The simplest head model that accommodates the layered tissues is comprised an inner sphere (brain) surrounded by 2 (ignoring CSF) or 3 (including CSF)
concentric spheres (see Fig. 1). For a dipole current source at brain location a with dipole moment m, the potential at the scalp surface location r may be written compactly
where f = a/r4 is the dipole eccentricity, r4 is the outer scalp radius, 0 is the angle between r and a, r is the radial unit vector, t is the tangential unit vector, and the Cn are constant coefficients (Salu et al. 1990). Current conservation ensures that the surface integral of the absolute potential $ induced by a dipolar current source is zero. This is reflected in (5.3) by the absence of a constant term that would be represented by n = 0. Thus the potential $ computed with (5.3) is implicitly referenced to infinity.
In numerical implementations of (5.3), the calculation of the Legendre polynomials Pn(x) is the rate limiting step. Faster implementation is available by noting the convergence properties of the series (Sun 1997).
5.3 Boundary Element Method
The simplest approach for accommodating realistic head geometry keeps the assumption that the head is comprised of four tissue layers: brain, CSF, skull and scalp, and that each layer is described by a single homogeneous and isotropic conductivity a, but relaxes the assumption of sphericity. Green's theorem may be used to write the solution to Poisson's equation as an integral equation for $
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