1 Quote as: Pascual - Marqui RD. Reply to Comments Made by R. Grave De Peralta Menendez and S.I. Gozalez Andino. International Journal of Bioelectromagnetism 1999, Vol. 1, No. 2, at: http://www.ee.tut.fi/rgi/ijbem/volume1/number2/html/pascual.htm REPLY TO COMMENTS MA DE BY R. GRAVE DE PERALTA MENENDEZ AND S.L. GOZALEZ ANDINO [ 10 ] Roberto Domingo Pascual - Marqui The KEY Institute for Brain - Mind Research, University Hospital of Psychiatry, 8029, Zurich, Switzerland - Lenggstr. 31, CH ] c) Quoting from [ 10 , comment “c)” : “The superposition principle cannot be applied to a non linear relationship...” My Reply c) : The principles of linearity and superposition for localization are illustrated in Figure 1 . These results (and their generalization) can be replicated by the interested reader, using software that has been available upon request to the 1b and 1a author since June 1998 ( http://www.keyinst.unizh.ch/loreta.htm ). Figures s at different depths. The deep show LORETA point spread functions for two source source is more blurred than the shallow source. The only way LORETA can resolve both simultaneously active sources is by increasing the strength of the deep source, as 1c shown in Figure (which is the same as Figure 4 in [1]). In general, LORETA can resolve two sources if they are sufficiently separated, and if their estimated strengths are comparable. This is the essence and main property of Low Resolution Brain it will always produce a blurred Electromagnetic Tomography (LORETA): (approximate) image of reality. Blurring will not always allow resolving all maxima. There was never any pompous claim of “high” or “optimum” resolution in LORETA. The main property of LORETA holds. 1.1. Quoting from [ 10 ] : “The study of all possible spread functions is equivalent to the analysis of all the resolution kernels [2],[3].” 1.1. My reply : Grave de Peralta Menendez and Gonzalez Andino quote ment is not to themselves for this statement. They have falsified the truth: this state ] . This statement can be found in my paper [1] (in ] [ 3 be found in their papers [ 2 and section “The resolution matrix”). Actually, Grave de Peralta Menendez and Gonzalez ion Andino have made statements quite to the contrary, scorning the informat contained in the point spread functions. For instance, in [ 2 ] they state: “The information contained on the impulse responses concerns exclusively single point sources,...” (Note: impulse response and point spread function have the same meaning.) - 1.2. [ 10 ] : “However, the analysis presented by Pascual Quoting from Marqui in [1] and [6] to evaluate the solutions is not really using the spread functions but a measure derived from them: The dipole localization error..." I present a ] n exhaustive analysis of all point spread [ 1 1.2. My reply : In functions, based on the feature which I consider most relevant to the aim of EEG inverse solutions: localization error. The “dispersion” of point spread functions was 10 Pag e 1 of

2 Quote as: Pascual Marqui RD. Reply to Comments Made by R. Grave De Peralta Menendez and S.I. Gozalez Andino. - International Journal of Bioelectromagnetism 1999, Vol. 1, No. 2, at: http://www.ee.tut.fi/rgi/ijbem/volume1/number2/html/pascual.htm defined, studied, and reported in [ 6 ] . The comp uter programs used and offered to the [ 1 reader in (available upon request to the author since June 1998: ] http://www.keyinst.unizh.ch/loreta.htm ), allow the full and complete exhaustive evaluation of a ll point spread functions, including amplitudes (see Figure 1 ). I define “first order localization errors” of an instantaneous 3D discrete linear inverse solution as the set of localization errors for each point spread function. My methodology for comparing solutions belonging to this class starts with the following two principles: (1) high first order localization errors indicate the inadequacy of a solution; (2) the converse is not true. One essential fact must stressed: while l ow errors do not indicate adequacy of a solution, they do constitute a necessary (but not sufficient) condition for adequacy of a solution. 2.1. Quoting from [ 10 ] : “About the “ futility of trying to design near ideal averaging kernels”...” 2.1. My reply : The averaging (or resolution) kernels are harmonic functions. This fact was published in [ 1 ] , and it proves that in a 3D solution space, the averaging kernels can not be optimized. Therefore, all efforts towards optimization in a 3D )”, ] 3 [ , ] solution space, as pub lished and “extensively discussed in the literature ( [ 8 ] , [ 2 have been futile. This fact of nature holds and cannot be changed for a 3D solution space. Any insistence in the rationality of optimization in 3D space is pointless. I wish to emphasiz e that the “curse of harmonic resolution kernels” was [ 1 ] . It was not reported in the papers by Grave de Peralta Menendez and reported in Gonzalez Andino, a fact that can be confirmed by reading carefully their self - quoted papers. For instance, in [ 3 ] they state: “A certain eccentricity value seems to exist below which all solutions fail to obtain adequately centered resolution kernels around the target point.” This statement is a far cry away from the full mathematical nic character of the resolution kernels reported characterization implied by the harmo ] . 1 [ in One word of caution with respect to the equivalence of resolution kernel and point spread function optimization: Resolution kernels can not be optimized in 3D ter. Point spread functions might be amenable space, because of their harmonic charac to optimization, since, in general, they are not harmonic. However, for optimization to take effect, one must find the proper functional. The WROP functionals in [3] may not full necessarily be the best ones. Other functionals for optimizing the resolution . This optimization, with ] [ 1 matrix exist, such as the one reported in equation 10, in the proper weight, produces LORETA, which satisfies “the minimum necessary rs. condition” of low first order localization erro : “We are pleased to see that in this paper [1], the 2.2. Quoting from [ ] 10 author coincides with us...” 2.2. My reply : First, it is worth emphasizing that the definition and interpretation of the averaging kernels for any linear inverse problem were p ublished by Backus and Gilbert [8]. This contribution was not made by Grave de Peralta Menendez and Gonzalez Andino, as they so pretentiously imply. Second, all averaging 10 Pag e 2 of

3 Quote as: Pascual Marqui RD. Reply to Comments Made by R. Grave De Peralta Menendez and S.I. Gozalez Andino. - International Journal of Bioelectromagnetism 1999, Vol. 1, No. 2, at: http://www.ee.tut.fi/rgi/ijbem/volume1/number2/html/pascual.htm kernel features emphatically proposed in ([2], [3], [4], [5]) are practically non - inf ormative in 3D space, due to the fact that the averaging kernels are harmonic [1]. 3.1. Quoting from 10 ] : “It is not true that we “ omitted an explicit equation of [ the inverse solution for the case of an unknown vector field” ...” : Substitut 3.1. My reply ion of the proper weights (given by the unidentified equation following equation 14 in [3]), into equation 14 in [3], is undefined. The authors did not specify in their paper the definition of the product of a Kronecker delta with a lead field. The correct form of equations was originally defined by Backus in Gilbert (see equation (4.10) in [8]), where such a product was specified. The correct explicit equations can also be found in [1]. [ 3.2. Quoting from 10 ] : “Finally, the author fails to realize that th e WROP method is not a particular inverse solution but an strategy...” 3.2. My reply : Grave de Peralta Menendez et al. proposed the WROP method in [3]. They claimed “optimum resolution”. They did not indicate how to choose the “so - Furthermore, they presented results that are not called” weights in WROP. reproducible by other researchers, since they did not specify the particular weights that were used in creating their Figures. Now the authors claim that the WROP searcher interested in testing concrete strategy is very general. For the re - method inverse solutions, the only practical issue is: which WROP weights should be used for - spherical) head geometry? realistic (non Whatever the case may be, the results in [1] show that the WROP strategy is doomed to fa ilure because in 3D space, optimization is pointless. Moreover, using the WROP strategy with a particular choice of weights [1] was shown to produce an inverse solution incapable of correct first order localization. As of this moment, a new software packa ge for the fair comparison of instantaneous 3D discrete linear inverse solutions (for current density) is available upon request to the author ( http://www.keyinst.unizh.ch/loreta.htm ). This package is b ased on a somewhat more realistic head model: the average human brain Talairach MRI atlas from McGill University. The approximate EEG lead field was computed numerically using the boundary element method (BEM). No use is made here of “spherical” approximat I Appendix includes some new, unambiguously specified, - ions. inverse solutions that can be found in the package. Also included here ( Appendix - II ) is the treatment of the regularization issue. U sing a 7 mm resolution grid for the cortical grey matter solution space, the mean localization errors for LORETA and minimum illustrates LORETA and Figure 2 norm were 11.45 and 18.61 mm, respectively. minimum norm images (non - regu larized and regularized) due to a point source, in the case of noisy measurements. Regularization was estimated via minimum cross - validation error. Once Grave de Peralta Menendez and Gonzalez Andino publish a completely ous 3D discrete linear inverse solution for specific and unambiguous instantane spherical head models, it will be included in the Talairach - current density in non package. 10 Pag e 3 of

4 Quote as: Pascual - Marqui RD. Reply to Comments Made by R. Grave De Peralta Menendez and S.I. Gozalez Andino. International Journal of Bioelectromagnetism 1999, Vol. 1, No. 2, at: http://www.ee.tut.fi/rgi/ijbem/volume1/number2/html/pascual.htm the : “Then, the conclusion of Pascual - Marqui [1], that “ ] [ 3.3. Quoting from 10 ined here constitutes a minimum necessary low localization error, in the sense def condition” even if apparently reasonable, is not justifiable on theoretical or simulation grounds.” 3.3. My reply : Grave de Peralta Menendez and Gonzalez Andino failed to remember, again, that the principles of li nearity and superposition hold (see my reply above). to comment “c)” [ 3.4. Quoting from ] : “Earlier conclusions about LORETA ( the main 10 properties of LORETA [9]) conjectured on the basis of the dipole localization error have proved to be false.” 3.4. My reply : “Blurring” is certainly equivalent to “distortion”. LORETA produces blurred images (low resolution) of reality (see my reply to comment “c)” ORETA hold. above), and therefore, the main properties of L References [1] Pascual - Marqui, R.D. “Review of methods for solving the EEG inverse International Journal of Bioelectromagnetism. No. 1, Vol.1, 1999. problem”. [2] Grave de Peralta Menendez R and Gonzalez Andino SL. “A critical analysis 48. 1998. - . Vol 4: 440 IEEE Trans. Biomed. Engn ear inverse solutions”. of lin [3] Grave de Peralta Menendez R, Hauk O, Gonzalez Andino, S, Vogt H and Michel. CM: “Linear inverse solutions with optimal resolution kernels applied to the Human Brain Mapping , Vol 5: 454 - electromagnetic tomography.” 67. 1997. [4] Grave de Peralta Menende R., Gonzalez Andino SL. “Distributed source models: Standard solutions and new developments”. In: Uhl C, ed. Analysis of Neurophysiological Brain Functioning . Heidelberg: Springer Verl ag. 1998. [5] Grave de Peralta Menendez R., Gonzalez Andino SL and Lütkenhönner B. “Figures of merit to compare linear distributed inverse solutions”. Brain Topography . Vol. 9. No. 2:117 - 124. 1996 Ilmoniemi and [6] Pascual - Marqui, R.D. “Reply to comments by Hämäläinen, - 28. Nunez. In ISBET Newsletter No. 6, December 1995. Ed: W. Skrandiws. 16 [7] Grave de Peralta Menendez R, Gonzalez Andino SL, (1998c). Basic limitations of linear inverse solutions: A case study. Proceedings of the 20th annual international conference of the Engineering and Biology Society (EMBS). [8] Backus G and Gilbert F: The resolving power of gross earth data. Geophys. J. R. Astr. Soc. 16:169 - 205, 1968. [9] Pascual Marqui, RD and Michel, CM (1994) LORETA: New Authentic 3D ages of the brain. In: ISBET Newsletter No. 5, November 1994. Ed: W. functional im Skrandies. 4 8. - [10] Grave de Peralta Menendez R and Gonzalez Andino SL. “Comments on "Review of methods for solving the EEG inverse problem" by R.D. Pascual - Marqui”. [11] C.R. Rao and S. K. Mitra. Theory and application of constrained inverse of matrices. SIAM J. Appl. Math., 1973, 24: 473 - 488. [12] Stone, M. Journal of the Royal Statistical Society, Series B, 1974, 36:111 - 147. 10 Pag e 4 of

5 Quote as: Pascual Marqui RD. Reply to Comments Made by R. Grave De Peralta Menendez and S.I. Gozalez Andino. - International Journal of Bioelectromagnetism 1999, Vol. 1, No. 2, at: http://www.ee.tut.fi/rgi/ijbem/volume1/number2/html/pascual.htm Appendix I Introduction resented in [1]. Here, the following This Appendix extends the results p additional instantaneous, 3D, discrete, linear solutions for the EEG inverse problem are considered: 1. Weighted minimum norm, with weights that normalize (in the sense) Lnorm − p the lead field. LORETA w ithout lead field normalization, with Laplacian having null current 2. density beyond boundary. 3. LORETA without lead field normalization, with singular Laplacian having arbitrary current density beyond boundary. 1. Methods The reader must refer to [1] for ba ckground and notation. One type of generalized minimum norm inverse problem is: T (1) Φ min,underconstraint: = JWJKJ } { J for any given positive definite matrix of dimension (3)(3) MM • . The solution is: W + −− 11 TT ˆ (2) == JTTWKKWK ,with: Φ + 1 T − − T 1 where Penrose pseudoinverse of - denotes the Moore KWK KWK . 1.1. Weighted minimum norm, with weights that normalize (in the Lnorm − sense) the lead field p In this case: 2 ε (3) Ω =⊗ WI ) ( 3 I where ε ≥ is an exponent, ⊗ - • 0 is the identity 33 denotes the Kronecker product, 3 • matrix with: - matrix, and Ω is a diagonal MM L (4) Υ Ω= p βββ • N vector defined by: 31 which denotes the L norm, for any 0 p > , of the ) ( p T TTT ,,..., (5) Υ = kkk ) ( N 12 ββββ t comments: Three importan corresponds to the classical “minimum norm”. = ε 0 The particular case 1. =2 corresponds to the classical “weighted minimum p , 2. = The particular case 1 ε norm”. 3. etry, i.e., it is not limited to spherical This version can be applied to any head geom heads where concepts such as “radius” exist. of Pag e 5 10

6 Quote as: Pascual Marqui RD. Reply to Comments Made by R. Grave De Peralta Menendez and S.I. Gozalez Andino. - International Journal of Bioelectromagnetism 1999, Vol. 1, No. 2, at: http://www.ee.tut.fi/rgi/ijbem/volume1/number2/html/pascual.htm 1.3. LORETA without lead field normalization, with Laplacian having null current density beyond boundary In this case: 2 = WB (6) where B implements a discrete spatial Laplacian operator. It should be the matrix emphasized that such a choice for B produces the smoothest possible inverse solution. 1 − This is because the inverse matrix, i.e. B , implements a discrete spati al smoothing operator. For a solution space given by a regular cubic 3D grid, with minimum inter - d point distance “ ”, the Laplacian operator used in practice is defined as: grid - 6 :, BAIAAI =−=⊗ with ) ( M 303 2 d (7) ifd 16, vv −= ) ( αβ αβ =∀= M A ,,1... [ ] αβ 0 otherwise 0, acian operator implicitly defined in [6] Equation (7) corresponds exactly to the Lapl (see Eq. (2) therein). The explicit definition of the Laplacian is included here (Equation (7)) for the benefit of readers that may be interested in implementing LORETA correctly. An important comment: This impleme nts LORETA, without weights, and with boundary condition different from that used previously. 1.4. LORETA without lead field normalization, with singular Laplacian having arbitrary current density beyond boundary In this case: T = WBB (8) B where the matrix implements the following discrete spatial Laplacian operator: 6 BAIAAI with =−=⊗ :, ( ) 303 M 2 d 1 − (9) AA1A , = diag ) ( M 011 16, vv ifd −= ) ( αβ ,,1... A M αβ =∀= [ ] 1 αβ 0, otherwise Equation (9) corresponds exactly to the Laplacian operator implicitly defined and used plicit definition of the Laplacian is included here in [6] (see Eq. (2’) therein). The ex (Equation (9)) for the benefit of readers that may be interested in implementing LORETA correctly. The actual boundary condition here is not exactly arbitrary, but s beyond the solution space are certain linear rather that the current density at voxel combinations of the nearest neighboring voxels within the solution space. 10 Pag e 6 of

7 Quote as: Pascual Marqui RD. Reply to Comments Made by R. Grave De Peralta Menendez and S.I. Gozalez Andino. - International Journal of Bioelectromagnetism 1999, Vol. 1, No. 2, at: http://www.ee.tut.fi/rgi/ijbem/volume1/number2/html/pascual.htm It should be emphasized that such a choice for B produces the smoothest + possible inverse solution. This is because the pseudo inverse , - matrix, i.e. B is singular, with B implements a discrete spatial smoothing operator. Also, note that three null eigenvalues, whose corresponding null eigenvectors are: Γ =⊗ 1I (10) • 033 1 is an where M 1 - eigenvectors correspond to the “spatial • vector of ones. The null DC”. is singular, the solution given by equation (2) is not valid. The general form B Because lemma 6.1 owever, an of solution for this type of problem can be found in in [11]. H equivalent simpler derivation will be developed here. The new problem, which will now be reformulated, can be solved in two steps. In the first step, the problem to be solve is: T =+ (11) min,underconstraint: ΦΓ JWJKJC ( ) } { 0 JC / J C , where C is a 3 • 1 Note that the minimization is , for some given with respect to vector of unknown coefficients, expressing the contribution of the DC level current Γ ≡ W0 . density. Note that, by definition 0 is: The solution to the problem expressed in equation (11) + TT ++ ˆ =− ΦΓ (12) JCWKKWKKC ( ) ( ) ( ) 0 which depends on C . The second step consists of solving the problem: T ˆˆ (13) min •• JCWJC ( ) ( ) C which is equivalent to: + TTTTT + (14) −− min ΦΓΦΓ CKKWKKC ( ) ) ( ) ( 00 C The solution is: 1 − TTTT ˆ = ΓΓΓΦ (15) CKNKKN ) ( 000 with: + T + (16) NKWK = ) ( Substituting 15 in 12 gives: − 1 + TTTTT ˆ (17) ΓΓΓΓΦ =− JWKNIKKNKKN ( ) 0000 N ) ( E T ˆ Γ Note that ≡ J0 . 0 The total current density estimator is: ˆ ˆˆ (18) Γ =+ JJC total 0 or equivalently: 1 − + TTTTT ˆ ΓΓΓΓΦ JWKNNKKNKKN =− ( ) 0000 total ) ( (19) − 1 TTTT ΓΓΓΓΦ + KNKKN ) ( 0000 e 10 of 7 Pag

8 Quote as: Pascual Marqui RD. Reply to Comments Made by R. Grave De Peralta Menendez and S.I. Gozalez Andino. - International Journal of Bioelectromagnetism 1999, Vol. 1, No. 2, at: http://www.ee.tut.fi/rgi/ijbem/volume1/number2/html/pascual.htm An important comment: This implements LORETA, without weights, and with boundary condition different from that used previously. Appendix II Introduction This Appendix extends the results presented in [ 1]. Here, regularized instantaneous, 3D, discrete, linear solutions for the EEG inverse problem are considered. Here I make two propositions. First, ordinary cross - validation is the method of comparative study of choice for selecting the regularization parameter. Second, in a - inverse solutions, the method of choice is the inverse solution with minimum cross validation error. All this work is based on the contributions of Stone [12]. There are two methodological principles involved in judging the performan ce of an inverse solution: 1. not good An inverse solution is “ - validation error. ” if it has high cross The converse is not true. 2. Very informally, the principle states that the selected model must produce the best prediction (in the true sense of objective pr ediction, as in e.g., the leave - one - out procedure). It is important to emphasize the “prediction error” as defined in the work of Stone [12] is a concept very different from the classical one used in “goodness of fit” and in “least squares”. 1. Methods T he reader must refer to [1] for background and notation. The regularized version of the generalized minimum norm inverse problem considered is: 2 T (20) min α Φ −+ KJJWJ { } J , and for any given • for any given positive definite matrix W of dimension (3)(3) MM α . This problem and its solution can be found in [6], equations (9), (9’), (10), (10’) 0 > therein. and Note that K e is the H Φ belong to the linear manifold , wher M H ( ) N N • average reference operator (or centering matrix) defined as: NN T T 1 1 1 1 I H (21) = − N • N N N N N N • K 1 =− rankN 1 In this case, vector composed of ones. 1 is an , N • where ( ) ( ) N ≡ H ΦΦ ≡ , and . KHK N N e 10 of 8 Pag

9 Quote as: Pascual - Marqui RD. Reply to Comments Made by R. Grave De Peralta Menendez and S.I. Gozalez Andino. International Journal of Bioelectromagnetism 1999, Vol. 1, No. 2, at: http://www.ee.tut.fi/rgi/ijbem/volume1/number2/html/pascual.htm The solution to the problem in equation (20) is: + 11 TT −− ˆ α Φ (22) JWKKWKH =+ ( ) N The cross - validation error is defined as: N 2 1 (23) Φ =− CVE KZ α [ ] [ ] ( ) ( ) ∑ i ii N = 1 i th th where element of the vector denotes the KZ , Φ denotes the i i element of Φ [ ] [ ] i i i KZ , a nd: i − 1 TT 11 −− ZWKKWKH =+ (24) α Φ ) ( i iiiNi − 1 ) { ( } } } { } { { th KHK ≡ where ; M K is obtained from K by deleting its ; i row; KH ∈ ( ) i − − iN iNi 1 1 ) ) { { } ( } { } ( } { th H ≡ . is obtained from Φ by deleting its ΦΦ i element; and Φ iNi 1 i − } } ( ) { } { { validation error is: - An equivalent equation for cross 2 1 − T 2 N Γ α Λ Y Γ I + ] ) ( [ N 1 − ( ) i α CVE (25) = ) ( ∑ 2 3 1 − N T 1 i = α Γ Λ Γ I + [ ) ( ] ii 1 TT − where •− ΓΛΓ tion; = KWK denotes the eigen - decomposi matrix Γ is an NN 1 ) ( −•− diagonal matrix with whose columns are eigenvectors; and Λ NN 11 is an ) ( ) ( - non null eigenvalues. Three important comments: These equatio ns are valid if the matrix 1. W does not depend on the measurement space, i.e., it must not depend on the position or number of electrodes. Equation (25) is valid for any 2. α ≥ 0 . ation as the “common 3. I have emphasized here the use of simple ordinary cross - valid yard stick” for comparing inverse solutions. Ordinary cross - validation corresponds exactly to the concept of “prediction error” based on “leave one out”. Furthermore, - inear, Bayesian, it can be calculated exactly for any inverse solution (linear, non l etc.). Generalized cross validation does not have these properties. - Figures 10 Pag e 9 of

10 Quote as: Pascual - Marqui RD. Reply to Comments Made by R. Grave De Peralta Menendez and S.I. Gozalez Andino. International Journal of Bioelectromagnetism 1999, Vol. 1, No. 2, at: http://www.ee.tut.fi/rgi/ijbem/volume1/number2/html/pascual.htm Figure 1. Rows are horizontal tomographic slices through a unit radius spherical head. First row indicates the actual test sources and dipolar ond to fourth rows show estimated moment. Sec LORETA current density for each field component (dipolar moments). Fifth row is strength of current density field. Test sources in (1a) and (1b) have unit strength. Note that the estimated deep source in (1a) is more blurr ed (about 17 times less amplitude) than the shallow source in (1b). LORETA can resolve both simultaneously active sources by increasing the strength of the deep source, as shown in (1c). LORETA Minimum Norm Non - regularized Regularized re 2. LORETA and minimum norm images (non regularized and regularized) corresponding to a - Figu 3, test point source at Talairach coordinates ( - - 46,43). Noise was added to the scalp voltages (ratio of - n was estimated via minimum cross variances of signal to noise (SNR) equal to 10). Regularizatio validation error. Anatomy is coded in black to white. Estimated current density in cortical grey matter is coded white (zero) to red (maximum). Location of maximum activity is indicated numerically and by black triangles o n the coordinate axes. Estimated maximum strength is indicated numerically (single number following Talairach coordinates). Exact localization (with blurring) is achieved only with or). regularized LORETA. Other notation: (L=left; R=right; A=anterior; P=posteri 10 Pag e 1 0 of

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