# Cortical Source Localization of Human Scalp EEG. Kaushik Majumdar Indian Statistical Institute Bangalore Center

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1 Cortical Source Localization of Human Scalp EEG Kaushik Majumdar Indian Statistical Institute Bangalore Center

2 Cortical Basis of Scalp EEG Baillet et al, IEEE Sig Proc Mag, Nov 2001, p 16

3 Mountcastle, Brain, 120: , 1997 Six Layer Cortex

5 Source Localization in Two Parts Part I : Forward Problem Part II : Inverse Problem

6 Forward Problem : Schematic Head Model Scalp EEG Channels Source Skull Brain

7 Source Models Dipole Source Model (parametric model) Distributed Source Model (nonparametric model)

8 Dipole Source Model

9 Majumdar, IEEE Trans Biomed Eng, vol 56(4), p , 2009 Distributed Source Model

10 Kybic et al, Phys Med Biol, vo 51, p , 2006 Forward Calculation Poisson s equation in the head (σ V) = f = J p

11 Hallez et al, J NeuroEng Rehab, 2007, open access Published Conductivity Values

12 Hallez et al, J NeuroEng Rehab, 2007, open access 6 Parameter Dipole Geometry

13 Potential at any Single Scalp Electrode Due to All Dipoles r is the position vector of the scalp electrode r dip - i is the position vector of the ith dipole d i is the dipole moment of the ith dipole

14 Potential at All Scalp Electrodes

15 For N Electrodes, p Dipoles, T Discrete Time Points

16 Generalization V = GD + n G is gain matrix, n is additive noise

17 EEG Gain Matrix Calculation For detail of potential calculations see Geselowitz, Biophysical J, 7, 1967, 1-11

18 Gain Matrix : Elaboration = m nm n n n m m n J J J J B B B B B B B B B B B B V V V

19 Boundary Elements Method for Distributed Source Model If 2 u = 0 on the complement of a smooth surface then u can be completely determined by its values and the values of its derivatives on that surface

20 Hallez et al, J NeuroEng Rehab, 2007, open access Nested Head Tissues BRAIN SKULL SCALP

21 Green s Function f = f 1 Ω i G( r) = 1 4π r Ω i v Ω i ( r) f G( r) = Ω i

22 Kybic, et al, IEEE Trans Med Imag, vol 24(1), p 12 28, 2005 Representation Theorem Ω be a connected, open, bounded subset of R 3 and Ω be regular boundary of Ω u : R 3 - Ω R is harmonic and r u(r) <, r( u/ r) = 0, then if p(r) = n u(r) the following holds:

23 Representation Theorem (cont) I is the identity operator and

24 Representation Theorem (cont) n G means n G, where n is normal to an interfacing head tissue surface

25 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 (openmeeggforgeinriafr/ )

26 Kybic, et al, IEEE Trans Med Imag, vol 24(1), p 12 28, 2005 Distributed Source : Gain Matrix

27 Gain Matrix (cont) Gain matrix is to be obtained by multiplying several matrices A, one for each layer in the single layer formulation

28 Hallez et al, J NeuroEng Rehab, 2007, open access Finite Elements Method

29 Further Reading on FEM Awada et al, Computational aspects of finite element modeling in EEG source localization, IEEE Trans Biomed Eng, 44(8), pp , 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 , 2004

30 Inverse Problem : Peculiarities Inverse problem is ill-posed, because the number of sensors is less than the number of possible sources Solution is unstable, ie, susceptible to small changes in the input values Scalp EEG is full of artifacts and noise, so identified sources are likely to be spurious

31 Mattout et al, NeuroImage, vol 30(3), p , Apr 2006 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 2 Wn + λ J 2 Wp λ is a regularization positive constant between 0 and 1

32 Geometric Interpretation V-BJ 2 Wn J 2 Wp Convex combination of the two terms with λ very small minu(j)

33 Derivation U(J) = V BJ 2 Wn + λ J 2 Wp = <W n (V BJ), W n (V BJ)> + λ<w p J, W p J> J U(J) = 0 implies (using <AB,C> = <B,A T C>) J = C p B T [BC p B T + C n ] -1 V where C p = (W T pw p ) -1 and C n = λ(w T nw n ) -1

34 Different Types When C p = I p (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) = C p B T [BC p B T + C n ] -1 V On this expectation maximization algorithm can be readily applied

35 Types (cont) If we take W p 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

36 Types (cont) Recursive MUSIC Mosher & Leahy, IEEE Trans Biomed Eng, vol 45(11), p , Nov 1998

37 Low Resolution Brain Electromagnetic Tomography (LORETA) Pascual-Marqui et al, Int J Psychophysiol, vol 18, p 49 65, 1994

38 Standardized Low Resolution Brain Electromagnetic Tomography (sloreta) U(J) = V BJ 2 + λ J 2 Ĵ = TV, where T = B T [BB T + λh] +, where H = I LL T /L T L is the centering matrix Pascual-Marqui at Math02htm

39 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 = B T [BB T + λh] + BJ

40 Form of H n n n n n n n n n n n n

41 If # Source = # Electrodes = n B and B T both will be n x n identity matrix with λ = = n n n n n n n n n n n n T λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ 1 1 1

42 sloreta Result

43 Single Trial Source Localization Averaging signals across the trials to increase the SNR cannot be done

44 Generated using OpenMeeg in INRIA Sophia Antipolis in 2008 Cortical Sources

45 Generated using OpenMeeg in INRIA Sophia Antipolis in 2008 Localization by MN

46 Majumdar, IEEE Trans Biomed Eng, vol 56(4), p , 2009 Clustering

47 Trial Selection Identify the time interval Identify channels Calculate cumulative signal amplitude in those channels Sort trials according to the decreasing amplitude

48 Phase Synchronization m=n =1

49 Synchronization (cont)

50 Synchronization (cont)

51 Synchronization (cont) E[n] = syn( x ( t), x ( t)) = mean( E( n) ) std( E( n) j k + )

52 Phase Synchronization and Power

53 The Best Time Epoch

54 Majumdar, IEEE Trans Biomed Eng, vol 56(4), p , 2009 Source Localization

55 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 46

56 Must Reading (cont) Grech et al, Review on solving the inverse problem in EEG source analysis, J Neuroeng Rehab, open access, available at 25

57 THANK YOU This presentation is available at

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