the m/eeg inverse problem gareth r. barnes. format what is an inverse problem prior knowledge- links...
TRANSCRIPT
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The M/EEG inverse problem
Gareth R. Barnes
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Format
• What is an inverse problem• Prior knowledge- links to popular algorithms.• Validation of prior knowledge/ Model
evidence
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Inverse problems aren’t difficult
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The forward problem
Lead fields (L)
Dipolar source
Model describesconductivity & geometry
Lead fields (determined by head model) describe the sensitivity of
an M/EEG sensor to a dipolar source at a particular location
Head Model
Prediction
Predicted
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
M/EEGsensors
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Inverse problem
Gen
erat
ed s
ourc
e ac
tivity
brain?
Gen
erat
ed s
ourc
e ac
tivity
Predicted
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Measured
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Forwardproblem
Current densityEstimate
M/EEG sensors
J
Y LJPrior info
Measurement (Y) Prediction ( )Y
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Inverse problem
Measurement (Y)
Gen
erat
ed s
ourc
e ac
tivity
brain?
Gen
erat
ed s
ourc
e ac
tivity
Predicted
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Measured
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Forwardproblem
Current densityEstimate
Prediction ( )
M/EEG sensors J WY
J
Y
Y LJ
Prior info
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' ' 1( )W CL R LCL Sensor Noise(known)
Lead field(known)
sour
ces
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
Gene
rated
sour
ce a
ctivit
y
sources
Source covariance matrix,One diagonal element per source
Inversion depends on choice of source covariance matrix (prior information)
Prior information
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Measured
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Predicted
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
5010
015
020
025
030
035
040
0-0
.2
-0.1
5
-0.1
-0.0
50
0.050.
1
0.15
estim
ated
resp
onse
- co
nditio
n 1
at -5
4, -2
7, 1
0 m
m
time
ms
PPM
at 1
88 m
s (1
00 p
erce
nt c
onfid
ence
)51
2 di
pole
sPe
rcen
t var
ianc
e ex
plai
ned
11.2
1 (1
0.26
)lo
g-ev
iden
ce =
53.
1
Gen
erat
ed s
ourc
e ac
tivity
Y (measured field)
Prior info (source covariance)
PREDICTED
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2
4
6
8
10
12
14
16
Inverse problem
Single dipole fit
J
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Measured
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Gen
erat
ed s
ourc
e ac
tivity
Y (measured field)
Prior info (source covariance)
PREDICTED
Inverse problem
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
5010
015
020
025
030
035
040
0-0
.25
-0.2
-0.1
5
-0.1
-0.0
50
0.050.
1
0.150.
2
0.25
estim
ated
resp
onse
- co
nditio
n 1
at 5
0, -2
6, 1
1 m
m
time
ms
PPM
at 1
88 m
s (1
00 p
erce
nt c
onfid
ence
)51
2 di
pole
sPe
rcen
t var
ianc
e ex
plai
ned
99.7
8 (9
1.33
)lo
g-ev
iden
ce =
112
.8
Predicted
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Single dipole fit
J
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2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
Measured
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Gen
erat
ed s
ourc
e ac
tivity
Y (measured field)
Prior info (source covariance)
PREDICTED
Inverse problem
Predicted
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
5010
015
020
025
030
035
040
0-0
.25
-0.2
-0.1
5
-0.1
-0.0
50
0.050.
1
0.150.
2
0.25
estim
ated
resp
onse
- co
nditio
n 1
at 5
0, -2
6, 1
1 m
m
time
ms
PPM
at 1
88 m
s (1
00 p
erce
nt c
onfid
ence
)51
2 di
pole
sPe
rcen
t var
ianc
e ex
plai
ned
99.7
8 (9
1.33
)lo
g-ev
iden
ce =
112
.8
Two dipole fit
J
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Gen
erat
ed s
ourc
e ac
tivity
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
Measured
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Y (measured field)
Prior info (source covariance)
PREDICTED
Inverse problem
Predicted
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
5010
015
020
025
030
035
040
0-4-3-2-101234
x 10
-3es
timat
ed re
spon
se -
cond
ition
1at
65,
-21,
11
mm
time
ms
PPM
at 1
88 m
s (1
00 p
erce
nt c
onfid
ence
)51
2 di
pole
sPe
rcen
t var
ianc
e ex
plai
ned
98.4
9 (9
0.16
)lo
g-ev
iden
ce =
93.
1
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
Minimum norm
J
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Gen
erat
ed s
ourc
e ac
tivity
Measured
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Y (measured field)
Prior info (source covariance)
PREDICTED
Inverse problem
246810121416
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4
6
8
10
12
14
160
0.1
0.2
0.3
0.4
0.5
0.6
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0.8
0.9
1
Predicted
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
5010
015
020
025
030
035
040
0-0
.03
-0.0
2
-0.0
10
0.01
0.02
0.03
estim
ated
resp
onse
- co
nditio
n 1
at 5
7, -2
6, 9
mm
time
ms
PPM
at 1
88 m
s (1
00 p
erce
nt c
onfid
ence
)51
2 di
pole
sPe
rcen
t var
ianc
e ex
plai
ned
99.1
8 (9
0.78
)lo
g-ev
iden
ce =
99.
9
Projectiononto lead field*
Beamformer
*Assuming no correlated sources
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2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
Measured
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Y (measured field)
Prior info (source covariance)
PREDICTED
Inverse problem
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
160
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Predicted
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
5010
015
020
025
030
035
040
0-0
.03
-0.0
2
-0.0
10
0.01
0.02
0.03
estim
ated
resp
onse
- co
nditio
n 1
at 5
7, -2
6, 9
mm
time
ms
PPM
at 1
88 m
s (1
00 p
erce
nt c
onfid
ence
)51
2 di
pole
sPe
rcen
t var
ianc
e ex
plai
ned
99.1
8 (9
0.78
)lo
g-ev
iden
ce =
99.
9
fMRI data
fMRI biased dSPM(Dale et al. 2000)
Maybe…
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2 4 6 8 10 12 14 16
2
4
6
8
10
12
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16
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Minimum norm
Dipole fit LORETA
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1
SAM,DICsBeamformer
?
2 4 6 8 10 12 14 16
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14
160
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0.8
0.9
1
fMRI biaseddSPM
Some popular priors
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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
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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/spm-beamforming-toolbox/
• Fieldtrip : http://fieldtrip.fcdonders.nl/• Brainstorm:http://neuroimage.usc.edu/brainstorm/• MNE: http://martinos.org/mne/stable/index.html
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Which priors should I use ?
• Compare to other modalities..
• Use model comparison… rest of the talk.
Singh et al. 2002
fMRI
MEGbeamformer
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5010
015
020
025
030
035
040
0-0
.2
-0.1
5
-0.1
-0.0
50
0.050.
1
0.15
estim
ated
resp
onse
- co
nditio
n 1
at -5
4, -2
7, 1
0 m
m
time
ms
PPM
at 1
88 m
s (1
00 p
erce
nt c
onfid
ence
)51
2 di
pole
sPe
rcen
t var
ianc
e ex
plai
ned
11.2
1 (1
0.26
)lo
g-ev
iden
ce =
53.
1
5010
015
020
025
030
035
040
0-0
.25
-0.2
-0.1
5
-0.1
-0.0
50
0.050.
1
0.150.
2
0.25
estim
ated
resp
onse
- co
nditio
n 1
at 5
0, -2
6, 1
1 m
m
time
ms
PPM
at 1
88 m
s (1
00 p
erce
nt c
onfid
ence
)51
2 di
pole
sPe
rcen
t var
ianc
e ex
plai
ned
99.7
8 (9
1.33
)lo
g-ev
iden
ce =
112
.8
5010
015
020
025
030
035
040
0-0
.03
-0.0
2
-0.0
10
0.01
0.02
0.03
estim
ated
resp
onse
- co
nditio
n 1
at 5
7, -2
6, 9
mm
time
ms
PPM
at 1
88 m
s (1
00 p
erce
nt c
onfid
ence
)51
2 di
pole
sPe
rcen
t var
ianc
e ex
plai
ned
99.1
8 (9
0.78
)lo
g-ev
iden
ce =
99.
9
Predicted
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
Y (measured field)How do we chose between priors ?
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
Variance explained
Predicted
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
Predicted
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
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8
10
12
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0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Predicted
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
5010
015
020
025
030
035
040
0-4-3-2-101234
x 10
-3es
timat
ed re
spon
se -
cond
ition
1at
65,
-21,
11
mm
time
ms
PPM
at 1
88 m
s (1
00 p
erce
nt c
onfid
ence
)51
2 di
pole
sPe
rcen
t var
ianc
e ex
plai
ned
98.4
9 (9
0.16
)lo
g-ev
iden
ce =
93.
1
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
Measured
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
11 % 96% 97% 98%
Prior
J
Y
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Measurement (Y)
Gen
erat
ed s
ourc
e ac
tivity
Gen
erat
ed s
ourc
e ac
tivity
Forwardproblem
Prediction ( )
Eyes
J
Y
Inverse problem
True recipe J
Estimatedrecipe
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Measurement (Y)
Gen
erat
ed s
ourc
e ac
tivity
Forwardproblem
Prediction ( )
J
Y
Inverse problem
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
Prior info (source covariance)
Diagonal elements correspondto ingredients
Use prior info (possible ingredients)
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Possible priors
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A
B
C
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Measurement (Y)
Gen
erat
ed s
ourc
e ac
tivity
Forwardproblem
Prediction ( )
J
Y
Inverse problem
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Prior info (source covariance)
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2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
?
Which is most likely prior (which prior has highest evidence) ?
A
B
C
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P(Y)
Complexity
Area under each curve=1.0
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
246810121416
2
4
6
8
10
12
14
160
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Space of possible datasets (Y)
Consider 3 generative models
Evidence
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Measurement (Y)
Gen
erat
ed s
ourc
e ac
tivity
Forwardproblem
Prediction ( )
J
YInverse problem
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
Prior info (source covariance)
246810121416
2
4
6
8
10
12
14
160
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
?
Cross validation or prediction of unknown data
A
B
C
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The more parameters in the model the more accurate the fit(to training data).
y
x
Polynomial fit example
4 parameter fit
2 parameter fit
Training data
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The more parameters the more accurate the fit to training data, but more complex model may not generalise to new (test) data.
y
x
Polynomial fit example
4 parameter fit
2 parameter fittest data
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Training data
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
x
-0.6 -0.4 -0.2 0 0.2 0.4-0.6
-0.4
-0.2
0
0.2
0.4
Measured
Pre
dict
ed
GSIID
Predicted
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Measured
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
O
Fit to training data
More complex model fits training data better
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-0.6 -0.4 -0.2 0 0.2 0.4-0.6
-0.4
-0.2
0
0.2
0.4
Measured
Pre
dict
ed
GSIID
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
x
Predicted
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Measured
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
O
Fit to test data
Simpler model fits test data better
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0.37 0.38 0.39 0.4 0.410
5
10
15
20
Mod
el e
vide
nce
Cross validation accuracy
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
160
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
160
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
160
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
160
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Cross validation error
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
Relationship between model evidence and cross validation
Random priorslo
g
Can be approximated analytically…
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5010
015
020
025
030
035
040
0-0
.2
-0.1
5
-0.1
-0.0
50
0.050.
1
0.15
estim
ated
resp
onse
- co
nditio
n 1
at -5
4, -2
7, 1
0 m
m
time
ms
PPM
at 1
88 m
s (1
00 p
erce
nt c
onfid
ence
)51
2 di
pole
sPe
rcen
t var
ianc
e ex
plai
ned
11.2
1 (1
0.26
)lo
g-ev
iden
ce =
53.
1
5010
015
020
025
030
035
040
0-0
.25
-0.2
-0.1
5
-0.1
-0.0
50
0.050.
1
0.150.
2
0.25
estim
ated
resp
onse
- co
nditio
n 1
at 5
0, -2
6, 1
1 m
m
time
ms
PPM
at 1
88 m
s (1
00 p
erce
nt c
onfid
ence
)51
2 di
pole
sPe
rcen
t var
ianc
e ex
plai
ned
99.7
8 (9
1.33
)lo
g-ev
iden
ce =
112
.8
5010
015
020
025
030
035
040
0-0
.03
-0.0
2
-0.0
10
0.01
0.02
0.03
estim
ated
resp
onse
- co
nditio
n 1
at 5
7, -2
6, 9
mm
time
ms
PPM
at 1
88 m
s (1
00 p
erce
nt c
onfid
ence
)51
2 di
pole
sPe
rcen
t var
ianc
e ex
plai
ned
99.1
8 (9
0.78
)lo
g-ev
iden
ce =
99.
9
Predicted
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
How do we chose between priors ?
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
Predicted
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
Predicted
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
246810121416
2
4
6
8
10
12
14
160
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Predicted
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
5010
015
020
025
030
035
040
0-4-3-2-101234
x 10
-3es
timat
ed re
spon
se -
cond
ition
1at
65,
-21,
11
mm
time
ms
PPM
at 1
88 m
s (1
00 p
erce
nt c
onfid
ence
)51
2 di
pole
sPe
rcen
t var
ianc
e ex
plai
ned
98.4
9 (9
0.16
)lo
g-ev
iden
ce =
93.
1
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
Prior
J
Y
Dip 1 Dip 2 BMF Min Norm05
10152025
Log modelevidence
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Muliple Sparse Priors (MSP), Champagne
1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
7
8
9
10
1l
1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
7
8
9
10
2l
l3
+
Prior to maximise model evidence
12345678910
1
2
3
4
5
6
7
8
9
10
Candidate Priors
1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
7
8
9
10
ln
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0 100 200 300 400 500 600 700-8
-6
-4
-2
0
2
4
6
8
estimated response - condition 1at 36, -18, -23 mm
time ms
PPM at 229 ms (100 percent confidence)512 dipoles
Percent variance explained 99.15 (65.03)log-evidence = 8157380.8
0 100 200 300 400 500 600 700-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
estimated response - condition 1at 46, -31, 4 mm
time ms
PPM at 229 ms (100 percent confidence)512 dipoles
Percent variance explained 99.93 (65.54)log-evidence = 8361406.1
0 100 200 300 400 500 600 700-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
estimated response - condition 1at 46, -31, 4 mm
time ms
PPM at 229 ms (100 percent confidence)512 dipoles
Percent variance explained 99.87 (65.51)log-evidence = 8388254.2
0 50 100 15010
2
104
106
108
Iteration
Free energyAccuracyComplexity
0 100 200 300 400 500 600 700-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
estimated response - condition 1at 46, -31, 4 mm
time ms
PPM at 229 ms (100 percent confidence)512 dipoles
Percent variance explained 99.90 (65.53)log-evidence = 8389771.6
AccuracyFree EnergyCompexity
Multiple Sparse priors
So now construct the priors to maximise model evidence(minimise cross validation error).
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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.
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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
• Wipf and Nagarajan, 2009• D. Wipf, S. Nagarajan• A unified Bayesian framework for MEG/EEG source imaging• NeuroImage, 44 (2009), pp. 947–966
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Thank you
• Karl Friston• Jose David Lopez• Vladimir Litvak• Guillaume Flandin• Will Penny• Jean Daunizeau
• Christophe Phillips• Rik Henson• Jason Taylor• Luzia Troebinger• Chris Mathys• Saskia Helbling
And all SPM developers
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Analytical approximation to model evidence
• Free energy= accuracy- complexity
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MEG
EEG
What do we measure with EEG & MEG ?
From a single source to the sensor: MEG