EEG/MEG Inverse Solution Driven by fMRI Yaroslav Halchenko CS @ - - PowerPoint PPT Presentation

EEG/MEG Inverse Solution Driven by fMRI

Yaroslav Halchenko CS @ NJIT

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Functional Brain Imaging

EEG – ElectroEncephaloGram MEG – MagnetoEncephaloGram fMRI – Functional Magnetic Resonance Imaging

• thers

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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

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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) = 2σ0v0( r) + 1 2π

• i

(σ+

i − σ− i )

• Si

v( r′) d/d3d Si

• b(

r) = b0( r) + µ0 4π

• i

(σ+

i − σ− i )

• Si

v( r′) d/d3 × d Si

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

b0( r) and v0( r):

• b0(

r) = µ0 4π q × d/d3 v0( r) = 1 4πσ0

• q ·

d/d3

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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

qi and ri

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.

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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 Sp, so

f(S) = W−1/2

S

(S − Sp)2

F ,

where WS is weighting matrix of source space.

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EEG/MEG Inverse: Solution

Taking in account prior information Sp

ˆ S = G#X + (I − G#G)Sp = Sp + G#(X − GSp),

where G# is the solution with no prior information Sp: Bayesian methodology to maximize the posterior p(S|X) assuming Gaussian prior on S (Baillet and Garnero, 1997), Wiener estimator with proper Cǫ and CS, Tikhonov regularization to trade-off goodness of fit and regularization term f(S) all lead to the solution

• f next general form

G# = WSGt(GWSGt + λWX)−1.

For the noiseless case simple Moore-Penrose pseudo-inverse

G† = WSGt(GWSGt)−1.

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EEG/MEG Inverse: Choices

WS = IN, WX = IM - the simplest regularized minimum norm

solution.

WX = C−1

ǫ

accounts for possible noise covariance structure.

WS = C−1

S

performs minimization in pre-whitened model space. Besides the data-driven factors WS can be used to incorporate different geometric parameters as

WS = (diag(GtG))−1 - normalizes columns of matrix G to

account for deep sources by penalizing the voxels which lay too close to the sensors.

WS incorporates spatial derivative of the image of first

• rder (Wang et al., 1992) or Laplacian

form (Pascual-Marqui et al., 1994).

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EEG/MEG Inverse: fMRI Driven Factors

WS = C−1

S , where CS is a covariance matrix derived from fMRI

data (Dale and Sereno, 1993).

WS = (IN + αCS)−1(Liu et al., 1998) to account for activations

not revealed by fMRI. Mattout et al. (2000) incorporate fMRI prior into f(S).

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EEG/MEG Inverse: fMRI Driven Factors

WS = C−1

S , where CS is a covariance matrix derived from fMRI

data (Dale and Sereno, 1993).

WS = (IN + αCS)−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 Sp, so

ˆ S = G#X + (I − G#G)Sp = Sp + G#(X − GSp),

where G# is the solution with no prior information Sp:

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fMRI Signal

Forward model for fMRI signal is F = SB, where B is a convolution matrix consisting of HRF for each voxel

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fMRI Prior

To incorporate some temporal information from fMRI signal, we find cross-covariance between the signal and HRF ¯

F = FBT

Assuming that fMRI activation produced current EEG signal, we can incorporate that information in fMRI prior Sp by normalizing by the power of EEG signal

Sp(t) = ¯ F(t) E(t) A¯ F(t)

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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.

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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-fi eld analysis : the unique minimum-norm least-squares estimation. IEEE Transactions

• n Biomedical Engineering, 39(7):665–675, 1992.

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The END

Ooops... To be continued

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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)

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