Figure 3.

Implementations of anisotropy in the brain. (a) An anisotropic conductivity tensor (Σ) can be decomposed into three conductivities (i.e., σ₁, σ₂, and σ₃) in three orthogonal directions (v₁, v₂, and v₃). It is common to treat tensors in white matter as prolate and tensors in cerebral spinal fluid (CSF) and grey matter as approximately degenerate. More recent approaches make no assumptions about the shape of the tensors and model Σ as a general ellipsoid. (b) We used a spherical shell with an inner and outer radius of 1 mm and 100 mm, respectively, to approximate a homogeneous, infinite conductive medium. The anisotropic tensor (Σ) was parameterized by its largest eigenvalue (σ₁), the ratio of σ₁ to σ₂ (w₁₂), and the ratio of σ₁ to σ₃ (w₁₃). The eigenvalues of the isotropic tensor (σ_iso) were all 1 S/m. We used a numerical approach to determine a scalar mapping (θ) between σ_iso and Σ (see equations 9–11). (c) θ could be approximated with a nonlinear analytic function (see equation 12).