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. Magnetic field MEG pick-up coil Electrical potential difference (EEG) scalp skull cortex Volume currents MEG measurement What we’ve got Forward problem Inverse problem pick-up coils 1pT Active Passive Local field potential (LFP) 1nAm What we want 1s 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) Source number The source covariance matrix Source number Measured data -13 x 10 6 4 2 0 -2 -4 -6 ? -6 -4 -2 0 2 4 Estimated data x 10 6 -13 Estimated position Dipole Fitting Prior source covariance -13 x 10 6 4 2 0 -2 -4 -6 -4 -2 0 2 4 x 10 6 Measured data/ Channel covariance True source covariance Estimated data/ Channel covariance matrix -13 Dipole fitting 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 (but see Kiebel et al. 2007), nonlinear minimization becomes unstable for more sources. Fisher et al. 2004 -8-8 -10 -10 00 Minimum norm 100 100 200 200 300 400 300 400 time ms ms time 500 500 - allow all sources to be active, but keep energy to a minimum PPMatat379 379ms ms(79 (79percent percentconfidence) confidence) PPM 512 dipoles 512 dipoles Percentvariance varianceexplained explained99.91 99.91(93.65) (93.65) Percent log-evidence==21694116.2 21694116.2 log-evidence Solution True (Single Dipole) Prior 600 600 700 700 Problem is that superficial elements have much larger lead fields 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 ? MEG sensitivity 1.0 0.5 8-13Hz band 0 “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 Coherence Cortical oscillations have local domains 0 12 24 Distance 30mm Bullock et al. 1989 Leopold et al. 2003. -10 0 100 200 300 400 time ms 500 Beamformer: if you assume no correlations between sources, can calculate a prior covariance matrix from the PPM at 379 ms (79 percent confidence) data 512 dipoles Percent variance explained 99.91 (93.65) log-evidence = 21694116.2 True Prior, Estimated From data 600 700 fMRI Oscillatory changes are co-located with haemodynamic changes MEG composite 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 correlated sources ? Singh et al. 2002 -1.5 0 100 200 400 300 time ms 500 600 Beamformer for correlated sources True Sources Prior (estimated from data) PPM at 229 ms (99 percent confidence) 512 dipoles Percent variance explained 99.81 (65.46) log-evidence = 8249519.0 70 Prior source covariance -13 x 10 6 4 2 0 -2 -4 -6 ? -4 -2 0 2 4 x 10 6 Measured data/ Channel covariance True source covariance Estimated data/ Channel covariance matrix -13 Dipole fitting Multiple Sparse Priors (MSP) True P(l) l -4 el1 Priors + el2 eln Estimated (based on data) = sensitivity (lead field matrix) (Covariance estimates are made in channel space) -4 -1 -1 8 10 -6 -1.5 -1.5 -1.5 -2 -8 0 100 200 -2 300 0400 time ms 500 100 6 10 600 200 -2 700 300 4000 time ms 500 100 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 4 log-evidence = 8361406.1 10 600 200 0 700 300 400 time ms 100 500 200 600 Free energy Accuracy PPM at 229 ms (100 percent confidence) Accuracy Free Energy 512 dipoles Percent variance explained 99.87 (65.51) Compexity Complexity 0 50 log-evidence = 8388254.2 100 Iteration 500 PPM at 229 ms (100 percent confidence) 512 dipoles Percent variance explained 99.90 (65.53) log-evidence = 8389771.6 2 10 300 400 time ms 700 150 6 512 dipoles Percent variance explained 99.81 (65.46) log-evidence = 8249519.0 512 dipoles Percent variance explained 99.90 (65.53) log-evidence = 8389771.6 Can use model evidence to choose between solutions 8400000 8350000 8300000 Free energy Model Evidence 8250000 8200000 8150000 BMF MSP So it is possible, but why bother ? Correct inversion algorithm Stimulus(3cpd,1.5º) Single trial data • Correct location information • Correct unmixing of sensor data = best estimate of source level time series • Higher SNR (~ sqrt (Nchans)) Duncan et al . 2010 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. Thanks to • • • • • • Vladimir Litvak Will Penny Jeremie Mattout Guillaume Flandin Tim Behrens Karl Friston and methods group
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