A Meshless Method for Solving the EEG Forward Problem

Laboratorio de Electrónica Industrial, Control e Instrumentación, Departamento de Electrotecnia, Facultad de Ingeniería, Universidad Nacional de La Plata, CC 91, 1900 La Plata, Argentina.
IEEE Transactions on Biomedical Engineering (Impact Factor: 2.35). 03/2005; 52(2):249-57. DOI: 10.1109/TBME.2004.840499
Source: PubMed


We present a numerical method to solve the quasistatic Maxwell equations and compute the electroencephalography (EEG) forward problem solution. More generally, we develop a computationally efficient method to obtain the electric potential distribution generated by a source of electric activity inside a three-dimensional body of arbitrary shape and layers of different electric conductivities. The method needs only a set of nodes on the surface and inside the head, but not a mesh connecting the nodes. This represents an advantage over traditional methods like boundary elements or finite elements since the generation of the mesh is typically computationally intensive. The performance of the proposed method is compared with the boundary element method (BEM) by numerically solving some EEG forward problems examples. For a large number of nodes and the same precision, our method has lower computational load than BEM due to a faster convergence rate and to the sparsity of the linear system to be solved.

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Available from: Arye Nehorai, Mar 13, 2014
    • "Difficulties in handling the geometric complexity of biological structures motivated the recent incorporation of meshfree methods in M/EEG research [2] [57]. The meshfree methods that have been proposed so far in this field require computational nodes distributed in the entire domain; therefore, though they avoid the mesh generation step in preprocessing, BEMs may outperform them from a computational cost per accuracy standpoint [57]. "
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    ABSTRACT: The estimation of neuronal activity in the human brain from electroencephalography (EEG) and magnetoencephalography (MEG) signals is a typical inverse problem whose solution process requires an accurate and fast forward solver. In this paper the method of fundamental solutions is, for the first time, proposed as a meshfree, boundary-type, and easy-to-implement alternative to the boundary element method (BEM) for solving the M/EEG forward problem. The solution of the forward problem is obtained by numerically solving a set of coupled boundary value problems for the three-dimensional Laplace equation. Numerical accuracy, convergence, and computational load are investigated. The proposed method is shown to be a competitive alternative to the state-of-the-art BEM for M/EEG forward solving.
    No preview · Article · Jul 2015 · SIAM Journal on Scientific Computing
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    • "boundaries at high quality and could potentially introduce mesh-related artifacts in the reconstructed neural activation pattern. Meshfree methods have been previously proposed for solving the EEG [4] and the MEG [5] forward problem. However, they are domain methods, thus they may be outperformed by BEM from a computational efficiency standpoint. "
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    ABSTRACT: Non-invasive estimation of brain activity via magnetoencephalography (MEG) involves an inverse problem whose solution requires an accurate and fast forward solver. To this end, we propose the method of fundamental solutions as a meshfree alternative to the boundary element method (BEM). The solution of the MEG forward problem is obtained, via the method of particular solutions, by numerically solving a boundary value problem for the electric scalar potential, derived from the quasi-stationary approximation of Maxwell's equations. The magnetic field is then computed by the Biot-Savart law. Numerical experiments have been carried out in a realistic single-shell head geometry. The proposed solver is compared with a state-of-the-art BEM solver. A good agreement and a reduced computational load show the attractiveness of the meshfree approach.
    Full-text · Article · Mar 2015 · IEEE Transactions on Magnetics
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    • "The 1D test case considers the problem of determining the electric potential generated by a dipole located at x = 0 inside the first subdomain of conductivity σ 1 extending from −R 1 to R 1 . A second subdomain of conductivity σ 2 is located in the intervals [−R 2 , −R 1 ] and [R 1 , R 2 ], where R 2 > R 1 > 0. To make the solution of the problem easier by removing the source singularity induced by the dipole, it is convenient to consider the potential φ induced by the dipole as the sum of two terms φ = φ d + Φ [36] [52], where φ d is the potential induced by the dipole in an infinite homogeneous medium, and Φ is a correction potential. The potential induced by a dipole at x = 0 with moment q d in an infinite medium of conductivity σ 1 is "
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    ABSTRACT: We present a meshless particle method for Poisson and diffusion problems on domains with discontinuous coefficients and possibly inhomogeneous boundary conditions. The method is based on a domain-decomposition approach with suitable interface and boundary conditions between regions of different diffusivities, and on using discretization-corrected particle strength exchange operators [B. Schrader, S. Reboux and I. F. Sbalzarini, J. Comput. Phys. 229, No. 11, 4159–4182 (2010; Zbl 05712956)]. We propose and compare two methods: The first one is based on an immersed interface approach, where interfaces are determined implicitly using a simplified phase-field equation. The second method uses a regularization technique to transform inhomogeneous interface or boundary conditions to homogeneous ones with an additional continuous volume contribution. After presenting the methods, we demonstrate their capabilities and limitations on several one-dimensional and three-dimensional test cases with Dirichlet and Neumann boundary conditions, and both regular and irregular particle distributions.
    Full-text · Article · Jan 2013 · SIAM Journal on Scientific Computing
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