EEG/MEG source reconstruction

EEG/MEGEEG/MEGsource reconstructionsource reconstruction

EEG/MEGEEG/MEGsource reconstructionsource reconstruction

Wellcome Dept. of Imaging Neuroscience, Institute of Neurology, UCL, London

Jérémie Mattout / Christophe Phillips / Karl FristonJérémie Mattout / Christophe Phillips / Karl Friston

jmattout

- source reconstruction = solving the EEG or MEG inverse problem.- two types of approach are proposed and they correspond to the two main models used in the litterature:Equivalent Current Dipoles&Distributed or Imaging methods- the talk focusses on the latter since the other part is Christophe work. Nevertheless, it enables to span all the needed pipeline and this approach is the one which corresponds to the 3D projection for SPM analysis.I guess you could also say that ECD methods will be mentionned in the DCM-ERP talk.

Estimating brain activity from scalp electromagnetic data

SourcesSources

‘Imaging’‘Imaging’‘Equivalent Current Dipoles’ (ECD)

‘Equivalent Current Dipoles’ (ECD)

Source Reconstruction


MEG dataMEG data

EEG dataEEG data

jmattout

Simple and general definition of the problem/objective:Reconstructing the sources or solving the inverse problem corresponds to localizing and quantifying the brain activity from the scalp electromagnetic data.It is a mathematically ill-posed inverse problem since it does not admit a unique solution, evenso one would have an infinite number of sensors!Prior information on the solution are required.- In ECD, a strong constraint usually consists of assuming a prior fixed number of active sources.- In 'Imaging', the solution space is constrained by the subject anatomy and various priors such as physiological, functional, mathematical knowledge can be introduced in a so-called regularization process.

Components of the source reconstruction process

Source modelSource model

Forward modelForward model

Inverse methodInverse method

RegistrationRegistration

‘Imaging’‘Imaging’‘ECD’‘ECD’

DataData AnatomyAnatomy

jmattout

The whole picture showing the four main components and steps of source reconstruction:- Source modeling:A source and its current density are representated by a current dipole which is parameterized by its location, orientation and amplitude.With ECD, one dipole models the activity of a large brain area whose extension is not estimated. With Imaging, dipoles are usually spread all over the cortical surface and enable one to quantify the local activity of a macro-column of pyramidal neurons (~1 million of them).- Forward modeling:To generate a full generative model of the data, one needs to model the geometrical and physical properties of the head tissues. As an example, a three sphere model can be used to represent the three layers: CSF, skull, scalp and each medium is assigned with a conductivity value (actually, it can be shown that only relative conductivity values between tissues are needed). Such a model can be solve analytically to produce the forward or lead-field matrix which is made of the potentials or fields produced by each source of the model.- Registration:To relate the reconstructed sources to the anatomy (particularly with Imaging) one needs to coregistrate the data (sensor locations) with the subject abatomy (typically given by the T1 MRI).- Inverse method:Using the coregistrated data, the source model and the forward model, source activity can be estimated. Due to the ill-posdness of the inverse problem, this reconstruction is non-trivial and constitutes the heart of the source reconstruction process.

Components of the source reconstruction process

Source modelSource model Forward modelForward model Inverse solutionInverse solutionRegistrationRegistration

jmattout

In SPM5, the Imaging pipeline requires processing the four main steps successively:- Source modelDefining a cortical mesh of dipoles for a given individual- Registrationis independent of the source model... so could be performed before- Forward modelrequires the two previous step to have been completed- Inverse solutionrequires the three previous componentsLet's consider those steps one by one...

Source model

TemplatesTemplatesIndividual MRIIndividual MRI

Compute transformation TCompute transformation T

Apply inverse transformation T-1Apply inverse transformation T-1

Individual meshIndividual mesh

- Individual MRI- Template mesh

input

- spatial normalization into MNI template- inverted transformation applied to the template mesh

- individual mesh

functions output

jmattout

The individual mesh is derived from a template mesh. Several mesh sizes are available (3000 ; 4000 ; 5500 and 7200 for now).- First the spatial deformation from the individual sMRI to the T1 MNI template is computed.- Second, the above transformation is inverted and the inverse is applied on the template mesh which is also in MNI space.The obtained mesh contains the same number of dipoles as the template one and there is a one to one mapping between the two.

Registration

Rigid transformation (R,t)Rigid transformation (R,t)

Individual MRI spaceIndividual MRI space

fiducials

Individual sensor spaceIndividual sensor space

fiducials

- sensor locations- fiducial locations(in both sensor & MRI space)- individual MRI

input

- registration of the EEG/MEG data into individual MRI space

- registrated data- rigid transformation

functions output

jmattout

A rigid transformation is applied to the sensor locations in order to registrate them into the individual sMRI space.To compute the Rotation and translation composing this transformation, one uses landmarks (or fiducials) whose position have to be known in both spaces.Those fiducials corresponds to the Nazion, the Left ear and the right ear.The sensor location and fiducial locations in the same space are typically acquired with a Polhemus system (EEG) or automatically recorded by the MEG.To know the fiducial location in sMRI space, one can use MRI markers or one have to be able to locate precisely their position on the MRIs a posteriori (by knowing where on the subject landmarks are usually chosen, usually less precised!).

Foward model

Compute foreach dipole

Compute foreach dipole

Individual MRI spaceIndividual MRI spaceModel of the

head tissue properties

Model of thehead tissue properties

+

Forward operatorForward operator

K

p

n

- sensor locations- individual mesh

input - single sphere- three spheres- overlapping spheres- realistic spheres

- forward operator K

functions

output

BrainStorm

jmattout

Having defined the source model and having coregistrated the data space and the source (MRI) space, one can compute the forward operator.In Imaging, since the dipole locations and orientations are fixed, the foward operator K is computed only once, prior to the inverse procedure.Several analytical methods will be available which are computed the way they are computed in BrainStorm software (freeware made by John Mosher, Richard Leahy from UCLA and Sylvain Baillet from CNRS in Paris...)if someone asks about it:- single sphere: the whole head is simply represented by one single sphere- three spheres...- overlapping spheres: the spheres are optimized for each sensors (one single or 3-sphere model) per sensor.- realistic spheres (Christophe approach)K is of dimension n x p, where n is the number of sensors and p the number of sources

Inverse solution (1) - General principles

1 dipole sourceper location

Cortical meshCortical mesh

General Linear Model

Y = KJ+ E[nxt] [nxp] [nxt][pxt]

n : number of sensorsp : number of dipolest : number of time samplesUnder-determined GLMUnder-determined GLM

Regularized solutionRegularized solution J : min( ||Y – KJ||2 + λf(J) )J

data fit priors

^

jmattout

We consider one dipole per location and is oriented perpendicularly to the surface (following the orientation of the cortical neuron dendrites).(I am thinking of proposing also a three-dipoles per location model, but not in the first release of the toolbox)In Imaging, the problem comes down to a GLM formulation that one has to invert.Contrarery to the fMRI GLM, this one is higly under-determined since p >> n.Regularization is required, which consists of minimizing a twofold criterion made of a data fit term and a prior term.To give rise to a realistic and reliable solution, the prior term needs to be defined carefully and the two terms have to be optimally relatively weighted.

Inverse solution (2) - Parametric empirical Bayes

2-level hierarchical model

E2 ~ N(0,Cp)

E1 ~ N(0,Ce)Y = KJ + E1

J = 0 + E2

Sensor levelSensor level

Source levelSource level

Gaussian variableswith unknown variance


Linear parametrization of the variances

Linear parametrization of the variances



Ce = 1.Qe1 + … + q.Qe

q

Cp = λ1.Qp1 + … + λk.Qp

kQ: variance components(,λ): hyperparameters

Q: variance components(,λ): hyperparameters

jmattout

SPM uses a Parametric empirical Bayesian approach, based on the following 2-level hierarchical model:- 1st level, the previous GLM- 2nd level, the unknown source parameters are considered as a gaussian random variable with zero mean (shrinkage prior) and unknown variance.Contrary to the classical Minimum Norm or Weighted Minimum Norm approach, the noise and source variances are considered unknown and are estimated together with the source amplitudes.Therfore, the variances are parameterized as a linear combination of variance components.- At the sensor level, several variance components can be considered such as an identity matrix (noise i.i.d), an empirical estimate of the noise variance (rest acquisition and/or data anti-averaging)- At the source level, all sorts of prior which can be expressed in terms of variance component can be incorporatedThe corresponding weights or hyperparameter are estimated in the PEB approach.


Bayesian inference on model parameters

Inference on J and (,λ)Inference on J and (,λ)

Model MModel MQe

1 , … , Qeq

Qp1 , … , Qp

k+J K

+,λ

F = log( p(Y|M) ) = log( p(Y|J,M) ) + log( p(J|M) ) dJ

E-step: maximizing F wrt J J = CJKT[Ce + KCJ KT]-1Y^

M-step: maximizing of F wrt (,λ) Ce + KCJKT = E[YYT]

Maximizing the log-evidenceMaximizing the log-evidence

data fit priors

Expectation-Maximization (EM)Expectation-Maximization (EM)

MAP estimateMAP estimate

ReML estimateReML estimate

jmattout

Given a model M, PEB provides inference on the parameters and hyperparameters of M.Note that M is defined by the source model (associated with J, the mesh and its particular size), the forward calculation attached to it and the considered variance components.Inference is made by maximizing the log-evidence (maybe you should show what is the evidence in the Bayes law !?). In the formulation of the log-evidence one recognizes the two terms of a regularization process.In SPM, this is solved using an iterative EM algorithm.The E-step provides the Maximum A Posteriori estimate of the source amplitudes.The M-step provides the ReML estimates of the hyperparameters, accounting for the loss of degrees of freedom due to the E-step and so that the predicted data covariance matrix should fit the empirical data covariance best.Importantly, the PEB approach enables the user to accomodate multiple priors and to weight their contribution optimally, according to the data (data-driven estimate of the hyperparameters, contrary to the classical and so-called L-curve approach).


Bayesian model comparison

B12 =p(Y|M1)p(Y|M2)

Model evidenceModel evidence

• Relevance of model M is quantified by its evidence p(Y|M) maximized by the EM scheme

Model comparisonModel comparison

• Two models M1 and M2 can be compared by the ratio of their evidence

Bayes factorBayes factor

Model selection using a‘Leaving-one-prior-out-strategy‘

Model selection using a‘Leaving-one-prior-out-strategy‘

jmattout

Finally, given a particular data set, PEB allows evaluating the relevance of the model M. Indeed, the higher the maximized log-evidence, the more relevant the model.Then, model comparison can be performed and two models can be quantatively compared using their Bayes factor (the ratio of their log-evidence). The Bayes factor can then be interprated probabilistically as proposed by Kass & Raftery (1995) or as done with DCMs (Penny 2004). For instance, if B12 = 1, there's no evidence in favor of any of the model. If B12 > 20, there is a strong evidence in favor of model M1.This becomes very useful for chosing the optimal sets of priors.Indeed, based on Bayes factor, one can adopt a 'Leaving one out strategy' and thus evaluate the effect of each prior and finally chose the prior model which yields the higher evidence.

Inverse solution (5) - implementation

- preprocessed data- forward operator- individual mesh- priors

input

- compute the MAP estimate of J- compute the ReML estimate of (,λ)- interpolate into individual MRI voxel-space

- inverse estimate- model evidence

functions output

- iterative forward and inverse computation

ECD approach

jmattout

Well, just read the slide more or less... with a mention of Christophe's approach that will be also available. Note that for the ECD approach, the foward operator will have to be recomputed along the iterative inverse process.Note also that interpolation from the mesh to the MRI voxels can be performed... then enabling to derive SPMs.Thanks to the use of a template mesh, this can be easily performed directly into the MNI template for inter-subject comparison or group analysis.

Conclusion - Summary

Data spaceData space MRI spaceMRI space

RegistrationRegistration

Forward modelForward model

EEG/MEG preprocessed data

EEG/MEG preprocessed data

PEB inverse solution

PEB inverse solution

SPMSPM

jmattout

Well, I guess you just need to rephrase and summarize the whole pipeline.Note that the approach yields equivalently to SPMs in individual or template space as well as to PPMs (Guillaume's talk on PPMs will just preceed yours on source localization).A group analysis is also possible, well everything which SPM_EEG offers can be applied...Bravo!

Estimating brain activity from scalp electromagnetic data

SourcesSources

‘Imaging’‘Imaging’‘Equivalent Current Dipoles’ (ECD)

‘Equivalent Current Dipoles’ (ECD)



MEG dataMEG data

EEG dataEEG data

jmattout

Simple and general definition of the problem/objective:Reconstructing the sources or solving the inverse problem corresponds to localizing and quantifying the brain activity from the scalp electromagnetic data.It is a mathematically ill-posed inverse problem since it does not admit a unique solution, evenso one would have an infinite number of sensors!Prior information on the solution are required.- In ECD, a strong constraint usually consists of assuming a prior fixed number of active sources.- In 'Imaging', the solution space is constrained by the subject anatomy and various priors such as physiological, functional, mathematical knowledge can be introduced in a so-called regularization process.

With the ECD solution :A priori fixed number of sources considered, (usually less than 5)

over-determined but nonlinear problem iterative fitting of the 6 parameters of each source,

(actually only 3 parameters, for the source location, are iteratively adjusted).

Drawback : How many ECDs a priori ?The number of sources limited : 6xNs < Ne

Advantage : Simple focused solution.But is a single (or 2 or 3 or…) dipole(s) representative of the cortical activity ?

ECD approach

222 ˆˆcost YKKYYKYYY

Y = KJ + EY = KJ + EProblem to solve:

jmattout


Global minimum

Local minimum

Local minimum

Value of parameter

Cost

funct

ion

The iterative optimisation procedure can only find a local minimum

the starting location(s) used can influence the solution found !

For an ECD solution, initialise the dipoles •at multiple random locations and repeat the fitting procedure cluster of solutions ?

•at a «guessed» solution spot, (also named « seeded-ECD »)

1D example of optimisation

problem:

ECD approach, cont’d

jmattout


• How many dipoles ? The more sources, the better the fit… in a mathematical sense !!!

• Is a dipole, i.e. a punctual source, the right model for a patch of activated cortex ?

• What about the influence of the noise ? Find the confidence interval.

• Is the seeded-ECD a good approach ? Given that you find what you put in…

ECD interpretation and limitation

jmattout


ECD application: epilepsy

First peak, above F4

jmattout


The head is NOT spherical: cannot use the exact analytical solution

because of model/anatomical errors.

Realistic model needs BEM solution: surfaces extraction computationnaly heavy errors for superficial sources

Could we combine the advantages of both solutions ?

Anatomically constrained spherical head models, or pseudo-spherical model.

Forward Problem: analytical vs. numerical solution

Scalp (or brain) surface

Best fitting sphere:centre and radii (scalp, skull, brain)

Spherical transformation of source locations

Leadfield for the spherical model

Anatomically constrained spherical model

Dipole:defined by its polar coordinates (Rd,IRM, d, d )

Fitted sphere:defined by its centre and radius, (cSph,RSph)

Scalp surface

Fitted sphere

Rd,IRM

Rscalp(d,d)

Direction (d,d)

cSph RSph

),(,,ddscalp

SphIRMdpSphd R

RRR

Anatomically constrained spherical model

Fitted sphere and scalp surface

Application: scalp surface

Application: cortical surface

EEG/MEG source reconstruction

Documents

Transcript of EEG/MEG source reconstruction