where r is the surface on the sides of the scalp patch, and t is a unit vector normal to r and therefore tangential to the scalp-air boundary. Assume that $4(r, 9, 4>) depends negligibly on r (on the grounds that the scalp is thin and a4 is high, at least compared to as), so that $4(r, 9, 4>) — V(9, 4>) leads to
where A4 is the cross-sectional area of the scalp patch. The boundary condition on on each side of the skull implies that the current flow through the skull is primarily radial, thus Is = Is. Given that, the potential within the skull patch must vary radially according to the function
Considering how the cross-sectional area of the patch A(r) varies as a function of r, and making use of the boundary condition on J^ at the skull-scalp boundary rs, shows that the potential difference across the skull is given approximately by
03 rs As
Making use of the boundary condition on $ leads to s
$s(rs) - $4(rs) = - — — (r4 - rs)(rs - rs)VSV (10.11)
0s rsrs which states that the potential difference across the skull is approximately proportional to the surface Laplacian. Finally, assuming that: 1) the potential drop across the CSF is small compared to that across the skull due to the low skull conductivity and high CSF conductivity, and the fact that the CSF layer is thinner than the skull, and 2) that the potential on the dura surface = $2(ri) = $1(r1) is large compared to the potential on the scalp surface $4(r4) = V, leads to
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