# Jeremie Mattout`s slides from SPM course

```EEG/MEG
source reconstruction
Jérémie Mattout / Christophe Phillips / Karl Friston
Wellcome Dept. of Imaging Neuroscience, Institute of Neurology, UCL, London
Estimating brain activity from scalp electromagnetic data
Sources
MEG data
Source
Reconstruction
‘Equivalent Current
Dipoles’ (ECD)
EEG data
‘Imaging’
Components of the source reconstruction process
Source model
‘ECD’
‘Imaging’
Forward model
Registration
Inverse method
Data
Anatomy
Components of the source reconstruction process
Source model
Registration
Forward model
Inverse solution
Source model
Compute transformation T
Individual MRI
Templates
Apply inverse transformation T-1
Individual mesh
input
- Individual MRI
- Template mesh
functions
- spatial normalization into MNI template
- inverted transformation applied to the template mesh
output
- individual mesh
Registration
fiducials
fiducials
Rigid transformation (R,t)
Individual sensor space
Individual MRI space
input
- sensor locations
- fiducial locations
(in both sensor & MRI space)
- individual MRI
functions
- registration of the EEG/MEG data into
individual MRI space
output
- registrated data
- rigid transformation
Foward model
p
Compute for
each dipole
K
+
n
Forward operator
Individual MRI space
Model of the
functions
input
- sensor locations
- individual mesh
- single sphere
- three spheres
- overlapping spheres
- realistic spheres
BrainStorm
output
- forward operator K
Inverse solution (1) - General principles
General Linear Model
1 dipole source
per location
Cortical mesh
Under-determined GLM
Regularized solution
Y = KJ+ E
[nxt]
[nxp][pxt] [nxt]
n : number of sensors
p : number of dipoles
t : number of time samples
||Y – KJ||2 + λf(J) )
J^ : min(
J
data fit
priors
Inverse solution (2) - Parametric empirical Bayes
2-level hierarchical model
Gaussian
Gaussian variables
variables
with
with unknown
unknown variance
variance
Y = KJ + E1 E1 ~ N(0,Ce)
J = 0 + E2
Sensor level
E2 ~ N(0,Cp)
Source level
Linear parametrization
of the variances
Ce = 1.Qe1 + … + q.Qeq
Cp = λ1.Qp1 + … + λk.Qpk
Q: variance components
(,λ): hyperparameters
Inverse solution (3) - Parametric empirical Bayes
Bayesian inference on model parameters
+
Model M
+
J
Q e1 , … , Q eq
Qp1 , … , Qpk
,λ
K
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)
E-step: maximizing F wrt J
M-step: maximizing of F wrt (,λ)
J^ = CJKT[Ce + KCJ KT]-1Y
MAP estimate
Ce + KCJKT = E[YYT]
ReML estimate
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‘
Inverse solution (5) - implementation
input
- preprocessed data
- forward operator
- individual mesh
- priors
functions
- compute the MAP estimate of J
- compute the ReML estimate of (,λ)
- interpolate into individual MRI voxel-space
ECD approach
- iterative forward and inverse computation
output
- inverse estimate
- model evidence
Conclusion - Summary
MRI space
Data space
Registration
Forward model
EEG/MEG
preprocessed data
PEB inverse
solution
SPM
```
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