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06_MEEG_Source_analys

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The M/EEG inverse problem
Gareth R. Barnes
Format
• What is an inverse problem
• Prior knowledge- links to popular algorithms.
• Validation of prior knowledge/ Model
evidence
Inverse problems aren’t difficult
The forward problem
Predicted
0.08
Prediction
0.06
0.04
0.02
M/EEG
sensors
0
Lead fields (determined by head
model) describe the sensitivity of
an M/EEG sensor to a dipolar
source at a particular location
-0.02
-0.04
-0.06
-0.08
Lead fields (L)
Head Model
Model describes
conductivity & geometry
Dipolar source
Measurement (Y)
Prediction ( Y )
Predicted
Measured
0.08
0.1
0.06
0.08
0.04
0.06
0.02
0.04
0
0.02
0
-0.02
-0.02
-0.04
-0.04
-0.06
Forward
problem
-0.06
-0.08
-0.08
Y  LJ
Generated source activity
M/EEG sensors
Prior info
brain
?
Current density
Estimate
J
Measurement (Y)
Prediction ( Y )
Predicted
Measured
0.08
0.1
0.06
0.08
0.04
0.06
0.02
0.04
0
0.02
0
-0.02
-0.02
-0.04
-0.04
-0.06
-0.06
-0.08
-0.08
brain
?
J  WY
Forward
problem
Generated source activity
M/EEG sensors
Y  LJ
Prior info
Current density
Estimate
J
Inversion depends on choice of source covariance matrix
(prior information)
' 1
W  CL ( R  LCL )
'
Sensor Noise
(known)
sources
2
Lead field
(known)
sources
4
6
8
Generated source activity
Source covariance
matrix,
One diagonal
element per source
10
12
14
Prior information
16
2
4
6
8
10
12
14
16
Single dipole fit
Y (measured field)
PREDICTED
Measured
Predicted
0.1
0.02
0.08
0.015
0.06
0.01
0.04
0.005
0.02
0
0
-0.005
400
-0.02
-0.04
-0.01
-0.015
-0.08
-0.02
350
-0.06
200
250
time ms
300
PPM at 188 ms (100 percent confidence)
512 dipoles
Percent variance explained 11.21 (10.26)
log-evidence = 53.1
-0.025
2
150
4
6
8
100
10
J
12
2
4
6
-0.2
50
16
-0.15
-0.1
-0.05
0
0.05
0.1
14
8
10
12
14
16
-0.1
-0.05
0
0.05
0.1
0.15
0.2
2
-0.15
4
-0.2
12
100
8
4
200
250
time ms
2
6
10
6
8
10
12
300
PPM at 188 ms (100 percent confidence)
512 dipoles
Percent variance explained 99.78 (91.33)
log-evidence = 112.8
150
350
-0.02
-0.02
-0.04
-0.04
400
-0.06
-0.25
50
Y (measured field)
Single dipole fit
Measured
14
16
14
16
PREDICTED
Predicted
0.1
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0
0
-0.08
-0.06
-0.08
J
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
12
100
8
4
200
250
time ms
2
6
16
2
4
6
8
300
PPM at 188 ms (100 percent confidence)
512 dipoles
Percent variance explained 99.78 (91.33)
log-evidence = 112.8
150
350
400
-0.06
-0.25
50
Y (measured field)
Two dipole fit
Measured
14
10
12
14
16
PREDICTED
0.1
Predicted
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0
0
-0.02
-0.02
-0.04
-0.04
-0.08
-0.06
-0.08
10
J
Measured
PREDICTED
Predicted
0.1
0.1
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0
0
-0.02
-0.0
400
-0.04
-0.06
-0.0
-0.08
200
250
time ms
300
PPM at 188 ms (100 percent confidence)
512 dipoles
Percent variance explained 98.49 (90.16)
log-evidence = 93.1
350
-0.0
22
44
150
66
88
100
10
10
12
12
J
-4
50
-3
16
16
-2
-1
0
1
2
14
14
3
4
Minimum norm
Y (measured field)
22
44
66
88
10
10
12
12
14
14
16
16
Beamformer
Y (measured field)
PREDICTED
Measured
Predicted
0.1
0.1
0.08
0.06
0.08
0.04
0.06
0.02
0.04
0
0.02
-0.02
0
-0.04
-0.02
400
-0.06
-0.08
-0.04
350
-0.06
4
6
8
10
12
14
16
0
0.1
0.2
0.4
10
8
150
0.3
12
0.5
0.6
6
100
0.7
0.8
0.9
-0.02
-0.01
0
0.01
1
0.02
-0.03
50
2
0.03
4
*Assuming no correlated sources
PPM at 188 ms (100 percent confidence)
512 dipoles
Percent variance explained 99.18 (90.78)
log-evidence = 99.9
14
200
250
time ms
2
16
at 57, -26, 9 mm
300
Projection
onto
lead field*
-0.08
fMRI biased dSPM
(Dale et al. 2000)
Y (measured field)
Measured
PREDICTED
Predicted
0.1
0.1
0.08
0.06
0.08
0.04
0.06
0.02
0.04
0
0.02
-0.02
0
-0.04
-0.02
-0.06
-0.04
400
-0.08
-0.06
200
250
time ms
estimated response - condition 1
at 57, -26, 9 mm
300
PPM at 188 ms (100 percent confidence)
512 dipoles
Percent variance explained 99.18 (90.78)
log-evidence = 99.9
350
-0.08
2
2
4
4
6
1
0.9
0.8
0.7
6
8
0.6
150
8
10
0.5
10
0.4
100
12
12
14
0.3
0.2
fMRI data
12
14
16
16
2
4
-0.02
10
-0.01
8
0
6
0.01
4
0.02
0.03
2
6
8
-0.03
50
14
16
10
0.1
12
14
16
0
Maybe…
Some popular priors
2
4
6
8
10
12
14
16
0
16
0.1
14
2
0.2
12
0.3
4
0.4
0.5
8
0.6
Minimum norm
8
SAM,DICs
Beamformer
10
6
6
10
0.7
4
0.8
12
0.9
2
14
1
16
2
4
6
8
10
12
14
16
1
2
2
0.9
4
4
0.8
6
6
Dipole fit
8
0.7
LORETA
0.6
8
0.5
10
10
12
12
14
14
0.2
16
0.1
0.4
16
2
4
6
8
10
12
14
0.3
16
2
1
2
0.9
4
0.8
fMRI biased
dSPM
8
10
12
14
16
2
6
4
0.5
8
10
0.4
10
12
0.3
6
?
8
12
10
0.2
14
14
12
0.1
16
2
4
6
8
10
12
14
16
0
0
4
0.6
8
6
2
0.7
6
4
16
2
14
16
4
6
8
10
12
14
16
Summary
• MEG inverse problem requires prior
information in the form of a source covariance
matrix.
• Different inversion algorithms- SAM, DICS,
LORETA, Minimum Norm, dSPM… just have
different prior source covariance structure.
• Historically- different MEG groups have
tended to use different algorithms/acronyms.
See
Mosher et al. 2003, Friston et al. 2008, Wipf and Nagarajan 2009, Lopez et al. 2013
Software
• SPM12: http://www.fil.ion.ucl.ac.uk/spm/software/spm12/
• DAiSS- SPM12 toolbox for Data Analysis in Source Space (beamforming,
minimum norm and related methods), developed by collaboration of UCL,
Oxford and other MEG centres.https://code.google.com/p/spmbeamforming-toolbox/
• Fieldtrip : http://fieldtrip.fcdonders.nl/
• Brainstorm:http://neuroimage.usc.edu/brainstorm/
• MNE: http://martinos.org/mne/stable/index.html
Which priors should I use ?
• Compare to other modalities..
fMRI
Singh et al.
2002
MEG
beamformer
• Use model comparison… rest of the talk.
x 10
-4
50
-3
-2
-1
0
-0.031
50
-0.02
2
3
4
-0.01
0
-3
100
100
150
300
200
250
time ms
300
200
250
time ms
6
8
10
0.02
0.015
0.01
0.005
0
96%
12
14
16
300
PPM at 188 ms (100 percent confidence)
512 dipoles
Percent variance explained 98.49 (90.16)
log-evidence = 93.1
150
PPM at 188 ms (100 percent confidence)
512 dipoles
Percent variance explained 99.18 (90.78)
log-evidence = 99.9
150
PPM at 188 ms (100 percent confidence)
512 dipoles
Percent variance explained 99.78 (91.33)
log-evidence = 112.8
estimated response - condition 1
at 65, -21, 11 mm
200
250
time ms
4
1
100
0.9
-0.25
0.01
50
0.8
2
Predicted
2
11 %
16
0.7
Y
14
4
Predicted
12
0.6
0.02
10
6
-0.2
14
8
-0.15
512 dipoles
Percent variance explained 11.21 (10.26)
log-evidence = 53.1
8
0.5
6
16
350
350
350
0.4
4
0.3
2
12
10
14
0.2
12
400
400
400
0
16
10
12
8
16
8
14
10
6
12
6
10
14
4
8
Variance
explained
-0.08 4
6
J
2
0.06
0.04
0.02
0
-0.005
-0.02
-0.01
-0.04
-0.015
-0.06
-0.08
97%
4
Prior
-0.06 2
0.1
-0.04
2
Measured
16
Y (measured field)
How do we chose between priors ?
0.1
0.08
0.06
0.04
0.02
0
-0.02
2
4
6
10
8
12
14
16
2
Predicted
4
6
8
10
12
0.08
-0.02
-0.025
-0.08
98%
14
16
0.1
0.08
Predicted
0
0.06
0
0.04
0
0.02
0
0
0
-0.02
0
-0.04
-
-0.06
-
-
Y
Prediction (
Measurement (Y)
Forward
problem
Generated source activity
Eyes
True recipe
J
)
Estimated
recipe
J
Use prior info (possible ingredients)
Y
Prediction (
2
4
6
8
10
Diagonal elements correspond
to ingredients
12
14
16
2
4
6
8
10
12
14
16
)
Forward
problem
Generated source activity
Measurement (Y)
J
2
4
6
8
10
12
14
16
16
14
12
10
8
6
4
C
2
2
4
6
8
10
12
B
14
16
2
4
6
8
10
12
14
16
2
4
2
6
4
8
6
10
8
12
10
A
14
12
16
2
14
4
6
8
10
12
14
16
16
2
Possible priors
4
6
8
10
12
14
Which is most likely prior (which prior has highest evidence) ?
Y
Prediction (
2
A
4
6
8
10
?
12
14
16
16
10
12
14
14
8
16
12
6
0
J
0.1
10
8
B
12
0.2
0.3
0.4
0.5
0.6
2
6
0.7
4
4
C
0.8
6
2
0.9
1
10
4
8
2
6
14
4
16
2
8
10
12
14
16
2
4
6
8
10
12
14
16
)
Forward
problem
Generated source activity
Measurement (Y)
Consider 3 generative models
2
4
6
8
10
12
14
16
2
0
16
4
0.1
14
6
0.2
0.3
12
0.4
10
8
10
0.5
8
12
0.6
6
14
0.7
10
12
14
16
1
8
0.9
6
0.8
P(Y)
4
2
2
4
16
2
Evidence
4
6
8
10
12
14
16
2
4
6
8
10
12
14
16
Area under
each curve=1.0
Space of possible datasets (Y)
Complexity
Cross validation or prediction of unknown data
Prediction (Y )
2
A
4
6
Forward
problem
Generated source activity
Measurement (Y)
8
10
?
12
14
16
16
10
12
14
14
8
16
12
6
0
J
0.1
10
8
B
12
0.2
0.3
0.4
0.5
0.6
2
6
0.7
4
4
C
0.8
6
2
0.9
1
10
4
8
2
6
14
4
16
2
8
10
12
14
16
2
4
6
8
10
12
14
16
Polynomial fit example
4 parameter fit
y
Training data
2 parameter fit
x
The more parameters in the model the more accurate the fit
(to training data).
Polynomial fit example
4 parameter fit
y
2 parameter fit
test data
x
The more parameters the more accurate the fit to training data,
but more complex model may not generalise to new (test) data.
Fit to training data
0.4
Training data
0.2
Predicted
0.08
0.06
0.04
2
0
-0.02
-0.04
-0.06
-0.08
Predicted
0.02
4
0
6
O
8
10
12
14
-0.2
GS
IID
16
2
4
6
8
10
12
14
16
2
4
-0.4
6
x
8
10
12
-0.6
-0.6
14
16
-0.4
-0.2
0
Measured
0.2
0.4
2
4
6
8
10
Measured
0.1
0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
More complex model fits training data better
12
14
16
Fit to test data
0.4
Predicted
0.2
0.08
0.06
0.04
2
0.02
4
-0.02
-0.04
-0.06
-0.08
Predicted
0
6
0
O
8
10
12
14
16
-0.2
2
GS
IID
-0.4
4
6
8
10
12
14
16
2
4
6
x
8
10
12
14
-0.6
-0.6
16
2
-0.4
-0.2
0
Measured
0.2
0.4
Measured
0.1
0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
Simpler model fits test data better
4
6
8
10
12
14
16
Relationship between model evidence and cross validation
2
4
1
6
8
2
0.9
4
0.8
Random priors
0.7
6
10
0.6
0.5
10
14
20
1
8
12
2
4
6
8
10
12
14
16
0.9
4
0.8
0.4
0.3
12
16
2
0.7
0.2
6
14
0.6
0.1
8
0.5
16
2
4
6
8
10
12
14
16
0
10
0.4
12
0.3
log Model evidence
0.2
14
15
0.1
16
2
4
6
8
10
12
14
0
16
1
10
2
0.9
4
0.8
0.7
6
0.6
8
0.5
10
0.4
12
0.3
0.2
14
5
0.1
16
2
4
6
8
10
12
14
16
0
1
0
0.37
2
0.9
4
0.8
0.7
0.4
0.39
0.38
Cross
Cross validation
validation accuracy
error
0.41
6
0.6
8
0.5
10
0.4
12
0.3
0.2
14
0.1
16
2
Can be approximated analytically…
4
6
8
10
12
14
16
0
-0.02
2
3
4
-0.01
0
-4
50
-3
-2
-1
0
-0.031
50
x 10
-3
100
100
100
150
300
200
250
time ms
300
200
250
time ms
12
0.02
0.015
0.01
0.005
0
Dip 2
14
16
300
PPM at 188 ms (100 percent confidence)
512 dipoles
Percent variance explained 98.49 (90.16)
log-evidence = 93.1
150
PPM at 188 ms (100 percent confidence)
512 dipoles
Percent variance explained 99.18 (90.78)
log-evidence = 99.9
150
PPM at 188 ms (100 percent confidence)
512 dipoles
Percent variance explained 99.78 (91.33)
log-evidence = 112.8
estimated response - condition 1
at 65, -21, 11 mm
200
250
time ms
0.9
-0.25
0.01
50
0.8
0.02
10
1
8
Predicted
2
-0.2
6
0.7
4
4
Dip 1
2
0.6
Y
16
6
-0.15
14
8
Predicted
12
0.5
10
350
350
350
0.4
8
0.3
6
0.2
4
16
10
14
12
14
400
400
400
0.1
2
12
0
12
16
16
10
14
8
12
8
10
10
6
8
6
6
Log model
evidence
14
4
0.06
0.04
0.02
0
-0.005
-0.02
-0.01
-0.04
-0.015
-0.06
-0.08
BMF
4
J
2
4
2
Prior
2
16
512 dipoles
Percent variance explained 11.21 (10.26)
log-evidence = 53.1
How do we chose between priors ?
2
4
6
8
10
12
14
16
2
Predicted
4
6
8
10
12
14
0.08
-0.02
-0.025
-0.08
30
20
10
0
Min Norm
16
0.1
Predicted
0.08
0
0.06
0
0.04
0
0.02
0
0
0
-0.02
0
-0.04
-
-0.06
-
-
Muliple Sparse Priors (MSP), Champagne
Candidate Priors
Prior to maximise model evidence
l1
l2

l3
ln
0
0
-2
-0.5
Multiple0 Sparse priors -0.5
-0.5
-1
So now construct
the priors to maximise model evidence
-1
-1
(minimise cross validation error).
-1.5
8
10
-1.5
-1.5
-4
-6
-2
-8
0
100
200
-2
300
0400
time ms
500
100
600
200
-2
700
300
4000
time ms
500
100
600
200
700
300
400
time ms
0
100
500
200
600
300
400
time
ms
700
500
6
10
PPM at 229 ms (100 percent confidence) PPM at 229 ms (100 percent confidence)
512 dipoles
512 dipoles
Percent variance explained 99.15 (65.03)Percent variance explained 99.93 (65.54)
log-evidence = 8157380.8
log-evidence = 8361406.1
4
Free
energy
Accuracy
Accuracy
Free Energy
Complexity
Compexity
PPM at 229 ms (100 percent confidence)
512 dipoles
Percent variance explained 99.87 (65.51)
log-evidence = 8388254.2
10
PPM at 229 ms (100 percent confidence)
512 dipoles
Percent variance explained 99.90 (65.53)
log-evidence = 8389771.6
2
10
0
50
100
Iteration
150
6
Conclusion
• MEG inverse problem can be solved.. If you
have some prior knowledge.
• All prior knowledge encapsulated in a source
covariance matrix.
• Can test between priors (or develop new
priors) using cross validation or Bayesian
framework.
References
•
•
•
•
Mosher et al., 2003
J. Mosher, S. Baillet, R.M. Leahi
Equivalence of linear approaches in bioelectromagnetic inverse solutions
IEEE Workshop on Statistical Signal Processing (2003), pp. 294–297
•
•
•
•
Friston et al., 2008
K. Friston, L. Harrison, J. Daunizeau, S. Kiebel, C. Phillips, N. Trujillo-Barreto, R.
Henson, G. Flandin, J. Mattout
Multiple sparse priors for the M/EEG inverse problem
NeuroImage, 39 (2008), pp. 1104–1120
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Wipf and Nagarajan, 2009
D. Wipf, S. Nagarajan
A unified Bayesian framework for MEG/EEG source imaging
NeuroImage, 44 (2009), pp. 947–966
Thank you
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Karl Friston
Jose David Lopez
Vladimir Litvak
Guillaume Flandin
Will Penny
Jean Daunizeau
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Christophe Phillips
Rik Henson
Jason Taylor
Luzia Troebinger
Chris Mathys
Saskia Helbling
And all SPM developers
Analytical approximation to model
evidence
• Free energy= accuracy- complexity
What do we measure with EEG & MEG ?
From a single source to the sensor: MEG
MEG
EEG
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