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## Paul L. Nunez and Ramesh Srinivasan

Print publication date: 2006

Print ISBN-13: 9780195050387

Published to Oxford Scholarship Online: May 2009

DOI: 10.1093/acprof:oso/9780195050387.001.0001

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# (p.579) APPENDIX J The Spline Laplacian

Source:
Electric Fields of the Brain
Publisher:
Oxford University Press

The surface Laplacian algorithm is the principal high-resolution EEG method described in this book. Our discussion applies specifically to the New Orleans spline Laplacian algorithm developed by the Brain Physics Group at Tulane University. Various versions were created and tested during the period 1988–1995. The first version relied on planar interpolation based on the seminal work of the French group (Perrin et al. 1987). The discussion here refers to our latest version which involves interpolation of potential in three-dimensional space. While other spline Laplacian algorithms (often based on interpolation on a sphere) are expected to behave similarly, we only have experience with the New Orleans algorithm. However, the Melbourne dura imaging algorithm (Cadusch et al. 1992), which is based on a 3-sphere model, uses interpolation on a sphere. As shown in chapters 8 and 10, the New Orleans spline Laplacian and Melbourne dura imaging algorithms yield nearly identical spatial patterns when applied to either EEG or simulated data obtained with the 4-sphere head model, provided dense sampling is used (say more than 100 electrodes or simulated surface samples). With 131 electrodes, the electrode-by-electrode correlation coefficients comparing the two methods are typically in the 0.95 range. With 64 electrodes, the correlation coefficients fall to roughly the 0.80 to 0.85 range.

Chapter 8 presents the theoretical background to the surface Laplacian, while chapters 9 and 10 consider the use and interpretation of the surface Laplacian with experimental EEG data. The surface Laplacian is the second spatial derivative of the scalp potential estimated along a geometric model of the scalp surface that passes through the electrode positions. To estimate the surface Laplacian, a continuous potential distribution Φ(x, y, z) is estimated on the model scalp surface from the (p.580) discrete potentials V i(t) measured at i=1, 2,…n electrode sites distributed (mostly) over the upper surface of the head.

Splines are apparently the best choice for the interpolation of EEG data. The surface Laplacian is estimated by applying the surface Laplacian operator to the interpolating function. This spline Laplacian estimate is then evaluated at each electrode position and any other location on the model scalp. Spherical splines (Perrin et al. 1989; Wahba 1981) have come into widespread use to interpolate potentials for topographic maps, by approximating the electrode positions on a sphere. In section 1 of this appendix we provide the formulation of a three-dimensional spline interpolation that can be used to interpolate potentials on any surface that approximates head shape and passes through each electrode. The details of calculation of the surface Laplacian on a spherical surface are given in section 3. Better approximations to the scalp surface, perhaps derived from MRI can be used with this algorithm potentially to improve Laplacian estimates. For instance, the New Orleans spline Laplacian algorithm has been applied to prolate spheroidal and ellipsoidal surfaces (Law et al. 1993). MATLAB (Natick, MA) programs to implement the interpolation on arbitrary surfaces and the surface Laplacian on a spherical surface are provided here and may be downloaded from www.electricfieldsofthebrain.com.

# 1 Three-Dimensional Spline Interpolation

A three-dimensional interpolation of the potential distribution Φ at one instant in time can be obtained from n electrode sites using a third-order thin-plate spline, defined as

(J.1.1)
$Display mathematics$
where (x i, y i, z i) are the Cartesian coordinates of the electrode sites and (x, y, z) are the interpolated coordinates. Table J-1 provides a set of coordinates for 111 electrodes on a spherical surface. In principle, these coordinates can be on an arbitrary surface that passes through the electrode positions. While we have tested this spline on prolate spheroidal and ellipsoidal surfaces, our experience with spherical surfaces is much more extensive. That is, we have run several thousand simulations over 12 years using different noise levels, sampling densities, and head models (for forward solutions) with the spline algorithm based on a spherical surface. This version appears to be quite robust.

The basis function P(x, y, z) is the sum in (J.1.1) whose terms are

(J.1.2)
$Display mathematics$
(p.581)

Table J-1 The positions of 111 electrodes on a sphere of radius 9.2 cm are listed

Electrode

X (cm)

Y (cm)

Z (cm)

1

−5.014

6.901

−3.447

2

−6.691

6.240

−0.962

3

−7.511

5.066

1.598

4

−7.864

2.556

4.033

5

−6.687

1.422

6.156

6

−4.875

0.000

7.802

7

−2.052

−1.490

8.843

8

−6.901

5.014

−3.447

9

−8.224

4.011

−0.962

10

−8.664

2.649

1.598

11

−8.269

0.000

4.033

12

−6.687

−1.422

6.156

13

−3.944

−2.865

7.802

14

−8.112

2.636

−3.447

15

−9.037

1.431

−0.962

16

−9.060

0.000

1.598

17

−9.037

−1.431

−0.962

18

−8.617

−2.800

1.598

19

−7.864

−2.556

4.033

20

−5.531

−4.018

6.156

21

−8.112

−2.636

−3.447

22

−8.224

−4.011

−0.962

23

−7.330

−5.326

1.598

24

−6.689

−4.861

4.033

25

−6.901

−5.014

−3.447

26

−6.691

−6.240

−0.962

27

−5.578

−7.139

1.598

28

−4.861

−6.689

4.033

29

−3.418

−5.921

6.156

30

−1.507

−.637

7.802

31

0.784

−2.411

8.843

32

−5.014

−6.901

−3.447

33

−4.574

−7.924

−0.962

34

−3.247

−8.459

1.598

35

−2.556

−7.864

4.033

36

−0.715

−6.800

6.156

37

1.507

−4.637

7.802

38

−2.636

−8.112

−3.447

39

−2.059

−8.915

−0.962

40

−0.632

−9.038

1.598

41

0.000

−8.269

4.033

42

2.113

−6.502

6.156

43

0.000

−8.530

−3.447

44

0.638

−9.127

−0.962

45

2.038

−8.828

1.598

46

2.556

−7.864

4.033

47

3.279

−8.542

−0.962

48

4.530

−7.847

1.598

49

4.861

−6.689

4.033

50

4.574

−5.080

6.156

51

3.944

−2.865

7.802

52

2.536

0.000

8.843

53

3.317

−7.859

−3.447

54

5.633

−7.210

−0.962

55

6.626

−6.179

1.598

56

6.689

−4.861

4.033

57

6.246

−2.780

6.156

58

4.875

0.000

7.802

59

5.565

−6.465

−3.447

60

7.495

−5.248

−0.962

61

8.143

−3.971

1.598

62

7.864

−2.556

4.033

63

6.837

0.000

6.156

64

7.234

−4.520

−3.447

65

8.702

−2.827

−0.962

66

8.949

−1.418

1.598

67

8.269

0.000

4.033

68

8.401

−1.481

−3.447

69

9.150

0.000

−0.962

70

8.949

1.418

1.598

71

7.864

2.556

4.033

72

6.246

2.780

6.156

73

3.944

2.865

7.802

74

0.784

2.411

8.843

75

8.401

1.481

−3.447

76

8.702

2.827

−0.962

77

8.143

3.971

1.598

78

6.689

4.861

4.033

79

4.574

5.080

6.156

80

1.507

4.637

7.802

81

7.312

4.394

−3.447

82

7.495

5.248

−0.962

83

6.626

6.179

1.598

84

4.861

6.689

4.033

85

2.113

6.502

6.156

86

5.597

6.438

−3.447

87

5.633

7.210

−0.962

88

4.530

7.847

1.598

89

2.556

7.864

4.033

90

3.333

7.852

−3.447

91

3.279

8.542

−0.962

92

2.038

8.828

1.598

93

0.000

8.269

4.033

94

−0.715

6.800

6.156

95

−1.507

4.637

7.802

96

−2.052

1.490

8.843

97

0.638

9.127

−0.962

98

−0.632

9.038

1.598

99

−2.556

7.864

4.033

100

−3.418

5.921

6.156

101

−3.944

2.865

7.802

102

0.000

8.530

−3.447

103

−2.059

8.915

−0.962

104

−3.247

8.459

1.598

105

−4.861

6.689

4.033

106

−5.531

4.018

6.156

107

−2.636

8.112

−3.447

108

−4.574

7.924

−0.962

109

−5.578

7.139

1.598

110

−6.689

4.861

4.033

111

0.000

0.000

9.200

(p.582) (p.583) where
(J.1.3)
$Display mathematics$
is the square of the distance between the interpolation point (x, y, z) and the electrode position (x i, y i, z i). The osculating function
(J.1.4)
$Display mathematics$
serves to smooth the resulting distribution. The interpolation has been tested on spherical and ellipsoidal surfaces (Srinivasan et al. 1996; Law et al. 1993). The p and q coefficients depend on the potential as given in section 2 of this appendix.

The parameter w serves to distribute the loading of the interpolation function over a finite-sized region of width w rather than a point. In EEG applications it makes sense to select w = 0.5−1.0 cm, which is the typical effective electrode size (with gel or other conductive contact material). The simulations shown here and examples in chapters 8−10 were calculated with w = 1.0 cm. The effect of loading the spline over a finite size region is to spatially low-pass filter the potential distribution reducing the contribution of very high spatial frequencies, thereby tending to minimize spatial aliasing (Srinivasan et al. 1996).

# 2 Method of Solution for the Three-Dimensional Spline

Law et al. (1993) published a general method of solution for the three dimensional spline of different orders. We present here the solution for the third-order spline, which is implemented as MATLAB code.

(p.584) Let Q = [q 1, q 2…,q 10]T, P = [p 1, p 2,…,p n]T represent the spline coefficients in (J.1.1) and (J.1.4) and V = [V 1, V 2,…,V n]T are the scalp potentials (with any reference strategy) at n electrode positions. Q and P are solutions to the following matrix equations (Law et al. 1993):

(J.2.1)
$Display mathematics$
(J.2.2)
$Display mathematics$
where K is a square matrix whose elements are
(J.2.3)
$Display mathematics$
and E is an n by 10 matrix whose n rows are given by
(J.2.4)
$Display mathematics$
The solutions to (J.2.1) and (J.2.2) for the coefficient vectors P and Q are
(J.2.5)
$Display mathematics$
(J.2.6)
$Display mathematics$
The solution is presented in two MATLAB functions in figs. J-1 and J-2. The algorithm is separated into two separate functions for computational speed. The first function calculates quantities that only depend on the electrode positions. The second function solves the matrix equations (J.2.5) and (J.2.6). The use of these two functions to interpolate potentials at another set of coordinates (x s, y s, z s) is given as a function in fig. J-3.

# 3 Solution for the Surface Laplacian on a Spherical Surface

The surface Laplacian operator is defined as

(J.3.1)
$Display mathematics$
where u 2 and u 3 are two spatial coordinates on the surface, and the coordinate u 1 is normal to the surface. Spheres, prolate spheroids, and ellipsoids can all be described in their own coordinate systems using two surface coordinates with the other coordinate held fixed. The parameters h 1, h 2, and h 3 are scale factors that depend on the coordinate system (Morse and Feshbach 1953). Law et al. (1993) provide the details for (p.585)

Figure J-1 MATLAB program to calculate matrices K, E, A, K −1, and A −1.

(p.586)

Figure J-2 MATLAB program to calculate matrices P and Q.

Figure J-3 MATLAB program to interpolate potentials on any surface using the 3-dimensional spline.

(p.587) calculating surface Laplacians on prolate spheroidal and ellipsoidal surfaces. In spherical coordinates (θ, φ), the surface Laplacian is written as
(J.3.2)
$Display mathematics$
Alternatively, the surface Laplacian can be calculated by first calculating the three-dimensional Laplacian:
(J.3.3)
$Display mathematics$
and then subtracting the radial Laplacian:
(J.3.4)
$Display mathematics$
The surface Laplacian operator is applied to the interpolating function (J.1.1) to obtain the spline interpolation for the surface Laplacian. For computational speed and numerical stability, we have made use of the definition (J.3.2) to calculate the surface Laplacian of the osculating function Q(x, y, z) and the definition (J.3.4) to calculate the surface Laplacian of the basis function P(x, y, z). To calculate the surface Laplacian for a set of potential measurements V at locations (x, y, z), the spline interpolation coefficients P and Q are calculated using the functions given in figs. J-1 and J-2. Figure J-4 provides a function that uses these spline coefficients to evaluate the surface Laplacian at a set of positions on the sphere (x s, y s, z s). Since the accuracy of the interpolation is best at the electrode locations, we normally evaluate the surface Laplacian at electrode locations.

Table J-2 provides two examples using potentials due to a single radial dipole source in a four concentric spheres model of the head. The 4-sphere model of the head is described in appendix G. The potentials were calculated at 111 simulated electrode sites whose (x, y, z) coordinates are given in table J-1. Head model parameters are spherical shell radii (r 1, r 2, r 3, r 4) = (8, 8.1, 8.6, 9.2) cm and conductivity ratios are (σ12, σ13, σ14) = (0.2, 40, 1). The dipoles were placed directly under the vertex electrode (111). In example 1, the dipole is 4.2 cm below the scalp surface, while in example 2 the dipole is 2.2 cm below the scalp surface. Scalp potentials and analytic surface Laplacians were calculated at the 111 simulated electrode sites using the 4-sphere model. The spline Laplacian was calculated from the potentials at these sites using the functions in figs. J-1, J-2, and J-4. The spline Laplacian is very strongly correlated to the analytic surface Laplacian in both examples. Note that in EEG applications we are only interested in relative values of the spline Laplacian over the surface.

(p.588)

Figure J-4 MATLAB program to calculate surface Laplacian on a sphere at electrode positions.

(p.589)

Table J-2 Examples of the spline Laplacian. Two examples due to a radial dipole source in a four concentric spheres model are presented

Example 1

Example 2

Electrode

Potential

Analytic Laplacian

Spline Laplacian

Potential

Analytic Laplacian

Spline Laplacian

1

−0.838

−0.017

−0.020

−0.819

−0.012

−0.023

2

−0.574

−0.020

−0.019

−0.639

−0.017

−0.018

3

−0.185

−0.023

−0.028

−0.349

−0.024

−0.060

4

0.384

−0.023

−0.019

0.125

−0.035

−0.005

5

1.193

−0.007

−0.009

0.922

−0.046

−0.071

6

2.273

0.059

0.055

2.302

−0.033

−0.015

7

3.446

0.239

0.237

4.694

0.298

0.189

8

−0.838

−0.017

−0.020

−0.819

−0.012

−0.026

9

−0.574

−0.020

−0.019

−0.639

−0.017

−0.018

10

−0.185

−0.023

−0.024

−0.349

−0.024

−0.029

11

0.384

−0.023

−0.021

0.125

−0.035

−0.018

12

1.193

−0.007

−0.009

0.922

−0.046

−0.072

13

2.273

0.059

0.061

2.302

−0.033

0.036

14

−0.838

−0.017

−0.020

−0.819

−0.012

−0.016

15

−0.574

−0.020

−0.020

−0.639

−0.017

−0.021

16

−0.185

−0.023

−0.025

−0.349

−0.024

−0.034

17

−0.574

−0.020

−0.019

−0.639

−0.017

−0.014

18

−0.185

−0.023

−0.024

−0.349

−0.024

−0.034

19

0.384

−0.023

−0.020

0.125

−0.035

−0.012

20

1.193

−0.007

−0.011

0.922

−0.046

−0.093

21

−0.838

−0.017

−0.019

−0.819

−0.012

0.008

22

−0.574

−0.020

−0.021

−0.639

−0.017

−0.025

23

−0.185

−0.023

−0.022

−0.349

−0.024

−0.019

24

0.384

−0.023

−0.022

0.125

−0.035

−0.028

25

−0.838

−0.017

−0.019

−0.819

−0.012

−0.003

26

−0.574

−0.020

−0.019

−0.639

−0.017

−0.007

27

−0.185

−0.023

−0.023

−0.349

−0.024

−0.020

28

0.384

−0.023

−0.023

0.125

−0.035

−0.031

29

1.193

−0.007

−0.008

0.922

−0.046

−0.069

30

2.273

0.059

0.056

2.302

−0.033

−0.014

31

3.446

0.239

0.240

4.694

0.298

0.212

32

−0.838

−0.017

−0.019

−0.819

−0.012

−0.001

33

−0.574

−0.020

−0.021

−0.639

−0.017

−0.019

34

−0.185

−0.023

−0.026

−0.349

−0.024

−0.044

35

0.384

−0.023

−0.020

0.125

−0.035

−0.014

36

1.193

−0.007

−0.010

0.922

−0.046

−0.084

37

2.273

0.059

0.055

2.302

−0.033

−0.004

38

−0.838

−0.017

−0.018

−0.819

−0.012

0.005

39

−0.574

−0.020

−0.016

−0.639

−0.017

0.017

40

−0.185

−0.023

−0.025

−0.349

−0.024

−0.035

41

0.384

−0.023

−0.020

0.125

−0.035

−0.013

42

1.193

−0.007

−0.010

0.922

−0.046

−0.087

43

−0.838

−0.017

−0.018

−0.819

−0.012

0.007

44

−0.574

−0.020

−0.020

−0.639

−0.017

−0.014

45

−0.185

−0.023

−0.026

−0.349

−0.024

−0.044

46

0.384

−0.023

−0.020

0.125

−0.035

−0.014

47

−0.574

−0.020

−0.018

−0.639

−0.017

0.001

48

−0.185

−0.023

−0.020

−0.349

−0.024

0.000

49

0.384

−0.023

−0.025

0.125

0.035

−0.048

50

1.193

−0.007

−0.004

0.922

−0.046

−0.044

51

2.273

0.059

0.060

2.302

−0.033

0.013

52

3.446

0.239

0.233

4.694

0.298

0.160

53

−0.838

−0.017

−0.019

−0.819

−0.012

0.002

54

−0.574

−0.020

−0.023

−0.639

−0.017

−0.035

55

−0.185

−0.023

−0.022

−0.349

−0.024

−0.014

56

0.384

−0.023

−0.024

0.125

−0.035

−0.040

57

1.193

−0.007

−0.009

0.922

−0.046

−0.076

58

2.273

0.059

0.054

2.302

−0.033

−0.008

59

−0.838

−0.017

−0.018

−0.819

−0.012

0.006

60

0.574

−0.020

−0.016

−0.639

−0.017

0.015

61

−0.185

−0.023

−0.025

−0.349

−0.024

−0.034

62

0.384

−0.023

−0.019

0.125

−0.035

−0.003

63

1.193

−0.007

−0.007

0.922

−0.046

−0.060

64

−0.838

−0.017

−0.021

−0.819

−0.012

−0.016

65

−0.574

−0.020

−0.020

−0.639

−0.017

0.016

66

−0.185

−0.023

−0.024

−0.349

−0.024

−0.032

67

0.384

−0.023

−0.025

0.125

−0.035

−0.041

68

−0.838

−0.017

−0.018

−0.819

−0.012

−0.002

69

−0.574

−0.020

−0.020

−0.639

−0.017

−0.020

70

−0.185

−0.023

−0.024

−0.349

−0.024

−0.033

71

0.384

−0.023

−0.019

0.125

−0.035

−0.002

72

1.193

−0.007

−0.009

0.922

−0.046

−0.071

73

2.273

0.059

0.060

2.302

−0.033

0.020

74

3.446

0.239

0.241

4.694

0.298

0.219

75

−0.838

−0.017

0.020

−0.819

−0.012

−0.021

76

−0.574

−0.020

−0.020

−0.639

−0.017

−0.025

77

−0.185

−0.023

−0.025

−0.349

−0.024

−0.036

78

0.384

−0.023

−0.024

0.125

−0.035

−0.038

79

1.193

−0.007

−0.004

0.922

−0.046

−0.031

80

2.273

0.059

0.055

2.302

−0.033

0.006

81

−0.838

−0.017

−0.020

−0.819

−0.012

−0.022

82

−0.574

−0.020

−0.018

−0.639

−0.017

−0.007

83

−0.185

−0.023

−0.022

−0.349

−0.024

−0.017

84

0.384

−0.023

−0.025

0.125

−0.035

−0.044

85

1.193

−0.007

−0.011

0.922

−0.046

−0.082

86

−0.838

−0.017

−0.020

−0.819

−0.012

−0.027

87

−0.574

−0.020

−0.025

−0.639

−0.017

−0.056

88

−0.185

−0.023

−0.020

−0.349

−0.024

−0.005

89

0.384

−0.023

−0.019

0.125

−0.035

−0.003

90

−0.838

−0.017

−0.021

−0.819

−0.012

−0.033

91

−0.574

−0.020

−0.019

−0.639

−0.017

−0.019

92

−0.185

−0.023

−0.028

−0.349

−0.024

−0.056

93

0.384

−0.023

−0.022

0.125

−0.035

−0.021

94

1.193

−0.007

−0.005

0.922

−0.046

−0.042

95

2.273

0.059

0.055

2.302

−0.033

−0.009

96

3.446

0.239

0.237

4.694

0.298

0.193

97

−0.574

−0.020

−0.022

−0.639

−0.017

−0.042

98

−0.185

−0.023

−0.021

−0.349

−0.024

−0.013

99

0.384

−0.023

−0.020

0.125

−0.035

−0.012

100

1.193

−0.007

−0.008

0.922

−0.046

−0.062

101

2.273

0.059

0.062

2.302

−0.033

0.048

102

−0.838

−0.017

−0.020

−0.819

−0.012

−0.031

103

−0.574

−0.020

−0.017

−0.639

−0.017

−0.006

104

−0.185

−0.023

−0.028

−0.349

−0.024

−0.056

105

0.384

−0.023

−0.022

0.125

−0.035

−0.023

106

1.193

−0.007

−0.011

0.922

−0.046

−0.086

107

−0.838

−0.017

−0.020

−0.819

−0.012

−0.030

108

−0.574

−0.020

−0.022

−0.639

−0.017

−0.036

109

−0.185

−0.023

−0.020

−0.349

−0.024

−0.007

110

0.384

−0.023

−0.020

0.125

−0.035

−0.007

111

4.033

0.403

0.425

7.050

2.381

2.390

The 4-sphere model parameters are radii (r 1, r 2, r 3, r 4) = (8, 8.1, 8.6, 9.2) cm and conductivity ratios (σ12, σ13, σ14) = (0.2, 40, 1) of the spheres. The sources are a radial dipole under the vertex electrode (111), either 4.2 cm (example 1) or 2.2 cm (example 2) below the outer sphere (scalp). The first column is the electrode number, with locations on the sphere specified in table J-1.

(p.590)