N

j=i where the function gm(x) is given by

The functions Pn(x) are the Legendre polynomials of order n, which form a complete set of basis functions on a spherical surface.6

6 The use of ordinary Legendre polynomials does not imply that the surface potential must have azimuthal symmetry. The variable x in Pn (x) represents the angle between electrode position Ti and the interpolation point r, so the claim is that (10.3) is capable of fitting the net scalp potential without inherent symmetry.

Recently we elaborated the idea put forth in Junghofer et al. (1999) that the spherical splines permit a better estimate of the average surface potential (Ferree 2006). Integrating (10.3) over the entire spherical scalp surface, and using that the integral of Pn(x) on —1 < x < +1 vanishes for n = 0 (Arfken 1995), leads to where r4 is the outer scalp radius. Thus the coefficient c0 is equal to the average of the interpolated potential over the sphere surface.

Current conservation implies that, for dipolar current sources in an arbitrary volume conductor, the surface integral of the absolute potential $ vanishes (Bertrand et al. 1985). Substituting V(r) = $(r) — $ref leads to

Based upon (10.6), we expect c0 to provide a reasonable estimate of $ref, which can be used to compute the absolute potentials using $i = Vi + $ref ~ Vi — c0. This favorable situation is limited by the fact that the spline fit is severely under-constrained on the inferior head surface, and is unlikely to be numerically accurate there. It is conceivable that the estimate $ref ~ —c0 is worse than the usual estimate $ref ~ —V, but further investigation proved otherwise. A more convincing theoretical argument and numerical simulations showing that spherical splines generally provide a better estimate of $ref are given in (Ferree 2006).

8.3 Surface Laplacian

Complementary to the scalp potential is the scalp surface Laplacian, the second spatial derivative of the potential. Practically speaking, the surface Lapla-cian solves the problem of the reference electrode because the second spatial derivative discards any overall constant (corresponding to the potential at the reference electrode relative to infinity). The calculation of the surface Laplacian is made separately at each time point. Physically, it is most directly related to the local current density flowing radially through the skull into the scalp. Because current flow through the skull is mostly radial, the scalp surface Laplacian remarkably provides an estimate of the dura potential (Nunez 1987). Numerical simulations using real data have shown that the surface Laplacian has 80-95% agreement with other dura imaging algorithms (Nunez and Srini-vasan 2006). This connection between the scalp surface Laplacian and dura potential is derived next.

The following derivations make three main assumptions: 1) the current flow through the skull is nearly radial, 2) the potential drops across the scalp and CSF are small, at least compared to that across the skull, and 3) the

Fig. 8. A patch of scalp for consideration of the surface Laplacian in Problem 1. The parameter r3 represents the outer skull surface, and r4 the outer scalp surface. Alternatively, by replacing 3 ^ 2 and 4 ^ 3, the same figure may be used to represent a patch of skull in Problem 2

potential on the brain surface is much larger in amplitude than that on the scalp surface, by close proximity to the dipolar sources. Referring to Fig. 8, we have

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