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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 85
PROCEEDINGS OF THE FIFTEENTH UK CONFERENCE OF THE ASSOCIATION OF COMPUTATIONAL MECHANICS IN ENGINEERING
Edited by: B.H.V. Topping
Paper 13

Flux-Continuous Schemes for Solving EEG Source Localization Problems

M. Pal1, D. Gupta2, M.G. Edwards1 and C.J. James2

1Civil and Computational Engineering Centre, University of Wales Swansea, United Kingdom
2Signal Processing and Control Group, ISVR, University of Southampton, United Kingdom

Full Bibliographic Reference for this paper
M. Pal, D. Gupta, M.G. Edwards, C.J. James, "Flux-Continuous Schemes for Solving EEG Source Localization Problems", in B.H.V. Topping, (Editor), "Proceedings of the Fifteenth UK Conference of the Association of Computational Mechanics in Engineering", Civil-Comp Press, Stirlingshire, UK, Paper 13, 2007. doi:10.4203/ccp.85.13
Keywords: flux-continuous schemes, Poisson's equation, finite element method, control volume distributed, electroencephalographic, independent component analysis, source localization.

Summary
Determining the active regions of the brain, given potential measurements at the scalp is a challenging problem in neuroscience. If accurate solutions to such problems could be obtained, it would help neurologists to gain non-invasive access to patient-specific cortical and sub-cortical activities. This would eventually help in the effective treatment of patients suffering from neural pathologies such as multi-focal epilepsy. Estimating the location and distribution of current sources within the brain from electroencephalographic (EEG) recordings is also known as the source localization problem. The solution to this problem is challenging, because it is ill-posed and lacks a unique solution. One of the recent promising approaches being used to decompose scalp EEG into more workable components involves a Blind Source Separation (BSS) technique called Independent Component Analysis (ICA).

Mathematically the problem involves computing the electrical potentials within the cranial volume due to a set of current sources and can be described by Poisson's equation with specified Neumann boundary conditions. The solution algorithm is described as finding the current dipole(s) that lead to a 'best fit' of the computed potentials (generated by a forward model and assuming a current-dipole model of the sources) and the measured electrode potentials (from EEG/ ICA decomposed EEG). ICA helps by estimating temporally independent activation maps, hence potentially simplifying the source analysis process and in reducing the search to a smaller number of dipole sources. Our aim in this paper is to improve the technique used for calculating the forward model in the EEG source localization problem. The forward problem involves calculating the distribution on the surface of the head (scalp), (except for at the electrode locations where the electrostatic potential is known), knowing the position, orientation and magnitude of an equivalent current dipole, as well as the geometry and the multiple electrical conductivities (skull, scalp, cerebral fluid) of head volume, . Currently the Finite Element Method (FEM) is used to solve the Poisson's equation. It is well known that FEM applied to problems involving discontinuities and high heterogeneities (as in this case) results in local conservation issues [1,2,3] and hence is not suitable to be applied here. Therefore, we propose to use a method which is flux-continuous and locally conservative for electric potential computation in heterogeneous media. This method is called as the Control-Volume Distributed Multi-point flux approximation CVD (MPFA) [3,4]. In this paper the CVD (MPFA) method is presented and applied to the EEG source localization problem. Comparisons between CVD and FEM for problems involving heterogeneity in cranial volume are presented together with advantages of the CVD method.

References
1
P. Hansbo, "Aspects of conservation in finite element flow computations", Comput. Methods Appl. Mech. Engrg, 117: 423-437, 1994. doi:10.1016/0045-7825(94)90127-9
2
R.C.Berger and S.E. Howington, "Discrete fluxes and mass balance in finite elements", Journal of Hydraulic Engineering. Vol 128, No. 1, 87-92, 2002. doi:10.1061/(ASCE)0733-9429(2002)128:1(87)
3
M.G. Edwards and C.F. Rogers, "Finite volume discretization with imposed flux continuity for the general tensor pressure equation", Comput. Geo. no.2, 259-290, 1998.
4
M.G. Edwards, "Unstructured, control-volume distributed, full-tensor finite Volume schemes with flow based grids", Comput.Geo, no.6, 433-452, 2002.

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