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The electroencephalography (EEG) forward problem, the computation of the electric potential generated by a known electric current source configuration in the brain, is a key step of EEG source analysis. In this problem, it is often desired to model the anisotropic conductivity profiles of the skull and of the white matter. These profiles, however, cannot be handled by standard surface integral formulations and the use of volume finite elements is required. Leveraging on the representation theorem using an anisotropic fundamental solution, this paper proposes a modified symmetric formulation for solving the EEG forward problem by a surface integral equation which can take into account anisotropic conductivity profiles. A set of numerical results is presented to corroborate theoretical treatments and to show the impact of the proposed approach on both canonical and real case scenarios., Comment: Physics in medicine and biology (2018)

The symmetric formulation of the electroencephalography (EEG) forward problem is a well-known and widespread equation thanks to the high level of accuracy that it delivers. However, this equation is first kind in nature and gives rise to ill-conditioned problems when the discretization density or the brain conductivity contrast increases, resulting in numerical instabilities and increasingly slow solutions. This work addresses and solves this problem by proposing a new regularized symmetric formulation. The new scheme is obtained by leveraging on Calderon identities which allow to introduce a dual symmetric equation that, combined with the standard one, results in a second kind operator which is both stable and well-conditioned under all the above mentioned conditions. The new formulation presented here can be easily integrated into existing EEG imaging packages since it can be obtained with the same computational technology required by the standard symmetric formulation. The performance of the new scheme is substantiated by both theoretical developments and numerical results which corroborate the theory and show the practical impact of the new technique.

Source localization based on electroencephalography (EEG) has become a widely used neuroimagining 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 introduce a new BEM scheme taking into account the anisotropies of the white matter. The boundary nature of the formulation allows for an efficient discretization and modelling of the fibrous nature of the white matter as one-dimensional basis functions, limiting the computational impact of their modelling. We compare our scheme against widely used formulations and establish its correctness in both canonical and realistic cases., Comment: Update the title of the paper to reflect the title of the published version