Cortical Source Localization of Human Scalp EEG
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Cortical Source Localization of Human Scalp EEG
Kaushik Majumdar
Indian Statistical Institute
Bangalore Center
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Cortical Basis of Scalp EEG
Baillet et al., IEEE Sig. Proc. Mag., Nov 2001, p. 16.
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Six Layer Cortex
Mountcastle, Brain, 120:701-722, 1997.
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Head Tissue Layers
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Source Localization in Two Parts
Part I : Forward Problem
Part II : Inverse Problem
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Forward Problem : Schematic Head Model
Brain
Skull
Scalp
Source
EEG Channels
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Source Models
Dipole Source Model
(parametric model)
Distributed Source Model
(nonparametric model)
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Dipole Source Model
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Distributed Source Model
Majumdar, IEEE Trans. Biomed. Eng., vol. 56(4), p. 1228 – 1235, 2009.
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Forward Calculation
Poisson’s equation in the head
pfV J )(
Kybic et al., Phys. Med. Biol., vo. 51, p. 1333 – 1346, 2006
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Published Conductivity Values
Hallez et al., J. NeuroEng. Rehab., 2007, open access.
http://www.jneuroengrehab.com/content/4/1/46
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6 Parameter Dipole Geometry
Hallez et al., J. NeuroEng. Rehab., 2007, open access. http://www.jneuroengrehab.com/content/4/1/46
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Potential at any Single Scalp Electrode Due to All Dipoles
r is the position vector of the scalp electrode
rdip - i is the position vector of the ith dipole
di is the dipole moment of the ith dipole
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Potential at All Scalp Electrodes
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For N Electrodes, p Dipoles, T Discrete Time Points
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Generalization
V = GD + n
G is gain matrix, n is additive noise.
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EEG Gain Matrix Calculation
For detail of potential calculations see Geselowitz, Biophysical J., 7, 1967, 1-11.
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Gain Matrix : Elaboration
m
nmnnn
m
m
n
J
J
J
J
BBBB
BBBB
BBBB
V
V
V
.
.
.
.
.
.
.
.
.
.
.
...........
.
.
...........
.
.
3
2
1
321
2232221
1131211
2
1
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Boundary Elements Method for Distributed Source Model
If
on the complement of a smooth surface then
can be completely determined by its values and the values of its derivatives on that surface.
02 u
u
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Nested Head Tissues
Hallez et al., J. NeuroEng. Rehab., 2007, open access. http://www.jneuroengrehab.com/content/4/1/46
BRAIN SKULL
SCALP
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Green’s Function
iiff 1.
rr
4
1)( G
)()( rr Gfvii
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Representation Theorem
Ω be a connected, open, bounded subset of R3 and ∂Ω be regular boundary of Ω. u : R3- ∂Ω → R is harmonic and r|u(r)| < ∞, r(∂u/∂r) = 0, then if p(r) = ∂nu(r) the following holds:
Kybic, et al., IEEE Trans. Med. Imag., vol. 24(1), p. 12 – 28, 2005.
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Representation Theorem (cont)
I is the identity operator and
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Representation Theorem (cont)
∂nG means n.▼G, where n is normal to an interfacing head tissue surface.
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Justification
Holes in the skull (such as eyes) account for up to 2 cm of error in source localization.
The closer a source is to the cortical surface the more its effect tends to smear.
If size of mesh triangles is of the order of the gaps between the surfaces the errors go up rapidly.
Implemented in OpenMeeg – an open source software (openmeeg.gforge.inria.fr/ ).
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Distributed Source : Gain Matrix
Kybic, et al., IEEE Trans. Med. Imag., vol. 24(1), p. 12 – 28, 2005.
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Gain Matrix (cont)
Gain matrix is to be obtained by multiplying several matrices A, one for each layer in the single layer formulation.
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Finite Elements Method
Hallez et al., J. NeuroEng. Rehab., 2007, open access. http://www.jneuroengrehab.com/content/4/1/46
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Further Reading on FEM
Awada et al., “Computational aspects of finite element modeling in EEG source localization,” IEEE Trans. Biomed. Eng., 44(8), pp. 736 – 752, Aug 1997.
Zhang et al., “A second-order finite element algorithm for solving the three-dimensional EEG forward problem,” Phys. Med. Biol., vol. 49, pp. 2975 – 2987, 2004.
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Inverse Problem : Peculiarities
Inverse problem is ill-posed, because the number of sensors is less than the number of possible sources.
Solution is unstable, i.e., susceptible to small changes in the input values. Scalp EEG is full of artifacts and noise, so identified sources are likely to be spurious.
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Weighted Minimum Norm Inverse
V = BJ + E, V is scalp potential, B is gain matrix and E is additive noise.
Minimize U(J) = ||V – BJ||2Wn + λ||J||2Wp
λ is a regularization positive constant between 0 and 1.
Mattout et al., NeuroImage, vol. 30(3), p. 753 – 767, Apr 2006
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Geometric Interpretation
||V-BJ||2Wn ||J||2Wp
minU(J)
Convex combination
of the two terms with
λ very small.
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Derivation
U(J) = ||V – BJ||2Wn + λ||J||2Wp
= <Wn(V – BJ), Wn(V – BJ)> + λ<WpJ, WpJ>
ΔJU(J) = 0 implies (using <AB,C> = <B,ATC>)
J = CpBT[BCpBT + Cn]-1V
where Cp = (WTpWp)-1 and Cn = λ(WT
nWn)-1.
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Different Types
When Cp = Ip (the p x p identity matrix) it reduces to classical minimum norm inverse solution.
In terms of Bayesian notation we can write E(J|B) = CpBT[BCpBT + Cn]-1V. On this expectation maximization algorithm can be readily applied.
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Types (cont)
If we take Wp as a spatial Laplacian operator we get the LORETA inverse formulation.
If we derive the current density estimate by the minimum norm inverse and then standardize it using its variance, which is hypothesized to be due to the source variance, then that is called sLORETA.
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Types (cont)
Recursive – MUSIC
Mosher & Leahy, IEEE Trans. Biomed. Eng., vol. 45(11), p. 1342 – 1354, Nov 1998.
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Low Resolution Brain Electromagnetic Tomography (LORETA)
Pascual-Marqui et al., Int. J. Psychophysiol., vol. 18, p. 49 – 65, 1994.
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Standardized Low Resolution Brain Electromagnetic Tomography (sLORETA)
U(J) = ||V – BJ||2 + λ||J||2
Ĵ = TV, where
T = BT[BBT + λH]+, where
H = I – LLT/LTL is the centering matrix.
Pascual-Marqui at http://www.uzh.ch/keyinst/NewLORETA/sLORETA/sLORETA-Math02.htm
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sLORETA (cont)
Ĵ is estimate of J, A+ denotes the Moore-Penrose inverse of the matrix A, I is n x n identity matrix where n is the number of scalp electrodes, L is a n dimensional vector of 1’s.
Hypothesis : Variance in Ĵ is due to the variance of the actual source vector J.
Ĵ = BT[BBT + λH]+BJ.
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Form of H
nnnn
nnnn
nnnn
11...........
111...............
...............
1...........
111
1
1...........
1111
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If # Source = # Electrodes = n
B and BT both will be n x n identity matrix.
with λ = 0
nnnn
nnnn
nnnn
T
1...........
...............
...............
...........1
...........1
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sLORETA Result
http://www.uzh.ch/keyinst/loreta.htm
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Single Trial Source Localization
Averaging signals across the trials to increase the SNR cannot be done.
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Cortical Sources
Generated using OpenMeeg in INRIA Sophia Antipolis in 2008.
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Localization by MN
Generated using OpenMeeg in INRIA Sophia Antipolis in 2008.
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Clustering
Majumdar, IEEE Trans. Biomed. Eng., vol. 56(4), p. 1228 – 1235, 2009.
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Trial Selection
Identify the time interval. Identify channels. Calculate cumulative signal amplitude in
those channels. Sort trials according to the decreasing
amplitude.
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Phase Synchronization
1nm
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Synchronization (cont.)
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Synchronization (cont.)
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Synchronization (cont.)
E[n] =
|))((||))((|))(),(( nEstdnEmeantxtxsyn kj
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Phase Synchronization and Power
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The Best Time Epoch
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Source Localization
Majumdar, IEEE Trans. Biomed. Eng., vol. 56(4), p. 1228 – 1235, 2009.
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Must Reading
Baillet, Mosher & Leahy, “Electromagnetic brain mapping,” IEEE Sig. Proc. Mag., p. 14 – 30, Nov 2001.
Hallez et al., “Review on solving the forward problem in EEG source analysis,” J. Neuroeng. Rehab., open access, available at http://www.jneuroengrehab.com/content/4/1/46
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Must Reading (cont)
Grech et al., “Review on solving the inverse problem in EEG source analysis,” J. Neuroeng. Rehab., open access, available at http://www.jneuroengrehab.com/content/5/1/25
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THANK YOU
This presentation is available at http://www.isibang.ac.in/~kaushik