1. EEG/MEG source reconstruction
Jérémie Mattout / Christophe Phillips / Karl Friston
Wellcome Dept. of Imaging Neuroscience, Institute of Neurology, UCL, London

2. Estimating brain activity from scalp electromagnetic data
Sources
MEG data
Source Reconstruction
‘Equivalent Current Dipoles’ (ECD)
‘Imaging’
EEG data

3. Components of the source reconstruction process
Source model
‘ECD’
‘Imaging’
Forward model
Registration
Inverse method
Data
Anatomy

4. Components of the source reconstruction process
Source model
Registration
Forward model
Inverse solution

5. Source model Compute transformation T Apply inverse transformation T-1
Individual MRI
Templates
Apply inverse transformation T-1
Individual mesh
input
functions
output
Individual MRI
Template mesh
spatial normalization into MNI template
inverted transformation applied to the template mesh
individual mesh

6. Registration Rigid transformation (R,t) Individual MRI space
fiducials
Individual sensor space
fiducials
Rigid transformation (R,t)
input
functions
output
sensor locations
fiducial locations
(in both sensor & MRI space)
individual MRI
registration of the EEG/MEG data into individual MRI space
registrated data
rigid transformation

7. head tissue properties
Foward model
Individual MRI space
Model of the
head tissue properties
Compute for
each dipole p + K n
Forward operator
functions
input
single sphere
three spheres
overlapping spheres
realistic spheres
output
sensor locations
individual mesh
forward operator K
BrainStorm

8. Inverse solution (1) - General principles
General Linear Model
1 dipole source
per location
Cortical mesh
Y = KJ+ E
[nxt]
[nxp]
[pxt]
n : number of sensors
p : number of dipoles
t : number of time samples
Under-determined GLM J
: min( ||Y – KJ||2 + λf(J) ) ^
Regularized solution
data fit
priors

9. Inverse solution (2) - Parametric empirical Bayes
2-level hierarchical model
Y = KJ + E1
J = 0 + E2
E2 ~ N(0,Cp)
E1 ~ N(0,Ce)
Gaussian variables
with unknown variance
Gaussian variables
with unknown variance
Sensor level
Source level
Ce = 1.Qe1 + … + q.Qeq
Cp = λ1.Qp1 + … + λk.Qpk
Linear parametrization of the variances
Q: variance components
(,λ): hyperparameters

10. + Inverse solution (3) - Parametric empirical Bayes Qe1 , … , Qeq
Bayesian inference on model parameters
Model M
Qe1 , … , Qeq
Qp1 , … , Qpk + J K
,λ
Inference on J and (,λ)
Maximizing the log-evidence
F = log( p(Y|M) ) =  log( p(Y|J,M) ) + log( p(J|M) ) dJ
data fit
priors
Expectation-Maximization (EM)
J = CJKT[Ce + KCJ KT]-1Y ^
E-step: maximizing F wrt J
MAP estimate
M-step: maximizing of F wrt (,λ)
Ce + KCJKT = E[YYT]
ReML estimate

11. Inverse solution (4) - Parametric empirical Bayes
Bayesian model comparison
Model evidence
Relevance of model M is quantified by its evidence p(Y|M) maximized by the EM scheme
Model comparison
Two models M1 and M2 can be compared by the ratio of their evidence
B12 =
p(Y|M1)
p(Y|M2)
Bayes factor
Model selection using a
‘Leaving-one-prior-out-strategy‘

12. Inverse solution (5) - implementation
input
functions
output
iterative forward and inverse computation
ECD approach
preprocessed data
- forward operator
individual mesh
priors
- compute the MAP estimate of J
compute the ReML estimate of (,λ)
interpolate into individual MRI voxel-space
inverse estimate
model evidence

13. EEG/MEG preprocessed data
Conclusion - Summary
Data space
MRI space
Registration
Forward model
EEG/MEG preprocessed data
PEB inverse solution
SPM

15. Estimating brain activity from scalp electromagnetic data
Sources
MEG data
Source Reconstruction
‘Equivalent Current Dipoles’ (ECD)
‘Imaging’
EEG data

16. Problem to solve: Y = KJ + E ECD approach 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 ?

17. ECD approach, cont’d
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 »)
Global minimum
Local minimum
Value of parameter
Cost function
1D example of optimisation problem:

18. ECD interpretation and limitation
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…

19. ECD application: epilepsy
First peak, above F4

21. Forward Problem: analytical vs. numerical solution
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.

22. Anatomically constrained spherical model
Scalp (or brain) surface
Best fitting sphere:
centre and radii (scalp, skull, brain)
Spherical transformation of source locations
Leadfield for the spherical model

23. Anatomically constrained spherical model
Dipole:
defined by its polar coordinates (Rd,IRM, qd, fd )
Fitted sphere:
defined by its centre and radius, (cSph,RSph)
Direction (qd,fd)
Fitted sphere
Rscalp(qd,fd)
Rd,IRM
cSph
RSph
Scalp surface

24. Fitted sphere and scalp surface
Application: scalp surface
Fitted sphere and scalp surface

25. Application: cortical surface