On the modeling of brain fibers in the EEG forward problem via a new family of wire integral equations

Source localization based on electroencephalography (EEG) has become a widely used neuroimaging technique. However its precision has been shown to be very dependent on how accurately the brain, head and scalp can be electrically modeled within the so-called forward problem. The construction of this model is traditionally performed by leveraging Finite Element or Boundary Element Methods (FEM or BEM). Even though the latter is more computationally efficient thanks to the smaller interaction matrices it yields and near-linear solvers, it has traditionally been used on simpler models than the former. Indeed, while FEM models taking into account the different media anisotropies are widely available, BEM models have been limited to isotropic, piecewise homogeneous models. In this work we augment standard BEM with a new wire integral equation to account for the anisotropy of the white matter. The new formulation combines the efficiency of BEM discretization of the boundaries only and modeling of the fibrous nature of the white matter as one-dimensional basis functions which limits the computational impact of their modeling. We compare our scheme against widely used formulations and establish its correctness in both canonical and realistic cases.