A common formalism for the Integral formulations of the forward EEG problem

The forward electroencephalography (EEG) problem involves finding a potential V from the Poisson equation ∇·(σ∇V)=f, in which f represents electrical sources in the brain, and σ the conductivity of the head tissues. In the piecewise constant conductivity head model, this can be accomplished by the boundary element method (BEM) using a suitable integral formulation. Most previous work uses the same integral formulation, corresponding to a double-layer potential. We present a conceptual framework based on a well-known theorem (Theorem 1) that characterizes harmonic functions defined on the complement of a bounded smooth surface. This theorem says that such harmonic functions are completely defined by their values and those of their normal derivatives on this surface. It allows us to cast the previous BEM approaches in a unified setting and to develop two new approaches corresponding to different ways of exploiting the same theorem. Specifically, we first present a dual approach which involves a single-layer potential. Then, we propose a symmetric formulation, which combines single- and double-layer potentials, and which is new to the field of EEG, although it has been applied to other problems in electromagnetism. The three methods have been evaluated numerically using a spherical geometry with known analytical solution, and the symmetric formulation achieves a significantly higher accuracy than the alternative methods. Additionally, we present results with realistically shaped meshes. Beside providing a better understanding of the foundations of BEM methods, our approach appears to lead also to more efficient algorithms.