HBM 2008 - #298 M-PM
Functional localization in MEG and EEG using EM estimation on a state-space model with spatial and time smoothness constraint
Antonio Molins , Matti S. Hmlinen
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1
2,4,5
, Emery N. Brown
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Harvard Medical School, Cambridge, MA 2 Harvard/MIT Division for Health Sciences and Technology, Cambridge, MA 3 Brain and cognitive Sciences, MIT, Cambridge, MA 4 Massachusetss General Hospital, Boston, MA 5 MGH-MIT-HMS Athinoula A. Martinos Center for Biomedical Imaging, Charlestown, MA Simulation study We simulated the auditory evoked bilaterally using a forward model from one schizophrenic subject. Noise covariance was computed from pre-stimulus measurements, and sources were located 5mm apart on average. We simulate activity in the auditory cortex only (using Freesurfer for the parcelation). Source activity was uncorrelated, and SNR was set to 5. The nal dataset included activity from 587 xed orientation dipoles, recorded in 306 MEG channels and 68 EEG channels. Data included 5 right side stimulations and 4 left side stimulations.
Abstract Source current estimation from electromagnetic (MEG and EEG) signals is an ill-posed problem that often produces blurry or inaccurately positioned estimates. Because of the high temporal resolution of both MEG and EEG scanners and the temporal dynamics of the observed neural signal, state-space formulations can achieve superior performance than static methods. Bayesian estimation can be used then to obtain superior estimates, and the EM algorithm can be used to better characterize the dynamics (see poster 362 M-PM for more details). Here we present a model including stimulus time information in evoked response studies using MEG and EEG. Motivation As explained in poster 362 M-PM. Kalman smoothing paired with EM estimation can produce better estimates of the source current distribution than static methods. In evoked studies though, the fast dynamics of the input e ect make di cult to learn the underlying dynamics of the observed activity while keeping the fast paced input e ect, as illustrated below. A simple 1D model Consider the simple 1D linear dynamical model with 1D inputs, where the input takes only 0 or 1 values, we know the measurement noise covariance , and we want to estimate the unknown process noise, state transition value a, and the input e ect :
Stimulus 1 - gated average
Stimulus 2 - gated average
Simulated sources R responding sources L responding sources Inactive sources
Estimation The EM/KS algorithm was used to estimate the currents generating the simulated data in four di erent scenarios: A- , and were estimated from the data. This is the nal objective of the algorithm. Band are estimated, and is not (zero input e ect). This is the case presented in poster 362 M-PM, applied to our simulated data. Cand are estimated, are set to its actual value. This tests the e ect of letting be estimated by EM. Dis estimated, and are set to its actual values. This tests the e ect of letting be estimated by EM. Results Stimulus gated average of the resulting puted for the di erent scenarios: Actual activity are com-
If we knew the values of the parameters, and need to estimate , we can use the Kalman Smoother to recover the posterior expected value of the activity given the observations :
If the parameters the of dynamics update are not known, we can estimate them using the EM algorithm, and our estimates are then:
If we neglect either the dynamic term , or the input e ect terms , in our estimation, we get the following stimulus gated average estimates of the activity: No dynamics Estimated dynamics
KS-EM No input e ect: KS-EM, no input e ect KS-EM , known KS-EM, known and Time from stimulus 1 (ms) The conditional log-likelihood showed convergence of the EM algorithm for all 4 cases, achieving superior values when the input e ect was included. Convergence was typically observed after ~12 iterations. Conclusions and future work We introduce the coe cients that describe the state transition matrix as a linear combination of a orthonormal set of basis functions , and the input e ect is now of size , where is our number of sources. The measurement noise is supposed Gaussian and its covariance is estimated from the recorded data, the process -We derived a closed form of the M update for constrained state-transition matrix and noise is white and Gaussian, and all the parameters are time-invariant. time-dependent input e ect. Estimation: Kalman Smoothing and EM updates Using the EM algorithm we estimate both the unknown parameters of the system and the underlying activity iteratively: E step: We use the Rauch-Tung-Striebel algorithm to perform o ine xed-interval smoothing of the data with the given set of parameter estimates. M step: A closed form solution for each of the parameters gives the new update. The o -diagonal values of are xed to 0, and only the parameters are updated. Ai are xed and set to exponential functions of cortical distances with di erent decay rates. -We presented a new estimation algorithm to do inverse source localization in evoked EEG + MEG studies using EM.
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Estimated input e ect
-5 0 5 -5 0 5
State-space Model We model the EEG-MEG data generation process using a generalization of the simple 1D model previously described:
Time fr...