Mathematical Considerations Boundary Conditions

Consider a four-layer head model with conductivity parameters aa, where a = 14, and define the conductivity of air to be a5 = 0. Let $a be the potential in layer a, and let n be the outward-oriented normal to the boundary surface S. In each layer, a is constant and the homogenous solution is correct. The solutions to Poisson's equation in each layer are joined together by appropriate boundary conditions.

The first condition boundary condition is that the normal current density J^ be continuous across each boundary:

where the normal derivative is defined d$/dn = V$a • n. From Maxwell's equation a second boundary condition may be shown: continuity of the parallel component of the electric field. Assuming no sources or sinks on the surface, this is equivalent to continuity of the potential $ across each boundary:

as may be shown by drawing a rectangular loop with one side in each layer, and integrating the electric field around this loop.

The magnetic field B obeys similar boundary conditions involving discontinuities in j (Jackson 1975). These are not relevant to biological tissue, because to high accuracy j = j0, the magnetic susceptibility of vacuum (Plonsey 1969). Never must we consider discontinuities in j or boundary conditions on B in the usual sense. Yet boundary effects do enter at tissue discontinuities. In passing from (3.16) to (3.17) we assumed a to be constant. Without that assumption we have additional contributions to B arising from discontinuities in a. These contributions are identically zero for a spherical head model, but nonzero in general.

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