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arxiv: 2604.20448 · v3 · submitted 2026-04-22 · 🧮 math.NA · cs.NA

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Forward--Inverse Interplay in FEM-Based EEG Source Imaging: Distributional Signatures of Advanced Source Models and Inverse Solvers

Santtu S\"oderholm , Joonas Lahtinen , Sampsa Pursiainen

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Pith reviewed 2026-05-10 00:02 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords EEG source imagingfinite element methodsource modelsinverse methodsEarth Mover's Distancedistributional signaturesbrain localizationforward modeling
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The pith

The success of an inverse solver in EEG source imaging depends on how closely the forward source model matches the solver's assumptions about the spatial distribution of brain activity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines the interplay between different forward source models and inverse solvers in finite-element-method simulations of EEG data. It finds that source models designed for point-like sources perform better when paired with inverse methods that expect similar concentrated activity patterns. Using quantitative distributional metrics on realistic head models, the study identifies distinct signatures in the reconstructed activity that arise from this pairing. Understanding this dependence matters because it can guide the selection of modeling approaches to improve the reliability of locating brain sources from scalp measurements. The work uses divergence-conforming and local subtraction source models along with methods such as standardized hierarchical adaptive L1 regression and Kalman filtering.

Core claim

Advanced source models implemented via the finite element method produce unique distributional signatures in inverse solutions, with a clear dependence such that point-like source models align successfully with inverse solvers presupposing single-point activity, as quantified by Earth Mover's Distance, depth bias scatter plots, amplitude distributions, and focality measures.

What carries the argument

Distributional quantitative measures such as Earth Mover's Distance and depth bias scatter plots that capture how different source model implementations affect the spatial characteristics of inverse reconstructions.

Load-bearing premise

The chosen distributional metrics and the specific implementations of the divergence-conforming and local subtraction models are representative enough to reveal the general forward-inverse interplay.

What would settle it

Repeating the experiments on real measured EEG data with independently known source locations to check if the dependence between source model and inverse success rate persists.

Figures

Figures reproduced from arXiv: 2604.20448 by Joonas Lahtinen, Sampsa Pursiainen, Santtu S\"oderholm.

Figure 1
Figure 1. Figure 1: Electrodes positioned on the scalp of the ICBM152 2009a head model [17], according to the international 10–10 electrode positioning standard. only by the improvements in measurement technology, but by the rise of advanced signal processing methods [15], [16]. Mathematical modelling of the EEG sources also has a history of decades [18]. With a computational equivalent current dipole model in place [19], [20… view at source ↗
Figure 2
Figure 2. Figure 2: displays an examples of a column norms of a L com￾puted with DUNEuro and Zeffiro Interface. The DUNEuro lead fields have been computed using the Whitney basis functions, while Zeffiro uses H(div) interpolation in the source space. The norm of the Whitney-based L behaves as expected in [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: A display of the column norms ∥L∥2 of lead fields L produced with Whitney-based DUNEuro (2a, 2b) and H(div)-based Zeffiro Interface (2c, 2d). Subfigures 2a and 2c display a cortical view while 2b and 2d show a deeper cross-section in the RA-plane of the Right￾Anterior-Superior (RAS) coordinate system, roughly at the height of the thalamus. The DUNEuro field is strong near the EEG electrodes depicted as bla… view at source ↗
Figure 4
Figure 4. Figure 4: Depth of true source plotted against the depth of estimation done by sLORETA and SHAL1R. The thin black line shows the optimal agreement between the true and estimated depth when the localization error is zero. The dark gray solid line displays the linear regression, and the dashed gray curves are the 95 % confidence intervals. struction, as the difference between the Whitney and Local sub￾traction reconst… view at source ↗
Figure 3
Figure 3. Figure 3: Close up of reconstructions of a synthetic cortical dipole with sLORETA, SHAL1R, SKF and DS using DUNEuro’s Whitney basis function implementation (column 1), DUNEuro’s Local subtraction (column 2) and Zeffiro Interface’s H(div) (column 3). Column 4 shows the difference between Whitney and Local subtraction reconstructions, while column 5 does the same for H(div) and Local subtraction. SHAL1R reconstruction… view at source ↗
Figure 5
Figure 5. Figure 5: Earth Mover’s Distances for estimated sources. inverse method combination in [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
read the original abstract

Electroencephalography (EEG) source imaging aims to infer brain activity from electrical potentials measured on the scalp. This is a difficult problem because many different source patterns can explain the same measurements. The result depends strongly on two things: the forward model and the inverse method. In this work, we study how these two parts work together. We focus not only on where the activity is located, but also on how the reconstructed activity is distributed in space. We suggest that different source models create different signatures in the reconstructed activity. We use realistic head models and compute forward solutions with the finite element method using Zeffiro Interface and DUNEuro. We test different source models, including 2 implementations of a divergence-conforming model, and one implementation of Local subtraction approach. For inverse methods, we use advanced methods such as standardized hierarchical adaptive L1 regression (SHAL1R), standardized Kalman filtering (SKF), and classical dipole scanning. To understand the complex interplay between the forward and inverse approaches, we analyze the inverse source localization results using distributional quantitative measures, including Earth Mover's Distance and depth bias scatter plot, and qualitatively assess the amplitude distribution and focality. The results show that there is a strong dependence between the choice of source model and the success rate of a given inverse method: a source model that corresponds well with a single point-like source is a good match with an inverse method that presupposes such a source.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a simulation study of the forward-inverse interplay in FEM-based EEG source imaging. Using realistic head models computed via Zeffiro Interface and DUNEuro, it compares three source-model variants (two divergence-conforming implementations and one local-subtraction approach) against three inverse solvers (SHAL1R, SKF, and classical dipole scanning). Reconstruction quality is assessed with distributional metrics including Earth Mover's Distance, depth-bias scatter plots, amplitude distributions, and focality. The central claim is that a strong dependence exists between source-model choice and inverse-method performance, with point-like forward models pairing effectively with point-assuming inverses.

Significance. If the reported pairings hold under scrutiny, the work usefully demonstrates that distributional signatures can reveal assumption-matching effects between forward and inverse components, moving beyond scalar localization error. The empirical use of multiple advanced source-model implementations and quantitative metrics (EMD, depth bias) provides concrete evidence for the interplay in a controlled FEM setting. This could guide model selection in numerical EEG pipelines, though the simulation-only design limits direct translation to measured data.

major comments (2)
  1. [Methods] Methods section: the simulation protocol does not specify the number of Monte Carlo realizations, exact noise models, source-location sampling strategy, or quantitative differences between the two divergence-conforming implementations. Without these details it is impossible to judge whether the observed dependence is robust or sensitive to post-hoc choices.
  2. [Results] Results and Discussion: the claim of 'strong dependence' and differential 'success rates' rests on internal consistency within each source-model family; the manuscript should add a sensitivity test using source distributions that deviate from all tested models (e.g., extended patches or multi-dipole configurations) to show that the reported ranking is not an artifact of the closed simulation loop.
minor comments (2)
  1. [Abstract] Abstract: the phrasing '2 implementations of a divergence-conforming model' is repeated without clarifying their numerical distinctions (e.g., basis-function choice or boundary handling); a single sentence of differentiation would improve clarity.
  2. [Results] Figure captions and text: the depth-bias scatter plots and EMD values are presented without error bars or statistical comparison tests; adding these would strengthen the quantitative claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below and will revise the manuscript to incorporate the suggested improvements for clarity and robustness.

read point-by-point responses

  1. Referee: [Methods] Methods section: the simulation protocol does not specify the number of Monte Carlo realizations, exact noise models, source-location sampling strategy, or quantitative differences between the two divergence-conforming implementations. Without these details it is impossible to judge whether the observed dependence is robust or sensitive to post-hoc choices.

    Authors: We agree that these details are essential for reproducibility and for allowing readers to assess the robustness of the reported forward-inverse dependence. In the revised manuscript we will expand the Methods section to specify the number of Monte Carlo realizations performed, the precise noise model (including SNR and distribution), the source-location sampling strategy, and the quantitative differences between the two divergence-conforming implementations (e.g., basis-function order and stabilization techniques). revision: yes

  2. Referee: [Results] Results and Discussion: the claim of 'strong dependence' and differential 'success rates' rests on internal consistency within each source-model family; the manuscript should add a sensitivity test using source distributions that deviate from all tested models (e.g., extended patches or multi-dipole configurations) to show that the reported ranking is not an artifact of the closed simulation loop.

    Authors: We acknowledge that the current results are obtained within the tested source-model families and that an external sensitivity test would further strengthen the claim. In the revised manuscript we will add a dedicated sensitivity subsection that includes simulations with extended source patches and multi-dipole configurations, reporting the corresponding distributional metrics (EMD, depth-bias, focality) to demonstrate that the observed model-matching effects are not an artifact of the closed loop. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical simulation study with independent benchmarks

full rationale

The paper reports results from FEM-based numerical simulations comparing source models (divergence-conforming variants, local subtraction) and inverse solvers (SHAL1R, SKF, dipole scanning) using metrics such as Earth Mover's Distance and depth-bias plots. The central observation of dependence between source-model choice and inverse success is an empirical finding from these controlled experiments, not a mathematical derivation. Ground-truth sources are generated within the simulation framework, but the study explicitly benchmarks against classical dipole scanning as an external reference method. No equations are presented that reduce predictions to fitted parameters defined from the same data, and no load-bearing self-citations or uniqueness theorems are invoked to force the conclusions. The work is self-contained as a comparative simulation study.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard FEM assumptions for the forward problem and the validity of the chosen inverse solvers as black-box methods; no new entities are postulated.

axioms (2)
  • domain assumption Finite element discretization of the EEG forward problem on realistic head models accurately represents volume conduction.
    Invoked when using Zeffiro Interface and DUNEuro to compute forward solutions.
  • domain assumption The distributional measures (Earth Mover's Distance, depth bias) are appropriate proxies for reconstruction quality and source model suitability.
    Used to quantify signatures and success rates without explicit justification in the abstract.

pith-pipeline@v0.9.0 · 5577 in / 1467 out tokens · 31548 ms · 2026-05-10T00:02:01.031520+00:00 · methodology

discussion (0)

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Reference graph

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