Ebook Brain source localization using EEG signal analysis: Part 2

Part 2 book “Brain source localization using EEG signal analysis” has contents: EEG inverse problem III - Subspace-based techniques, EEG inverse problem IV- Bayesian techniques, EEG inverse problem V - Results and comparison, future directions for EEG source localization.

chapter seven EEG inverse problem III Subspace-based techniques Introduction Over the past few decades, a variety of techniques have been developed for brain source localization using noninvasive measurements of brain activities, such as EEG and magnetoencephalography (MEG) Brain source localization uses measurements of the voltage potential or magnetic field at various locations on the scalp and then estimates the current sources inside the brain that best fit these data using different estimators The earliest efforts to quantify the locations of the active EEG sources in the brain occurred more than 50 years ago when researchers began to relate their electrophysiological knowledge about the brain to the basic principles of volume currents in a conductive medium [1–3] The basic principle is that an active current source in a finite conductive medium produces volume currents throughout the medium, which lead to potential differences on its surface Given the special structure of the pyramidal cells in the cortical area, if enough of these cells are in synchrony, volume currents large enough to produce measurable potential differences on the scalp will be generated The process of calculating scalp potentials from current sources inside the brain is generally called the forward problem If the locations of the current sources in the brain are known and the conductive properties of the tissues within the volume of the head are also known, the potentials on the scalp can be calculated from the electromagnetic field principles Conversely, the process of estimating the locations of the sources of the EEG from measurements of the scalp potentials is called the inverse problem Source localization is an inverse problem, where a unique relationship between the scalp-recorded EEG and neural sources may not exist Therefore, different source models have been investigated However, it is well established that neural activity can be modeled using equivalent current dipole models to represent well-localized activated neural sources [4,5] Numerous studies have demonstrated a number of applications of dipole source localization in clinical medicine and neuroscience research, and many algorithms have been developed to estimate dipole locations 91 92 Brain source localization using EEG signal analysis [6,7] Among the dipole source localization algorithms, the subspace-based methods have received considerable attention because of their ability to accurately locate multiple closely spaced dipole sources and/or correlated dipoles In principle, subspace-based methods find (maximum) peak locations of their cost functions as source locations by employing certain projections onto the estimated signal subspace, or alternatively, onto the estimated noise-only subspace (the orthogonal complement of the estimated signal subspace), which are obtained from the measured EEG data The subspace methods that have been studied for MEG/EEG include classic multiple signal classification (MUSIC) [8] and recursive types of MUSIC: for example, recursive-MUSIC (R-MUSIC) [6] and recursively applied and projected-MUSIC (RAP-MUSIC) [6] Mosher et al [4] pioneered the investigation of MEG source dipole localization by adapting the MUSIC algorithm, which was initially developed for radar and sonar applications [8] Their work has made an influential impact on the field, and MUSIC has become one of most popular approaches in MEG/EEG source localization Extensive studies in radar and sonar have shown that MUSIC typically provides biased estimates when sources are weak or highly correlated [9] Therefore, other subspace algorithms that not provide large estimation bias may outperform MUSIC in the case of weak and/or correlated dipole sources when applied to dipole source localization In 1999, Mosher and Leahy [6] introduced RAP-MUSIC It was demonstrated in one-dimensional (1D) linear array simulations that when sources were highly correlated, RAP-MUSIC had better source resolvability and smaller root mean-squared error of location estimates as compared with classic MUSIC In 2003, Xu et al [10] proposed a new approach to EEG three-dimensional (3D) dipole source localization using a nonrecursive subspace algorithm called first principle vectors (FINES) In estimating source dipole locations, the present approach employs projections onto a subspace spanned by a small set of particular vectors (FINES vector set) in the estimated noise-only subspace instead of the entire estimated noise-only subspace in the case of classic MUSIC The subspace spanned by this vector set is, in the sense of principal angle, closest to the subspace spanned by the array manifold associated with a particular brain region By incorporating knowledge of the array manifold in identifying FINES vector sets in the estimated noise-only subspace for different brain regions, the present approach is able to estimate sources with enhanced accuracy and spatial resolution, thus enhancing the capability of resolving closely spaced sources and reducing estimation errors In this chapter, we outline the MUSIC and its variant, the RAP-MUSIC algorithm, and the FINES as representatives of the subspace techniques in solving the inverse problem with brain source localization Because we are primarily interested in the EEG/MEG source localization problem, we have restricted our attention to methods that not Chapter seven:  EEG inverse problem III 93 impose specific constraints on the form of the array manifold For this reason, we not consider methods such as estimation of signal parameters via rotational invariance techniques (ESPRIT) [11] or root multiple signal classification-MUSIC (ROOT-MUSIC), which exploits shift invariance or Vandermonde structure in specialized arrays Subspace methods have been widely used in applications related to the problem of direction of arrival estimation of far-field narrowband sources using linear arrays Recently, subspace methods started to play an important role in solving the issue of localization of equivalent current dipoles in the human brain from measurements of scalp potentials or magnetic fields, namely, EEG or MEG signals [6] These current dipoles represent the foci of neural current sources in the cerebral cortex associated with neural activity in response to sensory, motor, or cognitive stimuli In this case, the current dipoles have three unknown location parameters and an unknown dipole orientation A direct search for the location and orientation of multiple sources involves solving a highly nonconvex optimization problem One of the various approaches that can be used to solve this problem is the MUSIC [8] algorithm The main attractions of MUSIC are that it can provide computational advantages over least squares methods in which all sources are located simultaneously Moreover, they search over the parameter space for each source, avoiding the local minima problem, which can be faced while searching for multiple sources over a nonconvex error surface However, two problems related to MUSIC implementation often arise in practice The first one is related to the errors in estimating the signal subspace, which can make it difficult to differentiate “true” from “false” peaks The second is related to the difficulty in finding several local maxima in the MUSIC algorithm because of the increased dimension of the source space To overcome these problems, the RAP-MUSIC and FINES algorithms were introduced In the remaining part of this chapter, the fundamentals of matrix subspaces and related theorems in linear algebra are first outlined Next, the EEG forward problem is briefly described, followed with a detailed discussion of the MUSIC, the RAP-MUSIC, and the FINES algorithms 7.1  Fundamentals of matrix subspaces 7.1.1  Vector subspace Consider a set of vectors S in the n-dimension real space r n S is a subspace of r n if it satisfies the following properties: • The zero vector ϵ S • S is closed under addition This means that if u and v are vectors in S, then their sum u + v must be in S • S is closed under scalar multiplication This means that if u is a vector in H and c is any scalar, the product cu must be in S 94 Brain source localization using EEG signal analysis 7.1.2  Linear independence and span of vectors Vectors a1 , a ,…, a n ∈ r m are linearly independent if none of them can be written as a linear combination of the others: n ∑α a = j j implies α(1:n) = j=1 (7.1) Given a1 , a ,…, a n ∈ r m, the set of all linear combinations of these ­vectors is a subspace S ∈ r m : S = span {a1 , a ,…, a n } (7.2) 7.1.3  Maximal set and basis of subspace If the set φ = {a1, a2,…, an} represents the maximum number of independent vectors in r m , then it is called the maximal set If the set of vectors ϕ = {a1, a2,…, ak} is a maximal set of subspace S, then S = span {a1, a2, …, ak} and ϕ is called the basis of S If S is a subspace of r m , then it is possible to find various bases of S All bases for S should have the same number of vectors (k) The number of vectors in the bases (k) is called the dimension of the subspace and denoted as k = dim (S) 7.1.4  The four fundamental subspaces of A ∈ r m×n Matrix A ∈ r m×n has four fundamental subspaces defined as follows The column space of A is defined as: C(A) = span {c1 ,, c n } C( A ) ∈ r m (7.3) The nullspace of A is defined as: N (A) = {x ∈ r n : Ax = 0} N (A ) ∈ r n (7.4) Chapter seven:  EEG inverse problem III 95 The column space of AT is defined as: C(A T ) = span{r1 , …, rm } C( A T ) ∈ r n (7.5) The nullspace of AT is defined as: N (A T ) = {y ∈ r m : A T y = 0} N (A T ) ∈ r m (7.6) The column space and row space have equal dimension r = rank (A) The nullspace N(A) has the dimension n – r, N(AT) has the dimension m – r, and the dimensions of the four fundamental subspaces of matrix A ϵ Rm×n are given as follows: dim[C(A)] + dim[N (A)] = (r ) + (n − r ) T dim[C(A )] + dim[N (A )] = (r ) + (m − r ) =n (7.7) T =m The row space C(AT) and nullspace N(A) are orthogonal complements (Figure 7.1) The orthogonality comes directly from the equation Ax = C(AT) C(A) Row space all ATy dim r Column space all Ax dim r Rn dim n – r N(A) x in nullspace Ax = Rm y in nullspace of AT ATy = dim m – r N(AT) Figure 7.1  Dimensions and orthogonality for any m by n matrix A of rank r [12] 96 Brain source localization using EEG signal analysis Each x in the N(A) is orthogonal to all the rows of A as shown in the following equation: Ax =  (row 1)     0 ← x is orthogonal to row         x  =  0      (row m)    0 ← x is orthogonal to row m       (7.8) The column space C(A) and nullspace of AT are orthogonal complements The orthogonality comes directly from the equation ATy = 0 Each y in the nullspace of AT is orthogonal to all the columns of A as shown in the following equation: AT y = (column 1)    0 ← y is orthogonal to collumn         y =  0      (column n)    0 ← y is orthogonal to column n       (7.9) 7.1.5  Orthogonal and orthonormal vectors Subspaces S1, …, Sp in r m are mutually orthogonal if xTy = whenever x ∈ s i and y ∈ s j for i ≠ j (7.10) Orthogonal complement subspace in r m is defined by s ⊥ = {y ∈ r m : y T x = for all x ∈ s} dim(s ) + dim(s ⊥ ) = m (7.11) (7.12) Matrix Q ∈ r m×m is said to be orthogonal if QT Q = Im×m, where I is the identity matrix If Q = q1 q  q m  is orthogonal, then qi forms the orthonormal basis for r m Chapter seven:  EEG inverse problem III 97 Theorem [12]: If V1 ∈ r m×r has orthogonal column vectors, then there exists V2 ∈ r m×( m−r ) such that C(V1) and C(V2) are orthogonal 7.1.6  Singular value decomposition Singular value decomposition (SVD) is a useful tool in handling the problem of orthogonality SVD deals with orthogonality through its intelligent handling of the matrix rank problem Theorem [12]: If A is a full rank real m-by-n matrix and m > n, then there exists orthogonal matrices: U = [u1 , , u m ] ∈ r m×m V = [v , , v n ] ∈ r n×n U T AV = Σ = diag (σ1 , , σn ) ∈ r m×n where σ1 ≥ σ2 ≥  ≥ σn ≥ (7.13) n A = UΣV T =   ∑σ u v i i T i i=1 where σi is the singular value of A, and ui and vi are the left and right s­ ingular vectors, respectively It is easy to verify that AV = UΣ and ATU = VΣ, where Av i = σiui   A T ui = σi v i  i = : min{m, n} (7.14) If A is rank deficient with rank (A) = r, then σ1 ≥ σ2 ≥  ≥ σr > σr+1 =  = (7.15) C(A) = span{u1 ,…, u r } ∈ r m , N (A) = span{v r+1 ,…, v n } ∈ r n   C(A T ) = span{v ,…, v r } ∈ r n , N (A T ) = span{u r+1 ,…, u m } ∈ r m (7.16) 7.1.7  Orthogonal projections and SVD Let S be a subspace of r n P ∈ r n×n is the orthogonal projection onto S if Px ϵ S for each vector x in Rn The projection matrix P satisfies two properties: P2 = P PT = P (7.17) 98 Brain source localization using EEG signal analysis From this definition, if x ∈ r n , then Px ∈ S and (I − P)x ∈ S⊥ Suppose A = UΣVT is the SVD of A of rank r and  r ] V = [ Vr U = [ Ur U r m−r r r ] V n−r (7.18) There are several important orthogonal projections associated with SVD: Vr VrT = projection of x ∈ r n onto C(A T ) rV  rT = projection of x ∈ r n onto C(A T )⊥ = N (A) V U r U Tr = projection of x ∈ r m onto C(A)  rU  Tr = projection of x ∈ r m onto C(A)⊥ = N (A T ) U (7.19) 7.1.8  Oriented energy and the fundamental subspaces Define the unit ball (UB) in r m as: { UB = q ∈ r m } q =1 (7.20) Let A be an m × n matrix, then for any unit vector q ∈ r m; the energy Eq measured in direction q is defined as: n Eq [A] = ∑ (q a ) T k (7.21) k =1 The energy ES measured in a subspace S ⊂ r m S ⊂ Rm is defined as follows: n ES [A] = ∑ P (a ) S k =1 k 2 (7.22) where Ps(ak) denotes the orthogonal projection of ak onto S Theorem: Consider the m × n matrix A with its SVD defined as in the SVD theorem, where m ≥ n, then Eui [A] = σi2 (7.23) Chapter seven:  EEG inverse problem III 99 If the matrix A is rank deficient with rank = r, then there exist directions in Rm that contain maximum energy and others with minimum and no energy at all Proof For the proof, see [13] Corollary max Eq∈UB [A] = Eu1 [A] = σ12 Eq∈UB [A] = Eur [A] = σr2 Eq∈UB [A] = Eur+i [A] = for i = 1,2, ,n r max ES⊂r m [A] = ESr [A] = U ∑σ i (7.24) i =1 n ES⊂r m [A] = E r ⊥ SU ( ) [A] = ∑σ i i= r +1 = 7.1.9  The symmetric eigenvalue problem Theorem (Symmetric Schur decomposition): If R ∈ r n×n is symmetric (ATA), then there exists an orthogonal V ∈ r n×n such that V T RV = Λ = diag(λ1 , , λn ) (7.25) Moreover, for k = 1:n, RV(:,k) = λkV(:,k) Proof For the proof, see Golub and Van Loan [12] Theorem: If A ∈ r m×n is symmetric-rank deficient with rank = r, then Λ R = VΛ V T = (V1 V2 )  0V1T    0V1T  V1 = (v , v , , v r ) ∈ r n×r V2 = (v r +1 , v r +2 , , v n ) Λ = diag (λ1 , λ2 , , λr ) ∈ r r×r span(v , v , , v r ) = C(R) = C(R T ) span( v r +1 , v r +2 , , v m ) = C(R)⊥ = N (R T ) = N (R) Proof For the proof, see Golub and Van Loan [12] (7.26) 100 Brain source localization using EEG signal analysis There are important relationships between SVD of A ∈ r m×n (m ≥ n) and Schur decomposition of symmetric matrices (A T A) ∈ r n×n and (AA T ) ∈ r m×m If UT AV = diag (σ1, …, σn) is the SVD of A, then the eigendecomposition of ATA is (7.27) V T (A TA)V = diag(σ12 , , σn2 ) ∈ r n×n and the eigendecomposition of AAT is U T (AA T )U = diag(σ12 , , σn2 , 0, , m ) ∈ r m×m (7.28) Let the eigendecomposition of rank r symmetric correlation matrix R s ∈ r m×m be given by  Λ s1 R s = Vs Λ s Vs T = (Vs1 Vs )  0Vs1T    0Vs T2  Vs1 = (v , v , , v r ) ∈ r m×r Vs = (v r+1 , v r+2 , , v m ) (7.29) Λ s1 = diag (λ1 , λ2 , , λr ) ∈ r r×r span(v , v , , v r ) = C(R s ) = C (R Ts ) span(v r+1 , v r+2 , , v m ) = C(R s )⊥ = N (R Ts ) = N (R s ) If the data matrix is corrupted with additive white Gaussian noise of variance σnoise , then the eigendecomposition of the full-rank noisy correla2 tion matrix R x = R s + σnoise Im is given as:  Λ s + σnoise R x = Vs Λ x VsT = (Vs1 Vs )   VsT1    Im−r VsT2  σnoise (7.30) The eigenvectors Vs1 associated with the r largest eigenvalues span the signal subspace or principal subspace The eigenvectors Vs2 associated with the smallest (m – r) eigenvalues, Vs2, span the noise subspace 7.2  The EEG forward problem The EEG forward problem is simply to find the potential g(r, rdip , d) at an electrode positioned on the scalp at a point having position vector r, due to a single dipole with dipole moment d = dedip (with magnitude d and 210 Appendix B Temporal projector evaluation for i = 1 to length(D) define time window of interest apply Hanning operator and filtering endfor Optimize spatial priors over subjects   Creating spatial basis in source space   for i = 1 to number of patches calculate modified source space for left hemisphere ∋ SMSP(LH) = Kmod × q and q is derived from smoothening Green’s function calculate for the right hemisphere depending on vert matrix calculate for bilateral endfor Inversion if OP-MSP with greedy search is selected then number of patches ← length of source space for i = 1 to number of patches define new source space by modification introduced earlier endfor Calculate multivariate Bayes, which provides Bayesian optimization of multivariate linear model Empirical priors are accumulated elseif OP-MSP with automatic relevance determination (ARD) is selected then Use restricted maximum likelihood (ReML) to estimate the covariance components from sample covariance matrix Design spatial priors using the number of patches and modified hyperparameters for i = 1 to number of patches if hyperparameter lies within threshold then define new source space based on modified hyperparameters and spatial priors endif endfor accumulate empirical priors based on modified leadfield UL and spatial prior Optimized patches for number of patches 200 to 2100 all the parameters are updated observe the results if Optimization of hyperparameters is successful in terms of free energy and localization error then Plot the solution and compare with previous solution Appendix B 211 else Optimize the patches and repeat from previous step endif endfor Plotting Plot the solution and observe the free energy and localization error if free energy is high localization error is low endif else start again optimizing the parameter Pseudocode for implementation protocol of MSP (ARD and greedy search) Input: Preprocessed EEG data D ∈ ℜ380×991×3, gain/lead-field matrix Output: Inversion of EEG data at certain co-ordinates in 3D activation map Define: Time window of interest ← [start time stop time] SF for source priors ←(0 ≤ SF ≤ 1) Number of patches ← 100 Low pass and high pass frequency ← any value that suits Standard deviation for Gaussian temporal correlation ← Checking lead field and optimization for spatial projector for i = 1 to length(D) check the lead field and get number of dipoles and channels check for null spaces and remove it endfor Average lead-field computation for i = 1 to number of modes initialize average lead field and compute regularized inverses for lead field for j = 1 to eliminate redundant virtual channels eliminate low SNR spatial modes by SVD(UL×ULT) ∋ UL = reduced lead field normalize the lead field as UL = UL/scaling factor endfor endfor Temporal projector evaluation for i = 1 to length(D) define time window of interest apply Hanning operator and filtering endfor 212 Appendix B Optimize spatial priors over subjects Creating spatial basis in source space for i = 1 to number of patches calculate modified source space for left hemisphere ∋ SMSP(LH) = Kmod × q and q is derived from smoothening Green’s function calculate for the right hemisphere depending on vert matrix calculate for bilateral endfor Inversion if MSP with greedy search is selected then number of patches ← length of source space for i = 1 to number of patches define new source space by modification introduced earlier endfor Calculate multivariate Bayes, which provides Bayesian optimization of multivariate linear model Empirical priors are accumulated elseif MSP with ARD is selected then Use ReML to estimate the covariance components from sample covariance matrix Design spatial priors using number of patches and modified hyperparameters for i = 1 to number of patches if hyperparameter lies within threshold then define new source space based on modified hyperparameters and spatial priors endif endfor accumulate empirical priors based on modified lead-field UL and spatial prior Plotting Plot the solution and observe the free energy and localization error if free energy is high localization error is low endif else start again optimizing the projectors and repeat through Step Pseudocode for implementation protocol for minimum norm estimation (MNE) Input: Preprocessed EEG data D ∈ ℜ380×991×3, gain/lead-field matrix Appendix B 213 Output: Inversion of EEG data at certain co-ordinates in 3D activation map Define: Time window of interest ←[start time stop time] SF for source priors ←(0 ≤ SF ≤ 1) Number of patches ← 100 Low pass and high pass frequency ← any value that suits Standard deviation for Gaussian temporal correlation ← Checking lead field and optimization for spatial projector for i = 1 to length(D) check the lead field and get number of dipoles and channels check for null spaces and remove it endfor Average lead-field computation for i = 1 to number of modes initialize average lead field and compute regularized inverses for lead field for j = 1 to eliminate redundant virtual channels eliminate low SNR spatial modes by SVD(UL×ULT) ∋ UL = reduced lead field normalize the lead field as UL = UL/scaling factor endfor endfor Temporal projector evaluation for i = 1 to length(D) define time window of interest apply Hanning operator and filtering endfor Optimize spatial priors over subjects Create source component create minimum norm prior based on modified lead-field UL and source component Inversion for i = 1 to number of patches Define spatial priors as done in previous steps Accumulate empirical priors endfor Plotting Plot the solution and observe the free energy and localization error if free energy is high localization error is low endif else start again optimizing the projectors and repeat through Step 214 Appendix B Pseudocode for implementation protocol for low-resolution brain electromagnetic tomography (LORETA) Input: Preprocessed EEG data D ∈ ℜ380×991×3, gain/lead-field matrix Output: Inversion of EEG data at certain co-ordinates in 3D activation map Define: Time window of interest ←[start time stop time] SF for source priors ←(0 ≤ SF ≤ 1) Number of patches ← 100 Low pass and high pass frequency ← any value that suits Standard deviation for Gaussian temporal correlation ← Checking lead field and optimization for spatial projector for i = 1 to length(D) check the lead field and get number of dipoles and channels check for null spaces and remove it endfor Average lead-field computation for i = 1 to number of modes initialize average lead field and compute regularized inverses for lead field for j = 1 to eliminate redundant virtual channels eliminate low SNR spatial modes by SVD(UL×ULT) ∋ UL = reduced lead field normalize the lead field as UL = UL/scaling factor endfor endfor Temporal projector evaluation for i = 1 to length(D) define time window of interest apply Hanning operator and filtering endfor Optimize spatial priors over subjects create minimum norm prior based on modified lead-field UL and source component add smoothness component in source space T ∋ SLOR = K mod ×Qp × K mod Inversion for i = 1 to number of patches Define spatial priors as done in previous steps Accumulate empirical priors endfor Appendix B 215 Plotting Plot the solution and observe the free energy and localization error if free energy is high localization error is low endif else start again optimizing the projectors and repeat through Step Pseudocode for implementation protocol for beamforming Input: Preprocessed EEG data D ∈ ℜ380×991×3, gain/lead-field matrix Output: Inversion of EEG data at certain co-ordinates in 3D activation map Define: Time window of interest ←[start time stop time] SF for source priors ← (0 ≤ SF ≤ 1) Number of patches ← 100 Low pass and high pass frequency ← any value that suits Standard deviation for Gaussian temporal correlation ← Checking lead field and optimization for spatial projector for i = 1 to length(D) check the lead field and get number of dipoles and channels check for null spaces and remove it endfor Average lead-field computation for i = 1 to number of modes initialize average lead field and compute regularized inverses for lead field for j = 1 to eliminate redundant virtual channels eliminate low SNR spatial modes by SVD(UL×ULT) ∋ UL = reduced lead field normalize the lead field as UL = UL/scaling factor endfor endfor Temporal projector evaluation for i = 1 to length(D) define time window of interest apply Hanning operator and filtering endfor Optimize spatial priors over subjects for i = 1 to number of sources normalized power is calculated using modified lead-field UL source power is calculated 216 Appendix B power for all sources is calculated by dividing source power to normalized power endfor The power from all sources is normalized to generate T source component ∋ SBeamformer = K mod ×Qp × K mod Inversion for i = 1 to number of patches Define spatial priors as done in previous steps Accumulate empirical priors endfor Plotting Plot the solution and observe the free energy and localization error if free energy is high localization error is low endif else start again optimizing the projectors and repeat through Step Index A Accuracy, 125, 128 of inversion methods, 126 Action potentials (APs), 8, Adaptive segmentation, 27 Addition of noise, 138 Alpha (α), 23 band, 25 waves, 23 Alzheimer disease, 10, 17, 27 Ampere’s law, 38, 39 Analytical head modeling, 50–51 Analytical models, 49, 51 Anatomical imaging modalities, 51 ANSYS, 55 APs, see Action potentials ARD, see Automatic relevance determination Artifact removal, 25–26, 144 Artificial noise, 138 Astrocytes, Automatic relevance determination (ARD), 128–130 Averaging, 144 Axon, B Ballistocardiogram artifact (BCG artifact), 25 Bayesian-based models, 133, 181 Bayesian approaches, Bayesian formulation, 71 Bayesian framework-based MSP, 194 Bayesian inverse problems, 30 Bayesian techniques; see also Subspacebased techniques derivation of free energy, 121–125 EEG source localization, 113 flowchart for implementation of MSP, 132 generalized Bayesian framework, 113–118 MSP, 119–121 optimization of cost function, 126–132 selection of prior covariance matrices, 118–119 variations in MSP, 132–134 Bayes’ theorem, 113–114 BCG artifact, see Ballistocardiogram artifact BCI, see Brain computer interfacing Beamformer activation map for, 148 algorithms, 176, 179 pseudocode for implementation protocol for, 215–216 results for, 145–161, 195–199 BEM, see Boundary element method BESA, see Brain Electrical Source Analysis Beta (β), 23 band, 25 waves, 23 Blood oxygenation level-dependent (BOLD), 18 BOLD, see Blood oxygenation level-dependent Boundary conditions, 42–43 Boundary element method (BEM), 1, 49, 55–59 Brain activity, 192 anatomical features, cells, development, hemispheres, metabolism analysis, 18 217 218 Brain (Continued) rhythms, 24 surface, tissues, waves, 23 waves in EEG signal analysis, 24 Brain applications EEG for, 17–33 fMRI for, 17–33 MEG for, 17–33 Brain computer interfacing (BCI), 27, 28 Brain disorder, economic burden due to, 10–12 neuroimaging techniques for, 9–10 Brain Electrical Source Analysis (BESA), 55 Brain source localization, 2–3, 10, 32, 102 potential applications, 12 software used for, 207–208 techniques, 11, 176 Brainstorm software packages, 57 “Butterworth” low-pass filter, 144 C Canonical correlation, 105 Cerebral cortex, 4, 6, 44, 93, 102 lobes of, Cerebrum, Chi rhythm, 25 Classical techniques, 63, 163, 165, 172, 193–194 eLORETA, 72–73 FOCUSS, 73–75 founding method, 65–66 inverse problems, 63–64 LORETA, 68–70 MNE, 66–68 regularization methods, 64–65 sLORETA, 70–72 Column space, 94 Complexity, 125, 128 Computational complexity parameter, 171 Computational time, 203 Computed tomography (CT), 19, 51 Computed tomography fluoroscopy head model (CTF head model), 139 Conductive medium, 91 Constrained Laplacian, Control theory, 30 Convergence, 83 Correlation matrix, 103 Index Cortical surface, 119 Covariance components, 126 Covariance matrix, 118 CT, see Computed tomography CTF head model, see Computed tomography fluoroscopy head model Current density (J), 115, 120 Current dipole, 1, 2, 30, 44, 55, 65, 81, 83, 91, 93, 107 Curvature of free energy, 129 D Dale methods, 72 Data downsampling, 143 Data epoching, 143 Data plots, 138 Delta (δ), 23 band, 25 waves, 24 Dendrite tree, Depression, 10, 11–12, 17 Designed algorithm, 205 Deterministic inverse problems, 30 Diagonal matrix, 83, 84 Diencephalon, Dimension, 94 Dimensionless scaling factors, 74 Dipole approximation and conductivity estimation, 44–45 current, 1, 2, 30, 44, 55, 65, 81, 83, 91, 93, 107 source localization algorithms, 92 Direct method, 53 Dirichlet boundary conditions, 42–43 Discrete Laplace operator, 68 E EEG, see Electroencephalography EEG forward problem, 100–102 EEG inverse problem I, 63–75 EEG inverse problem II, 79–88 EEG inverse problem III, 91–110 EEG inverse problem IV, 113–134 EEG ill-posed problem, EEG inverse problem, 1, 10, 28 EEG forward problem I, 37–45 EEG forward problem II, 49–59 EEG inverse problem V, 137 classical algorithms, 193–195 Easycap EEG electrode layout, 194 Index ENOBIO electrode layout, 195 Maxwell’s equations in, 37–39 RAP-MUSIC, 193 real-time EEG data, 139–161, 199–200 reduced channel source localization, 192 results for MNE, LORETA, beamformer, MSP, and modified MSP, 195–199 results for synthetic data, 176–192 synthetic EEG data, 137–139 EEG source localization, 10, 113, 203 future directions, 204 inverse solution for, 31–32 potential applications of, 32–33 significance of research with potential applications, 205 Electrical generators, 10 Electrical signals, 4, Electric field, 39 Electrocardiography, 22 Electroencephalography (EEG), 1, 10, 27, 68, 203 applications, 27–28 for brain applications, 17 recording signals, 21 rhythms, 23–25 sensors, 63 signal preprocessing, 25–27 signals, 20 source analysis, 28–31 10–20-electrode system, 22 Electromagnetic(s) field, 7, Maxwell’s equation for, 37 signals in brain, 10 Electromyography, 22, 25 Electrooculograms, 142–143 Electrooculography (EOG), 22, 143 Elekta Neuromag Vector View 306 Channel Meg system, 141 eLORETA, see Exact LORETA EM, see Expectation maximization Embryonic developments, Empirical mode decomposition, 26 Enobio cap, 194 Entropy, 122 EOG, see Electrooculography Ependymal cells, Epilepsy, 10–12, 17 Epithalamus, ERPs, see Event-related potentials 219 ESPRIT, see Estimation of signal parameters via rotational invariance technique Estimation of signal parameters via rotational invariance technique (ESPRIT), 2, 92 Event-related potentials (ERPs), 22, 143 signal for three-trial data, 146, 159, 161, 163, 165, 167, 169, 171, 173 Event, 114 Exact LORETA (eLORETA), 2, 32, 66, 69, 72–73 Expectation maximization (EM), 126 Experimental design, Stimuli, 140 F Face dataset, 140 Faraday’s law, 39 FDM, see Finite difference method FEM, see Finite element method Fieldtrip software packages, 53, 55, 57 Filtering, 25–26, 143–144 FINES, see First principle vectors Finite difference method (FDM), 1, 49, 52–53 Finite element method (FEM), 1, 49, 53–55, 57 head modeling, 53 Finite volume method, 57 First principle vectors (FINES), 92 subspace algorithm, 108–110 Fisher scoring over free energy variation, 127–128 fMRI, see Functional magnetic resonance imaging Focal underdetermined system solution (FOCUSS), 2, 73–75, 79, 83 Forebrain, 3, Forward model generation, 138 Forward problem, 28–31, 37, 91 boundary conditions, 42–43 potential derivation for, 41 Four-shell homogeneous spherical head model, 80 Fredholm integral equation, 64 Free energy, 121, 128, 203 accuracy and complexity, 125 derivation, 121 inverse of posterior covariance, 124 KL divergence, 122 second-order Taylor series expansion, 123 220 Frontal lobes, 3, 5–7 Functional brain imaging, 17 Functional magnetic resonance imaging (fMRI), 10, 17 for brain applications, 17–33 Functional neuroimaging techniques, 9–10 Fundamental subspaces of A ∈ R m×n matrix, 94–96 G Gain matrix, 102, 138 Galerkin’s method, 54 Gamma (γ), 23 band, 25 waves, 23 Gaussian distribution, 121 Gaussianity check, 27 Gaussian white noise, 138 Gauss–Jordan elimination, 59 Gauss’ law, 38, 39 for magnetism, 38, 39 Generalized Bayesian framework, 113–118 Generalized linear model (GLM), 65 Generalized probability distribution, 120 Glial cells, GLM, see Generalized linear model Global optimization algorithms, Gradient of free energy, 129 Graph, Laplacian matrix, 58, 118 Gray matter, Greedy search algorithm (GS algorithm), 128, 130–132 Green’s function, 58 Green’s theorem for BEM, 57–59 GS algorithm, see Greedy search algorithm H Harmonic functions, 68 Head modeling approaches, 49, 204 analytical head modeling, 50–51 BEM, 55–59 FDM, 52–53 FEM, 53–55 numerical head models, 51–52 volume conductor head models, 49–50 Head position indicator coils (HPI coils), 141 Hemispheres, 5–6 Hindbrain, 3, Index Homogenous Neumann boundary condition, 43 Horizontal electrooculography, 143 HPI coils, see Head position indicator coils Human brain anatomy and neurophysiology, 3–9 Hybrid algorithms, 2, 11, 79 Hybrid techniques hybrid WMN, 79–80 recursive sLORETA-FOCUSS method, 82–84 shrinking LORETA-FOCUSS method, 84–86 SSLOFO technique, 86–88 WMN-LORETA method, 80–82 Hybrid weighted minimum norm (Hybrid WMN), 2, 79–80 Hyperparameter, 118, 120, 125, 128 I Identity matrix, 82 Ill-posed EEG system, 63, 64 Ill-posed problems, 29 Independent topographies model (ITs model), 107 Interbrain, see Diencephalon Intersection, 114 Inverse problems, 28–29, 91, 102 for EEG, 30–31 FINES subspace algorithm, 108–110 methodologies in, 30 MUSIC algorithm, 103–107 RAP-MUSIC, 107–108 Inverse solution for EEG source localization, 31–32 Inversion techniques, 64, 157, 159 Iterative weighted method, 80 ITs model, see Independent topographies model J Joint probability distribution, 122 K Kappa rhythm, 25 Kronecker product, 68 k-step’s weighted matrix (Wk), 80 Kullback–Leibler divergence (KL divergence), 122 Kurtosis, 27 Index L Lambda rhythm, 25 Laplace equation, 42 LAURA, 193 Lead-field matrix, 65, 74, 102, 103, 138 Lead-field vectors, 106 Least squares, 2, 65, 93 Levenberg–Marquardt simplex searches, 1–2, 31 Linear arrays, 93 Linear independence and span of vectors, 94 Loading and conversion of data, 143 Localization error, 203 LORETA, see Low-resolution brain electromagnetic tomography LORETA-FOCUSS, see Recursive standardized LORETA–focal underdetermined system solution Low-resolution brain electromagnetic tomography (LORETA), 2, 32, 66, 68–70, 79, 86, 113, 137, 176, 179, 193, 214 activation map from random trial, 192 model, 118 pseudocode for implementation protocol for, 214–215 results for, 145–161, 195–199 for schizophrenic patients, 11 M Magnetic field, 39 Magnetic resonance imaging (MRI), 10, 19, 20, 51, 56, 58 Magnetoencephalography (MEG), 1, 10, 17, 68, 91, 139; see also Electroencephalography (EEG) for brain applications, 17–33 Mathematical background for EEG forward problem I, 37 dipole approximation and conductivity estimation, 44–45 Maxwell’s equations in EEG inverse problem, 37–39 potential derivation, 41–43 quasi-static approximation for head modeling, 40–41 MATLAB®, 53, 57, 83 Matrix subspaces, 93 221 fundamental subspaces of A ∈ R m×n matrix, 94–96 linear independence and span of vectors, 94 maximal set and basis of subspace, 94 oriented energy and fundamental subspaces, 98–99 orthogonal and orthonormal vectors, 96–97 orthogonal projections and SVD, 97–98 SVD, 97 symmetric eigenvalue problem, 99–100 vector subspace, 93 Maximal set and basis of subspace, 94 Maxwell’s equations, 37 in EEG inverse problem, 37–39 quasi-static approximations for, 40 M/EEG data acquisition, 141–143 preprocessing protocol, 143–144 source localization, 20 MEG, see Magnetoencephalography Memory retention and recalling analysis, 27 Mesencephalon, see Midbrain Microglia, Midbrain, 3, Minimization algorithms, 64 Minimum norm estimation (MNE), 2, 63, 65, 66–68, 70, 113, 137, 176, 179, 193, 212 pseudocode for implementation protocol, 212–213 results for, 145–161, 195–199 Minimum norms, 65 Minor rhythms, 24 MNE, see Minimum norm estimation Modified multiple sparse priors (Modified MSP algorithm), 113, 180, 181, 190–192 results for, 145–161, 195–199 Moore–Penrose pseudoinverse of matrix, 69 MRI, see Magnetic resonance imaging MSP, see Multiple sparse priors Multimodal data, 139 Multinormal distributions, 116 function, 121 Multiple signal classification (MUSIC), 2, 32, 92, 103–107 Multiple sparse priors (MSP), 113, 119–121, 137, 176, 194, 209 flowchart for implementation, 132 222 Multiple sparse priors (Continued) implementation protocol, 211–212 pseudocode for modified, 209–211 results for, 145–161 variations in, 132–134 Multiple sparse priors, Multivariate Gaussian distribution, 116 MUSIC, see Multiple signal classification Mutually exclusive events, 114 N Near-infrared spectroscopy, 10 Negative variational free energy, 121 Negentropy, 27 Nelder–Mead downhill simplex searches, 1–2, 31 NETSTATION software packages, 53 Neumann boundary conditions, 42–43, 54 Neuroimaging techniques, 1, 17 for brain disorders, 9–10 EEG for brain applications, 17–33 fMRI for brain applications, 17–33 MEG for brain applications, 17–33 with respect to temporal and spatial resolution, 19 Neurons, living neuron in culture, structure, Neuroscience research, 91 Noise, 18 covariance, 118, 129 subspace, 105 Noninvasive imaging techniques, 10 Noninvasive measurement of brain activities, 91 Nonparametric recursive algorithm, 73 Nonrecursive subspace algorithm, 108–109 Nucleus, Nullspace, 94 Null subspace of matrix, 105 Number of patches (Np), 128 Number of sources (Ns), 130 Numerical head models, 1, 51–52 Numerical models, 49, 57 O Object saving, 138 Occipital lobes, 3, 5–6, Oligodendrocytes, 1D linear array simulations, 92 OpenMEEG software packages, 57 Index Optimal basis set, 26 Optimization of cost function, 126 ARD, 128–130 GS algorithm, 130–132 ReML, 127 Oriented energy and fundamental subspaces, 98–99 Orthogonality, 94 Orthogonal projections, 97–98 operator, 108 Orthogonal vector, 96–97 Orthonormal basis of eigenvectors, 104 Orthonormal vector, 96–97 P Parametric methods, Parietal lobes, 3, 5–6, Parkinson disease, 10, 12 Participants, 139 Pascual-Marqui, R D., 68–69 Patches, 119, 126, 133–134 PET, see Positron emission tomography Phi rhythms, 24 Poisson’s equation, 42, 45, 50–52, 56, 101 Positron emission tomography (PET), 10, 17 Posterior covariance matrix, 126 Posteriori distribution, 30 Posterior probability, 114 Postsynaptic potentials (PSPs), 8–9 Preprocessing stage, 143 Principal component analysis, 26 Principal subspace, 105 Prior covariance matrices selection, 118–119 Prior probability, 114 Probability distribution, 204 Product commercialization, 12 Programming languages, 57 Prosencephalon, see Forebrain Pseudocodes for classical and modern techniques, 209 for implementation protocol for beamforming, 215–216 for implementation protocol for LORETA, 214–215 for implementation protocol for MNE, 212–213 for implementation protocol of MSP, 211–212 for modified multiple sparse priors implementation protocol, 209–211 PSPs, see Postsynaptic potentials Pyramidal neurons, Index Q Quantitative analysis of EEG (qEEG), 27 R R-MUSIC, see Recursive-MUSIC Rank-deficient matrix, 63 RAP-MUSIC, see Recursively applied and projected-multiple signal classification Rayleigh–Ritz method, 55 Real-time EEG data, 139, 193, 204; see also Synthetic EEG data comparison between various methods, 158, 160, 162, 164, 166, 168, 170, 172, 174, 175 details, 139–144 discussion of results, 161–176 flowchart for, 144, 145 reduced channels results, 199–200 results, 144–161 Recursive-MUSIC (R-MUSIC), 2, 92 Recursively applied and projected-multiple signal classification (RAPMUSIC), 2, 32, 92, 107–108, 193 Recursive sLORETA-FOCUSS method, 2, 82–84 Recursive standardized LORETA–focal underdetermined system solution (LORETA-FOCUSS), 32 Recursive technique, 79 Reduced electrodes technique validation, 204 Regularization, 74 methods, 65 parameter, 118 ReML, see Restricted maximum likelihood Research purposes, EEG, 27 Resolution matrix (R), 81–82, 86 Restricted maximum likelihood (ReML), 127 hyperparameters, 128 Rhombencephalon, see Hindbrain Root multiple signal classification-MUSIC (ROOT-MUSIC), 93 S Sample space, 114 Scalp-to-skull/scalp-to-brain conductivity ratio, 45 Schizophrenia, 10–12, 17 223 Scrambled faces, 140, 141 Second-order Taylor series expansion, 123 Segmentation, 27 Selection of Prior Covariance Matrices, 126 Sensors, 102–103 74-channel Easycap EEG cap, 141, 194 Shrinking LORETA-FOCUSS method, 2, 79, 84–86 Signal preprocessing, 25 filtering and artifact removal, 25–26 segmentation, 27 Signal-processing methods, 20 Signal-to-noise ratio (SNR), 137, 204 Signal subspace, 105 correlation, 106 SimBio software packages, 55 Simple straightforward linear algebra techniques, 59 Simply free energy, 121 Simulated annealing, Singular value decomposition (SVD), 64, 97–98 SVD-based subspace methods, 65 Skewness, 27 sLORETA, see Standardized LORETA sLORETA-FOCUSS algorithm, 83, 86 SNR, see Signal-to-noise ratio Soma, Source analysis, EEG, 28 forward and inverse problems, 29–31 Source covariance, 129 Source estimation algorithm, 121 Source generation, 137–138 Source localization, 28, 193 Spatial weighting elements, 74 Spatiotemporal ITs, 107 SPM, see Statistical parametric mapping SSLOFO, see Standardized shrinking LORETA-FOCUSS Stacked matrix, 129 Standardized LORETA (sLORETA), 2, 32, 66, 69, 70–72, 79 algorithm, 83 for schizophrenic patients, 11 Standardized shrinking LORETA-FOCUSS (SSLOFO), 2, 86–88 Statistical parametric mapping (SPM), 57, 139 Statistics, 30 Stimuli, 140 experimental design, 140 M/EEG data acquisition, 141–143 M/EEG preprocessing protocol, 143–144 224 Stochastic sampling techniques, 115 Stress, 12, 17 Subspace correlation, 105 factorization, Subspace-based techniques, 92; see also Bayesian techniques dipole source localization algorithms, 92 EEG forward problem, 100–102 fundamentals of matrix subspaces, 93–100 inverse problem, 102–110 MEG, 91 Subtraction method, 53 Superposition rule, 45 Support vector machine, 26 SVD, see Singular value decomposition SymBEM software packages, 57 Symmetric BEM, 57 Symmetric eigenvalue problem, 99–100 Symmetric Schur decomposition theorem, 99 Synthetic EEG data, 137, 193; see also Real-time EEG data flowchart for, 140 localization error, 176 protocol for synthetic data generation, 137–139 results for, 176, 195–199 SNR = dB, 181–185 SNR = 10 dB, 179–181 SNR = −20 dB, 187–192 SNR = dB, 176–179 SNR = −5 dB, 185–187 specifications, 138–139 T Tau rhythm, 25 Telencephalon, Temporal domain approach, 26 Index Temporal lobes, 3, 5–6, 10–20-electrode placement system, 21, 22, 25 Theta (θ), 23 band, 25 waves, 24 Three-dimensional (3D), 92 anatomical data, 51 coordinates, 138 discretized Laplacian matrix, 68 Tikhonov regularization, 65 Tissue conductivity, 45 Total current density (J), 41 Tumor analysis, 12 Two-dimensional Fourier transform (2D Fourier transform), 140 U Underdetermined systems, 63 V Variations in MSP, 132–134 Vector subspace, 93 Vert matrix, 138 Volume conductor head models, 49–50 W Waveform generation for each source, 138 Weighted minimum norm–LORETA (WMN-LORETA), 2, 32, 80–82 Weighted minimum norm (WMN), 65, 68, 79 Weight matrix, 81, 83, 84, 86 White matter, 3, 45 WMN-LORETA, see Weighted minimum norm–LORETA WMN, see Weighted minimum norm World Health Organization, 10–11 ... 92 Brain source localization using EEG signal analysis [6,7] Among the dipole source localization algorithms, the subspace-based methods have... 47(9), pp 124 8– 126 0, 20 00 22 J C Mosher and R M Leahy, Recursively applied MUSIC: A framework for EEG and MEG source localization, IEEE Transactions on Biomedical Engineering, vol 45, pp 13 42 1354,... of prior components C, which ultimately refers to 120 Brain source localization using EEG signal analysis selection of prior assumptions [22 ] Hence, the prior covariance matrix for MSP is generated