- The paper demonstrates that forward models and inverse solvers must be matched to source extent, with sHAL1R and Local subtraction achieving near-zero EMD for point sources while performance worsens under H(div).
- Experiments comparing Whitney, H(div), and Local subtraction models with sLORETA, sHAL1R, SKF, and dipole scanning show that reconstruction focality, robustness, and depth bias vary substantially across pairings.
- The findings show why depth regression alone is insufficient: H(div) produced near-perfect depth slopes but wider reconstructions, highlighting distributional metrics such as Earth Mover’s Distance for model selection.
Overview and motivation
EEG source imaging is a coupled problem: a forward model maps candidate brain activity to scalp potentials, and an inverse solver reconstructs activity from measurements. The paper by Söderholm, Lahtinen, and Pursiainen (2604.20448) argues that these two components cannot be evaluated in isolation. Rather than reporting only localization error, the authors characterize the distributional properties of reconstructions—spatial spread, smoothness, focality, and depth bias—and show that these depend strongly on the interaction between the choice of FEM source model and the assumptions embedded in the inverse method.
The study is motivated by two strands of prior work: the theoretical unbiasedness of standardized inverse methods (sLORETA and its descendants) under noiseless data, and recent evidence that forward modelling choices such as anatomical peeling alter not just accuracy but the spatial character of reconstructions. The availability of the Local subtraction approach for singular dipolar sources in FEM provides a mathematically principled reference point against which other interpolation schemes can be compared.
Experimental design
All experiments use the ICBM152 2009a multicompartment head model with the point electrode model (PEM). Lead field matrices are computed via the FEM transfer matrix approach in DUNEuro and Zeffiro Interface. Three source models are compared:
- Whitney basis functions (face-intersecting and edgewise), implemented in DUNEuro
- Divergence-conforming H(div) basis functions, implemented in Zeffiro Interface
- Local subtraction, implemented in DUNEuro
Four inverse solvers are tested: sLORETA, the sparsity-promoting standardized hierarchical adaptive L1 regression (sHAL1R), spatiotemporal standardized Kalman filtering (SKF), and classical dipole scanning (DS).
Two experiments structure the evaluation. Experiment I reconstructs a cortical source under inversion crime to assess smoothness and spread of the estimates, using anisotropic brain conductivity. Experiment II places 100 randomly positioned Cartesian sources per 5 mm interval along the superior axis, spanning 0–60 mm from the inner skull surface (1200 sources total); here the synthetic data use anisotropic conductivity while inversion uses an isotropic model on the same mesh, deliberately avoiding inversion crime while retaining mesh-induced model mismatch. DS is omitted from Experiment II because it is primarily sensitive to forward–inverse model discrepancy, and SKF is omitted because its performance depends on agreement between the evolution model and the true temporal dynamics—both omissions reflect deliberate scoping rather than methodological weakness of those solvers.
Evaluation uses two distributional metrics: the Earth Mover's Distance (EMD) between true and estimated source positions, which quantifies reconstruction spread, and depth bias scatter plots with linear regression against perfect depth agreement, exploiting the theoretical unbiasedness of standardized estimators.
Lead field characteristics
A structural difference emerges at the level of the lead fields themselves. The Whitney-based lead field column norms behave as expected from reciprocity-based construction: potential decays smoothly with distance from the electrodes toward the thalamus. The H(div) lead field from Zeffiro Interface is markedly more irregular, decays more slowly away from the sensors, and exhibits magnitudes roughly five orders of magnitude larger; the authors note an unwarranted field maximum near the eye region that required clipping at the 95th percentile for visualization. This irregularity foreshadows the downstream behavior: a source model whose lead field does not cleanly separate distinct sources will confound any inverse method that presumes well-separated point activity.
Cortical reconstruction behavior (Experiment I)
Reconstructions of a single cortical dipole reveal a clear hierarchy of robustness. sHAL1R is the most robust of the four solvers: its reconstruction is extremely focal around the true position, and switching between Whitney and Local subtraction lead fields produces a zero difference in the reconstruction. This is a strong claim—it implies that for sparse, point-like ground truth, sHAL1R's sparsity assumption dominates whatever differences exist between those two forward interpolations. By contrast, SKF achieves slightly greater focality than sLORETA and DS but is considerably less robust to changes in the lead-field interpolation scheme, showing large relative differences between Local subtraction and both alternatives.
The H(div) source model degrades all methods except SKF: sLORETA, sHAL1R, and DS produce wider reconstructions under H(div) than under Local subtraction, while SKF alone yields a more focal estimate. The interpretation offered is that H(div) creates a region of nearly identical sources at the cortical level, weakening source separability—a regime better matched to inverse methods that assume distributed activity.
Depth bias and EMD results (Experiment II)
The depth bias analysis quantifies how closely each forward–inverse pair approaches ideal unbiased localization. Regression slopes against perfect depth agreement are:
| Source model |
sLORETA slope |
sHAL1R slope |
| Whitney |
0.85 |
1.03 |
| Local subtraction |
0.85 |
1.02 |
| H(div) |
1.00 |
0.99 |
sHAL1R agrees near-perfectly with optimal depth under Whitney and Local subtraction, while sLORETA systematically underestimates depth (slope 0.85) for both. Notably, the best regression lines for both solvers come from the H(div) model—an apparent contradiction with Experiment I, where H(div) produced the widest, most ambiguous cortical reconstructions. Depth agreement alone therefore does not capture reconstruction quality, which is precisely the paper's argument for distributional metrics.
The EMD analysis resolves this tension and delivers the paper's sharpest numerical contrast. With sHAL1R, the bulk of EMD values sit near 2 mm for Whitney and drop to essentially 0 mm for Local subtraction, with only superficial-source outliers; the average EMD under H(div) rises to around 5 mm. For sLORETA, EMD values cluster between roughly 60–80 mm (Whitney) and 50–60 mm ($H(\mathrm{div}}$), regardless of whether Local subtraction or Whitney is used. Two implications follow directly. First, the near-zero sHAL1R/Local-subtraction EMD indicates no alternative dipole can produce similar data—the forward model is unambiguous for point sources, so residual ambiguity in practice should stem from modelled noise rather than the lead field. Second, sLORETA's EMD decreases with increasing source depth, suggesting an inflated ability to reconstruct deep sources relative to superficial ones—an unrealistic characterization of deep, patch-like activity given that standardized methods are theoretically strongest for cortical sources.
Discussion
The central conclusion is a matching principle: a source model corresponding well to a single point-like source pairs successfully with an inverse method that presupposes such a source. Local subtraction emerges as the most accurate point-source forward model, evidenced jointly by sHAL1R's and DS's high performance under it. Conversely, the weaker source separation of the H(div)0 model suits distributed-activity solvers such as sLORETA and SKF, with SKF providing the most concise cortical estimate under H(div)1. The authors explicitly concede that a source model designed for patch-like activity is lacking: Local subtraction causes sub-optimal behavior for sLORETA, so no currently tested combination serves distributed sources well.
Limitations and open questions
Several constraints bound the findings. Experiment I is conducted under full inversion crime, so its conclusions about focality and robustness do not directly transfer to mismatched realistic settings; Experiment II mitigates this via anisotropic-to-isotropic mismatch but retains the same mesh, leaving discretization-consistent model error unexamined. The evaluation covers a single head model (ICBM152 2009a) and one electrode configuration, so population-level variability in the reported signatures is unknown. SKF's omission from the depth/EMD analysis leaves its depth behavior uncharacterized, and DS's omission from Experiment II means the matching principle is validated for DS only in the noiseless cortical case. Finally, the observation that H(div)2 simultaneously yields the best depth regression but the worst EMD raises an open question about which metric should govern forward model selection when the true source extent is uncertain.
Conclusion
This work reframes EEG source imaging evaluation around distributional signatures rather than pointwise accuracy alone. Its principal result—that forward source models and inverse solvers must be matched in their assumptions about source extent—is supported by a concrete quantitative demonstration: sHAL1R with Local subtraction achieves near-zero EMD across depths, while the same solver degrades substantially under H(div)3, and sLORETA remains poor under every point-source model. The identified gap—a source model suited to patch-like activity paired with correspondingly matched inverse methods—remains unresolved by this study.