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CANDLE: Cortical Null-Space Decomposition for Noninvasive Brain Source Imaging

Shuntaro Suzuki, Yuiga Wada, Komei Sugiura

arXiv:2610.07824v1cs.LGeess.SPq-bio.NC

TL;DR

ESI is ill-posed, and learning-based approaches can struggle to account for subject-specific cortical geometry. CANDLE learns a prior over the unobservable null-space component using a T1-derived geometric constraint and a whole-brain simulator. It outperformed prior methods on simulated estimation and generalized to two empirical tasks without further tuning.

  • Problem

    ESI is non-unique because cortical source activity is higher-dimensional than EEG/MEG observations, and common-space learned priors may not account for individual cortical geometry.

    Source: §1 · p. 1, §2.1 · p. 3

  • Method

    CANDLE derives a subject-specific lead field from T1 MRI and learns the unobservable null-space source component while preserving geometric constraints.

    Source: §1 · p. 2, §2.1 · p. 3

  • Results

    Without further tuning, CANDLE outperformed prior ESI methods on simulated source estimation and generalized to intracranial stimulation localization and epileptogenic-zone estimation.

    Source: §1 · p. 2, §4 · p. 9

  • Takeaways & Limitations

    The results support simulated-data-trained source imaging on subject-specific geometries across simulated estimation and two empirical tasks.

    Source: §1 · p. 2, §4 · p. 9

    The current implementation uses a fixed EEG montage and the Hagmann/Lausanne source parcellation.

    Source: §4 · p. 9

Abstract

from arXiv · show

Electrophysiological source imaging (ESI) aims to estimate cortical source activity from noninvasive electrophysiological measurements such as electroencephalogram (EEG). However, ESI is fundamentally ill-posed because source activity is substantially higher-dimensional than sensor observations, resulting in non-unique solutions. Recent learning-based approaches address this ambiguity by learning data-driven source priors, yet they often struggle to generalize across subject-specific cortical geometries. To address this, we propose CANDLE, a learning-based ESI model that estimates source activity on subject-specific cortical geometries. CANDLE learns a prior over the null space induced by the source-to-sensor mapping derived from T1-weighted MRI, restricting learning to unobservable source components while preserving geometric constraints. To train CANDLE, we develop a whole-brain simulator spanning over 1,100 subject-specific cortical geometries with source configurations derived from over 26,000 statistical brain maps. Trained exclusively on simulated data, CANDLE outperformed prior ESI methods on simulated source activity estimation and generalized to two empirical tasks: (i) intracranial stimulation localization from simultaneously recorded scalp EEG and (ii) epileptogenic zone estimation from presurgical interictal EEG. Our project page is available at https://candle-esi.pages.dev}{https://candle-esi.pages.dev.

1 INTRODUCTION

ESI promises cortical source estimates from noninvasive EEG/MEG, but its inverse is non-unique and learning-based priors often fail to account for individual cortical geometry. CANDLE combines a subject-specific null-space prior with a whole-brain simulator and reports improved simulated and empirical performance without tuning.

1 INTRODUCTION

  • EEG/MEG enable noninvasive, high-temporal-resolution monitoring but have limited sensor-level spatial resolution; ESI reconstructs cortical sources, whose higher dimensionality makes solutions non-unique.Source: §1 · p. 1
  • Manual regularization and source constraints limit ESI applicability and reproducibility, while common-space learned priors may miss individual geometry and require subject-specific post-training.Source: §1 · pp. 1–2
  • Existing simulators were not designed to jointly capture anatomical variability and realistic source configurations, a gap the new simulator addresses.Source: §1 · p. 2
  • CANDLE uses a T1-derived lead field to preserve subject-specific geometry and learns a data-driven prior over the unobservable null-space source component.Its denoising model analytically recovers the range-space component, while a second model estimates the null-space component.Source: §1 · p. 2, Figure 1 · p. 2
  • The whole-brain simulator generates source activity across more than 1,100 subject-specific geometries, with source regions derived by clustering over 26,000 statistical brain maps.Source: §1 · p. 2
  • Without subject-specific tuning, CANDLE outperformed prior ESI methods on simulated source estimation and generalized to stimulation localization and epileptogenic-zone estimation.Source: §1 · p. 2

2 METHODOLOGY

CANDLE estimates cortical source activity from EEG while explicitly incorporating each subject’s T1-derived cortical geometry. It learns unobservable source components and is trained with a whole-brain simulator spanning subject-specific geometries and neurobiologically grounded source patterns.

2 METHODOLOGY

  • Given EEG and a subject’s T1-weighted MRI, CANDLE estimates that subject’s cortical source activity.The MRI-derived lead-field matrix maps cortical sources to EEG and provides the subject-specific geometric constraint.Source: §2 · pp. 2–3, §2.1 · p. 3

2.1 PROPOSED METHOD: CANDLE

  • CANDLE decomposes source activity into components in the range space and null space of the lead-field operator, learning only the unobservable null-space component.In the noiseless case, the range-space component can be recovered from EEG using the lead-field pseudoinverse.Source: §2.1 · p. 3
  • Because low-SNR EEG can make direct pseudoinverse recovery unstable or diffuse, CANDLE denoises EEG before estimating the range-space component and models the null-space component separately.The formulation uses fθ to denoise EEG, L† to estimate the range-space component, and gγ to estimate the null-space component.Source: §2.1 · p. 3

2.2 SIMULATOR CONSTRUCTION

  • The simulator couples Jansen–Rit neural mass models across structurally connected cortical regions and synthesizes EEG through subject-specific lead fields for 1,113 MRI-derived geometries.It models EEG as X = LY + ϵ, with additive Gaussian noise.Source: §2.2 · p. 4, Figure 2 · p. 4
  • Source configurations come from 26,273 statistical brain maps associated with cognitive terms, clustered into representative patterns and sampled to modulate cortical excitatory gain.The maps yield 200 representative clusters, each perturbed four times to produce 800 distinct source regions.Source: §2.2 · pp. 4–5, Figure 2 · p. 4

3 EXPERIMENTS

CANDLE was evaluated against established ESI methods, with ConvDip and DeepSIF as direct learning-based baselines; empirical models were trained only on simulations and applied without further tuning. Tests covered simulated source estimation, two empirical localization tasks, and ablations of the architecture, losses, and simulator.

3.2 INTRACRANIAL STIMULATION LOCALIZATION

  • Spatial dispersion of 54.1 mm and peak distance of 32.9 mm were lowest overall; CANDLE was best at depths beyond 10 mm.For shallow sites (0–10 mm), its peak distance was 32.48 mm, behind GBF at 29.20 mm; overall improvements were not significant versus sLORETA and GBF.Source: Figure 4 · p. 6, §3.2 · pp. 7–8

3.1 SIMULATED SOURCE ACTIVITY ESTIMATION

  • HD95 37.3 mm and Dice 0.48 beat the second-best baseline by 2.5 mm and 0.15 points, respectively.CANDLE also achieved the highest F1 scores across IoU thresholds; improvements over all baselines were significant (p < 0.001).Source: §3.1 · p. 6

3.3 EPILEPTOGENIC ZONE ESTIMATION

  • HD95 of 61.7 mm and centroid distance of 35.4 mm were lowest among methods, improving on the second-best baseline by 0.9 mm and 3.4 mm.Improvements over LCMV were significant for both metrics, and HD95 improved significantly over GBF.Source: Figure 5 · p. 7, §3.3 · p. 8

3.4 ABLATION STUDIES

  • Removing either the denoising model or null-space estimator degraded performance; removing the null-space estimator had the larger impact.Peak distance increased by 1.4 mm without the denoising model and 4.8 mm without the null-space estimator; the latter also significantly worsened spatial dispersion, peak distance, and HD95.Source: §3.4 · p. 9
  • Removing the denoising loss degraded all four metrics, while removing the null-space loss degraded three, with centroid distance as the exception.Peak distance increased by 13.1 mm without the denoising loss and 1.3 mm without the null-space loss; the authors attribute the denoising loss's greater role to its position before range–null space decomposition.Source: §3.4 · p. 9
  • A fixed fsaverage head model degraded all four metrics, while random source regions degraded three; both simulator variants raised peak distance.Peak distance increased by 0.7 mm for the fixed head model and 0.5 mm for random source regions; the findings support anatomical diversity and neurobiologically grounded source regions in training.Source: §3.4 · p. 9

4 CONCLUSION AND LIMITATIONS

CANDLE combines subject-specific geometry with a prior over unobservable source activity and outperformed prior methods on simulated and empirical tasks without further tuning. The current implementation is limited to a fixed EEG montage, a specified cortical parcellation, and one neural mass model.

4 CONCLUSION AND LIMITATIONS

  • Without further tuning, CANDLE outperformed prior ESI methods on simulated source estimation and two empirical tasks: stimulation localization and epileptogenic zone estimation.The simulator generates neurobiologically grounded source activity across more than 1,100 subject-specific cortical geometries.Source: §4 · p. 9
  • The implementation uses a fixed EEG montage, Hagmann/Lausanne parcellation, and a single Jansen–Rit model, limiting its current configurations and modeled dynamics.The authors identify broader sensor configurations, parcellations, electrophysiological modalities, and neural dynamics as future directions.Source: §4 · p. 9

AI USE STATEMENT

  • Generative AI tools assisted with methodology, implementation, literature retrieval, and editing, but not synthetic data generation, mathematical claims, hypotheses, or results interpretation.Authors reviewed and tested AI-assisted implementations and independently verified AI-assisted literature retrieval.Source: AI USE STATEMENT · p. 10

ETHICS STATEMENT

  • The study used only previously collected human datasets and collected no new participant data.The data were used under applicable terms and the original studies' ethical approvals.Source: ETHICS STATEMENT · p. 10

REPRODUCIBILITY STATEMENT

  • The paper reports that implementation code and trained checkpoints are included in supplementary materials.Reproduction details cover the model, training, preprocessing, head-model construction, and neural mass model parameters.Source: REPRODUCIBILITY STATEMENT · p. 10

A RELATED WORKS

Prior ESI methods address the inverse problem with explicit source constraints or learned data-driven priors. Existing whole-brain simulators often prioritize dynamics over the anatomical variability and spatial organization needed for anatomically realistic multi-subject ESI training.

A RELATED WORKS

  • Conventional ESI methods impose source-space constraints, from equivalent dipoles in MU-SIC to depth weighting in sLORETA (Pascual-Marqui, 2002) and dSPM.FAST-IRES (Sohrabpour et al., 2020) promotes sparsity under a temporal basis; GBF (Wang et al., 2026) regresses subject-specific geometry-derived eigenmodes.Source: A RELATED WORKS · p. 14
  • Learning-based ESI acquires data-driven priors; ConvDip maps EEG topographies to sources, while DeepSIF transfers from synthetic training to real recordings.Most such methods use common template spaces and may require post-training adaptation to individual anatomy.Source: A RELATED WORKS · p. 14
  • Whole-brain simulators embed neural mass models in connectomes, but often abstract away anatomical variability and cortical spatial organization relevant to multi-subject ESI training.Examples include BrainPy, The Virtual Brain, and Virtual Brain Inference.Source: A RELATED WORKS · p. 14

B.1 LEAD-FIELD MATRIX CALCULATION

CANDLE constructs the lead-field matrix from each subject’s anatomical image using a three-layer boundary element model. Its source space is parcellated into 994 subject-specific cortical regions, defining the mapping from cortical activity to EEG.

B.1 LEAD-FIELD MATRIX CALCULATION

  • A watershed-based procedure segments white matter, inner skull, outer skull, and scalp from the anatomical image to build a surface-based head model and three-layer BEM.The assigned conductivities are 0.3, 0.006, and 0.3 S/m for brain, skull, and scalp, respectively.Source: §B.1 · p. 14
  • The three-layer BEM reduces computational cost while retaining anatomical detail relevant to EEG propagation.The choice follows a cited finding that greater volume-conduction model complexity has negligible effects on empirical EEG source localization.Source: §B.1 · p. 15
  • The Hagmann/Lausanne parcellation defines 994 cortical regions, each about 1.5 cm2, and the resulting subject-specific lead field maps cortical activity to EEG.The selected region scale is motivated by estimates that detectable scalp potentials require synchronized activity over approximately 6–10 cm2 of cortex.Source: §B.1 · p. 15

B.2 ADDITIONAL METHODOLOGICAL DETAILS

CANDLE combines denoising and null-space estimation models with a sequence-modeling backbone. The section specifies its training configuration, validation-based checkpoint selection, and EEG and anatomical preprocessing.

B.2 ADDITIONAL METHODOLOGICAL DETAILS

  • CANDLE uses Kimi Delta Attention with a gated attention module, inverted temporal normalization, and M stacked blocks for its denoising and null-space estimation models.The two model architectures are shown in Figure 7.Source: §B.2 · p. 15, Figure 7 · p. 15
  • Training used 20 epochs, batch size 64, sequence length 500, 64 EEG channels, and 994 cortical regions, with approximately 11.6 M parameters and 15.30 G FLOPs.The optimizer was AdamW with learning rate 1.0 × 10−4 and weight decay 1.0 × 10−2.Source: §B.2 · p. 15
  • The checkpoint with the lowest validation-set HD95 was selected for final test-set evaluation.The validation set was used for model selection, while the test set was reserved for final evaluation.Source: §B.2 · p. 15
  • EEG was downsampled to 500 Hz, band-pass filtered from 0.5 to 45 Hz, common-average referenced, and z-score normalized; FreeSurfer MRI processing supplied cortical geometry for the lead-field matrix.Training required approximately 1.8 hours on the stated hardware.Source: §B.2 · p. 15

B.3 JANSEN–RIT MODEL

The whole-brain simulator uses Jansen–Rit neural mass models to generate cortical activity, couples regional models using structural connectivity, and implements source configurations through local synaptic-gain modulation.

B.3 JANSEN–RIT MODEL

  • The Jansen–Rit model generates mesoscopic cortical activity through second-order synaptic dynamics among pyramidal neurons and excitatory and inhibitory interneurons.Regional observable activity is defined as the difference between excitatory and inhibitory postsynaptic potentials.Source: §B.3 · p. 16
  • One neural mass model is instantiated per cortical region, with pyramidal populations coupled according to Hagmann–Lausanne structural connectivity to form whole-brain source activity.The interregional input uses normalized connectivity weights, with Wii = 0.Source: §B.3 · p. 16
  • Sampled source configurations are implemented by locally modulating excitatory synaptic gain Ai, while other Jansen–Rit parameters remain unchanged.This generates activity by increasing the excitatory postsynaptic response in neurobiologically defined regions rather than directly injecting artificial waveforms.Source: §B.3 · p. 16
  • Simulations used stochastic Heun integration at 2,000 Hz, discarded a 1-second burn-in, analyzed the subsequent 10 seconds, and then downsampled activity to 500 Hz.Three independent trials were generated for each source configuration under each of three noise conditions, giving 30 seconds per noise condition.Source: §B.3 · p. 17

C EVALUATION METRICS

The evaluation metrics quantify spatial distance, overlap, and dispersion for source estimates, with spatial distances measured in subject-specific anatomical space.

C EVALUATION METRICS

  • HD95 (95th-percentile Hausdorff distance) evaluates the ground-truth and estimated source regions using directed nearest-neighbor distances, with Q0.95 denoting the 95th percentile.Source: C EVALUATION METRICS · p. 17
  • Dice score quantifies spatial overlap between the estimated and ground-truth source regions.Source: C EVALUATION METRICS · p. 17
  • Spatial dispersion measures the estimated activity's spatial spread relative to the stimulation site, weighting source distances by activity at the time of maximum source activity.The distance for each cortical source is measured from its location to the stimulation location.Source: C EVALUATION METRICS · p. 17
  • F1 scores are computed across IoU thresholds from 0.1 to 0.9 in 0.1 increments, counting an estimate correct when its IoU exceeds the threshold; centroid distance is Euclidean distance between estimated and ground-truth region centroids.All spatial distances are computed in subject-specific anatomical space.Source: C EVALUATION METRICS · p. 17
  • For intracranial stimulation localization, peak distance is the Euclidean distance from the known stimulation site to the cortical source with maximum estimated activity.Source: C EVALUATION METRICS · p. 17

D ADDITIONAL EXPERIMENTAL RESULTS

Additional results include qualitative examples across simulated and empirical tasks, alongside complete quantitative comparisons. CANDLE was best or second-best across reported metrics, with one stated exception for shallow stimulation sites.

D ADDITIONAL EXPERIMENTAL RESULTS

  • In two additional simulated test samples, CANDLE's estimated source region closely matched ground-truth location and extent, while conventional and learning-based baselines were diffuse or displaced.Figure 8 compares EEG input, ground truth, and estimates from eight ESI methods.Source: D ADDITIONAL EXPERIMENTAL RESULTS · p. 17, Figure 8 · p. 18
  • For two additional resection cases, CANDLE's dominant source region lay within or adjacent to the resection cavity; additional figures show stimulation estimates from four patient runs.Figure 10 includes right- and left-hemisphere resections; Figure 9 displays estimated SPES locations in three views.Source: D ADDITIONAL EXPERIMENTAL RESULTS · p. 18, Figure 10 · p. 19, Figure 9 · p. 19
  • CANDLE achieved the best or second-best value in every metric across Tables 2–4; GBF had lower peak distance for the shallowest SPES sites (0–10 mm).The tables provide full quantitative results for simulated source estimation, intracranial stimulation localization, and epileptogenic zone estimation.Source: D ADDITIONAL EXPERIMENTAL RESULTS · p. 18, Table 2 · p. 20, Table 3 · p. 20
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