LETTERS TO THE E D I T O R
171
cant difference between time delays measured using h 2 and r 2 methods. The correspondents suggest that non-linear methods are useful because more general conclusions can be drawn about the relationships between signals. If an insignificant association value is found using a non-linear method, they conclude that the signals are not associated either non-linearly or linearly. However we would argue that this is only true for the particular relationship measured by the non-linear method used. For example, h 2 is defined for signals related by a smooth function (Fernandes de Lima et al. 1990) and it is possible that signals may be related by other functions. We agree entirely that signal analysis methods, which provide information not available from conventional visual inspection of the EEG, offer new insights of theoretical and clinical importance in epilepsy. The motivation behind this study was to apply signal analysis techniques in the setting of a clinical EEG department. In this context and the patient population studied, we were not able to show that any of the techniques used provided more information on interhemispheric time differences. However, this does not mean that signal analysis methods do not have wide application, including the quantitative analysis of seizures.
Reference Fernandes de Lima, V.M.F., Pijn, J.P., Nunes Filipe, C. and Lopes da Silva, F.H. The role of hippocampal commissures in the interhemispheric transfer of epileptiform afterdischarges in the rat: a study using linear and non-linear regression analysis. Electroenceph, clin. Neurophysiol., 1990, 76: 520-539.
P.J. Allen S.J.M. Smith Department of Clinical Neurophysiology, National Hospital for Neurology and Neurosurgery, Queen Square, London WC1N 3BG (U.K.)
EEG92021
Comments on article by Biggins et al. In reading the paper entitled "Artifactually high coherences result from using spherical spline computation of scalp current density" by Biggins et al. (1991) which has appeared in this journal, I found that it is based on the erroneous formula for the scalp current density h(x) which appeared in Perrin et al. (1989) and which was corrected in the same journal: Corrigendum (1990). This has some implications on the results and, I think, on the conclusions.
40- sum of weights
3
°
!
~
20" 100
-
-100.0' 0'.5 ' I i 0 ' 1'.5 '210 ' 2'.5 ' 310 curvilinear distance from F5 (radians) Fig. 1. For "target electrode" F5, the graph displays, for each distance x from F5, the sum of the weights of the electrodes whose distance to F5 is less or equal to x. The sum of weights is within 10% of the target electrode weight as soon as the 4 nearest electrodes are included.
Using the exact formula for h(x): 1 h(x)=
~
2n+l
n2'ff_l n m - l ( n + l ) m - 1
pn(x),
I tried to redo some of the computations. The coherence computations on simulated EEG data could not be duplicated since first, the simulated EEG data are only available at the authors' laboratory and second, the steps followed to compute these coherences were not detailed enough. On the other hand, it was possible to compute the "matrix of weights" S (Biggins' Eqs 12 and 13) which depends only on the electrode positions. For an interior (i.e., not belonging to the boundary of the scalp area covered by electrodes) "target electrode" it can be shown, with the correct formula for h(x) (using m equals 4), that the weight of the target electrode is always the highest, and that the weights tend to decrease with increasing distance from the target electrode. The fact that the pattern of weights presents regular alternations in sign is quite normal for finite differences formulas estimating derivatives (Laplacian). One may check (Fig. 1) that the sum of weights tends to zero rapidly with increasing distance from the target electrode. If the coherences are computed from the correct SS', one sees (Fig. 2) that they do in fact decrease with increasing distance from the target electrode, and that the average coherence values are always smaller than those given in Biggins' Table III. For example, for splines with m equals 4, the mean value in Biggins' Table III should be almost halved (0.25 + 0.19 instead of 0.49 _+0.19). In addition, as expected, the decrease is even more pronounced when one uses splines with m equals 3 instead of splines with m equals 4 (Fig. 2). These previous remarks point to the following comment: the spline interpolation formulas were chosen because they seemed to be adequate to the smoothness of the spatial distributions of simulated evoked potentials (Perrin et al. 1989). They were not devised to
172
LETTERS TO THE EDITOR
0.7]
coherence
0.6
•
m=4
[] m=3
0.5. 0.4"
De Munck, J.C. A Mathematical and Physical Interpretation of the Electromagnetic Field of the Brain. Thesis. Amsterdam University, 1989. Perrin, F., Pernier, J., Bertrand, O. and Echallier, J.F. Spherical splines for scalp potential and current density mapping. Electroenceph, clin. Neurophysiol., 1989, 72: 184-187.
0.3. 0.2.
F. Perrin
0.1.
INSERM, U280, 151 Cours Albert Thomas, 69424 Lyon Cedex 03 (France)
0.01 60% interelectrode distance (percent of nasion-inion distance) Fig. 2. Average coherence for 4 different inter-electrode distances (expressed as a percentage of the nasion to inion distance) and for two spherical splines of different degree of smoothness (m equals 4 and 3).
represent random data, as was the case with the simulated E E G data which were constructed by the authors to have, ideally, a spatial coherence of zero. In fact, at the limit, when the number of electrodes becomes large, such spatial distributions are discontinuous in space, have meaningless spatial derivatives and therefore meaningless Laplacians. This suggests that it would perhaps have been more appropriate to have done these tests of coherence on more realistic simulations of E E G which do not have completely random spatial distributions, i.e., whose spatial coherence is non-zero. All this leads to the following open question: given a scalp point at which one wants to estimate a quantity (potential, SCD . . . . ), what is the scalp spatial extent around that point on which data should be collected, and which spatial sampling should be used? The answer may depend on the quantity which is estimated and may differ for EEG, for evoked potentials and even among different types of evoked potentials. In other words, to what extent and under what conditions, local interpolation methods may present some valfle in either the evoked potential or the E E G situation? Objective studies could be done, for example, with simulated E E G data obtained from electrical models containing random dipoles (DeMunck 1989). Local interpolation methods may be devised through the use of a moving spatial window or more economically by a partition of the space domain.
References Biggins, C.A., Fein, G., Raz, J. and Amir, A. Artifactually high coherences result from using spherical spline computation of scalp current density. Electroenceph. clin. Neurophysiol., 1991, 79: 413-419. Corrigendum EEG 02274. Electroenceph. clin Neurophysiol., 1990, 76: 565.
EEG 92063
Spline computation of scalp current density and coherence: a reply to Perrin Dr. F. Perrin has noted that the computation of the spherical spline scalp current density (SCD) in our paper, "Artifactually high coherences result from using spherical spline computation of scalp current density" (Biggins et al. 1991), was based on the incorrect formula which was published in their original manuscript (Perrin et al. 1989). Using the correct formula (which had been published as a corrigendum after our manuscript was in press), he recomputed the coefficients of the spline linear weighting matrix (the matrix S in our paper) and commented that with the new weights, interelectrode coherence did in fact go down as interelectrode distance increased. While the use of the corrected equation does improve the picture, we believe caution in using splines to estimate the SCD is still strongly warranted. Although SCD coherences decrease with increasing interelectrode distance, especially if the value of m is set to 3, the spline method of estimating the SCD still artificially inflates coherence compared to the Hjorth method. This can be seen in the following Table I, which reproduces Table I from our earlier paper, now using the correct formula for calculating the spline SCD. For the spline SCD, average coherences ranged from 0.05 to 0.82, with a mean of 0.29, and with over 26% of the coherences exceeding 0.4. This is much higher than the average coherence with the Hjorth method. These coherences were produced with m = 4, but the spline function still inflates coherence with m -- 3. At this value, average coherences ranged from 0.05 to 0.78, with a mean of 0.21. We note, however, that compared to m = 3, Perrin et al. (1989) found that setting m equal to 4 produced a lower root mean square error between 'true' potential values obtained from a 3-concentric shell model and interpolated val-
171
cant difference between time delays measured using h 2 and r 2 methods. The correspondents suggest that non-linear methods are useful because more general conclusions can be drawn about the relationships between signals. If an insignificant association value is found using a non-linear method, they conclude that the signals are not associated either non-linearly or linearly. However we would argue that this is only true for the particular relationship measured by the non-linear method used. For example, h 2 is defined for signals related by a smooth function (Fernandes de Lima et al. 1990) and it is possible that signals may be related by other functions. We agree entirely that signal analysis methods, which provide information not available from conventional visual inspection of the EEG, offer new insights of theoretical and clinical importance in epilepsy. The motivation behind this study was to apply signal analysis techniques in the setting of a clinical EEG department. In this context and the patient population studied, we were not able to show that any of the techniques used provided more information on interhemispheric time differences. However, this does not mean that signal analysis methods do not have wide application, including the quantitative analysis of seizures.
Reference Fernandes de Lima, V.M.F., Pijn, J.P., Nunes Filipe, C. and Lopes da Silva, F.H. The role of hippocampal commissures in the interhemispheric transfer of epileptiform afterdischarges in the rat: a study using linear and non-linear regression analysis. Electroenceph, clin. Neurophysiol., 1990, 76: 520-539.
P.J. Allen S.J.M. Smith Department of Clinical Neurophysiology, National Hospital for Neurology and Neurosurgery, Queen Square, London WC1N 3BG (U.K.)
EEG92021
Comments on article by Biggins et al. In reading the paper entitled "Artifactually high coherences result from using spherical spline computation of scalp current density" by Biggins et al. (1991) which has appeared in this journal, I found that it is based on the erroneous formula for the scalp current density h(x) which appeared in Perrin et al. (1989) and which was corrected in the same journal: Corrigendum (1990). This has some implications on the results and, I think, on the conclusions.
40- sum of weights
3
°
!
~
20" 100
-
-100.0' 0'.5 ' I i 0 ' 1'.5 '210 ' 2'.5 ' 310 curvilinear distance from F5 (radians) Fig. 1. For "target electrode" F5, the graph displays, for each distance x from F5, the sum of the weights of the electrodes whose distance to F5 is less or equal to x. The sum of weights is within 10% of the target electrode weight as soon as the 4 nearest electrodes are included.
Using the exact formula for h(x): 1 h(x)=
~
2n+l
n2'ff_l n m - l ( n + l ) m - 1
pn(x),
I tried to redo some of the computations. The coherence computations on simulated EEG data could not be duplicated since first, the simulated EEG data are only available at the authors' laboratory and second, the steps followed to compute these coherences were not detailed enough. On the other hand, it was possible to compute the "matrix of weights" S (Biggins' Eqs 12 and 13) which depends only on the electrode positions. For an interior (i.e., not belonging to the boundary of the scalp area covered by electrodes) "target electrode" it can be shown, with the correct formula for h(x) (using m equals 4), that the weight of the target electrode is always the highest, and that the weights tend to decrease with increasing distance from the target electrode. The fact that the pattern of weights presents regular alternations in sign is quite normal for finite differences formulas estimating derivatives (Laplacian). One may check (Fig. 1) that the sum of weights tends to zero rapidly with increasing distance from the target electrode. If the coherences are computed from the correct SS', one sees (Fig. 2) that they do in fact decrease with increasing distance from the target electrode, and that the average coherence values are always smaller than those given in Biggins' Table III. For example, for splines with m equals 4, the mean value in Biggins' Table III should be almost halved (0.25 + 0.19 instead of 0.49 _+0.19). In addition, as expected, the decrease is even more pronounced when one uses splines with m equals 3 instead of splines with m equals 4 (Fig. 2). These previous remarks point to the following comment: the spline interpolation formulas were chosen because they seemed to be adequate to the smoothness of the spatial distributions of simulated evoked potentials (Perrin et al. 1989). They were not devised to
172
LETTERS TO THE EDITOR
0.7]
coherence
0.6
•
m=4
[] m=3
0.5. 0.4"
De Munck, J.C. A Mathematical and Physical Interpretation of the Electromagnetic Field of the Brain. Thesis. Amsterdam University, 1989. Perrin, F., Pernier, J., Bertrand, O. and Echallier, J.F. Spherical splines for scalp potential and current density mapping. Electroenceph, clin. Neurophysiol., 1989, 72: 184-187.
0.3. 0.2.
F. Perrin
0.1.
INSERM, U280, 151 Cours Albert Thomas, 69424 Lyon Cedex 03 (France)
0.01 60% interelectrode distance (percent of nasion-inion distance) Fig. 2. Average coherence for 4 different inter-electrode distances (expressed as a percentage of the nasion to inion distance) and for two spherical splines of different degree of smoothness (m equals 4 and 3).
represent random data, as was the case with the simulated E E G data which were constructed by the authors to have, ideally, a spatial coherence of zero. In fact, at the limit, when the number of electrodes becomes large, such spatial distributions are discontinuous in space, have meaningless spatial derivatives and therefore meaningless Laplacians. This suggests that it would perhaps have been more appropriate to have done these tests of coherence on more realistic simulations of E E G which do not have completely random spatial distributions, i.e., whose spatial coherence is non-zero. All this leads to the following open question: given a scalp point at which one wants to estimate a quantity (potential, SCD . . . . ), what is the scalp spatial extent around that point on which data should be collected, and which spatial sampling should be used? The answer may depend on the quantity which is estimated and may differ for EEG, for evoked potentials and even among different types of evoked potentials. In other words, to what extent and under what conditions, local interpolation methods may present some valfle in either the evoked potential or the E E G situation? Objective studies could be done, for example, with simulated E E G data obtained from electrical models containing random dipoles (DeMunck 1989). Local interpolation methods may be devised through the use of a moving spatial window or more economically by a partition of the space domain.
References Biggins, C.A., Fein, G., Raz, J. and Amir, A. Artifactually high coherences result from using spherical spline computation of scalp current density. Electroenceph. clin. Neurophysiol., 1991, 79: 413-419. Corrigendum EEG 02274. Electroenceph. clin Neurophysiol., 1990, 76: 565.
EEG 92063
Spline computation of scalp current density and coherence: a reply to Perrin Dr. F. Perrin has noted that the computation of the spherical spline scalp current density (SCD) in our paper, "Artifactually high coherences result from using spherical spline computation of scalp current density" (Biggins et al. 1991), was based on the incorrect formula which was published in their original manuscript (Perrin et al. 1989). Using the correct formula (which had been published as a corrigendum after our manuscript was in press), he recomputed the coefficients of the spline linear weighting matrix (the matrix S in our paper) and commented that with the new weights, interelectrode coherence did in fact go down as interelectrode distance increased. While the use of the corrected equation does improve the picture, we believe caution in using splines to estimate the SCD is still strongly warranted. Although SCD coherences decrease with increasing interelectrode distance, especially if the value of m is set to 3, the spline method of estimating the SCD still artificially inflates coherence compared to the Hjorth method. This can be seen in the following Table I, which reproduces Table I from our earlier paper, now using the correct formula for calculating the spline SCD. For the spline SCD, average coherences ranged from 0.05 to 0.82, with a mean of 0.29, and with over 26% of the coherences exceeding 0.4. This is much higher than the average coherence with the Hjorth method. These coherences were produced with m = 4, but the spline function still inflates coherence with m -- 3. At this value, average coherences ranged from 0.05 to 0.78, with a mean of 0.21. We note, however, that compared to m = 3, Perrin et al. (1989) found that setting m equal to 4 produced a lower root mean square error between 'true' potential values obtained from a 3-concentric shell model and interpolated val-
