The Finite Element Method in EEG / MEG Source Analysis

Electroand magnetoencephalography (EEG/MEG)-based source reconstruction of cerebral activity (the EEG/MEG inverse problem) is an important tool both in clinical practice and research and in cognitive neuroscience. Methods for solving the inverse problem are based on solutions to the corresponding forward problem, i.e., simulation of EEG/MEG fields for a given primary source in the brain with a volume-conduction model of the head. The associated differential equations for the forward problem are the quasi-static Maxwell equations. The primary sources are electrolytic currents within the dendrites of the large pyramidal cells of activated neurons in the human cortex, generally formulated as a mathematical point current dipole. Such focal brain activation can be observed in epilepsy (interictal spikes), or it can be induced by a stimulus in neurophysiological or neuropsychological experiments, e.g., somatosensory or auditory evoked fields. Realistic volume-conductor modeling for accurate solution of the forward problem begins with segmentation of the tissues of the head; conductivity values are assigned in a second step. The tissues vary in conductivity and can also be inhomogeneous, e.g., the human skull, and anisotropic (with conductivity showing directional dependence), e.g., the skull and brain. The finite element (FE) method is often used for the forward problem, because it allows realistic representation of the complicated head volume conductor. In the case of a point current dipole in the brain, the singularity of the potential at the source position can be treated with the “subtraction dipole model”; the model divides the total potential into the analytically known singularity potential and the singularity-free correction potential, which can then be approximated numerically with an FE approach [5]. For the correction potential, existence and uniqueness proofs of a weak solution in a zero-mean function space and statements about FE convergence properties have been given [5]. Beyond the subtraction dipole model are direct FE approaches to the total potential; these approaches are computationally less expensive, but also mathematically less sound if the point dipole is seen as the most realistic source model. They use either partial integration over the point source on the right-hand side of the weak formulation, approximating the source singularity by means of a projection in the function space of the FE trial-functions (partial integration dipole model; [4]), or approximation of the point dipole by an even smoother monopolar primary source distribution (St. Venant dipole model; [4]). A prerequisite for FE modeling is the generation of a mesh that represents the geometric and electric properties of the volume conductor. An effective meshing strategy will achieve both acceptable forward problem accuracy and reasonable computation times and memory usage. Surface-based Delaunay tetrahedral tesselations are often used because of their ability to represent tissue boundaries in a smooth and regular way [3,5]. Hexahedral elements are also used; the hexahedra exploit the spatial discretization inherent in segmented medical tomographic data, and good performance has been achieved with them in recent accuracy studies [4,5]. A geometry-adapted node-shifting approach was developed to avoid the stair-like approximation of curved tissue boundaries that occurs with regular hexahedra; its use has led to significant reductions in field topographic and magnitude errors, despite the detrimental effects of deformed elements [5]. Adaptive methods preclude the use of lead field bases (see below) and lose efficiency in solving the inverse problem. An important question is how to handle the computational complexity of FE modeling with regard to the inverse problem. The longtime stateof-the-art approach was to solve an FE equation system for each anatomically and physiologically meaningful dipolar source (each source results in one FE right-hand side vector). Iterative solvers were used, among them the successive over-relaxation or the preconditioned conjugate gradient (CG) method, with preconditioners like Jacobi (Jacobi–CG) or incomplete Cholesky. More recently, algebraic multigrid (AMG) solvers, used as a preconditioner for the CG method, have proved more efficient than solvers tried earlier. Specifically, large speedups have been achieved with a parallel AMG–CG method for an anisotropic FE head model as compared with a standard Jacobi–CG method on a single processor [1]. Still, repeated solution of FE equation systems with a constant geometry matrix for thousands of right-hand sides (the sources) was the most time-consuming part of the inverse localization process and limited the resolution of the models. Another very efficient concept for reducing the computational complexity of the problem is reciprocity. The reciprocity theorem for the electric case states that the field of the lead vectors is the same as the current field produced by feeding a reciprocal current to the lead. This means that we can switch the role of the sensors and the dipole locations. Recently, for efficient computation of the FE-based EEG and MEG forward problem, an even easier principle, as-sociativity with respect to matrix multiplication, was applied [2]. Using this principle, which is closely The Finite Element Method in EEG/MEG Source Analysis