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The Local Subtraction Approach For EEG and MEG Forward Modeling

Malte B. Höltershinken, Pia Lange, Tim Erdbrügger, Yvonne Buschermöhle, Fabrice Wallois, Alena Buyx, Sampsa Pursiainen, Johannes Vorwerk, Christian Engwer, Carsten H. Wolters

TL;DR

This work presents the localized subtraction approach for EEG and MEG forward modeling to dramatically reduce the computational burden of subtraction-based methods while preserving their mathematical rigor. By localizing the singularity near the dipole with a patch $\Omega^\infty$ and cutoff $\chi$, the FEM right-hand sides become sparse, enabling efficient forward solves. The authors derive a weak formulation for the EEG correction potential, develop a boundary-subtraction strategy for MEG based on Geselowitz-type ideas, implement the method in the DUNEuro toolbox, and validate it against analytical solutions in multilayer sphere models. Across extensive tests, the localized subtraction approach matches or surpasses existing methods in accuracy and is orders of magnitude faster, especially for EEG, making it highly suitable for large-scale source analyses. The work thus provides a practical, scalable alternative for realistic head models with complex conductivity, while maintaining fidelity near the dipole source.

Abstract

EDIT: A revised version of this article has been published in the SIAM Journal on Scientific Computing, see https://epubs.siam.org/doi/full/10.1137/23M1582874. In the revised version, the name of the approach was changed from "localized subtraction" to "local subtraction". In FEM-based EEG and MEG source analysis, the subtraction approach has been proposed to simulate sensor measurements generated by neural activity. While this approach possesses a rigorous foundation and produces accurate results, its major downside is that it is computationally prohibitively expensive in practical applications. To overcome this, we developed a new approach, called the local subtraction approach. This approach is designed to preserve the mathematical foundation of the subtraction approach, while also leading to sparse right-hand sides in the FEM formulation, making it efficiently computable. We achieve this by introducing a cut-off into the subtraction, restricting its influence to the immediate neighborhood of the source. In this work, this approach will be presented, analyzed, and compared to other state-of-the-art FEM right-hand side approaches. We perform validation in multi-layer sphere models where analytical solutions exist. There, we demonstrate that the local subtraction approach is vastly more efficient than the subtraction approach. Moreover, we find that for the EEG forward problem, the local subtraction approach is less dependent on the global structure of the FEM mesh when compared to the subtraction approach. Additionally, we show the local subtraction approach to rival, and in many cases even surpass, the other investigated approaches in terms of accuracy. For the MEG forward problem, we show the local subtraction approach and the subtraction approach to produce highly accurate approximations of the volume currents close to the source.

The Local Subtraction Approach For EEG and MEG Forward Modeling

TL;DR

This work presents the localized subtraction approach for EEG and MEG forward modeling to dramatically reduce the computational burden of subtraction-based methods while preserving their mathematical rigor. By localizing the singularity near the dipole with a patch and cutoff , the FEM right-hand sides become sparse, enabling efficient forward solves. The authors derive a weak formulation for the EEG correction potential, develop a boundary-subtraction strategy for MEG based on Geselowitz-type ideas, implement the method in the DUNEuro toolbox, and validate it against analytical solutions in multilayer sphere models. Across extensive tests, the localized subtraction approach matches or surpasses existing methods in accuracy and is orders of magnitude faster, especially for EEG, making it highly suitable for large-scale source analyses. The work thus provides a practical, scalable alternative for realistic head models with complex conductivity, while maintaining fidelity near the dipole source.

Abstract

EDIT: A revised version of this article has been published in the SIAM Journal on Scientific Computing, see https://epubs.siam.org/doi/full/10.1137/23M1582874. In the revised version, the name of the approach was changed from "localized subtraction" to "local subtraction". In FEM-based EEG and MEG source analysis, the subtraction approach has been proposed to simulate sensor measurements generated by neural activity. While this approach possesses a rigorous foundation and produces accurate results, its major downside is that it is computationally prohibitively expensive in practical applications. To overcome this, we developed a new approach, called the local subtraction approach. This approach is designed to preserve the mathematical foundation of the subtraction approach, while also leading to sparse right-hand sides in the FEM formulation, making it efficiently computable. We achieve this by introducing a cut-off into the subtraction, restricting its influence to the immediate neighborhood of the source. In this work, this approach will be presented, analyzed, and compared to other state-of-the-art FEM right-hand side approaches. We perform validation in multi-layer sphere models where analytical solutions exist. There, we demonstrate that the local subtraction approach is vastly more efficient than the subtraction approach. Moreover, we find that for the EEG forward problem, the local subtraction approach is less dependent on the global structure of the FEM mesh when compared to the subtraction approach. Additionally, we show the local subtraction approach to rival, and in many cases even surpass, the other investigated approaches in terms of accuracy. For the MEG forward problem, we show the local subtraction approach and the subtraction approach to produce highly accurate approximations of the volume currents close to the source.
Paper Structure (24 sections, 7 theorems, 76 equations, 27 figures, 6 tables)

This paper contains 24 sections, 7 theorems, 76 equations, 27 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. The EEG forward problem
  4. Definition of the problem
  5. Deriving a weak formulation
  6. FEM discretization
  7. The MEG forward problem
  8. Definition of the problem
  9. The boundary subtraction approach
  10. Implementation
  11. Experimental setup
  12. Results and Discussion
  13. Patch construction
  14. Comparison to other potential approaches
  15. Accuracy comparison
  16. ...and 9 more sections

Key Result

Lemma 1

Let $\Omega^\infty \subset \Omega$ and $\chi \in H^1(\Omega)$ be such that $\chi$ is identical to $1$ on $\Omega^\infty$ and $x_0 \in \Omega^\infty$. Let $\widetilde{\Omega} \subset \Omega\setminus\Omega^\infty$ be a region such that $\partial\Omega^\infty$ is a subset of $\partial\Omega \cup \parti

Figures (27)

  • Figure 1: 2-dim. visualization of a realization of the construction described in lemma \ref{['distributional_derivative_u_infty_times_chi']}. The whole square is $\Omega$. In the yellow region we have $\chi = 1$, in the white region we have $\chi = 0$.
  • Figure 1: Histogram visualizing the relative frequency of distance-edge length ratios during the computation of surface and transition integrals (\ref{['surface_flux_meg']} and \ref{['transition_flux_meg']}) in a tetrahedral mesh. To generate this image, $1000$ dipoles at an eccentricity of $0.99$ were considered.
  • Figure 1: Relative error in the EEG case for 1000 tangential dipoles at 0.99 eccentricity for different numbers of vertex extensions during patch construction, computed using mesh_brain (see Figure \ref{['meshes_refinement']} (b)). The rightmost yellow boxplot shows the errors when employing the analytical subtraction approach from beltrachini_analytic_subtraction.
  • Figure 1: Accuracy comparison of EEG forward simulations using the analytical subtraction, multipolar Venant, and localized subtraction potential approaches for tangential dipoles at different eccentricities using mesh_init (see Figure \ref{['meshes_refinement']} (a)). The $y$-axis shows the relative error. The physiologically relevant sources at 1-2 mm distance from the CSF are highlighted.
  • Figure 2: Visualization of the volume currents as computed by the localized subtraction approach. The colors of the lines indicate the magnitude of the current and the colors in the background show the conductivity of the volume conductor. The numerical simulation was performed in the multilayer sphere model mesh_init shown in Figure \ref{['meshes_refinement']} (a), for a radial dipole with a 1 mm distance to the conductivity jump. Details on this mesh can be found in the Results section. The current was visualized by generating 5000 points in a sphere of radius 2.5 mm around the dipole position and applying a Runge-Kutta method, using the streamtrace filter implemented in the ParaView software ahrens_paraview.
  • ...and 22 more figures

Theorems & Definitions (12)

  • Lemma 1
  • Lemma 2
  • Lemma
  • proof
  • Lemma B1
  • proof
  • Lemma C1
  • proof
  • Lemma C2
  • proof
  • ...and 2 more