Brain Topography, Volume 8, Number 4, 1996

355

Spatial Sampling and Filtering of EEG with Spline Laplacians to Estimate Cortical Potentials Ramesh Srinivasan**, Paul L. Nunez ^, Don M. Tucker**, Richard B. Silberstein #, and Peter J. Cadusch #

Summary: The electroencephalogram (EEG) is recorded by sensors physically separated from the cortex by resistive skull tissue that smooths the potential field recorded at the scalp. This smoothing acts as a low-pass spatial filter that determines the spatial bandwidth, and thus the required spatial sampling density, of the scalp EEG. Although it is better appreciated in the time domain, the Nyquist frequency for adequate discrete sampling is evident in the spatial domain as well. A mathematical model of the low-pass spatial filtering of scalp potentials is developed, using a four concentric spheres (brain, CSF, skull, and scalp) model of the head and plausible estimates of the conductivityof each tissue layer. The surface Laplacian estimate of radial skull current density or cortical surface potential counteracts the low-pass filtering of scalp potentials by shifting the spatial spectrum of the EEG, producing a band-passed spatial signal that emphasizes local current sources. Simulations with the four spheres model and dense sensor arrays demonstrate that progressively more detail about cortical potential distribution is obtained as sampling is increased beyond 128 channels. Key words: Spatial nyquist; Laplacian; Splines.

Introduction An important goal for studies of brain function is the accurate characterization of the brain's electrical fields recorded at the scalp surface. Mapping the potential field is the most typical approach in studies of EEG topography, typically recorded with less than 32 electrodes, and often confounded by the assumption that a reference site at the mastoid, nose, or body may be assumed to remain at zero potential. The potential fields of brain sources are *Institute of Cognitive and Decision Sciences, Department of Psychology, University of Oregon, Eugene, OR, USA. +Electrical Geodesics, Inc., Eugene, OR, USA. ^Brain Physics Group, Department of Biomedical Engineering, Tulane University, New Orleans, LA, USA. #Centre for Applied Neurosciences and Department of Physics, Swinburne University of Technology, Melbourne, Australia. Accepted for publication: January 30,1996. This research was supported by a National Research Service Award (NRSA) from National Institutes of Mental Health (NIMH # 1-F32-MH11004-01),a grant from the National Institutes of Health (N1H # 1R01NS243314), NIMH grants MH42129 and MH42669, by NIMH Small Business Innovation and Research (SBIR) grants R44 50409 and R44 51069 to Electrical Geodesics, Inc., and by a grant from the Pew Memorial Trusts and the James S. McDonnell Foundation to support the Center for the Cognitive Neuroscience of Attention. The authors also wish to thank Michael Murias for assistance with the data collection and Lynn McDougal for help with the illustrations. Correspondence and reprint requests should be addressed to Dr. Ramesh Srinivasan, Department of Psychology, 1227 - University of Oregon, Eugene, OR 97403 - 1227, USA. Copyright 9 1996 Human Sciences Press, Inc.

volume-conducted through head tissue, so that no reference site can be assumed to remain at zero potential (Nunez 1981). The potential field can be mapped with respect to the average reference (Bertrand et al. 1985), but this requires an adequate sampling of the head surface, including inferior as well as superior regions (Tucker et al. 1994). The surface Laplacian measure provides a reference-independent estimate of radial skull current density (Katznelson 1981; Nunez et al. 1994), but the sensitivity of this measure to high spatial frequency information in the EEG places additional demands on spatial sampling. In this paper, we provide simulations and dense sensor array measurements to estimate the required spatial sampling density of the human scalp EEG.

Methods Spatial Filtering by Head Volume Conduction Four concentric spheres, which represent brain, cerebrospinal fluid (CSF), skull, and scalp is a simple physical model of the volume conduction properties of the head. This model has been introduced as an improvement of the three concentric spheres model of the head, partly to account for the possibility of significant CSF volume conduction effects in older subjects. It is also the head model used in cortical imaging methods which have been shown to closely approximate surface Laplacian techniques (Nunez et al. 1993, 1994). A sche-

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there is no outward current flow from the scalp surface. In the simplest case, the source term on the right hand side of equation I is chosen to be a dipole with current I and length (pole separation) d located on the z-axis (0 = 0) of the spherical coordinate system at some arbitrary depth rz below the surface of the brain

Dipole

B

r

~

__

Scalp Skull

I~ = I--~-d~(r-rz)6(cosO -1)

(3)

atTI

Figure 1. Four concentric spheres model of the head. The four spheres are indicated in this schematic as 1 = scalp; 2 = skull; 3 = CSF; 4 = brain. The radii of the spheres are measured in centimeters. The scalp and brain assumed to have equal conductivity. M o d e l parameters: rscal p = 9.2, rskul I = 8.7, rcs f = 8 . 2 , rbrai n = 8.0, Gbrain/Gcs f = 0.2, ~brain/~skuil = 8 0 .

mafic of the model with typical values of the radii and tissue conductivity ratios of the spherical shells is given in figure 1. The fundamental assumptions of the model is that Ohm's law applies in each region. In this case the potential distribution depends on the magnitudes and locations of the current sources and the thickness and conductivity of the spherical shells (Nunez 1981). There is substantial variability in the thickness of the skull and scalp and head size in the adult population. In addition, skull thickness appears to vary across different regions of the head (Law 1993). Nevertheless, the four concentric spheres model is a valuable tool because it is readily solved, easily simulated, and has been demonstrated to provide reasonable estimates (typically within 10-20 %) of scalp potentials for brain current sources in comparisons with more realistic finite element models (Yan et al. 1991). The potential distribution due to known cortical (brain) sources in an infinite medium is given by Poisson's equation r~V2~ (r, 0, ~ ) = Is (r, 0, 0 )

(1)

The inhomogeneous medium introduces boundary conditions of continuous potential, ~, and continuous radial current density

The dipole may be oriented either radial or tangential to the spherical surfaces coresponding (for example) to macrocolumns consisting predominantly of pyramidal cells oriented parallel to each other in the gyri and sulci respectively. Generally, radial dipoles probably make the largest contribution to spontaneous EEG (Nunez 1995). Restriction to radial dipoles results in a spherically s y m m e t r i c potential distribution in the four spheres. Despite this restriction, the general results On spatial filtering obtained here are applicable even if scalp potentials result from mixtures of radial and tangential dipoles. The solution is expressed in each sphere (i) as

Vi(r'0'qb)=E A 9n / r + Bni rri n=0 \ ri J

(cos0)

(4)

where Pn are the Legendre polynomials and boundary conditions are applied to obtain the solutions as a set o f recursion relations for the coefficients (Srinivasan 1995). The terms corresponding to n=0 have zero contribution as a consequence of current conservation. Several physical quantities relevant to the analysis contained here can be calculated from this model. The scalp surface potential Vs (0,~) can be calculated as

co

0o

Vs (0,•) = E H~P~ (cos 0 ) = E (A~ + B~) P~ (cos 0 ) (5) n=l

n=l

The scalp surface Laplacian L s (0,~) then follows by applying the Laplacian operator to the series obtaining

Ls(0,~) =

I) H~P~(cosO) ~n(n+ 2 n=l

(6)

r scalp

The cortical surface potential Vc (0,q~) is obtained as Jr = ~ - -

Or

(2)

at the the boundaries between the spheres. In addition,

co

y (a n n=l

po(cosO)

(7)

Spatial S a m p l i n g

357

0.8 0.7 =

,

Vscalp(0,0) = fs' Gscalp(0,0,0', 0')B(0', 0')dS'

0.6

--"

A

0.5

~

B

rio)

Any source distribution B(0,0), can be expressed as a sum over spherical harmonics

~. 0.4 0.3

n

B(0,0) = ~ L BnmYnm(0,0) n=lm=-n

~= 0.2

(11)

0.1

2

3

4

5

6

7

8

9

10

11

Spherical Harmonic Degree

12

I~

14

Substituting the expansion equation 11 into equation !0 and integrating over the spherical region of source distribution results in

15

(n)

Figure 2. Spatial transfer functions of h e a d volume conductor. Relative magnitude of each spatial frequency c o m p o n e n t (n). Cases A-C include a 2 mm CSF layer with Obrain/o-cs f : 0.2 and D has no CSF layer. A: o-brain/O-skull-- 40. B: O-brain/o-skul I = 80. C : o-brain/o-skull = 120. D: Obrain/O-skul I = 80. The transfer function is proportional to the magnitude of surface potential resulting from source distributions forming single spherical harmonic functions on the cortical surface.

Vscalp(0,0)=~ ~

, 4rr HnBnmYnm(0,0)

n=l m = - n

zn + 1

(12)

The scalp potential can also be expanded as oo

Vscalp(0,0) = L ~ SnmYnm(0,0) n=lm=-n

(13)

so that a spatial frequency domain transfer function for scalp potential can be defined as and the radial current density in the middle of the thickness of the skull JK(0,0 ) is

JK(0,0) -- ~ skull L rskull n=l

xn-1 / rskull, + B 2 / ~

A2[----R/,

r-

/

hn+2]

/

JPn(cOs0' (8)

where R = rskun/2 + rcsf/2. The EEG is generated by the distribution of sources over the entire brain. The scalp surface potential due to a radial dipole at an arbitrary location (0',0') is readily obtained from equation 5 and the addition theorem for spherical harmonics Ynm(O,O) as

Gscalp(0'0'o''0')= E 4Tr H n E y . n m (0,.0,)Ynm(0.0)(9) n=l

2n+1

m=-n

Equation 9 is the Green's function for Poisson's equation for scalp potential in four concentric spheres. If the current distribution (source strength per unit area) at a fixed depth is B(rz, O, 0), for instance macrocolumnar sources in the gyri ( rz =7.8 cm), the surface potential is obtained by multiplying the source distribution by the Green's function and integrating over the source distribution

Tscalp(n)= Snm = ~ -4re H -~ n B•m

(14)

This transfer function is plotted in figure 2 for different values of skull resistivity and and CSF thickness. In all cases, the volume conduction causes the wellknown low-pass spatial filtering of scalp potentials. The filtering property is not strongly dependent on the specific value of skull conductivity, but only on a high conductivity ratio of brain to skull. This characteristic is only slightly modified by the inclusion of a normal CSF layer. Examining low order (n = 1-3) spherical harmonics, with examples plotted in figure 3, it is apparent that the unprocessed EEG should be relatively homogeneous over the scalp, which is in fact experimentally observed (Friedrich et al. 1991; Silberstein and Cadusch 1992; Nunez 1995). Spatial S a m p l i n g of Scalp Potentials

An important practical issue in EEG recordings is the choice of reference electrode. The reference may be on the head, chest or neck. The neck reference is the effective reference for any choice of reference below the head (Nunez 1981). This is a consequence of the fact that very little current is expected to flow from the brain down the neck to the body. Thus reference recording involves

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potential. One straightforward resolution is to employ the average reference, i.e., to estimate the instantaneous average potential and to subtract this quantity from each channel. If the number of samples is sufficiently large, and the head surface is completely sampled the average reference simply removes the DC component of the spatial signal. In the ideal case of reference independent sampling on a closed surface, such as a sphere, the DC signal is zero. If recording and reference electrode are placed close together, a bipolar recording, which estimates local tangential scalp current density, is obtained. This can be appreciated by writing equation 15 in coordinate independent form

Vbipolar(~'~)= Vscalp(~-~+ ~'~0) - Vscalp(~'~)

(17)

Here Q = (0,dp) is a position on the sphere and f~0 is a displacement. If f20 is small, the resulting bipolar operation is proportional to taking a directional derivative along the curve contained on the scalp surface joining the two electrodes

Y31

Y33

Figure 3. Selected spherical harmonics of degree n=l-3. Solid lines indicate positive values while dashed lines indicate negative values. The spherical harmonics with index m negative are rotated 90 degrees from these plots.

performing a difference operation between two points on the head volume conductor:

Vref (0 ,~) = Vscalp(0 ,qb)- Vscalp(0 ref,~}ref)

gbipolar oc Vscalp(~ + ~'~0 ~0 ) - Vscalp(~-~) ~ ~-~ vscalp (~)

(18)

The gradient of the surface potential gives components of the electric field normal and tangential to the scalp surface. The tangential components are

~' tangential = ee - - 1 c9Vscalp + ~, 1 0 Vscalp rscalp 00 rscalp sin0 0~

(19)

(15)

For a given source distribution B(rz,0,~ ), we obtain

Then the scalp surface tangential current density follows from Ohm's law

Jtangential = ~ scalp Etan gential Vref (0 ,qb) = s s 4% gngnm [gnm (0, (~) _ Ynm(0 ref,~ref)] rt=0 m=-n -2n + 1 (16) The spherical harmonics are periodic functions as shown in figure 3. In the idealized case of "reference with respect to infinity" the contribution of spherical harmonic components is zero for electrodes placed on the nodal lines of the harmonic functions. The reference recording equation 16 simply shifts the location of the nodal lines depending on the choice of reference. In the realistic case of irregular head geometry and inhomogeneous or anisotropic tissue properties, the transformation of the potential distribution by choice of reference will have a more complicated effect on the recorded

(20)

In the simplest case of a single radial dipole located on the z axis the potential distribution only varies in the 0 direction. Since current flows from higher to lower potential, the tangential current density also only has a 0 component. If the bipolar pair is placed along an isopotential line zero potential is recorded. If the bipolar pair is placed across the isopotentiat lines, the bipolar potential approximates the tangential curent density as: 89 J0 ~ ~ scalp uipolar A0

(21)

Substituting the definition of VSCALP (equation 5) into

Spatial Sampling

359

0.9

Jx = Jo cos 4, Jy = Jo sin ~

0.8

(25)

0.7

The bipolar pair records a potential which is proportional to the dot product between the total current vector and the orientation of the bipolar pair (B) in the local coordinate system

0.5 9

0.5

0.4

,~ o.~ 0.2 0.1

Vbipolar -

0 1

2

3

4

5

6

7

8

9

10

II

12

13

14

Figure 4. Spatial transfer function for bipolar recording. The transfer function is the relative magnitude of spatial frequency components (n) for a white noise source using the four concentric spheres model with parameter values indicated in figure 1.

equation 19 we obtain (22)

H a__Pn(cOs0) rscal p nd~=l

90

The derivatives of the Legendre polynomials Pn(cOs O) with respect to O can be calculated and the following series expansion derived n-1

0-~ P~(cos0)=

E (2m + 1)Pm (cos0)sin0

(23)

m=0

Substituting equation 23 into 22 and interchanging sums results in the following expression for scalp current density induced by a dipole located on the z-axis of the brain sphere

J--Jo % =-%

scalp

E (2m + 1

rscalp m=l

Hn k.n=m+l

(26)

15

Spherical Harmonic Degree ( n )

J0 - :

1 (~.g) scalp

cos0 ) sin0 ]

(24) In this expression we can shift the coordinate system to pass between the bipolar pair and obtain the same result. In general, the location of sources is not known, and can be expected to be distributed, so that the bipolar pair will also receive contributions from current sources from various directions, each of which contributes a current vector at the bipolar pair. Since the electrodes are closely spaced we can approximate the local geometry by a plane. Then the vectors are readily summed by expressing each vector in terms of its x and y components

Equation 24 is the vector Green's function for tangential current density at the pole of the scalp surface sphere for sources in the brain. The total current vector from superficial sources distributed over a sphere in the brain with magnitude B(rz, O, q~) is obtained by multiplying the Green's function by the source distribution and integrating over the source distribution. These integrals were evaluated using Monte Carlo integration, which replaces each integral by the product of the surface area of the sphere and the average of the integrand estimated by sampling points drawn randomly from a uniform distribution (International Mathematical and Statistical Libraries - IMSL routine RNUNF). The integrals were evaluated by drawing samples until the variance in the mean estimate was less than 1%. The magnitude of the spatial frequency components of the transfer function for scalp tangential current density is evaluated by allowing the source distribution to be spatial white noise, i.e., having equal power at all spatial frequencies. The spherical harmonic expansion coefficients of a source distribution B(rz, O, ~), Bnm depend on the orientation of the coordinate system. However it has been established that n

[32 _ 4~ E Bnm Bnm 2n+1 m=-n

(27)

are invariant under rotations of the coordinate system (Mcleod and Coleman 1980; Mcleod 1980). The proof follows from the Addition theorem for spherical harmonics. Spatial white noise source distribution is then defined b y [3n2 = 1 for all n, but practical computation restricted this study to a maximum n of 15. The magnitude of the bipolar potential then estimates the magnitude of the tangential current vector: Vbipoiar oc ~

(28)

which is calculated for realistic values of the four spheres model in figure 4. The bipolar recording is sensitive to somewhat higher spatial frequencies than the raw potential. The orientation of the bipolar pair will result in the

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6

=

3 2

E 1 0

1

2

3

4

5

6

7

8

9

10

11

12

15

14

15

Spherical Harmonic Degree ( o )

Figure 5. Spline filtering function. Numerical estimate of filtering function Zi defined by equation A-9 for r=9.2 and w=1.0.

selective suppression of modes, similar to the specific choice of reference. Clinical electroencephalographers are often able to emphasize different sources by changing the bipolar pair or reference. Spline Interpolation of EEG Topography Multichannel EEG recordings provide discrete samples of the topography of scalp potential. A three dimensional spline interpolant function from n samples can be defined as (Perrin et al. 1987, Law et al. 1993) rt

V(x, y, Z) = 2 piK2(x- xi, y - yi, z - zi)+ Q2(x, y, z )

(29)

i=1

where the basis function P2(x,y,z) is represented by the sum whose terms are K2(x - xi, Y - Yi, z - zi) = (d2) 2 log(d 2 + w 2) d 2 = (x- xi) 2 + ( y - yi) 2 + (z- Zi) 2

(30)

and the osculating function Q2(x,y,z) is given as Qz(x,y, z) = ql + q2x+ q3Y+q4 x2 +q5xy

(31)

+ q6y 2 + qTz+ qszx + q9zy + ql0 z2 The coefficients Pi and qi depend on the data, and are calculated by solving a matrix equation. The purpose of this representation is to produce topographic maps and to obtain surface Laplacian estimates by differentiation. In either case, the samples are usually assumed to come from a "best-fit" sphere to the subjects head, where the fitting is performed by a nonlinear regression routine (Law et al. 1993).

The spline is rewritten in Appendix A as a spherical harmomc expansion, using approporiate expansions. The form of the spline given by equation A-2 and A-10 or A-12 allows for direct comparison of the three dimensional spline interpolation with the simple least-squares spherical harmonic expansion (Shaw 1989) and the Wahba spherical splines (Wahba 1980; Perrin et ai. 1989), which are the alternative interpolation schemes used in EEG topography. Both of these methods use an abrupt cutoff in the harmomc expansion determined by the number of electrodes and computer limitations. By contrast, the logarithmic basis function used by the spline is an infinite order spherical harmonic expansion with an intrinsic filtering function analogous to filtering techniques applied in the time domain to prevent aliasing. The coefficients ~ , defined in equation A-9, shown in figure 5 for w =1.0 cm, provides the filtering function that serves to smooth the potential map and minimize aliasing errors that may occur as a consequence of undersampling. The choice of w is typically the effective electrode size (with gel) as it serves to distribute the loading of the interpolation function over a finite sized area rather than a point (Law et al. 1993), thereby determining the smoothing filter. This particular filtering technique may be effective because the fall-off in amplitude at higher spatial frequencies is similar to that for a white noise cortical source distribution. The spatial spectra of scalp potential corresponding to a white noise cortical source distribution has already been calculated as shown in figure 2. The same distributions were sampled with several grids including the 64 and 128 channel GeodesicO sensor nets (Electrical Geodesics, Inc., Eugene, Oregon), and the 64 channet SCAN (Swinburne Centre for Applied Neurosciences) spring loaded helmet (Tucker 1993, Silberstein 1995). The 64 and 128 channel Geodesic nets have mean interelectrode distances of 3.8 and 2.7 cm respectively and subtend an angle of roughly 130 degrees from the subjects vertex. The 64 channel proprietary helmet has mean interelectrode distance of 2.7 cm subtending an angle of 90 degrees from vertex. The 64 and 128 channel Geodesic nets provide a relatively uniform sampling, while the helmet provides a less uniform sampling (interlectrode distances from 1.3 to 3.7 cm) but includes the International 10-20 electrode positions. To provide contrast with the helmet, a urnform grid corresponding to interelectrode distance of 2.0 cm subtending an angle of 90 degrees from vertex is also investigated here. The spline approximated spatial power spectra are estimated by evaluating the integral

s~ = .~0~"XJ0f2~Vspline (0,*) Y~m(0,~)r 2 sin0

dq~ dO (32)

The integrals are evaluated simltaneously for all the spherical harmonics by Monte Carlo integration, which

Spatial Sampling

361 1 0'

POTENTIAL

0.7 o bJ

o

[] 64 net

o.5 0.4

N

0.3 Z

s

o

- - ~

0.2

0.1

CSF

0 1

5

5

7

9

11

13

lo;

15

I 8.1

r

SKULL I

82

,

r 8.3

,

1 84

~ I 8.5

R F 8,6

SCALP r

I 8.7

i

I 88

r

P 8,9

,

~ 9.8

i

G 9.

, 1 9.2

RAOI AL 0 I S T A N C E

Spherical Harmonic Degree ( n )

Figure 7. Attenuation of potential with distance a b o v e a radial dipole. In the simulation the four concentric spheres model parameters of figure 1 were used.

0.9 L a p l a c i a n Estimates from t h e Spline M o d e l 0.7 0,6

0.5

[ ] 64 SC#,N

o.4 "11111

[ ] z.o cm

o.,

:,

1

:

3

. . . . .

5

7

9

11

13

,,

15

Spherical Harmonic Degree ( n )

Figure 6. Spline interpolated spatial power spectra. Scalp sampies calculated using 4 spheres model and a white noise source distribution; 64 and 128 channel sensor nets, 64 channel helmet and a uniformly distributed grid with mean interelectrode distance of 2.0 cm are indicated.

replaces the integrals b y the sums

Snm

--

N z]~ max EVspline (0 n,qbn) Y;m(0n,~n) N n=l

(33)

The integration points are drawn from a uniform distribution. The integral is restricted to the sampling space (@max)since the electrodes effectively constrain the spline model over this range. The estimated spatial spectra associated with each electrode array is shown in figure 6. The spline further smooths the scalp potential when a limited number of samples are available. The spline representation becomes stable in uniform grids with sampiing density less than 2.5 cm. However, the spline model consistently overestimates the power in modes n=2, as a consequence of the smoothing filter Zj (figure 5).

Although topographic maps improve the information available in EEG studies, their use is normally limited by the poor spatial resolution. Simply increasing the number of electrodes is not an adequate solution. This can be appreciated by noting that scalp potential and cortical surface potential are quite different physical quantities (equations 5 and 6), so that simply increasing the number of scalp samples will not yield an accurate representation of cortical surface potential. It has been demonstrated in both theoretical and experimental studies that an estimate of the surface Laplacian can improve the spatial resolution of the EEG, providing reasonable estimates of cortical surface potentials (Nunez et al. 1994; Nunez 1995; Gevins et al. 1994). The physical basis for the relationship between cortical potential and scalp surface Laplacian has been derived earlier (Katznelson 1981; Nunez et al. 1994) where the following approximate relationship, based on Ohm's law was obtained V1 ~ V2 + Pk dkdsL s Ps

(34)

Here V 1 and V 2 are the potentials on the inner and outer surfaces of the skull Ls is the scalp surface Laplacian, Pk and Ps are skull and scalp resisitivity, and d k and ds are skull and scalp thickness. Figure 7 shows the theoretical potential due to a radial dipole in the cortex as a function of radial location in CSF, skull, and scalp. The dipole is located in the cortex (r=7.8) and the potential is normalized with respect to its magnitude at the inner CSF surface (r=8.0). Although potential falls through the CSF layer, it is so thin that the drop is small and thus cortical suface potential is in the same range as V1. By contrast, the potential falls by a factor of 500 between the inner and outer surfaces of the

362

Srinivasan ef al.

1.2

1.0

/S

i z:

"

o

DISTANCE (CM)

-.2

5 [email protected],54.04 0 [STANCE (CMJ

55.05.56

Figure 8. Theoretical fall-off of different physical variables from a dipole source. The four variables are scalp potential (Vs}, cortical potential (Vc), radial skull current density (JK) and scalp surface Laplacian (Ls). Left: radial dipole. Right: tangential dipole. The four concentric spheres model parameters of figure 1 were used here.

skull. Potential varies slowly through scalp thickness, so that scalp potential Vs may be considered roughly equivalent to the potential on the outer surface of the skull V1. Thus, cortical potential can be roughly estimated from surface potentials with the surface Laplacian Ls by equation 34. Further, the severe attenuation of cortical potential implied by figure 7 suggests that Vs due to localized sources can be neglected and that Ls is approximately proportional V1. The ratio of cortical to scalp potential in EEG data is typically 2 to 4 for distributed sources and much larger for localized sources (Nunez 1981). The scalp surface Laplacian Ls, radial skull current density JK, cortical potential Vo and scalp potential, Vs, can be directly computed for a radial or tangential dipole source using the techniques given in Methods. Figure 8 demonstrates the fall-off of these quantities with distance from a radial and tangential dipole demonstrating the close relationship between the scalp Laplacian, cortical potential and radial skull current density. The approximation improved as the quantities are averaged over regions of tissue under electrodes of larger size satisfying the "large-scale" approximation inherent in equation 34 (Katznelson 1981). As dipoles go deeper this approximate relationship is even more accurate, but the magnitude of the Laplacian becomes much smaller relative to the potential, so that the signal to noise ratio becomes small. The implication is that surface Laplacian estimates are mainly sensitive to superficial cortical sources. The principal motivation for obtaining the spline model of the instantaneous scalp potential distribution, discussed in Methods, is to estimate the Laplacian. Once the spline coefficients Pi and qi have beeen calculated, the surface Laplacian can be directly estimated from the second spatial derivative of the spline function (Law et al. 1993). The spline generated surface Laplacian estimate may have significantly larger errors than scalp potential w h e n only a small n u m b e r of samples are

available. Figure 9 demonstrates the spline generated Laplacian spatial spectra estimate from the same grids shown for scalp surface potential in figure 6, with a white noise cortical source distribution. Inadequate spatial sampling can introduce errors in the apparent spatial spectra. The effect of the spline is to shift power from higher to lower spatial frequencies in the Laplacian estimate as a consequence of the filtering characteristic discussed earlier. As the number of samples increases the spatial spectra are similar to the analytic Laplacian spatial transfer function given in figure 10 for the same cases examined for scalp surface potentials in figure 2. Spatial Spectra of EEG

The theoretical arguments developed here were obtained for the case of an idealized white noise neocortical source distribution. In any EEG experiment, it is hypothesized that a specific source distribution (or in the dynamic case, a temporal sequence of source distributions) generates the recorded potentials. This source configuration need not be white noise and is subject to more complex tranformations by volume conduction than elicited by the 4 spheres model. Thus it is instructive to examine the potential and Laplacian spatial spectra obtained from EEG data using both the 64 and 128 channel sensor nets and the 64 channel helmet. Eyes closed resting EEG (alpha rhythm) was obtained from 27 subjects using the 64 channel sensor net recorded as 20 one-second epochs, as part of an event related potential study (Tucker et al. 1994). To provide contrast, 100 two-second eyes closed EEG epochs were recorded with a 128 channel sensor net from 5 subjects, and 3 minutes of continuous eyes closed EEG were recorded from one subject with the 64 channel helmet. Atl of the data were edited for artifact and surface Laplacian estimates were obtained from the sptine model~ The spa-

Spatial Sampling

363

POTENTIAL LAPLACIAN

0.5

0.6 [ ] 64 net

0.5

o.4

[ ] t 28 net

9 0.4

[ ] 64 net

0.3 [ ] 64SCAN

~. 0.2

0.1 O, I

3

5

7

9

13

11

i

l

i

3

15

J

,

5

~1 7 i

l

,

~

9

,

,

11

i

i

33

~

i

15

Spherical Harmonic Degree ( n )

Spherical Harmonic Degree ( n )

LAPLACIAN

0.45 T 0.4

[] 645CAN

0.35

[ ] 2.0 cm

0.25 =

,= T-

0.2

[ ] 128 net

o.15 0.1

~

o 0.05 ~

e~

0 . . . . . . . . . 1

5

5

7

9

', : ', ', ', : : 11 15 15

I

3

5

7

9

11

13

15

Spherical Harmonic Degree ( n )

Spherical Harmonic Degree ( n )

Figure 9. Spline Laplacian spatial power spectra. Scalp samples calculated using four spheres model and a spatial white noise source distribution. 64 and 128 channel nets, 64 channel helmet and a uniformly distributed grid with mean intere[ectrode distance of 2cm are indicated.

Figure 11. Potential and Laplacian spatial amplitude spectrums for alpha rhythm recorded with the 64 and 128 channel nets and the 64 channel helmet. To compare the different data sets, the amplitudes have been reexpressed as the proportion of total amplitude at each spatial frequency (n).

0.45 0.4

=

0.25

6

0.2

....m~=..~.=

~

B

--+--C .

~"

0.1

tial spectra of the alpha rhythm data are shown for each electrode array in figure 11. For the 64 and 128 channel sensor nets the spectra shown are averaged across the subjects. Comparing the Laplacian spectrum for the dense 128 channel net and 64 SCAN helmet with the spectra obtained with a white noise (figure 9), we note that the alpha rhythm source distribution has more relative power at lower spatial frequencies than white noise.

0.05 0

I

1

I

I

I

I

]

I

I

]

]

I

I

I

Spherical Harmonic Degree ( n )

Figure 10. Spatial transfer functions of analytic Laplacians. Relative magnitude of each spatial frequency component [n). Cases A-C include a 2 mm CSF layer with O'BRAIN/O-CSF=0.2and D has no CSF layer. A: Obrain/Oskull=40. B: o - b r a i n / o - s k u l l : 8 0 . C: o-brain/Oskull:120. D:O'brain/O-skulI = 8 0 .

Discussion Scalp EEG recordings are low-pass filtered by the poorly conducting skull, producing highly smoothed potential maps that provide only crude estimates of the underlying source distribution. Bipolar recordings provide measures of local sources that are unsuitable for topographic mapping, as they represent the projection of

364

scalp surface tangential current vectors along the orientation of the bipolar pair. While the Laplacian estimate provides a reference independent estimate of the radial current density in the skull, low spatial frequency information is filtered in the Laplacian measure, so that a characterization of the potential field is also required. The three-dimensional spline model provides a means of obtaining both potential maps and a spatial model for the surface Laplacian estimation. This approach provides an intrinsic spatial filter, determined by the effective surface area of the electrode in contact with the scalp, which minimizes aliasing errors due to the finite sampling of the surface potentials. For scalp potential field mapping, a stable spatial representation is obtained with a sampling density (mean interelectrode distance) of 2.7 cm, provided by both the 128- channel sensor net and the 64-charmel helmet. An even higher sampling density may be required to maximize the benefits of surface Laplacian estimates, with improvements noted with a sampling density (interelectrode distance) up to 2.0 cm (180 electrodes). However, we cannot claim that very high electro de densities will provide substantial improvement in accuracy in real heads, where effects of variable tissue properties may be significant. Several calculations were performed in this paper to estimate the required sampling density for accurate potential and Laplacian mapping from scalp EEG. These calculations were all based on a four concentric sphere model of the head which is a coarse physical approximation to the human head, but which has been found to be in approximate agreement with more realistic (but also approximate) finite element models of the human head (Yan et al. 1991). The finite element models can account for departure of head shape from sphericity and inhomegeniety within each tissue layer, e.g., the eye sockets. By contrast to the surface Laplacian estimate, cortical imaging methods are potentially able to use electrical and geometric information about the head volume conductor to provide more accurate estimates of cortical potential using these discrete models (Gevins et al. 1994). However, the accuracy of these methods is limited by our relative ignorance of tissue resistivities. Improvements in head modelling would certainly modify the estimate of required sampling density. In our estimation, this would likely increase the optimal number of channels, if an accurate head model could be developed for each EEG recording.

Appendix A Representation of the 3-Dimensional Sptine as a Spherical Harmonic Expansion

For spherical surfaces, the osculating function Q2(r, 0, d~) can be expressed as:

Srinivasanet al.

Q2(r,0,~) = ql +q2 r sin0 cos~ +q3 r sin• sin~ +q4 r2 sin20 cos2~ +q5 r2 sin20 cos0? sin~ +q6 r2 sin20 sin2d~ +q7 r cos0 +q8 r2 cos0 sin0 cos~ +q9 r2 COS0 sin0 sin~ +ql0 r2 COS20 (A-l) which is easily formulated in terms of spherical harmonics Q2(r,0,qb) = qt ~

+ q4 r2 3 +q6r2

3 +'-110 T )

Y00(0 ,}) - q3 r ~ Y l _ l (0,(~)

+ q7r ~ Y l o

(0,~) - q 2 r ~ Y l l

(O,qb)+ 2qsr2 ~-~Y2-2 (0 ,d~)

r2 ~-~-5 (2q10 - q6- q4 )Y20(0 ,qb) - q9r2 ~ 5 y2q (O,q~)+ ~--qsr2 ~ Y 2 1 ( 0 , ~ ) + 2r2 ~ 5 (q4- qs)Y22(0,*)

(A-2) The distance d can be expressed in terms of the angular distance between the interpolation point (8, ~), and the electrode (0i, ~i), 7 i= c~176

as

i c~

+ sinOi sin0 (cos(~i -~b))] (A,3)

d 2 = 21)2(1 - cosy i)

(A-4)

Then the basis function P2(r, 0, d?)can be written in spherical coordinates as P2(r,O,~) = 4r 4 ln(2r 2) n 2 Pi (2r2 (1 - cos ~' i )) 2in (2r2 (1 - cos 7 i) + w2) (ln 10) 2 i=1

(A-5) Here the base 10 logarithms have been replaced by natural logarithms. For w2>0, the logarithm can be replaced by its Taylor series expansion, which converges for all Yi,

r 4r4 ln(2r2) + pi ln(l+ ~02 / + s

(-1)J(c~ Yi)J j

(A-6)

Spatial Sampling

365

Any function of the form (cos 0)n can be expanded in a finite series of Legendre polynomials for even and odd powers as

P2(r, 0,q~) - 4r41n(2r2) + Pi In 102 z.., i=l

/ w21? iw2/

- 4P0 (cos? i) In 1 +

3

(COS0)2n= ~ A~mP2m(COS0 ) m=0 (COS0)2n+l = Z A @ 2m+lP2)m+ln(COS m=0

2P2(c~ 3

(A-7)

O9

27

i)ln 1+

2~-

+

1(cos y i ) ln(1 + 2~- +

+ Z Z j P j ( c o s y i) j=0

(A-10) where With the addition theorem for spherical harmonics and the definition

1 3 A~ = ~2n+1 Ap = 2n+3 n (4m + 1)(2n - 2m + 2) A2m+2 = (4m- 3)(2n + 2m + 1) A2m

n

(A-8)

n (4m+3)(2n- 2m+ 2) n Aam+3 = (4m- 1)(2n + 2m + 3) A2m+l

/

rl

t- 2Z

(2n-l) 1+

n=l

4r 4 ln(2r 2) (ln 10)2

W2 /

-]

PooYoo(0'*) + y Z PlkYlk(0'qb)+ ~5 P2kY2k(0,(~) k=-i =(

W2/2n-2

(2n- 2) 1+ 2-72)

+

~ , 47rZj J Z PjkYjk(0,qb)

A~m

+ 2Z

Aznrn

Z__a f 2 "~2n ( 2 2n-1 n=m2n[]_+ ~7r2j n=m(an- 1), l+W~12r 2) + ~

A2nm-1

//

-2 ~

2 )2n-1

n=m(2n-1)/1+~r2~

References

Asnm

~=m+l(2n- 2)(1+

Z2m_l = - ~

27;5k= j

(A-12)

oo

Zzm = +

(A-11)

ln[l+ 2~T j

A~

n=l 2n 1 + 2 7 )

P2 (r'@'(~) =

(

( n-I

oo

A~

i=1

this expression becomes

Substituting and rearranging terms, we introduce the following definitions

Zo ;

Pjk = Z PiYjk(0i,~i)

w2 / 2n-2

27)

A2nm-1

2)(1 + W2 t2n-2 n=m+l(2n2~) n

A2n-1 ( 2 ~2n-3 n=m+l(2n- 3)[1 + ~ T ) (A-9) and write the basis function as

Bertrand, O., Perrin, F., and Pernier, J. A theoretical justification of the average reference in topographic evoked potential studies. Electroencephalographyand clinical Neurophysiology, 1985, 62: 462-464. Friedrich, R., Fuchs, A., and Haken, H., Spatiotemporal EEG patterns. In: Rhythms in Physiological Systems. H. Haken, and H.P. Koepchen (eds.), Springer Verlag, Berlin, 1991. Gevins, A.S., Le, J., Martin, N.K., and Reutter, B. High resolution EEG: 124-channel recording, spatial deblurring, and MRI integration methods. Electroencephalography and Clinical Neurophysiology, 1994, 90:337-358 Katznelson, R.D. EEG recording, electrode placement, and aspects of generator localization. Ch. 6 in Electric Fields of the Brain: The Neurophysics of EEG, P.L. Nunez, Oxford University, New York, 1981. Law, S.K. Thickness and resistivity variations over the upper surface of the human skull. Brain Topography, 1993, 6:

366

99-110. Law, S.K., Nunez, P.L. and Wijesinghe, R.S. High Resolution EEG using spline generated surface Laplacians on spherical and ellipsoidal surfaces. IEEE Transactions on Biomedical Engineering, 1993, 40(2): 145-152. McLeod, M.G. and Coleman, P.J. Spatial power spectra of the crustal geomagnetic field. Physics of the Earth and Planetary Interiors, 1980, 23: P5-P19. McLeod, M.G. Orthogonality of spherical harmonic coefficients. Physics of the Earth and Planetary Interiors, 1980, 23: P1-P4. Nunez, P.L. Electric Fields of the Brain: The Neurophysics of EEG. Oxford University, New York, 1981. Nunez, P.L. Silberstein, R.B., Cadusch, P.J., and Wijesinghe, R.S. Comparison of high resolution methods having different theoretical bases. Brain Topography, 1993, 5: 361-364. Ntmez, P.L. Silberstein, R.B., Cadusch, P.J., Wijesinghe, R.S., Westdorp, A.F., and Srinivasan, R. A theoretical and experimental study of high resolution EEG based on surface Laplacians and cortical imaging. Electroencephalography and Clinical Neurophysiology, 1994, 90: 40-57. Nunez, P.L. Neocortical Dynamics and Human EEG Rhythms. Oxford University, New York, 1995. Perrin, F., Bertrand, O., and Pernier, J. Scalp current density mapping:value and estimation from potential data. IEEE Transactions on Biomedical Engineering, 1987, 34(4): 283289. Perrin, F., Pernier, J., Bertrand, O., and Echallier, J.F. Spherical splines for scalp potential and current density mapping.

Srinivasan et al.

Electroencephalography and Clinical Neurophysiology, 1989, 72: 184-187. Shaw, G.R. Spherical Harmonic Analysis of the Electroencephalogram. Ph. D. Thesis, University of Alberta, Edmonton, 1989. Silberstein, R.B. and Cadusch, P.J. Measurement processes and spatial principal components analysis. Brain Topography, 1992, 4(4): 267-276. Silberstein, R.B. Steady-state visually evoked potentials, brain resonances, and cognitive processes. Ch. 6 in Neocortical Dynamics and Human EEG Rhythms, P.L. Nunez, Oxford University, New York, 1995. Srinivasan, R, A Theoretical and Experimental Study of Neocortical Dynamics. Ph.D. Dissertation, Tulane University, New Orleans, 1995. Tucker, D.M. Spatial sampling of head electrical fields: The geodesic sensor net. Electroencephalography and Clinical Neurophysiology, 1993, 87: 154-163. Tucker, D. M., Liotti, M., Potts, G. F., Russell, G. S., and Posner, M.I. Spatiotemporal Analysis of Brain Electrical Fields. Human Brain Mapping, 1994, 1: 134-152. Wahba, G. Spline interpolation and smoothing on a sphere. SIAM Jounal on Scientific and Statistical Computing, 1980, 2:5-16. Yan, Y., Nunez, P.L.and Hart, R.T. A finite element model of the human head: Scalp potentials due to dipole sources. Medical and Biological Engineering and Computing, 1991, 29: 475-481.

355

Spatial Sampling and Filtering of EEG with Spline Laplacians to Estimate Cortical Potentials Ramesh Srinivasan**, Paul L. Nunez ^, Don M. Tucker**, Richard B. Silberstein #, and Peter J. Cadusch #

Summary: The electroencephalogram (EEG) is recorded by sensors physically separated from the cortex by resistive skull tissue that smooths the potential field recorded at the scalp. This smoothing acts as a low-pass spatial filter that determines the spatial bandwidth, and thus the required spatial sampling density, of the scalp EEG. Although it is better appreciated in the time domain, the Nyquist frequency for adequate discrete sampling is evident in the spatial domain as well. A mathematical model of the low-pass spatial filtering of scalp potentials is developed, using a four concentric spheres (brain, CSF, skull, and scalp) model of the head and plausible estimates of the conductivityof each tissue layer. The surface Laplacian estimate of radial skull current density or cortical surface potential counteracts the low-pass filtering of scalp potentials by shifting the spatial spectrum of the EEG, producing a band-passed spatial signal that emphasizes local current sources. Simulations with the four spheres model and dense sensor arrays demonstrate that progressively more detail about cortical potential distribution is obtained as sampling is increased beyond 128 channels. Key words: Spatial nyquist; Laplacian; Splines.

Introduction An important goal for studies of brain function is the accurate characterization of the brain's electrical fields recorded at the scalp surface. Mapping the potential field is the most typical approach in studies of EEG topography, typically recorded with less than 32 electrodes, and often confounded by the assumption that a reference site at the mastoid, nose, or body may be assumed to remain at zero potential. The potential fields of brain sources are *Institute of Cognitive and Decision Sciences, Department of Psychology, University of Oregon, Eugene, OR, USA. +Electrical Geodesics, Inc., Eugene, OR, USA. ^Brain Physics Group, Department of Biomedical Engineering, Tulane University, New Orleans, LA, USA. #Centre for Applied Neurosciences and Department of Physics, Swinburne University of Technology, Melbourne, Australia. Accepted for publication: January 30,1996. This research was supported by a National Research Service Award (NRSA) from National Institutes of Mental Health (NIMH # 1-F32-MH11004-01),a grant from the National Institutes of Health (N1H # 1R01NS243314), NIMH grants MH42129 and MH42669, by NIMH Small Business Innovation and Research (SBIR) grants R44 50409 and R44 51069 to Electrical Geodesics, Inc., and by a grant from the Pew Memorial Trusts and the James S. McDonnell Foundation to support the Center for the Cognitive Neuroscience of Attention. The authors also wish to thank Michael Murias for assistance with the data collection and Lynn McDougal for help with the illustrations. Correspondence and reprint requests should be addressed to Dr. Ramesh Srinivasan, Department of Psychology, 1227 - University of Oregon, Eugene, OR 97403 - 1227, USA. Copyright 9 1996 Human Sciences Press, Inc.

volume-conducted through head tissue, so that no reference site can be assumed to remain at zero potential (Nunez 1981). The potential field can be mapped with respect to the average reference (Bertrand et al. 1985), but this requires an adequate sampling of the head surface, including inferior as well as superior regions (Tucker et al. 1994). The surface Laplacian measure provides a reference-independent estimate of radial skull current density (Katznelson 1981; Nunez et al. 1994), but the sensitivity of this measure to high spatial frequency information in the EEG places additional demands on spatial sampling. In this paper, we provide simulations and dense sensor array measurements to estimate the required spatial sampling density of the human scalp EEG.

Methods Spatial Filtering by Head Volume Conduction Four concentric spheres, which represent brain, cerebrospinal fluid (CSF), skull, and scalp is a simple physical model of the volume conduction properties of the head. This model has been introduced as an improvement of the three concentric spheres model of the head, partly to account for the possibility of significant CSF volume conduction effects in older subjects. It is also the head model used in cortical imaging methods which have been shown to closely approximate surface Laplacian techniques (Nunez et al. 1993, 1994). A sche-

356

Srinivasan et ai.

there is no outward current flow from the scalp surface. In the simplest case, the source term on the right hand side of equation I is chosen to be a dipole with current I and length (pole separation) d located on the z-axis (0 = 0) of the spherical coordinate system at some arbitrary depth rz below the surface of the brain

Dipole

B

r

~

__

Scalp Skull

I~ = I--~-d~(r-rz)6(cosO -1)

(3)

atTI

Figure 1. Four concentric spheres model of the head. The four spheres are indicated in this schematic as 1 = scalp; 2 = skull; 3 = CSF; 4 = brain. The radii of the spheres are measured in centimeters. The scalp and brain assumed to have equal conductivity. M o d e l parameters: rscal p = 9.2, rskul I = 8.7, rcs f = 8 . 2 , rbrai n = 8.0, Gbrain/Gcs f = 0.2, ~brain/~skuil = 8 0 .

mafic of the model with typical values of the radii and tissue conductivity ratios of the spherical shells is given in figure 1. The fundamental assumptions of the model is that Ohm's law applies in each region. In this case the potential distribution depends on the magnitudes and locations of the current sources and the thickness and conductivity of the spherical shells (Nunez 1981). There is substantial variability in the thickness of the skull and scalp and head size in the adult population. In addition, skull thickness appears to vary across different regions of the head (Law 1993). Nevertheless, the four concentric spheres model is a valuable tool because it is readily solved, easily simulated, and has been demonstrated to provide reasonable estimates (typically within 10-20 %) of scalp potentials for brain current sources in comparisons with more realistic finite element models (Yan et al. 1991). The potential distribution due to known cortical (brain) sources in an infinite medium is given by Poisson's equation r~V2~ (r, 0, ~ ) = Is (r, 0, 0 )

(1)

The inhomogeneous medium introduces boundary conditions of continuous potential, ~, and continuous radial current density

The dipole may be oriented either radial or tangential to the spherical surfaces coresponding (for example) to macrocolumns consisting predominantly of pyramidal cells oriented parallel to each other in the gyri and sulci respectively. Generally, radial dipoles probably make the largest contribution to spontaneous EEG (Nunez 1995). Restriction to radial dipoles results in a spherically s y m m e t r i c potential distribution in the four spheres. Despite this restriction, the general results On spatial filtering obtained here are applicable even if scalp potentials result from mixtures of radial and tangential dipoles. The solution is expressed in each sphere (i) as

Vi(r'0'qb)=E A 9n / r + Bni rri n=0 \ ri J

(cos0)

(4)

where Pn are the Legendre polynomials and boundary conditions are applied to obtain the solutions as a set o f recursion relations for the coefficients (Srinivasan 1995). The terms corresponding to n=0 have zero contribution as a consequence of current conservation. Several physical quantities relevant to the analysis contained here can be calculated from this model. The scalp surface potential Vs (0,~) can be calculated as

co

0o

Vs (0,•) = E H~P~ (cos 0 ) = E (A~ + B~) P~ (cos 0 ) (5) n=l

n=l

The scalp surface Laplacian L s (0,~) then follows by applying the Laplacian operator to the series obtaining

Ls(0,~) =

I) H~P~(cosO) ~n(n+ 2 n=l

(6)

r scalp

The cortical surface potential Vc (0,q~) is obtained as Jr = ~ - -

Or

(2)

at the the boundaries between the spheres. In addition,

co

y (a n n=l

po(cosO)

(7)

Spatial S a m p l i n g

357

0.8 0.7 =

,

Vscalp(0,0) = fs' Gscalp(0,0,0', 0')B(0', 0')dS'

0.6

--"

A

0.5

~

B

rio)

Any source distribution B(0,0), can be expressed as a sum over spherical harmonics

~. 0.4 0.3

n

B(0,0) = ~ L BnmYnm(0,0) n=lm=-n

~= 0.2

(11)

0.1

2

3

4

5

6

7

8

9

10

11

Spherical Harmonic Degree

12

I~

14

Substituting the expansion equation 11 into equation !0 and integrating over the spherical region of source distribution results in

15

(n)

Figure 2. Spatial transfer functions of h e a d volume conductor. Relative magnitude of each spatial frequency c o m p o n e n t (n). Cases A-C include a 2 mm CSF layer with Obrain/o-cs f : 0.2 and D has no CSF layer. A: o-brain/O-skull-- 40. B: O-brain/o-skul I = 80. C : o-brain/o-skull = 120. D: Obrain/O-skul I = 80. The transfer function is proportional to the magnitude of surface potential resulting from source distributions forming single spherical harmonic functions on the cortical surface.

Vscalp(0,0)=~ ~

, 4rr HnBnmYnm(0,0)

n=l m = - n

zn + 1

(12)

The scalp potential can also be expanded as oo

Vscalp(0,0) = L ~ SnmYnm(0,0) n=lm=-n

(13)

so that a spatial frequency domain transfer function for scalp potential can be defined as and the radial current density in the middle of the thickness of the skull JK(0,0 ) is

JK(0,0) -- ~ skull L rskull n=l

xn-1 / rskull, + B 2 / ~

A2[----R/,

r-

/

hn+2]

/

JPn(cOs0' (8)

where R = rskun/2 + rcsf/2. The EEG is generated by the distribution of sources over the entire brain. The scalp surface potential due to a radial dipole at an arbitrary location (0',0') is readily obtained from equation 5 and the addition theorem for spherical harmonics Ynm(O,O) as

Gscalp(0'0'o''0')= E 4Tr H n E y . n m (0,.0,)Ynm(0.0)(9) n=l

2n+1

m=-n

Equation 9 is the Green's function for Poisson's equation for scalp potential in four concentric spheres. If the current distribution (source strength per unit area) at a fixed depth is B(rz, O, 0), for instance macrocolumnar sources in the gyri ( rz =7.8 cm), the surface potential is obtained by multiplying the source distribution by the Green's function and integrating over the source distribution

Tscalp(n)= Snm = ~ -4re H -~ n B•m

(14)

This transfer function is plotted in figure 2 for different values of skull resistivity and and CSF thickness. In all cases, the volume conduction causes the wellknown low-pass spatial filtering of scalp potentials. The filtering property is not strongly dependent on the specific value of skull conductivity, but only on a high conductivity ratio of brain to skull. This characteristic is only slightly modified by the inclusion of a normal CSF layer. Examining low order (n = 1-3) spherical harmonics, with examples plotted in figure 3, it is apparent that the unprocessed EEG should be relatively homogeneous over the scalp, which is in fact experimentally observed (Friedrich et al. 1991; Silberstein and Cadusch 1992; Nunez 1995). Spatial S a m p l i n g of Scalp Potentials

An important practical issue in EEG recordings is the choice of reference electrode. The reference may be on the head, chest or neck. The neck reference is the effective reference for any choice of reference below the head (Nunez 1981). This is a consequence of the fact that very little current is expected to flow from the brain down the neck to the body. Thus reference recording involves

358

Srinivasan et al.

potential. One straightforward resolution is to employ the average reference, i.e., to estimate the instantaneous average potential and to subtract this quantity from each channel. If the number of samples is sufficiently large, and the head surface is completely sampled the average reference simply removes the DC component of the spatial signal. In the ideal case of reference independent sampling on a closed surface, such as a sphere, the DC signal is zero. If recording and reference electrode are placed close together, a bipolar recording, which estimates local tangential scalp current density, is obtained. This can be appreciated by writing equation 15 in coordinate independent form

Vbipolar(~'~)= Vscalp(~-~+ ~'~0) - Vscalp(~'~)

(17)

Here Q = (0,dp) is a position on the sphere and f~0 is a displacement. If f20 is small, the resulting bipolar operation is proportional to taking a directional derivative along the curve contained on the scalp surface joining the two electrodes

Y31

Y33

Figure 3. Selected spherical harmonics of degree n=l-3. Solid lines indicate positive values while dashed lines indicate negative values. The spherical harmonics with index m negative are rotated 90 degrees from these plots.

performing a difference operation between two points on the head volume conductor:

Vref (0 ,~) = Vscalp(0 ,qb)- Vscalp(0 ref,~}ref)

gbipolar oc Vscalp(~ + ~'~0 ~0 ) - Vscalp(~-~) ~ ~-~ vscalp (~)

(18)

The gradient of the surface potential gives components of the electric field normal and tangential to the scalp surface. The tangential components are

~' tangential = ee - - 1 c9Vscalp + ~, 1 0 Vscalp rscalp 00 rscalp sin0 0~

(19)

(15)

For a given source distribution B(rz,0,~ ), we obtain

Then the scalp surface tangential current density follows from Ohm's law

Jtangential = ~ scalp Etan gential Vref (0 ,qb) = s s 4% gngnm [gnm (0, (~) _ Ynm(0 ref,~ref)] rt=0 m=-n -2n + 1 (16) The spherical harmonics are periodic functions as shown in figure 3. In the idealized case of "reference with respect to infinity" the contribution of spherical harmonic components is zero for electrodes placed on the nodal lines of the harmonic functions. The reference recording equation 16 simply shifts the location of the nodal lines depending on the choice of reference. In the realistic case of irregular head geometry and inhomogeneous or anisotropic tissue properties, the transformation of the potential distribution by choice of reference will have a more complicated effect on the recorded

(20)

In the simplest case of a single radial dipole located on the z axis the potential distribution only varies in the 0 direction. Since current flows from higher to lower potential, the tangential current density also only has a 0 component. If the bipolar pair is placed along an isopotential line zero potential is recorded. If the bipolar pair is placed across the isopotentiat lines, the bipolar potential approximates the tangential curent density as: 89 J0 ~ ~ scalp uipolar A0

(21)

Substituting the definition of VSCALP (equation 5) into

Spatial Sampling

359

0.9

Jx = Jo cos 4, Jy = Jo sin ~

0.8

(25)

0.7

The bipolar pair records a potential which is proportional to the dot product between the total current vector and the orientation of the bipolar pair (B) in the local coordinate system

0.5 9

0.5

0.4

,~ o.~ 0.2 0.1

Vbipolar -

0 1

2

3

4

5

6

7

8

9

10

II

12

13

14

Figure 4. Spatial transfer function for bipolar recording. The transfer function is the relative magnitude of spatial frequency components (n) for a white noise source using the four concentric spheres model with parameter values indicated in figure 1.

equation 19 we obtain (22)

H a__Pn(cOs0) rscal p nd~=l

90

The derivatives of the Legendre polynomials Pn(cOs O) with respect to O can be calculated and the following series expansion derived n-1

0-~ P~(cos0)=

E (2m + 1)Pm (cos0)sin0

(23)

m=0

Substituting equation 23 into 22 and interchanging sums results in the following expression for scalp current density induced by a dipole located on the z-axis of the brain sphere

J--Jo % =-%

scalp

E (2m + 1

rscalp m=l

Hn k.n=m+l

(26)

15

Spherical Harmonic Degree ( n )

J0 - :

1 (~.g) scalp

cos0 ) sin0 ]

(24) In this expression we can shift the coordinate system to pass between the bipolar pair and obtain the same result. In general, the location of sources is not known, and can be expected to be distributed, so that the bipolar pair will also receive contributions from current sources from various directions, each of which contributes a current vector at the bipolar pair. Since the electrodes are closely spaced we can approximate the local geometry by a plane. Then the vectors are readily summed by expressing each vector in terms of its x and y components

Equation 24 is the vector Green's function for tangential current density at the pole of the scalp surface sphere for sources in the brain. The total current vector from superficial sources distributed over a sphere in the brain with magnitude B(rz, O, q~) is obtained by multiplying the Green's function by the source distribution and integrating over the source distribution. These integrals were evaluated using Monte Carlo integration, which replaces each integral by the product of the surface area of the sphere and the average of the integrand estimated by sampling points drawn randomly from a uniform distribution (International Mathematical and Statistical Libraries - IMSL routine RNUNF). The integrals were evaluated by drawing samples until the variance in the mean estimate was less than 1%. The magnitude of the spatial frequency components of the transfer function for scalp tangential current density is evaluated by allowing the source distribution to be spatial white noise, i.e., having equal power at all spatial frequencies. The spherical harmonic expansion coefficients of a source distribution B(rz, O, ~), Bnm depend on the orientation of the coordinate system. However it has been established that n

[32 _ 4~ E Bnm Bnm 2n+1 m=-n

(27)

are invariant under rotations of the coordinate system (Mcleod and Coleman 1980; Mcleod 1980). The proof follows from the Addition theorem for spherical harmonics. Spatial white noise source distribution is then defined b y [3n2 = 1 for all n, but practical computation restricted this study to a maximum n of 15. The magnitude of the bipolar potential then estimates the magnitude of the tangential current vector: Vbipoiar oc ~

(28)

which is calculated for realistic values of the four spheres model in figure 4. The bipolar recording is sensitive to somewhat higher spatial frequencies than the raw potential. The orientation of the bipolar pair will result in the

360

Srinivasan et al.

6

=

3 2

E 1 0

1

2

3

4

5

6

7

8

9

10

11

12

15

14

15

Spherical Harmonic Degree ( o )

Figure 5. Spline filtering function. Numerical estimate of filtering function Zi defined by equation A-9 for r=9.2 and w=1.0.

selective suppression of modes, similar to the specific choice of reference. Clinical electroencephalographers are often able to emphasize different sources by changing the bipolar pair or reference. Spline Interpolation of EEG Topography Multichannel EEG recordings provide discrete samples of the topography of scalp potential. A three dimensional spline interpolant function from n samples can be defined as (Perrin et al. 1987, Law et al. 1993) rt

V(x, y, Z) = 2 piK2(x- xi, y - yi, z - zi)+ Q2(x, y, z )

(29)

i=1

where the basis function P2(x,y,z) is represented by the sum whose terms are K2(x - xi, Y - Yi, z - zi) = (d2) 2 log(d 2 + w 2) d 2 = (x- xi) 2 + ( y - yi) 2 + (z- Zi) 2

(30)

and the osculating function Q2(x,y,z) is given as Qz(x,y, z) = ql + q2x+ q3Y+q4 x2 +q5xy

(31)

+ q6y 2 + qTz+ qszx + q9zy + ql0 z2 The coefficients Pi and qi depend on the data, and are calculated by solving a matrix equation. The purpose of this representation is to produce topographic maps and to obtain surface Laplacian estimates by differentiation. In either case, the samples are usually assumed to come from a "best-fit" sphere to the subjects head, where the fitting is performed by a nonlinear regression routine (Law et al. 1993).

The spline is rewritten in Appendix A as a spherical harmomc expansion, using approporiate expansions. The form of the spline given by equation A-2 and A-10 or A-12 allows for direct comparison of the three dimensional spline interpolation with the simple least-squares spherical harmonic expansion (Shaw 1989) and the Wahba spherical splines (Wahba 1980; Perrin et ai. 1989), which are the alternative interpolation schemes used in EEG topography. Both of these methods use an abrupt cutoff in the harmomc expansion determined by the number of electrodes and computer limitations. By contrast, the logarithmic basis function used by the spline is an infinite order spherical harmonic expansion with an intrinsic filtering function analogous to filtering techniques applied in the time domain to prevent aliasing. The coefficients ~ , defined in equation A-9, shown in figure 5 for w =1.0 cm, provides the filtering function that serves to smooth the potential map and minimize aliasing errors that may occur as a consequence of undersampling. The choice of w is typically the effective electrode size (with gel) as it serves to distribute the loading of the interpolation function over a finite sized area rather than a point (Law et al. 1993), thereby determining the smoothing filter. This particular filtering technique may be effective because the fall-off in amplitude at higher spatial frequencies is similar to that for a white noise cortical source distribution. The spatial spectra of scalp potential corresponding to a white noise cortical source distribution has already been calculated as shown in figure 2. The same distributions were sampled with several grids including the 64 and 128 channel GeodesicO sensor nets (Electrical Geodesics, Inc., Eugene, Oregon), and the 64 channet SCAN (Swinburne Centre for Applied Neurosciences) spring loaded helmet (Tucker 1993, Silberstein 1995). The 64 and 128 channel Geodesic nets have mean interelectrode distances of 3.8 and 2.7 cm respectively and subtend an angle of roughly 130 degrees from the subjects vertex. The 64 channel proprietary helmet has mean interelectrode distance of 2.7 cm subtending an angle of 90 degrees from vertex. The 64 and 128 channel Geodesic nets provide a relatively uniform sampling, while the helmet provides a less uniform sampling (interlectrode distances from 1.3 to 3.7 cm) but includes the International 10-20 electrode positions. To provide contrast with the helmet, a urnform grid corresponding to interelectrode distance of 2.0 cm subtending an angle of 90 degrees from vertex is also investigated here. The spline approximated spatial power spectra are estimated by evaluating the integral

s~ = .~0~"XJ0f2~Vspline (0,*) Y~m(0,~)r 2 sin0

dq~ dO (32)

The integrals are evaluated simltaneously for all the spherical harmonics by Monte Carlo integration, which

Spatial Sampling

361 1 0'

POTENTIAL

0.7 o bJ

o

[] 64 net

o.5 0.4

N

0.3 Z

s

o

- - ~

0.2

0.1

CSF

0 1

5

5

7

9

11

13

lo;

15

I 8.1

r

SKULL I

82

,

r 8.3

,

1 84

~ I 8.5

R F 8,6

SCALP r

I 8.7

i

I 88

r

P 8,9

,

~ 9.8

i

G 9.

, 1 9.2

RAOI AL 0 I S T A N C E

Spherical Harmonic Degree ( n )

Figure 7. Attenuation of potential with distance a b o v e a radial dipole. In the simulation the four concentric spheres model parameters of figure 1 were used.

0.9 L a p l a c i a n Estimates from t h e Spline M o d e l 0.7 0,6

0.5

[ ] 64 SC#,N

o.4 "11111

[ ] z.o cm

o.,

:,

1

:

3

. . . . .

5

7

9

11

13

,,

15

Spherical Harmonic Degree ( n )

Figure 6. Spline interpolated spatial power spectra. Scalp sampies calculated using 4 spheres model and a white noise source distribution; 64 and 128 channel sensor nets, 64 channel helmet and a uniformly distributed grid with mean interelectrode distance of 2.0 cm are indicated.

replaces the integrals b y the sums

Snm

--

N z]~ max EVspline (0 n,qbn) Y;m(0n,~n) N n=l

(33)

The integration points are drawn from a uniform distribution. The integral is restricted to the sampling space (@max)since the electrodes effectively constrain the spline model over this range. The estimated spatial spectra associated with each electrode array is shown in figure 6. The spline further smooths the scalp potential when a limited number of samples are available. The spline representation becomes stable in uniform grids with sampiing density less than 2.5 cm. However, the spline model consistently overestimates the power in modes n=2, as a consequence of the smoothing filter Zj (figure 5).

Although topographic maps improve the information available in EEG studies, their use is normally limited by the poor spatial resolution. Simply increasing the number of electrodes is not an adequate solution. This can be appreciated by noting that scalp potential and cortical surface potential are quite different physical quantities (equations 5 and 6), so that simply increasing the number of scalp samples will not yield an accurate representation of cortical surface potential. It has been demonstrated in both theoretical and experimental studies that an estimate of the surface Laplacian can improve the spatial resolution of the EEG, providing reasonable estimates of cortical surface potentials (Nunez et al. 1994; Nunez 1995; Gevins et al. 1994). The physical basis for the relationship between cortical potential and scalp surface Laplacian has been derived earlier (Katznelson 1981; Nunez et al. 1994) where the following approximate relationship, based on Ohm's law was obtained V1 ~ V2 + Pk dkdsL s Ps

(34)

Here V 1 and V 2 are the potentials on the inner and outer surfaces of the skull Ls is the scalp surface Laplacian, Pk and Ps are skull and scalp resisitivity, and d k and ds are skull and scalp thickness. Figure 7 shows the theoretical potential due to a radial dipole in the cortex as a function of radial location in CSF, skull, and scalp. The dipole is located in the cortex (r=7.8) and the potential is normalized with respect to its magnitude at the inner CSF surface (r=8.0). Although potential falls through the CSF layer, it is so thin that the drop is small and thus cortical suface potential is in the same range as V1. By contrast, the potential falls by a factor of 500 between the inner and outer surfaces of the

362

Srinivasan ef al.

1.2

1.0

/S

i z:

"

o

DISTANCE (CM)

-.2

5 [email protected],54.04 0 [STANCE (CMJ

55.05.56

Figure 8. Theoretical fall-off of different physical variables from a dipole source. The four variables are scalp potential (Vs}, cortical potential (Vc), radial skull current density (JK) and scalp surface Laplacian (Ls). Left: radial dipole. Right: tangential dipole. The four concentric spheres model parameters of figure 1 were used here.

skull. Potential varies slowly through scalp thickness, so that scalp potential Vs may be considered roughly equivalent to the potential on the outer surface of the skull V1. Thus, cortical potential can be roughly estimated from surface potentials with the surface Laplacian Ls by equation 34. Further, the severe attenuation of cortical potential implied by figure 7 suggests that Vs due to localized sources can be neglected and that Ls is approximately proportional V1. The ratio of cortical to scalp potential in EEG data is typically 2 to 4 for distributed sources and much larger for localized sources (Nunez 1981). The scalp surface Laplacian Ls, radial skull current density JK, cortical potential Vo and scalp potential, Vs, can be directly computed for a radial or tangential dipole source using the techniques given in Methods. Figure 8 demonstrates the fall-off of these quantities with distance from a radial and tangential dipole demonstrating the close relationship between the scalp Laplacian, cortical potential and radial skull current density. The approximation improved as the quantities are averaged over regions of tissue under electrodes of larger size satisfying the "large-scale" approximation inherent in equation 34 (Katznelson 1981). As dipoles go deeper this approximate relationship is even more accurate, but the magnitude of the Laplacian becomes much smaller relative to the potential, so that the signal to noise ratio becomes small. The implication is that surface Laplacian estimates are mainly sensitive to superficial cortical sources. The principal motivation for obtaining the spline model of the instantaneous scalp potential distribution, discussed in Methods, is to estimate the Laplacian. Once the spline coefficients Pi and qi have beeen calculated, the surface Laplacian can be directly estimated from the second spatial derivative of the spline function (Law et al. 1993). The spline generated surface Laplacian estimate may have significantly larger errors than scalp potential w h e n only a small n u m b e r of samples are

available. Figure 9 demonstrates the spline generated Laplacian spatial spectra estimate from the same grids shown for scalp surface potential in figure 6, with a white noise cortical source distribution. Inadequate spatial sampling can introduce errors in the apparent spatial spectra. The effect of the spline is to shift power from higher to lower spatial frequencies in the Laplacian estimate as a consequence of the filtering characteristic discussed earlier. As the number of samples increases the spatial spectra are similar to the analytic Laplacian spatial transfer function given in figure 10 for the same cases examined for scalp surface potentials in figure 2. Spatial Spectra of EEG

The theoretical arguments developed here were obtained for the case of an idealized white noise neocortical source distribution. In any EEG experiment, it is hypothesized that a specific source distribution (or in the dynamic case, a temporal sequence of source distributions) generates the recorded potentials. This source configuration need not be white noise and is subject to more complex tranformations by volume conduction than elicited by the 4 spheres model. Thus it is instructive to examine the potential and Laplacian spatial spectra obtained from EEG data using both the 64 and 128 channel sensor nets and the 64 channel helmet. Eyes closed resting EEG (alpha rhythm) was obtained from 27 subjects using the 64 channel sensor net recorded as 20 one-second epochs, as part of an event related potential study (Tucker et al. 1994). To provide contrast, 100 two-second eyes closed EEG epochs were recorded with a 128 channel sensor net from 5 subjects, and 3 minutes of continuous eyes closed EEG were recorded from one subject with the 64 channel helmet. Atl of the data were edited for artifact and surface Laplacian estimates were obtained from the sptine model~ The spa-

Spatial Sampling

363

POTENTIAL LAPLACIAN

0.5

0.6 [ ] 64 net

0.5

o.4

[ ] t 28 net

9 0.4

[ ] 64 net

0.3 [ ] 64SCAN

~. 0.2

0.1 O, I

3

5

7

9

13

11

i

l

i

3

15

J

,

5

~1 7 i

l

,

~

9

,

,

11

i

i

33

~

i

15

Spherical Harmonic Degree ( n )

Spherical Harmonic Degree ( n )

LAPLACIAN

0.45 T 0.4

[] 645CAN

0.35

[ ] 2.0 cm

0.25 =

,= T-

0.2

[ ] 128 net

o.15 0.1

~

o 0.05 ~

e~

0 . . . . . . . . . 1

5

5

7

9

', : ', ', ', : : 11 15 15

I

3

5

7

9

11

13

15

Spherical Harmonic Degree ( n )

Spherical Harmonic Degree ( n )

Figure 9. Spline Laplacian spatial power spectra. Scalp samples calculated using four spheres model and a spatial white noise source distribution. 64 and 128 channel nets, 64 channel helmet and a uniformly distributed grid with mean intere[ectrode distance of 2cm are indicated.

Figure 11. Potential and Laplacian spatial amplitude spectrums for alpha rhythm recorded with the 64 and 128 channel nets and the 64 channel helmet. To compare the different data sets, the amplitudes have been reexpressed as the proportion of total amplitude at each spatial frequency (n).

0.45 0.4

=

0.25

6

0.2

....m~=..~.=

~

B

--+--C .

~"

0.1

tial spectra of the alpha rhythm data are shown for each electrode array in figure 11. For the 64 and 128 channel sensor nets the spectra shown are averaged across the subjects. Comparing the Laplacian spectrum for the dense 128 channel net and 64 SCAN helmet with the spectra obtained with a white noise (figure 9), we note that the alpha rhythm source distribution has more relative power at lower spatial frequencies than white noise.

0.05 0

I

1

I

I

I

I

]

I

I

]

]

I

I

I

Spherical Harmonic Degree ( n )

Figure 10. Spatial transfer functions of analytic Laplacians. Relative magnitude of each spatial frequency component [n). Cases A-C include a 2 mm CSF layer with O'BRAIN/O-CSF=0.2and D has no CSF layer. A: Obrain/Oskull=40. B: o - b r a i n / o - s k u l l : 8 0 . C: o-brain/Oskull:120. D:O'brain/O-skulI = 8 0 .

Discussion Scalp EEG recordings are low-pass filtered by the poorly conducting skull, producing highly smoothed potential maps that provide only crude estimates of the underlying source distribution. Bipolar recordings provide measures of local sources that are unsuitable for topographic mapping, as they represent the projection of

364

scalp surface tangential current vectors along the orientation of the bipolar pair. While the Laplacian estimate provides a reference independent estimate of the radial current density in the skull, low spatial frequency information is filtered in the Laplacian measure, so that a characterization of the potential field is also required. The three-dimensional spline model provides a means of obtaining both potential maps and a spatial model for the surface Laplacian estimation. This approach provides an intrinsic spatial filter, determined by the effective surface area of the electrode in contact with the scalp, which minimizes aliasing errors due to the finite sampling of the surface potentials. For scalp potential field mapping, a stable spatial representation is obtained with a sampling density (mean interelectrode distance) of 2.7 cm, provided by both the 128- channel sensor net and the 64-charmel helmet. An even higher sampling density may be required to maximize the benefits of surface Laplacian estimates, with improvements noted with a sampling density (interelectrode distance) up to 2.0 cm (180 electrodes). However, we cannot claim that very high electro de densities will provide substantial improvement in accuracy in real heads, where effects of variable tissue properties may be significant. Several calculations were performed in this paper to estimate the required sampling density for accurate potential and Laplacian mapping from scalp EEG. These calculations were all based on a four concentric sphere model of the head which is a coarse physical approximation to the human head, but which has been found to be in approximate agreement with more realistic (but also approximate) finite element models of the human head (Yan et al. 1991). The finite element models can account for departure of head shape from sphericity and inhomegeniety within each tissue layer, e.g., the eye sockets. By contrast to the surface Laplacian estimate, cortical imaging methods are potentially able to use electrical and geometric information about the head volume conductor to provide more accurate estimates of cortical potential using these discrete models (Gevins et al. 1994). However, the accuracy of these methods is limited by our relative ignorance of tissue resistivities. Improvements in head modelling would certainly modify the estimate of required sampling density. In our estimation, this would likely increase the optimal number of channels, if an accurate head model could be developed for each EEG recording.

Appendix A Representation of the 3-Dimensional Sptine as a Spherical Harmonic Expansion

For spherical surfaces, the osculating function Q2(r, 0, d~) can be expressed as:

Srinivasanet al.

Q2(r,0,~) = ql +q2 r sin0 cos~ +q3 r sin• sin~ +q4 r2 sin20 cos2~ +q5 r2 sin20 cos0? sin~ +q6 r2 sin20 sin2d~ +q7 r cos0 +q8 r2 cos0 sin0 cos~ +q9 r2 COS0 sin0 sin~ +ql0 r2 COS20 (A-l) which is easily formulated in terms of spherical harmonics Q2(r,0,qb) = qt ~

+ q4 r2 3 +q6r2

3 +'-110 T )

Y00(0 ,}) - q3 r ~ Y l _ l (0,(~)

+ q7r ~ Y l o

(0,~) - q 2 r ~ Y l l

(O,qb)+ 2qsr2 ~-~Y2-2 (0 ,d~)

r2 ~-~-5 (2q10 - q6- q4 )Y20(0 ,qb) - q9r2 ~ 5 y2q (O,q~)+ ~--qsr2 ~ Y 2 1 ( 0 , ~ ) + 2r2 ~ 5 (q4- qs)Y22(0,*)

(A-2) The distance d can be expressed in terms of the angular distance between the interpolation point (8, ~), and the electrode (0i, ~i), 7 i= c~176

as

i c~

+ sinOi sin0 (cos(~i -~b))] (A,3)

d 2 = 21)2(1 - cosy i)

(A-4)

Then the basis function P2(r, 0, d?)can be written in spherical coordinates as P2(r,O,~) = 4r 4 ln(2r 2) n 2 Pi (2r2 (1 - cos ~' i )) 2in (2r2 (1 - cos 7 i) + w2) (ln 10) 2 i=1

(A-5) Here the base 10 logarithms have been replaced by natural logarithms. For w2>0, the logarithm can be replaced by its Taylor series expansion, which converges for all Yi,

r 4r4 ln(2r2) + pi ln(l+ ~02 / + s

(-1)J(c~ Yi)J j

(A-6)

Spatial Sampling

365

Any function of the form (cos 0)n can be expanded in a finite series of Legendre polynomials for even and odd powers as

P2(r, 0,q~) - 4r41n(2r2) + Pi In 102 z.., i=l

/ w21? iw2/

- 4P0 (cos? i) In 1 +

3

(COS0)2n= ~ A~mP2m(COS0 ) m=0 (COS0)2n+l = Z A @ 2m+lP2)m+ln(COS m=0

2P2(c~ 3

(A-7)

O9

27

i)ln 1+

2~-

+

1(cos y i ) ln(1 + 2~- +

+ Z Z j P j ( c o s y i) j=0

(A-10) where With the addition theorem for spherical harmonics and the definition

1 3 A~ = ~2n+1 Ap = 2n+3 n (4m + 1)(2n - 2m + 2) A2m+2 = (4m- 3)(2n + 2m + 1) A2m

n

(A-8)

n (4m+3)(2n- 2m+ 2) n Aam+3 = (4m- 1)(2n + 2m + 3) A2m+l

/

rl

t- 2Z

(2n-l) 1+

n=l

4r 4 ln(2r 2) (ln 10)2

W2 /

-]

PooYoo(0'*) + y Z PlkYlk(0'qb)+ ~5 P2kY2k(0,(~) k=-i =(

W2/2n-2

(2n- 2) 1+ 2-72)

+

~ , 47rZj J Z PjkYjk(0,qb)

A~m

+ 2Z

Aznrn

Z__a f 2 "~2n ( 2 2n-1 n=m2n[]_+ ~7r2j n=m(an- 1), l+W~12r 2) + ~

A2nm-1

//

-2 ~

2 )2n-1

n=m(2n-1)/1+~r2~

References

Asnm

~=m+l(2n- 2)(1+

Z2m_l = - ~

27;5k= j

(A-12)

oo

Zzm = +

(A-11)

ln[l+ 2~T j

A~

n=l 2n 1 + 2 7 )

P2 (r'@'(~) =

(

( n-I

oo

A~

i=1

this expression becomes

Substituting and rearranging terms, we introduce the following definitions

Zo ;

Pjk = Z PiYjk(0i,~i)

w2 / 2n-2

27)

A2nm-1

2)(1 + W2 t2n-2 n=m+l(2n2~) n

A2n-1 ( 2 ~2n-3 n=m+l(2n- 3)[1 + ~ T ) (A-9) and write the basis function as

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