EEG/MEG Inverse Solution Driven by fmri Yaroslav Halchenko CS @ NJIT 1
Functional Brain Imaging EEG ElectroEncephaloGram MEG MagnetoEncephaloGram fmri Functional Magnetic Resonance Imaging others 2
Functional Brain Imaging(Pros.&Cons.) EEG ElectroEncephaloGram Great time resolution - mseconds Poor spatial resolution - up to 100 scalp channels Ambiguity of inverse solution MEG MagnetoEncephaloGram EEG dis-/ad- vantages Better sensitivity to deep sources and SNR Can t capture normally oriented sources fmri Functional Magnetic Resonance Imaging Great spatial resolution - mm Unknown coupling between haemodynamic and neuronal activity Poor temporal resolution - seconds 2
EEG/MEG Physical Model Quasi-stationary approximation of magnetic field using Maxwell equations, Biort-Savart equation and Green s theorem creates basis of forward modeling in EEG (v( r)) and MEG( b( r)) (σ + j + σ j )v( r) = 2σ 0v 0 ( r) + 1 2π (σ i + σi S ) v( r ) d/d 3 ds i i i b( r) = b0 ( r) + µ 0 4π (σ i + σi S ) v( r ) d/d 3 ds i i i Assumption that primary current exists only at a discrete point, i.e. it can be presented as a dipole with moment q located at r then we can get simple equations for b 0 ( r) and v 0 ( r): b0 ( r) = µ 0 4π q d/d 3 v 0 ( r) = 1 4πσ 0 q d/d 3 3
EEG/MEG Physical Model Linearity b( r) and v( r) depend linearly on strength of the dipole q and non-linearly on locations of the sources r. Non-linear optimization to find q i and r i Sample the space for possible source locations and present EEG/MEG signal X as simple as X = AQ, where Physics-geometric parameter A(M 3N) - combines information about lead fields for each sensor/source couple. Dipole moments Q(3N T ) - keeps time-series of dipole activations, which presents underlying brain activity in current location. Depending on the number of dipoles (N) we want to fit our signal by, we can have over- or under- determined system. 4
EEG/MEG Inverse: Formulation Noisy linear forward model X = AS + N, where N corresponds to the sensor noise. Least-squares error minimization with a regularization L(S) = W 1/2 X (X GS) 2 F + λf(s), The simplest regularization: minimal 2-nd norm deviation from the prior assumption of source model S p, so f(s) = W 1/2 S (S S p ) 2 F, where W S is weighting matrix of source space. 5
EEG/MEG Inverse: Solution Taking in account prior information S p Ŝ = G # X + (I G # G)S p = S p + G # (X GS p ), where G # is the solution with no prior information S p : Bayesian methodology to maximize the posterior p(s X) assuming Gaussian prior on S (Baillet and Garnero, 1997), Wiener estimator with proper C ɛ and C S, Tikhonov regularization to trade-off goodness of fit and regularization term f(s) all lead to the solution of next general form G # = W S G t (GW S G t + λw X ) 1. For the noiseless case simple Moore-Penrose pseudo-inverse G = W S G t (GW S G t ) 1. 6
EEG/MEG Inverse: Choices W S = I N, W X = I M - the simplest regularized minimum norm solution. W X = C 1 ɛ W S = C 1 S accounts for possible noise covariance structure. performs minimization in pre-whitened model space. Besides the data-driven factors W S can be used to incorporate different geometric parameters as W S = (diag(g t G)) 1 - normalizes columns of matrix G to account for deep sources by penalizing the voxels which lay too close to the sensors. W S incorporates spatial derivative of the image of first order (Wang et al., 1992) or Laplacian form (Pascual-Marqui et al., 1994). 7
EEG/MEG Inverse: fmri Driven Factors W S = C 1 S, where C S is a covariance matrix derived from fmri data (Dale and Sereno, 1993). W S = (I N + αc S ) 1 (Liu et al., 1998) to account for activations not revealed by fmri. Mattout et al. (2000) incorporate fmri prior into f(s). 8
EEG/MEG Inverse: fmri Driven Factors W S = C 1 S, where C S is a covariance matrix derived from fmri data (Dale and Sereno, 1993). W S = (I N + αc S ) 1 (Liu et al., 1998) to account for activations not revealed by fmri. Mattout et al. (2000) incorporate fmri prior into f(s). NEW Incorporate fmri prior in S p, so Ŝ = G # X + (I G # G)S p = S p + G # (X GS p ), where G # is the solution with no prior information S p : 8
fmri Signal Forward model for fmri signal is F = SB, where B is a convolution matrix consisting of HRF for each voxel 9
fmri Prior To incorporate some temporal information from fmri signal, we find cross-covariance between the signal and HRF F = FB T Assuming that fmri activation produced current EEG signal, we can incorporate that information in fmri prior S p by normalizing by the power of EEG signal S p (t) = F(t) E(t) A F(t) 10
To Be Reported Challenge: Real Data Experiment ( not quite yet :-( ) Possible: Realistic Data Simulations (real subjects data + artificial activations) Preliminary Results: Solution activations maps appear to be too smeared due to the fact that all locations have some non-0 prior. Iterative rerun of the algorithm with the results of previous run used as a prior for the next run seems to improve the results. 11
References Sylvain Baillet and Line Garnero. A bayesian approach to introducing anatomo-functional priors in the EEG/MEG inverse problem. IEEE Transactions on Biomedical Engineering, 44(5): 374 385, May 1997. Anders M. Dale and Martin I. Sereno. Improved localization of cortical activity by combining EEG and MEG with mri cortical surface reconstruction: A linear approach. Journal of Cognitive Neuroscience, 5(2):162 176, 1993. A.K. Liu, J.W. Belliveau, and A.M. Dale. Spatiotemporal imaging of human brain activity using functional MRI constrained magnetoencephalography data: Monte Carlo simulations. Proc Natl Acad Sci U S A, 95(15):8945 50, 1998. J. Mattout, L. Garnero, L. Gavit, and Benali H. Functional mri derived priors for solving the EEG/MEG inverse problem. In 12th International Conference on Biomagnetism, Helsinski, Finlande, 2000. R. D. Pascual-Marqui, C. M. Michel, and D. Lehman. Low resolution electromagnetic tomography: A new method for localizing electrical activity of the brain. International Journal of Psychophysiology, 18:49 65, 1994. C. Phillips, M. D. Rugg, and K.J. Friston. Systematic regularization of linear inverse solutions of the EEG source localization problem. NeuroImage, 17(1):287 301, 2002. J. Z. Wang, S. J. Williamson, and L. Kaufman. Magnetic source images determined by a lead-field analysis : the unique minimum-norm least-squares estimation. IEEE Transactions on Biomedical Engineering, 39(7):665 675, 1992. 12
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EEG/MEG Inverse: fmri Prior Problem: fmri alone can t provide information on dipole orientation Solution: Restrict EEG inverse to the cortex so dipoles orthogonal to the derived from anatomical MRI white matter surface. Integrate fmri information in the space nearby each dipole location (Phillips et al., 2002) 14