the m/eeg inverse problem and solutions gareth r. barnes
TRANSCRIPT
The M/EEG inverse problem and solutions
Gareth R. Barnes
Format
• The inverse problem• Choice of prior knowledge in some popular
algorithms• Why the solution is important.
Volume currents
Magnetic field
Electrical potential difference (EEG)
5-10nAmAggregate post-synaptic potentials
of ~10,000 pyrammidal neurons
cortex
skull
scalp
MEG pick-up coil
Inverse problem
1s
Active Passive
Local field potential (LFP)
MEG measurement
1nAm
1pT
pick-up coils
What we’ve got
What we want
Forward problem
Useful priors cinema audiences
• Things further from the camera appear smaller
• People are about the same size• Planes are much bigger than people
Where does the data come from ?
1pT
1s
Useful priors for MEG analysis
• At any given time only a small number of sources are active. (dipole fitting)
• All sources are active but overall their energy is minimized. (Minimum norm)
• As above but there are also no correlations between distant sources (Beamformers)
The source covariance matrix
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Source number
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Estimated dataEstimated position
Measured data
Dipole Fitting
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Estimated data/Channel covariance matrixMeasured data/
Channel covariance
Dipole fitting
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True source covariance
Prior source covariance
Fisher et al. 2004
Dipole fitting
Effective at modelling short (<200ms) latency evoked responses
Clinically very useful: Pre-surgical mapping of sensory /motor cortex ( Ganslandt et al 1999)
Need to specify number of dipoles, non-linear minimization becomes unstable for more sources.
Minimum norm- allow all sources to be active, but keep energy to a minimum
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Solution
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estimated response - condition 1at 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
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estimated response - condition 1at 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
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True(Single Dipole)
Problem is that superficial elements have much larger lead fields
MEG sensitivity
Basic Minimum norm solutions
Solutions are diffuse and have superficial bias (where source power can be smallest).
But unlike dipole fit, no need to specify the number of sources in advance.
Can we extend the assumption set ?
Coh
eren
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Distance
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8-13Hz band
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Cortical oscillations have local domains
Bullock et al. 1989
“We have managed to check the alpha band rhythm with intra-cerebral electrodes in the occipital-parietal cortex; in regions which are practically adjacent and almost congruent one finds a variety of alpha rhythms, some are blocked by opening and closing the eyes, some are not, some respond in some way to mental activity, some do not.” Grey Walter 1964
Leopold et al. 2003.
Beamformer: if you assume no correlations between sources, can calculate a prior covariance matrix from the
data
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True
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estimated response - condition 1at 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,Estimated From data
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Singh et al. 2002
ME
G c
ompo
site
fMR
IOscillatory changes are co-located with haemodynamic changes
Beamformers
Robust localisation of induced changes, not so good at evoked responses.
Excellent noise immunity.
Clincally also very useful (Hirata et al. 2004; Gaetz et al. 2007)
But what happens if there are correlatedsources ?
Beamformer for correlated sources
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estimated response - condition 1at -2, -21, 5 mm
time ms
PPM at 229 ms (99 percent confidence)512 dipoles
Percent variance explained 99.81 (65.46)log-evidence = 8249519.0
Prior(estimated from data)
True Sources
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Estimated data/Channel covariance matrixMeasured data/
Channel covariance
Dipole fitting
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True source covariance
Prior source covariance
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Muliple Sparse Priors (MSP)
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Estimated (based on data)
TrueP
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Priors
(Covariance estimates are made in channel space)
= sensitivity (lead field matrix)
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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
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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
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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
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Iteration
Free energyAccuracyComplexity
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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
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estimated response - condition 1at -2, -21, 5 mm
time ms
PPM at 229 ms (99 percent confidence)512 dipoles
Percent variance explained 99.81 (65.46)log-evidence = 8249519.0
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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 = 8389771.6Can use model evidence to choose between solutions
So it is possible,but why bother ?
Rols et al. 2001
Stimulus(1º)
Gamma oscillations in monkey
evoked
Inducedgammapower
Time-frequencypower
Time-frequencypower changefrom baseline
20
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0Stimulus(3cpd,1.5º)
Evoked(0-70Hz)
-3nAm
GammaPower
Adjamian et al. 2004 ,Hall et al. 2005, Adjamian et al. 2008
100%
0%
0.8
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nAm
^2/H
z
030 40 50 60Hz
30
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0.3 2s
Hz
Hadjipapas et al. 2009,Kawabata Duncan et al. 2009,
Different spectra, different underlying neuronal populations
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p<0.05
Power spectrum
Rank spectrum
Where does the data come from ?
1pT
1s
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 in a Bayesian framework.
• Exciting part is the millisecond temporal resolution we can now exploit.