meeg source reconstruction: problems & solutions - … · 2012-05-03 · m/eeg source...
TRANSCRIPT
M/EEG source reconstruction:
problems & solutions
SPM-M/EEG course
Lyon, April 2012
C. Phillips, Cyclotron Research Centre, ULg, Belgium http://www.cyclotron.ulg.ac.be
Head anatomy
Solution by Maxwell’s equations
Head model : conductivity layout Source model : current dipoles
Forward problem
From Maxwell’s equations find:
where M are the measurements and f(.) depends on:
•signal recorded, EEG or MEG
•head model, i.e. conductivity layout adopted
•source location
•source orientation & amplitude
M = f ( j , r )
Solving the forward problem
From Maxwell’s equations find:
with f(.) as
•an analytical solution
- highly symmetrical geometry, e.g. spheres,
concentric spheres, etc.
- homogeneous isotropic conductivity
•a numerical solution
- more general (but still limited!) head model
M = f ( j , r )
Solving the forward problem
Con’s: •Human head is not spherical •Conductivity is not homogeneous and isotropic.
Analytical solution
Example: 3 concentric spheres
Pro’s: •Simple model •Exact mathematical solution •Fast calculation
Usually “Boundary Element Method” (BEM) : • Concentric sub-volumes of homogeneous and
isotropic conductivity,
• Estimate values on the interfaces.
Numerical solution
Pro’s: •More correct head shape modelling (not perfect though!)
Con’s: •Mathematical approximations of solution numerical errors
•Slow and intensive calculation
Features of forward solution
Find forward solution (any):
with f(.)
• linear in
• non-linear in
If N sources with known & fixed location,
then
M = f ( j , r )
j = jx jy jzéë ùû
T
r = rx ry rzéë ùû
T
M = f r1 r2… rN[ ]( ) × J = L × J
SPM solutions
Source space & head model
•Template Cortical Surface (TCS), in MNI space.
•Canonical Cortical Surface (CCS) = TCS warped to subject’s anatomy
•Subject’s Cortical Surface (SCS) = extracted from subject’s own structural image (BrainVisa/FreeSurfer)
SPM solutions
Jérémie Mattout, Richard N. Henson, and Karl J. Friston, 2007, Canonical Source Reconstruction for MEG
SPM solutions
Source space & head model
•Template Cortical Surface (TCS), in MNI space.
•Canonical Cortical Surface (CCS) = TCS warped to subject’s anatomy
•Subject’s Cortical Surface (SCS) = extracted from subject’s own structural image (BrainVisa/FreeSurfer)
Forward solutions:
•Single sphere
•Overlapping spheres
•Concentric spheres
•BEM
•… (new things get added to SPM & FieldTrip)
Useful priors for cinema audiences
• Things further from the camera appear smaller
• People are about the same size
• Planes are much bigger than people
M/EEG inverse problem
forward computation
Likelihood & Prior
inverse computation
Posterior & Evidence
p(Y |q, M ) p(q | M )
p(Y | M )p(q |Y, M )
Probabilistic framework
Distributed or imaging model
Likelihood ,, LJNMYP LJY
Hypothesis M: distributed (linear) model, gain matrix L, Gaussian distributions
Parameters : (J,)
Priors ,0NJP
Sensor level: # sources
# so
urc
es
IID (Minimum Norm)
Maximum Smoothness (LORETA-like)
Source level:
I2
# sensors
# se
nso
rs
Minimum norm solution
Solution
0 100 200 300 400 500 600 700-10
-8
-6
-4
-2
0
2
4
6
8
10
estimated response - condition 1
at 52, -31, 11 mm
time ms
PPM at 379 ms (79 percent confidence)
512 dipoles
Percent variance explained 99.91 (93.65)
log-evidence = 21694116.2
0 100 200 300 400 500 600 700-10
-8
-6
-4
-2
0
2
4
6
8
10
estimated response - condition 1
at 52, -31, 11 mm
time ms
PPM at 379 ms (79 percent confidence)
512 dipoles
Percent variance explained 99.91 (93.65)
log-evidence = 21694116.2
Prior
True (focal source)
= “allow all sources to be active, but keep energy to a minimum”
Incorporating multiple constraint
Likelihood ,, LJNMYP LJY
Hypothesis M: hierarchical model, Gaussian distributions, gain matrix L, variance components Qi
Parameters : (J,σ,λ)
Priors ,0NJP
Sensor level:
Source level:
I2
# sensors
# se
nso
rs
For example: Multiple Sparse Priors (MSP)
…
k
kQQ 1
1
logl = N a,b( )
0 100 200 300 400 500 600 700-8
-6
-4
-2
0
2
4
6
8
estimated response - condition 1
at 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 1
at 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 1
at 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 energy
Accuracy
Complexity
0 100 200 300 400 500 600 700-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
estimated response - condition 1
at 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
Accuracy Free Energy Complexity
-6-4-20246x 1
0-1
3
-6
-4
-2
0
2
4
6
x 10-13
-6
-4
-2
0
2
4
6
x 10-13
Estimated data Estimated position
Measured data
?
Dipole fitting
Constraint: very few dipoles!
-6-4-20246x 1
0-1
3
-6
-4
-2
0
2
4
6
x 10-13
-6
-4
-2
0
2
4
6
x 10-13
Estimated data Prior source covariance
Measured data
?
Dipole fitting
True source covariance
Conclusion
• Solving the Forward Problem is not exciting but necessary…
…MEG or EEG? individual sMRI available? sensor location available?
• M/EEG inverse problem can be solved…
…If you provide some prior knowledge!
• All prior knowledge encapsulated in a covariance matrices (sensors & sources)
• Can test between models and priors (a.k.a. constraints) in a Bayesian framework.