Re onstru tionby MCRintwosour e s enario. . . 73

25 MCRinsomatosensoryexperiment . . . 75

26 A tive surfa e areas in

S

1

and

S

2

regions. . . 76

27 HHDinreal worldappli ations . . . 81

28 Basisve torsof tangent plane. . . 82

29 FEMformalism . . . 90

30 Sour es andsinks onat manifold . . . 92

31 Sour es andsinks onspheri al manifold . . . 93

32 vorti eson at manifold . . . 93

33 Vortexonspheri al manifold . . . 94

34 HHDon rabbitmanifold . . . 96

35 HHDon elephant manifold . . . 97

36 Tra king on rabbitmanifold. . . 98

37 HHDon orti almanifold . . . 99

38 Typi alECoG setup . . . 102

39 De ompositionof experimental data . . . 104

40 Divergen e representation . . . 105

41 HHDon ECoGGrid . . . 106

42 HHDfor seizureonset dete tion . . . 108

43 Two neonate orti al surfa es . . . 109

44 Displa ement eld between two orti al surfa es . . . 110

45 Dete tion ofgrowth seed . . . 111

46 Thereprodu ibilityof the growthseeds . . . 112

47 Tumor Dete tion . . . 114

48 Interpolated grid . . . 122

51 MEGGranger . . . 127 52 ECoGlo alization . . . 128 53 ECoGGranger . . . 129 54 EEGSlow. . . 131 55 BrainstormLaun h . . . 134 56 Plug-in GUI . . . 135 57 Opti al Flow Tab . . . 136 58 HHDTab . . . 137 59 VisualizationTab . . . 138

Te hniques for the observation of the

Hu-man brain

Exploration of the Human brain is of utmost intelle tual interest: de iphering

brain using brain is a hallenging task. Although a great deal has been learnt

aboutbrainanatomyand physiology,thefundamentalquestionshowbrainstore,

retrieve and pro esses information is still largely unknown and full dis overy of

these me hanisms is thefoundational purposeof neuros ien e.

When brain pro esses information, ele trophysiologi al urrents ow within

and outsideneural ells, thus produ ing ele tri and magneti elds thatare

a - essible toexternal measurements. Indeed,signsof this ele tri al neurala tivity

in the brain an be measured with ele trodes at the s alp or with very

sensi-tive magneti dete tors pla ed very near the s alp. The te hnique of ele tri al

measurements from the s alp is alled ele troen ephalography (EEG) [8℄.

His-tori al and re ent EEGsetups areshownin Figure 1. The te hnique measuring

magneti signalsgenerated byneural urrentsis alledMagnetoen ephalography

(MEG) [15 ℄.

Themagneti eldprodu ed byneural urrent sour esarevery weakandare

at least 8 orders of magnitude smaller than the earth stati magneti eld, as

shown Figure 2. Theseelds are urrently pi ked using series of magnetometers

Figure 1: (a) EEG setup in 1970's. (b) Modern EEG setup with qui k-x

ap.

isasensitivedete torofmagneti ux,whi hwasdevelopedbyJamesZimmerman

[114 ℄ inthe late 1960's.

The seminal, original MEG measurements were performed at MIT in May,

1971 byCohen. Alphawaves (ele tromagneti brainos illationsinthefrequen y

rangeof [8,12℄Hz)werere orded asshownFigure3 .a. Atypi al, state-of-the-art

MEGsetup using151 hannels isshown Figure3.b.

Brain imaging te hniques an be divided into two ategories: stru tural and

fun tional. Anatomi al stru tures an be investigated using omputer-aided

to-mography(CT)s ansandbettersousingmorere entmagneti resonan eimaging

approa hes(MRI).For fun tionalimaging beside neural ele tromagneti signals,

brainmetabolism,bloodowand volume(hemodynami s) anbea essedusing

Figure2: Comparisonofbrainsignalswithothersour esofele tromagneti

waves.

Figure 3: (a)First MEGre ording at MIT inside aspa eship like magneti

shielded room using single hannel SQUID. (b) MEG Setup at La

Pitié-Salpêtrière Hospital, Paris inside modern multilayer shielded room using

151 SQUIDs overing whole brain.

inthespatialdistributionoftheseprobes,astheyaretransportedand hemi ally

modied within the brain, an be imaged using positron emission tomography

(PET). These images an rea h a spatial resolutions ashigh as 3mm. However,

temporalresolutionislimitedtominutesbythedynami softhephysiologi al

As neurons be ome a tive,they indu e very lo alized hanges inblood ow and

oxygenationlevels that an be imaged asa orrelateof neural a tivity[65 ℄.

Hemodynami hanges anbedete ted using PET,fun tionalMagneti

Res-onan e Imaging (fMRI), and trans ranial opti al imaging methods. Of these,

fMRI is urrently the most widely used and an be readily performed using a

standard 1.5T lini alMRImagnetalthough anin reasing fra tionofstudiesare

now performed on higher eld (3-7T) ma hines for improved SNR and

resolu-tion. Fun tionalMRIstudiesare apable ofprodu ingspatialresolutionsashigh

as 2-4mm; however, temporal resolution is again limited by the relatively slow

hemodynami response, when ompared to ele tri al neural a tivity,to

approxi-mately one se ond. In addition to limited temporal resolution, interpretation of

fMRI data is hampered by the rather omplex relationship between the blood

oxygenationleveldependent (BOLD) hanges thataredete ted byfMRIandthe

underlying neural a tivity. Regions of BOLD hanges in fMRI images do not

ne essarily orrespond one-to-one withregionsof ele tri alneural a tivity[62 ℄.

Figure 4: Spatialand temporalresolution of dierent brain imaging

Introdu tion to MEG and EEG:

EEG and MEG measure the ombined a tivity of multiple areas of the brain

as a mixture of omplex signalpatterns. A primary obje tive is to interpret the

omplexpatternsofthemeasuredele tri potentialsandmagneti elds,interms

of therespe tivelo ationsand time- oursesoftheir underlying sour es. Thekey

to this taskisto design aphysi aland numeri al modelto a ount for theorigin

of the eld patterns aptured by MEG/EEG surfa e re ordings. Estimation of

theele tri andmagneti eldpatternsforagivenmodelofthevolume ondu tor

is a forward problem, following the nomen lature of modeling dataformation as

en ounteredinalargevarietyofappli ations(fromgeophysi stomedi alimaging)

[96℄.

The estimation of neural urrents from measured eld patterns is a typi al

inverseproblem. InEEGorMEGstudies,thesimplestwaytomodelthegeometry

of thehead is to usea single sphere approximationor on entri spheri al shells

ea h withhomogeneous isotropi ondu tivity[76℄.

Themainreasonwhy onsidering spheri algeometry istheavailabilityof

an-alyti al solutions, andtherefore fastimplementations, to solvethe forward

mod-eling problem. Howeveraspheri alapproximationofthehead omplex geometry

is likely to indu elarge sour e lo alization errors[72℄.

Using MRI, itis possible to provide more realisti geometri al models of the

head. Numeri alte hniques su h as theBoundary Element Method (BEM) and

Finite Element Method (FEM) provide the exibility of utilizing a realisti

ge-ometry [51 ℄.

EEGandMEGs alp patternsarequalitativelyorthogonal toea h other(see

gure 5), providing distin tive information about the underlying neural urrent

distributions. They therefore might be viewed as omplementary rather than

as ompeting modalities[24 ℄. Most state-of-the-art MEG fa ilities are equipped

for simultaneous a quisition of EEG and MEG data. Inverse methods for the

Figure 5: Left hand side gure represent the topographi sensitivity maps

of MEG and EEG for radial and tangential dipoles. Figure on right hand

side shows the orthogonalityof MEG and EEG eld patterns. patterns

Neural bases of brain ele tromagneti signature

MEGandEEG(MEEG)aretwote hniquesbasedonwhatGalvani,attheendof

the18th entury, alled"animalele tri ity", todaybetter knownas

ele trophys-iology [85 ℄. Despite the apparent simpli ity in the stru ture of the neural ell,

the biophysi s of neural urrent ow relies on omplex models of ioni urrent

generation and ondu tion [48 ℄. Roughly,when aneuronisex itedbyother

neu-rons via an aerent volley of a tion potentials, postsynapti potentials (PSPs)

are generated at its api al dendriti tree. When the ex itatory PSP's be ome

largerthan inhibitoryPSP's, theapi al dendriti membrane be omestransiently

depolarized and onsequently extra ellularly ele tronegative with respe t to the

ell soma and the basal dendrites. This potential dieren e auses a urrent to

ow throughthe volume ondu tor from the non-ex itedmembrane of thesoma

and basal dendrites to the api al dendriti tree sustaining the PSP's. Some of

the urrent takes the shortest route between the sour e and the sink by

travel-ling within the dendriti trunk (see gure 6). Conservation of ele tri harges

imposes that the urrent loop be losed with extra ellular urrents owing even

se ondary,return,or volume urrents.

Figure 6: The orientation of pyramidal neurons is normal to the ortex

surfa e. MEG signalspreferentially ree tthe urrent owfrompyramidal

ells oriented tangential tothe skull surfa e.

Bothprimaryandse ondary urrents ontributetomagneti eldsoutsidethe

headandtoele tri s alppotentials,butspatiallystru turedarrangementsof ells

are of ru ial importan e to the superposition of neural urrents su h that they

produ e measurable elds. Ma ro- olumnsoftens ofthousands ofsyn hronously

a tivatedlargepyramidal orti al neuronsarethus believedtobethemainMEG

and EEG generatorsbe ause of the oherent distribution of their large dendriti

trunks lo ally oriented in parallel, and pointing perpendi ularly to the orti al

surfa e. The PSPs generated among their dendrites are believed to be at the

sour e of most of thesignals dete ted in MEG and EEG be ause they typi ally

last longer than therapidly ring a tion potentials travelling along theaxons of

ex ited neurons. Indeed, al ulations su h as those shown in [44℄ suggest ea h

synapse along a dendrite may ontribute as little as a 20 fA.m urrent sour e,

probablytoo smalltomeasure inMEEG. Empiri alobservations insteadsuggest

of neuronal density and orti al thi kness suggest that the ortex has a

ma ro- ellular urrent density ofthe orderof 100

nA.mm

−2

[44℄. If we assumethatthe

ortexisabout4

mm

thi k,thenasmallpat hofsize 5

mm

x5

mm

wouldyield a net urrent of 10

nA.m

, onsistent with empiri al observations and invasive studies [44 ℄.

In MEEG studies, one is usually on erned with the uppermost layer of the

brain; the erebral ortex, whi h isa 2 to 6

mm

thi ksheet of graytissue where most of the measured neural a tivity takespla e. The se tion of ortex is

illus-trated inFigure 6. At least 10 billion neurons residein thewhole ortex tissue.

The total surfa e area of the ortex is about 2500

cm

2

,folded ina ompli ated

way,sothatittswithintheinnerskullvolume. Thetruespatialextent of

realis-ti urrent sour es asso iated withbrain a tivationvariesa ordingto the ause

of thea tivation. Typi ally sensory stimuli a tivate orti al areas starting from

a few

mm

2

up to afew

cm

2

,whereas for spontaneous a tivityand epilepti fo i

an involve ana tivation areaup totens of

cm

2

[95℄.

At a larger s ale, distributed networks of ollaborating and syn hronously

a tivated orti alma ro- olumnsaremajor ontributorstoMEGandEEGsignals

[80℄. This is ompatible with neuro-s ienti theories thatmodel basi ognitive

pro esses intermsof dynami allyintera ting ellassemblies[105 ℄.

Mostregionsofthe ortexaremappedfun tionally. Forexample,theprimary

somatosensory ortex re eives ta tile stimuli from theskin. Areas of thefrontal

lobe are on erned with the integration of mus ular a tivity. Primary motor

ortex isinvolved inthe movement of a spe i partof thebody. Largeareas of

ortexaredevotedtobodyparts,whi haremostsensitiveto tou h(e.g.,lips)or

to theparts where a urate ontrol ofmus les isneeded(e.g.,ngers).

Forward problem

suredele tri /magneti eldsandthe urrentdistributionwhi hprodu eit. This

relationship isknownasforward modelingwhi h translatesasalead-eld matrix

or a gain matrix that binds theamplitude of sour e urrents to the sensor data

as we shall detail below. If the primary sour e and the surrounding

ondu tiv-ity prole of tissues are known, the ele tri potential and magneti eld an be

al ulated fromMaxwell's equations (see[7℄ for a omprehensive reviewofMEG

forward and inversemodeling).

Maxwell's equations

In 1873,Maxwell showed thatele tromagneti elds an bedes ribedusing only

4 ve tor dierential equations [70℄:

∇ × E +

∂B

∂t

= 0,

(1)

∇ · B = 0,

(2)

∇ · E =

ρ

ǫ

0

,

(3)

∇ × B = µ

0

(J + ǫ

0

∂E

∂t

),

(4)

where

E

istheele tri eld,

B

themagneti eld,

ρ

the hargedensity,and

ǫ

0

and

µ

0

representthepermittivityandthepermeabilityoftheemptyspa ewithvalues

8.85

10

−12

Fm

−1

and

4π10

−7

Hm

−1

, respe tively (the magneti permeability

µ

0

of braintissues is onsidered identi al tothat ofthefree spa e).

Negle tingtheee tsof thetime-dependent termsisthequasi-stati

approx-imation ofMaxwell's equations. Thisdepends on thetypi al frequen y range of

the signals of interest and the properties of the medium. The frequen y of the

signals obtained from bio-ele tromagneti measurements in MEG and EEG are

typi ally below 1 KHz. It has therefore been veried that the physi s of MEG

and EEG are well des ribed using the quasi-stati approximation of Maxwell's

equations [44 ℄. Quasi-stati Maxwell'sequations an bewritten as:

∇ · B = 0,

(6)

∇ · E =

_ǫ

ρ

0

,

(7)

∇ × B = µ

0

J.

(8)

Equation (5) an further be satised by representing the ele tri eld

E

as thegradient of as alarfun tion

V

:

E

=

−∇V.

(9)

From (8 ), we obtain the relation between the urrent distribution

J(r

′

₎

at

point

r

′

and the magneti eld

B(r)

measuredat

r

whi h reads:

B(r) =

µ

0

4π

Z

J(r

′

)

×

r

− r

′

||r − r

′

_||

3

dv

′

_,

(10)

where

||.||

representsthe Eu lidean norm.

Thisrelationship (10 ) ispopularly known asBiot-SavartLaw.

The urrent distribution

J(r)

an be dividedinto two parts:

1. Primary urrent

J

p

_(r)

produ ed bythe neural urrent a tivity;

2. Volume urrent

J

v

_(r)

produ edinallthevolumetoprevent hargebuildup.

Primaryandse ondary urrentsareshowninFigure5.b. The urrentdistribution

J(r)

now an be representedas

J(r

′

) = J

p

(r

′

) + J

v

(r

′

) = J

p

(r

′

) + σ(r

′

)E(r

′

) = J

p

(r

′

)

_{− σ(r}

′

)

_{∇V (r}

′

),

(11)

where

σ(r

′

₎

is the ele tri al ondu tivity of the tissue at lo ation

r

′

, whi h we

will onsidertobeisotropi throughout thisthesis. SeeFigure7) wherethehead

onsistsof regionsof onstant ondu tivities

σ

i

, i = 1, 2, . . . , N + 1

.

Now we an rewrite the Biot-Savart equation (10) and use (11 ) to divide it

into twoparts: therstpart onsistsof

B

0

(r)

,themagneti elddueto primary urrentsonlywhilethese ondtermisduetothe ontributionofvolume urrents,

formed asasumofsurfa eintegralsoverthe brain-skull,skull-s alpands alp-air

boundaries. In fa t,wehave

B(r) = B

0

(r) +

µ

0

4π

X

ij

(σ

i

− σ

j

)

Z

S

ij

V (r

′

)

r

− r

′

||r − r

′

_||

× dS

′

ij

.

(12)

Figure7: Shell modelof the head.

This general equation states that the magneti eld an be al ulated if we

knowthe primary urrent distributionandthepotential

V (r

′

₎

on allthesurfa es

S

ij

. We an reate asimilar equationfor thepotential itself,yielding

(σ

i

+ σ

j

)V (r) = 2σ

0

V

0

(r)

−

1

2π

X

ij

(σ

i

− σ

j

)

Z

S

ij

V (r

′

)

r

− r

′

||r − r

′

_||

× dS

′

ij

,

(13)

where

V

0

(r)

isthe potential at

r

due to the primary urrent distribution. Ifwespe ifyaprimary urrent distribution

J

p

_(r

′

₎

,we an al ulateaprimary

potential and aprimary magneti eldasfollows

V

0

(r) =

1

4πσ

0

Z

J

p

(r

′

)

·

r

− r

′

||r − r

′

_||

× dS

′

ij

,

(14)

B

0

(r) =

µ

0

4π

Z

J

p

(r

′

)

_·

r

− r

′

||r − r

′

_||

× dS

′

ij

.

(15)

Theprimary potential is thenused to solve (13)for the potentials on all the

surfa es, and therefore ompletes the resolution of the forward problem. These

solve (12) for theexternal magneti elds. Unfortunately,the solution to (13 )is

analyti onlyinaspe ialshapesandellipti volume ondu torandmustotherwise

besolved numeri ally. Thisthesiswill onsiderusingspheri al headmodels only.

Inthe next two se tions,modelsfor neural urrent distribution will be

intro-du ed and subsequently modelsfor volume ondu tor will be dis ussed.

Modeling primary urrents

Consider a small pat h of a tive ortex

S(r

′

₎

entered at

r

′

and an observation

point

r

at some distan e from this pat h. The primary urrent distribution in this ase an be well representedbythe multipolar representation

Ω

n

S((r

′

₎₎

given by

Ω

n

_S((r

′

₎₎

=

1

n!

Z

r

′

_⊂S((r

′

₎₎

(r

′

_{− l)}

n

J

p

(r

′

)dr

′

,

(16)

where

l

isthe point ofexpansion for multipoles.

It is important to note that the brain a tivity does not a tually onsist of

dis rete sets of physi al urrent dipoles, but rather that the dipole is a

onve-nient representation for oherent a tivation ofa largenumberof pyramidal ells,

possiblyextendingovera fewsquare entimeters ofgraymatter.

If the primary urrent distribution is very fo al then it an be well

approxi-mated by anequivalent urrent dipole(ECD) dened as:

Ω

0

= q

≡

Z

J

p

(r

′

)dr

′

.

(17)

TheECD an be representedasa point sour e

J

p

(r

′

) = qδ(r

′

− l),

(18)

where

δ(r)

is the Dira delta distribution. Note that an ECD is a multipolar expansion oforder

0

.

Ifthe urrent distributionis not fo al,thenmultipolar expansionsarebetter

suited for themodeling ofneural sour es. The ontributions reported[74, 54,53℄

Figure 8: Current Distribution

S(r

′

₎

entered at

r

′

and measured at

r

. Adapted from [54℄. Head modeling

Spheri al head model

Headmodeling usingasspheri al approximationofits geometry hasbeenwidely

used in the MEG ommunity, the reason for its popular use is the simpli ity it

oerswithrespe tto omputationrequirements. Computings alppotentialsand

indu edmagneti eldsrequiresolvingtheforwardequations(13)and(12)

respe -tively for a parti ular sour e model. We have seenabove that when the surfa e

integralsare omputed overrealisti headshapes, theseequationsmustbesolved

numeri ally. However, analyti solutions exist for simplied geometries, su h as

when the head is assumed to onsist of a set of nested on entri homogeneous

spheri al shells representing brain, skull, and s alp respe tively. These models

are routinely used in most lini al and resear h appli ations to E/MEG sour e

lo alization. Figure 9des ribes aspheri al head model approximation. Consider

spheri al head,and a MEG systeminwhi h we only measurethe radial

ompo-nent of the external magneti eld, i.e., the oil surfa e of the magnetometer is

oriented orthogonally to a radial line from the enter of the sphere through the

enter of the oil. It is relatively straightforward to showthat the ontributions

of thevolume urrents vanish inthis ase, andwe areleft withonly theprimary

term. Taking the radial omponent ofthis eld forthe urrent dipole redu es to

theremarkablysimple form:

B

r

(r) =

r

r

· B(r) =

r

r

· B

0

(r) +

µ

0

4π

·

X

ij

(σ

i

− σ

j

)

Z

S

ij

V (r

′

)

r

r

r

_{− r}

′

||r − r

′

_||

× dS

′

ij

.

(19)

Figure 9: Spheri al head model,where a sphereis ttedtothe head

geom-etry.

In this same ase, it is very simple to show that the ontribution of volume

urrents will also redu eto zero. Hen e these ond term in 19 vanishes and this

equation writethe following simpler form:

B

r

(r) =

r

r

· B

0

(r) =

µ

0

4π

r

× r

′

r

_{||r − r}

′

_||

3

· q.

(20)

Noti e here that the magneti eld

B

r

(r)

is linear with respe t to the dipole moment

q

buthighly nonlinearwithrespe tto dipolelo ation:

r

q

.

In nutshell,

B

r

(r)

is zero everywhere outside the head if

q

points towards theradialdire tion

r

q

. Amore general resultisthatradially-orienteddipoles do not produ e anyexternal magneti eld outside a spheri ally symmetri volume

ondu tor, regardlessofthe sensororientation [89℄.

Importantly,this isnot the asefor EEGwhi h issensitive to radialsour es,

whi hdemonstratesoneofthe omplementarydieren esbetweenMEGandEEG

prin iples.

Realisti head model

In reality, the head has anisotropi tissue properties, is inhomogeneous and not

spheri albutsurprisingly,thespheri alapproximationworksreasonablywell,

par-ti ularly for MEG, whi h is less sensitive than EEG to volume urrents. These

latter are more ae ted than primary urrents by deviations from the idealized

model. By usingthe individual MRIdata fromthesubje t, itispossibleto

on-stru t amore detailed head model byisolating dierent regions ofinterest using

fully-automati segmentationte hniques[16 ℄. Figure10showstypi alsurfa eand

volume tessellations for usewith BEM and FEM(see [33℄ for a omplete review

of thehead geometriesusedinMEG).

Two typesofapproa hesareavailable for realisti head modeling:

1. BoundaryElementMethod(BEM)BEMisanumeri alte hniqueofsolving

linearpartialdierentialequationswhi hhavebeenformulatedina

bound-ary integral form. Normally in MEG, single-shell and three-shell BEM

methods are used. BEM methods still assume homogeneity and isotropy

within ea h region of the head. It therefore ignores, for example, the

on-du tivity anisotropy indu ed by white matter tra ts, where ondu tion is

higher along axonal bers ompared to a transverse dire tion. Similarly,

2. Finite Element Method (FEM)) FEM is a numeri al te hnique for nding

approximatesolutions ofpartialdierential equations(PDE).InFEM,

dis- retizationof thePDE isperformedintheentire head volume. Anisotropy

and heterogeneity in dierent tissue types an therefore be modeled and

therefore represents a very omprehensive approa h to solving theMEEG

forward problem.

Typi ally,BEM andFEM al ulationsareverytime onsumingandtheiruse

may be onsidered as impra ti al when in orporated as part of an iterative

in-verse solverfor urrent sour es. In fa t, through use of fastnumeri al methods,

pre- al ulation,andlook-uptablesandinterpolationofpre- al ulatedelds,both

FEM and BEM an be made quite pra ti al for appli ations in MEG and EEG

[31℄. One problemremains: thesemethods reauirethe ondu tivity propertiesof

head tissues be known. Most of head models used in the bio-ele tromagnetism

ommunity onsider typi al values for the ondu tivity of the brain, skull and

skin. Skull is typi ally assumed to be 40 to 90 times more resistive than brain

and s alp, whi h are assumed to have similar ondu tive properties. These

val-ues were measured in vitro from postmortem tissue samples, with ondu tivity

values that may be signi antly altered from those in in vivo tissues however.

Consequently,somere ent resear heortshavefo usedoninvivomeasurements

of tissue ondu tivity. Ele tri al Impedan e Tomography (EIT) pro eeds by

in-je ting a small urrent (1-10 mi roA) between pairs of EEG ele trodes and by

measuring the resulting potentials at all ele trodes. Given a model for the head

geometry, EIT solves an inverse problem by minimizing the error between the

measuredpotentialsontherestoftheEEGleadsandthemodel-based omputed

potentials, interms of parameters of the ondu tivity prole. Simulation results

with three or four-shell spheri al head models have demonstrated the feasibility

of this approa h thoughtheasso iated inverseproblem isalso fundamentally

ill-posed [32 ℄. These methods are readily extendible to realisti surfa e models as

of spatiallyvaryinganisotropi ondu tivity. A se ondapproa hto imaging

on-du tivity is to usemagneti resonan e. One te hnique uses the shielding ee ts

of indu ed eddy urrents on spin pre ession and ould in prin iple help

deter-mine the ondu tivity prole at anyfrequen y [113℄. The se ondte hnique uses

diusion-tensor imaging withMRI(DT-MRI) thatprobes themi ros opi

diu-sion properties of water mole ules within thetissues of the brain. The diusion

values an then be tentatively related to the ondu tivity of these tissues [100℄.

None ofthese MR-based te hniques have rea hed ommon pra tise by far . F

ur-ther, giventhepoorsignal-to-noiseratio(SNR)oftheMRinboneregions,whi h

is of riti alimportan e forthe forward EEGproblem, thepotential for fully3D

impedan etomography withMRremains spe ulative.

Figure10: (a) FEM modelingof the forward model; (b) BEM modelingof

Linear formulation

Theforwardproblemnow anbeexplainedusingthemodelsforsour esandhead

geometry dis ussed above. Themagneti eldand s alppotential measurements

are linear with respe t to the dipole moment

q

and nonlinear with respe t to its lo ation

r

′

. For larity, it is onvenient to separate the dipole magnitude

q =

_||q||

from itsorientation

u

= q/

||q||

,whi hwewrite inspheri al oordinates by

Θ = [φ, ρ]

. Let

b(r)

denote the magneti eld generated by a dipole having xed orientation

Θ

:

b(r) = g(r, r

q

, Θ)q,

(21)

where

g(r, r

q

, Θ)

isa leadeldsolutionof the magneti eldfor adipolehaving unit amplitude andorientation

Θ

.

ForN dipoleslo atedat

r

q

i

,their ombinedmagneti elds anbeexpressed

using linearsuperposition ofMaxwell's equations as

b(r) =

N

X

i=1

g(r, r

q

i

, Θ

i

)q

i

.

(22)

The simultaneous MEGmeasurements made at m sensors for N dipoles, an be

expressedas B

=











B(r

1

)

. . .

B(r

m

)











=











G(r

1

, r

q1

, Θ

1

)

. . .

G(r

1

, r

qN

, Θ

N

)

. . . . . . . . .

G(r

m

, r

q1

, Θ

1

) . . . G(r

m

, r

qN

, Θ

N

)





















q

1

. . .

q

p











.

(23)

It an be written inamatrix form as

B

=

G

(

{r

qi

, Θ

i

})

J

,

(24)

where

G(

{r

qi

, Θ

i

})

isthe

m

× N

gainmatrixrelatingN dipolestothem sensors. Ea h olumn ontains the ontribution of one dipoleto ea hsensorin thearray.

The matrix

J

ontains thesetof instantaneous amplitudes ofall thedipoles. Inthismodel,theorientationofthedipoleisnotafun tionoftime. Thistype

of modelis often referredto asa "xed" dipole model. Alternative models that

Inverse problem

To produ e estimates of the neural urrent sour es that generated the observed

MEGsignals,wemustsolve theasso iated quasi-stati ele tromagnetism inverse

problem. The inherent ill-posedness of this problem, oupled with the limited

number of spatialmeasurements available with urrent MEGand EEG systems,

(150-300 measurements) and signal-to-noise ratio (SNR) make this estimation

very hallenging [44℄.

The solutions to the neuromagneti inverse problem will depend on whi h

forwardmodelisused. Infa t,agiveninversealgorithmwillyieldslightlydierent

results if dierent forward models are used; hen e, the importan e of using an

a urate realisti forward model. However, these two problems are relatively

independent of one another. In the forward problem, we attempt to model the

lassi al physi s ofMEGandEEGasrealisti allyaspossible. In ontrast, inthe

inverse problem, we often deal with purely mathemati al on epts and a priori

assumptions that are in orporated in a sour e model. The independen e of the

inverse problem from the model's physi s allows one to use the same inverse

algorithm for MEG or EEG. On the other hand, many dierent estimates of

a tivity anbeobtainedforaparti ulardatasetusingdierentinversealgorithms

but sharing the same forward model. This brings us to the main issue with

neuromagneti inverseestimation: nonuniqueness. Thereisnounique solutionto

the physi ally and mathemati ally ill-posed neuromagneti inverse problem. In

fa t,aninnitenumberof urrentsour edistributions anintheorygenerateany

parti ular magneti eld measurement ve tor due to the existen e of magneti

silentsour es [47 ,44, 89℄.

In both MEG and EEG, silent sour es an be added to any given inverse

solution without hanging the forward eld and/or potential that the ombined

sour e generates. Thus, there are indeed an innite number of solutions that

explain anygiven MEG/EEGdataset equallywell. Therefore, a priori

though mathemati ally unique solutions an be obtained by postulating spe ial

sour e properties, physi al non-uniqueness is intrinsi to the neuromagneti

in-verse problem.

The two major approa hes to the estimation of neural urrent sour es are

"imaging" and "parametri /lo alization"methods.

Imaging methods typi ally onstrain sour es to a tessellated surfa e

represen-tation of the ortex, assume an elemental urrent sour e in ea h area element

(vertex) normal to the ortex surfa e, and solve the linear inverse problem that

relates these urrent sour es to the measured data. A urate tessellation of the

ortexrequiresontheorderof

10

5

elements. Sin ethemaximumnumberofMEG

sensors is about 300, theproblem is highlyunder-determined. By using

regular-ized linear methods based on minimizing a weighted

l

2

-norm on the image, we

an produ e unique stablesolutions.

Parametri /lo alization methods assume a spe i parametri form for the

sour es. Byfarthemost widelyusedmodels inMEGaremultiple- urrent-dipole

approa hes[112 ,90℄. Theseassumethatthenumberofneuralsour esisrelatively

smalland ea hsu iently fo althatthey anberepresentedbyafewequivalent

urrent dipoles with unknown lo ations and orientations. In both imaging and

parametri methods, the MEG/EEGforward problem an be written as

B

= G(θ)J + ǫ,

(25)

where

B

isthe

M

×time

ve torrepresentingMEGmeasurements,

J

isthe

N

×time

ve tor representing thedistribution urrents. For imaging methods, itisthe

am-plitude ofelementary urrentsat ea h orti al vertex. In parametri methods,it

isthevaluesofamplitudeparametersforea h urrentmodelelement.

G(θ)

isthe

M

_×N

leadeldmatrixrelatingadditionalparametersofthe urrentdistribution to the magneti eld measured by

M

sensors.

θ

gathers the parameters whi h the lead elds depend uppon, i.e., urrent sour es, lo ations

r

q

i

, orientations

Θ

i

andtheiramplitudes

q

i

. The

M

× time

noiseve tor

ǫ

representsa ombination of

a tivity ofheart andeyes, et ) onsensors.

Parametri methods

Parametri methods an be broadly lassied into "Dipole tting" and

"Beam-forming".

Dipole tting

Therstinversemethodforequation(25 )isbasedontheassumptionthatneural

a tivity an be modeled by a few sparse, elementary sour es

α

. The problem redu es to theestimationfromthedataof theparameters

θ

for

α

sour es, whi h are des ribed as their positions

r

q

i

, their orientations

Θ

i

and their amplitudes

q

i

(with

i

∈ [1, α]

). This may be written as an optimization problem of a ost

fun tion to be minimized.

Theestimateinthe least-squares (LS)sense writes:

J(θ)

_LS

= arg min

J

||B − G(θ)J||

2

F

(26)

where

||.||

F

denotes the Frobenius norm. Let

G

+

_(θ)

be the pseudo-inverse of

G(θ)

:

G

+

(θ) = US

+

V

t

,

(27) where

USV

t

is the singular value de omposition (SVD) of

G(θ)

and

S

+

is the

diagonal matrix ontaininginverse ofsingularvalues of

G(θ)

[39℄. Equation(26) an bewritten intheform:

J(θ)

_LS

=

_{||B − G(θ)[G}

+

(θ)B]

_||

2

_F

=

_{||(I − G(θ)G}

+

(θ))B

_||

2

_F

,

(28)

where

I

istheidentitymatrix of rank

α

. Thus, theLSproblem an be optimally solvedinthelimitedsetofnonlinearparameters

r

q

i

,

Θ

i

withaniterative minimiza-tion pro edure. The linear parameters in

q

i

are then optimally estimated from 26;see[75 ℄. MinimizationmethodsrangefromMarquardt-Levenbergand

Figure 11: (a) Dipole Fitting in axial view; (b) Dipole Fitting in oronal

view ( ) Dipoletting insagittal view.

Thisleast-squaresmodel aneitherbeestimatedfromdatafromasingletime

snapshot or atimewindow. When applied sequentiallyto a setof timesamples,

thisresultsina"movingdipole"model,sin ethelo ationisnot onstrained[112 ℄.

Alternatively, by using a ontiguous time blo k of data in the least-squares t,

the dipole lo ations an optionally be xed over the entire interval. The xed

and moving dipole models have both proven useful inboth EEG and MEG and

remain the most widely usedapproa hes to pro essing experimentaland lini al

data. A key problem with the LS method is that the number of sour es to be

used must be de ided a priori. Estimates an be obtained by looking at the

ee tive rank of thedatausing a SVDor through information-theoreti riteria,

su iently largenumberofsour es an be madetot anydataset, regardlessof

its quality. Furthermore, as the number of sour es in reases, the non- onvexity

of the ost fun tion results inin reased han es of trapping in undesirable lo al

minima. This latter problem an be approa hed using sto hasti or multistart

sear h strategies [50℄. The alternatives to LS des ribed below avoid the

non- onvexity issue by s anning a region of interest that an range from a single

lo ation to the whole brain volume for possible sour es. An estimator of the

ontributionofea hputativesour elo ationtothedata anbederivedeithervia

spatial ltering te hniques or signal lassi ation indi es. An attra tive feature

of these methods is that they do not require a prior estimate of the number of

underlying sour es.

Beamforming approa hes

A beamformer performsspatial ltering on data from asensor array to

dis rim-inate between signals arriving from a lo ation of interest and those originating

elsewhere. Beamformingoriginated inradar and sonar signalpro essing but has

sin e found appli ations in diverse elds ranging from astronomy to biomedi al

signalpro essing [103℄.

Mat h lter

Thesimplestspatiallter,amat hedlter,isobtainedbynormalizingthe olumns

of the lead eld matrix and transposing this normalized di tionary. The spatial

lter for lo ation

r

i

is given by

W

_i

(T )

=

G

T

: i

kG

: i

k

F

.

(29)

Thisapproa hessentiallyproje tsthe dataontothe olumnve torsofthe

di tio-nary. Although thisguarantees thatwhen onlyone sour e isa tive,theabsolute

the spatial resolution of the lter is so low given the high orrelation between

di tionary olumns. Thisapproa h an beextendedto fast re ursive algorithms,

su h as mat hing pursuit and its variants, whi h sequentially proje t the data

or residual to the non-used di tionary olumns to obtain fastsuboptimal sparse

estimates.

Multiple signal lassi ation (MUSIC)

The MUSIC algorithm was adopted from spe tral analysis, Dire tion of

Ar-rival(DOA)estimation te hniquesand modiedfor spatiallteringof MEGdata

[75, 73℄. TheMUSIC ost fun tion isgivenby

W

_i

(T )

=

I

− U

_s

U

T

_s

G

_{: i}

2

2

kG

: i

k

2

2

=

P

⊥

_U

s

G

: i

2

2

kG

: i

k

2

2

,

(30) where

B

= USV

T

isthesingularvaluede ompositionofthedata,

U

s

isamatrix with the rst

d

s

right singular ve torsthat formthe signalsubspa e, and

G

: i

is thegainve torforthedipolelo atedat

r

i

andwithorientation

θ

i

(obtainedfrom anatomy or using thegeneralized eigenvalue de omposition). The operator

P

⊥

U

s

is an orthogonal proje tion operator onto the datanoise subspa e. TheMUSIC

mapis there ipro alofthe ost fun tionat all lo ationss anned. Thismap an

be usedto guidea re ursive parametri dipoletting algorithm. The number

d

s

is usuallyset byan expert user.

For more omplete explanation of subspa e methods likeMUSICsee [55℄.

Linearly onstrained minimum-varian e (LCMV)

Beamformers,asusedintheeldofbrainimaging,arespatiallteringalgorithms

that s anea h sour e-point independently to passsour e signalsat a lo ation of

interestwhilesuppressinginterferen efromotherregionsusingonlythelo algain

ve tors and the measured ovarian e matrix. One of the most basi and often

Figure12: A typi alMUSIC s an for epilepti spikes.

to a unitygain onstraint:

min

W

_{i :}

tr W

i

:

Σ

B

W

T

i

:

subje tto

W

i

:

G

: i

= I,

(31)

where

Σ

B

isthe data ovarian e matrix,

G

: i

isthe

d

b

by3gainmatrix ofthe

i

th

sour e point, and

W

i

:

is the 3 by

d

b

spatial ltering matrix [104 ℄. The solution to this problemis given by

W

_i

(T )

= G

T

_{: i}

Σ

B

−1

G

: i

−1

G

T

_{: i}

Σ

B

−1

.

(32)

The parametri sour e a tivity at the

i

th

a tivity. Thisbeamformingapproa h anbeextendedtoamoregeneral Bayesian

graphi al model that uses event timing information to model evoked responses,

whilesuppressinginterferen eandnoise sour es[115℄. Thisapproa husesa

vari-ationalBayesian EMalgorithm to ompute thelikelihoodofadipoleatea hgrid

lo ation.

Imaging methods

Imagingapproa hestotheMEGinverseproblem onsistofmethodsforestimating

the amplitudesof a dense setof dipoles distributed at xed lo ationsand

orien-tation within the head volume. In this ase, sin e thelo ations and orientation

are xed, only the linear parameters need to be estimated and theinverse

prob-lem redu estoalinearone withstrongsimilaritiesto thoseen ounteredinimage

restoration and re onstru tion. By putting lo ations and orientation onstraint

theequation (25)be omes

B

= GJ + ǫ.

(33)

Herethegainmatrix

G

isxedandonlydipoleamplitudes

J

havetobeestimated. The most basi approa h onsists of distributing dipoles over a predened

volumetri grid similar to theones usedin s anning approa hes. However, sin e

primary sour es are essentially restri ted to ortex, the image an be plausibly

onstrainedto sour eslyingonthe orti alsurfa e,asextra tedfroman

anatom-i al MR images of the subje t [22℄. Following segmentation of the MR volume,

dipolar sour esarepla ed at ea h node ofa triangular tessellationof thesurfa e

ofthe orti almantle. Sin ethepyramidal ellsthatprodu e themeasuredelds

areorientednormalto thesurfa e,we anfurther onstrainea hofthese

elemen-tal dipolar sour esto benormal to the surfa e. The highly onvoluted nature of

the human ortex requires that a high-resolution representation ontains of the

order of ten to one hundred thousand dipole "pixels". The inverse problem is

this hasbeen a omplished through the use of regularization or Bayesian image

estimation methods.

Bayesian formulation

Bayesian approa h to neuronmagneti inverse problem was rst introdu ed by

Clarkein1989[14℄. IntheBayesianformalism,theneuromagneti inverseproblem

isdened astheproblemofestimating thematrix

J

of dipoleamplitudesat ea h tessellation element from the spatio-temporal data matrix

B

, whi h arerelated inthenoiseless aseby

B

= GJ

. The

i

-throwof

J

ontains theamplitude image a rossthe ortexattime

i

. FromBayestheorem,theposteriorprobability

p(J

|B)

for theamplitude matrix

J

onditioned onthedata

B

is given by

p(J

|B) =

p(B

|J)p(J)

p(B)

,

(34)

where

p(B

|J)

gives the forward probability density of getting magneti eld

B

onditioned on

J

.

p(J)

is a prior distribution ree ting our knowledge of the statisti al properties of the unknown image. While Bayesian inferen eoers the

potentialforafullstatisti al hara terizationofthesour esthroughtheposterior

probability, images are typi ally estimated in pra ti e by maximization of the

posterior or log-posterior probability.

Theestimation of

J

inthe maximum a posteriori (MAP)sense isgiven by

ˆ

J

M AP

= arg max

J

p(B

|J)p(J).

(35)

The log-likelihood of (35)isgiven by

ˆ

J

M AP

= arg max

J

(log[p(B

|J)] + log[p(J)]).

(36)

Typi ally, MEG and EEG data are assumed to be orrupted with additive

Gaussian noise that we assume here to be spatially identi ally distributed over

all sensors (generalization is straightforward). The log-likelihood is thensimply

given,within a onstant, by

ln[p(B

_{|J)] = −}

√

1

2σ

2

||B − GJ||

2

The priorisa probabilisti modelthatdes ribesour expe tations on erningthe

statisti alpropertiesofthesour eforwhi hwewillassumeanexponentialdensity

p(J) =

1

z

exp[

−βf(J)],

(38)

where

z

and

β

and

f (J)

depends on the image

J

. This form en ompasses both multivariate Gaussian models and the lass of Gibbs distributions or Markov

random eld models [13℄. Combining the log-likelihood and log-prior gives the

generalformofthenegativelog-posteriorwhoseminimizationyieldsthemaximum

a posteriori estimate:

ˆ

J

M AP

= arg min

J

||B − GJ||

2

F

+ λf (J),

(39) where

λ = 2βσ

2

.

λ

is the regularization parameter. The parameter

λ

should be onsidered as a regularization parameter tuning between the prior

f (J)

and t to the data. If

λ = 0

estimation of the urrent distribution be omes simply least squares. This type of solution to the inverse problems was introdu ed by

Tikhonovin[97℄.

Choi e of the regularization parameter

λ

There are many approa hes to estimate the valueof

λ

. We summarize a few as explained below:

1.L-Curve: Whenplotted onalog-logs ale,theparametri urveofoptimal

valuesof

||W||

anddatat

||B − GJ||

oftentakesonanLshape. Forthisreason, the urve is alledanL- urve[45 ℄. The value of

λ

intheL- urve riterion isthe value of

λ

that gives the solution losest to the orner of the L- urve, asshown inFigure 13 .

2. Generalized ross validation (GCV) is an alternative method for

estimating the regularization parameter

λ

[107 ℄, that has a number of desirable statisti al properties. Consider

f (λ) =

||B − GJ||

T race(I

_{− GG}

†

₎

=

V (λ)

Figure13: Typi alL- urve for lassi shaw inverse problem.

The numerator in (14) is the data mist in the least squares sense and the

dominator measures the loseness of the data resolution matrix to the identity

matrix. In the GCV method, we pi k the value of

λ

that minimizes (14), as shown inFigure14 .

Linear estimators

Thesimplest approa h to(39) isto onsider priordistributionof sour e

ammpli-tudes

J

to be Gaussian withzeromean. Introdu e

f (J) = tr[JC

−1

_J

J

t

],

(41)

where

C

−1

J

is the inverse ovarian e matrix of sour es. If we break this inverse matrix as,

C

−1

J

= WW

t

,then(39 ) an be written inthefollowing manner:

ˆ

J

M AP

= arg min

J

||B − GJ||

2

Figure14: Typi al GCV- urve for lassi shawinverse problem.

The MAPestimator now takesthefollowing simple linearform:

ˆ

J

t

_{M AP}

= WW

t

G

t

(GWW

t

G

t

+ λI)

−1

B.

(43)

Inthis ase,

J

ˆ

M AP

alsofollowsaGaussiandistribution. (39 )isnormallyknownas zeroth orderTikhonovregularized solutionof

J

[97 , 26 ℄,where theregularization parameter

λ

anbeestimatedfromanyofthete hniquesexplainedintheprevious se tion.

Properties of the sour e ovarian e matrix

Sour e ovarian e is the last parameter of the model whi h will ondition the

1. The identity matrix,whi h yields lassi alminimum-normestimators [97℄.

Themajorassumptioninusingtheidentitymatrixisthatsour eamplitudes

J

areindependent andidenti ally distributed.

In Figure 15 ,a omparison is shown between LCMV beamformer and the

minimum-norm solution to the inverse problem, showing that though the

minimum-norm solution is widespread, the peak of maximum intensity is

in the right pla e in this median nerve stimulation experiment, where we

expe t a tivitywithin primarysomatosensory areas.

Figure 15: Comparison of LCMVand minimum norm.

2. A diagonal matrix whose elements are given by the norm of the elements

of the orresponding olumn in the lead-eld matrix (i.e.,

W

ii

=

||g

i

||

2

with

g

i

the

i

th

olumn of

G

). This solution is a forward-eld normalized solution.

3.

W

whi h isbasedontherelationship between sour e neighbors[108 ℄. The matrix

W

is given by

W

ij

=



















1

if

i = j,

−

n

1

if

j

∈ N (i),

0

otherwise

,

where

N (i)

denestherstorderneighborof

i

th

sour eand

n =

Card[N (i)]

.

4.

W

is diagonal with elements equal to some estimate of the sour e power at thatlo ation,whi hmaybe omputed fromtheoutputofabeamformer

or MUSICs anevaluated for ea hdipolepixel [69 ℄or weighted fromother

fun tional imagingmodalitiessu hasfMRI, PET, orSPECT [64 , 21 ℄.

Thesemethodshavetheadvantagetobefastandoverallrobusttowardsnoise

[106 ℄. They provide estimates where the enter of gravity of thea tivity is very

lose to the true sour e. However, results are often very smooth spatially and

do not allowfor estimation of thespatial extent ofthe a tivity. This problemof

spatial extent andits solutionwill beaddressed indetailsinChapter 2.

Nonlinear estimators of sour e amplitudes

Itispossibletoobtainsparserimageestimatesofthe urrentdistributionbyusing

alternative (non-quadrati ) ost fun tions

f (J)

in (39). Norms and semi-norms on sour e amplitude priors with values

p

≤ 2

in (42 ) have been investigated. Solutions will be ome in reasingly sparse as

p

is redu ed. For the spe ial ase of

p = 1

, the problem an be slightly modied to be re ast asa linear program. Thisisa hievedbyrepla ingthequadrati log-likelihoodtermwithasetof

under-determined linear inequality onstraints, where the inequalities ree t expe ted

mismat hesinthettothedataduetonoise. The

l

1

- ost anthenbeminimized

overthese onstraintsusing a linearsimplexalgorithm. Properties of linear

pro-grammingproblems guaranteethatthereexistsanoptimalsolutionfor whi hthe

number of non-zero pixels does not ex eed the number of onstraints, or

equiv-alently the number of measurements. Sin e the number of pixels far outweighs

thenumberofmeasurements, thesolutionsarethereforeguaranteedtobesparse.

This idea an betaken even further byusing thequasi-norm for valuesof

p < 1

. In this ase, it is possible to show that there existsa value

0 < p < 1

for whi h

Anotherapproa hdened liquishrelationshipsbetweenneighborhoodsour es.

The whole network of sour esmay be des ribed asdistributed within a Markov

Random Field(MRF), this relationship wasexploited in [5,84 ℄. Akeyproperty

ofMRFsisthattheirjointstatisti aldistribution anbe onstru tedfromasetof

potentialfun tionsdenedonalo alneighborhoodsystem[83℄. Thus,theenergy

fun tion

f (J)

for the prior an beexpressedas

f (J) = L

N

X

i=1

[α

i

J(i) + γ

i

[

X

j∈N

(i)

(J(i)

_{− J(j))}

2

]

Q

]

(44)

where

L

isthenumberoftimesamples,

α

i

and

γ

i

determinestheweightingfa tors between neighborhood sour es.

Q

is theindex of theamplitude of the neighbor-hoodgroup.

N (i)

neighborhoodofthesour e

i

isdenedasthe

9

losestneighbors to thesour e. Therstterminequation(44 )expressessparsitywhilethese ond

one favorsfo alsour esdistributions.

TheMRF-basedimagepriorsleadtonon- onvex[5℄andinteger[83℄

program-mingproblemsin omputingtheMAPestimate. Computational osts anbevery

highfor thesemethodssin e althoughthe priorshave omputationallyattra tive

neighborhood stru tures, the posteriors be ome fully oupled through the

likeli-hood term. Furthermore, to deal with non- onvexity and integer programming

issues,someformofdeterministi orsto hasti annealingalgorithmsmustbeused

[35℄.

Con lusion

Theex ellenttimeresolutionofMEGprovidesusauniquewindowonthe

dynam-i s of humanbrain fun tions. Though the limitedspatial resolution remains the

problemforthismodality,adequatemodelingandmodernsignalpro essing

meth-odsproveMEGasadependablefun tionalimagingmodality. Potentialadvan es

inforward modeling in lude better hara terization of theskull,s alp and brain

ods for ombining MEG with other fun tional modalities and exploiting signal

Introdu tion

Theequivalent urrent dipolemodelisdire tlyinterpretable asa urrentelement

restri tedto the orti alsurfa erepresenting apointsour e. However, oneof the

per eived key limitations of this model is that, distributed sour es may not be

adequately represented. This problem was one of the prime motivations to the

development of imaging approa hes. An alternative solutionis to remain within

the model-based framework but to broaden the model to allow parametri

rep-resentations ofdistributed sour es. Themultipolar expansionprovides a natural

framework for generating these models [79, 36 ℄. Multipolar expansions are

de-rived fromspheri al harmoni softhemagneti s alarpotential. Iftheexpansion

point is hosen near the enter of a distributed sour e, then the ontribution of

higher-ordertermswilldroporapidlyasthedistan efromsour estothesensors

in reases. Using this framework we expand the setof sour esto in lude urrent

dipoles and rst-order urrent multipoles. These sour es are able to represent

theeldfromadistributedsour e morea uratelythan by urrent dipolemodel,

thoughstillbenetingfroma ompa t,low-dimensionalform[78 ℄. Multipolar

ex-pansionsofmagneti s alarpotentialsoriginate fromgeneralspheri al harmoni s

solution ofthe Poissonequation.

In this thesis, we proposed an approa h for estimating the spatial extent of

orti al urrent sour es using a hybrid methodology alled Multipole Corti al