... | ... | @@ -15,6 +15,14 @@ The idea of the approach following St. Venant is to replace the dipolar source b |
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For the Whitney source modeling approach, a source term with a higher regularity is
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chosen. The primary current $`j_p`$ is discretized in a vector-valued function space consisting of functions with square-integrable divergence.
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For a detailed description, please see
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Miinalainen, T, Rezaei, A, Us, D, Nüßing, A, Engwer, C, Wolters, CH & Pursiainen, S 2019, 'A realistic, accurate and fast source modeling approach for the EEG forward problem', NeuroImage, vol. 184, no. 1, pp. 56-67. [https://doi.org/10.1016/j.neuroimage.2018.08.054](https://doi.org/10.1016/j.neuroimage.2018.08.054)
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# Subtraction
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The subtraction approach relies on the assumption that there exists a small area around the dipole location, where the conductivity is constant. As a result, the potential and the conductivity can be split into two contributions, a singularity contribution and a correction part: $`u = u_\infty + u_{corr}`$, and $`\sigma = \sigma_\infty + \sigma_{corr}`$.
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While $`u_\infty`$ can be computed analytically, the insertion of the above decomposition into the EEG forward problem results in a Poisson equation for the correction potential with a right hand side $`- \nabla \cdot (\sigma_{corr} \nabla u_\infty)`$ and inhomogeneous Neumann boundary conditions.
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For more details, see, e.g., the following publications:
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Drechsler F, Wolters CH, Dierkes T, Si H, Grasedyck L. A full subtraction approach for finite element method based source analysis using constrained Delaunay tetrahedralisation. Neuroimage. 2009 Jul 15;46(4):1055-65. doi: . [https://doi.org/10.1016/j.neuroimage.2009.02.024](https://doi.org/10.1016/j.neuroimage.2009.02.024)
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Wolters, CH, Köstler, H, Möller, C, Härdtlein, J, Grasedyck, L and Hackbusch, W. 2007. Numerical Mathematics of the Subtraction Method for the Modeling of a Current Dipole in EEG Source Reconstruction Using Finite Element Head Models. SIAM J. Sci. Comput. 30, 1 (November 2007), 24–45. [https://doi.org/10.1137/060659053](https://doi.org/10.1137/060659053). |