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REVIEW 4 major objections 5 minor 3 references

Efficient Representations of Cardiac Spatial Heterogeneity in Computational Models

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A parameter grid spacing of about 1.6 cm reproduces cardiac action-potential-duration profiles within 5 percent error during discordant alternans.

desk verdict A clean interpolation benchmark for cardiac tissue grids, but the 1.6 cm guidance only holds under exact parameter values; the experimental leap is unproven. read the letter →

arxiv 2412.06802 v1 pith:XPTMFBA6 submitted 2024-11-24 q-bio.TO

Alejandro Nieto Ramos , Elizabeth M. Cherry This is my paper · ORCID
classification q-bio.TO MSC 92C3065M06
keywords cardiacelectrophysiologyspatialheterogeneityactionpotentialdurationdiscordantalternanscoarse-gridparameterizationpiecewiselinearinterpolationconstantFenton-Karmamodel
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks how coarsely a cardiac tissue model can be parameterized in space while still reproducing the electrical behavior of fully resolved heterogeneous tissue. Using the Fenton-Karma cable model, the authors create smooth spatial gradients in individual model parameters and then assign parameter values only on coarser grids, using either nearest-point or linear interpolation. They find that grid spacings around 1.0 to 1.6 cm keep the average relative error in action potential duration below 5 percent across many parameters and gradient shapes, even during discordant alternans. If true, this means matching computational cardiac models to spatially variable experimental data could be done with relatively few sample points rather than per-pixel parameter identification.

What carries the argument

The central mechanism is the coarse parameter grid: a uniform subset of the computational grid at which model parameter values are assumed to be known exactly, with all other points assigned values by either piecewise-constant nearest-neighbor assignment or piecewise-linear interpolation. The accuracy measure is the average relative error over space between the true action-potential-duration profile and the approximated profile, tested against a threshold of 5 percent. The dynamical test bed is discordant alternans, a state in which action potential duration alternates between two values on successive beats with the phase of alternation varying across space, produced by pacing the Fenton-Karma cable at a chosen cycle length.

What would settle it

Run the same discordant-alternans simulations with the parameter values at the coarse grid points corrupted by a small amount of noise, for example 5 percent, and recompute the mean relative action-potential-duration error; if spacings near 1.6 cm no longer stay below 5 percent error, the central claim would fail.

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Extended reading notes

Core claim

The central claim is that, for a one-dimensional Fenton-Karma cardiac cable, a parameter grid spacing of roughly 1.6 cm reproduces the true spatial profiles of action potential duration within 5 percent mean relative error during discordant alternans, with required spacings ranging from 1.0 to 6.4 cm depending on the parameter, the spatial gradient function, and the cable length. This holds for both piecewise-constant and piecewise-linear interpolation, with the linear version performing slightly better. The result is robust across eight different model parameters and four nonsymmetric smooth spatial gradient functions, and in many cases the longer cable tolerates even coarser spacing.

Load-bearing premise

The result assumes the exact parameter values are known at the coarse grid points and that such values could be obtained from experimental data, so if those values are noisy or wrong, the 5 percent accuracy at 1.6 cm spacing is not guaranteed.

Editorial extensions

If this is right

  • Heterogeneous cardiac tissue could be represented in computational models by parameter values known at only a small number of sample points, roughly one point every 1.6 cm, rather than at every computational node.
  • The similarity of piecewise-constant and piecewise-linear interpolation suggests that simple nearest-neighbor assignment may suffice, which is convenient for interpreting experimental data that arrive on irregular grids.
  • Longer cables generally allowed coarser parameter grids, with the maximum tolerated spacing reaching 12.8 cm for one parameter, so larger tissue preparations may be represented even more efficiently.
  • The convergence trend, although not perfectly monotonic, indicates that the 5 percent error threshold can be used as a practical stopping rule when choosing the parameter grid spacing for a given experiment.
  • Because the approach works in a complex dynamical state like discordant alternans, the authors argue it would work even more readily in simpler dynamical states encountered in cardiac mapping.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural testable extension is to apply the same coarse-grid parameterization to two-dimensional tissue surfaces where optical mapping data are recorded, to see whether the roughly 1.6 cm spacing survives transverse diffusion and more complex wavefront shapes.
  • If coarse-point values must be estimated from noisy optical mapping signals rather than known exactly, the relevant quantity becomes a joint constraint on grid spacing and parameter-estimation uncertainty, not spacing alone.
  • The one parameter that failed the 5 percent test on the longer cable, due to an alternans flip, suggests that fidelity may be bounded by the dynamical stability of the state itself rather than by interpolation error alone.
  • The error often decreased with cable length for fixed spacing, which hints that the governing factor may be the relationship between the parameter gradient scale and the spatial wavelength of the alternans profile, not the physical length per se.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. This paper asks how coarsely one can specify spatially heterogeneous model parameters in a one-dimensional Fenton-Karma cable while still reproducing action potential duration (APD) profiles during discordant alternans. For each of eight model parameters, the authors impose a smooth spatial gradient (sigmoid, quadratic, cubic, or sinusoid, each also reflected), assume the exact parameter value is known at the points of a coarse parameter grid, and then assign values on the full computational grid by piecewise-constant or piecewise-linear interpolation. They compute APD profiles on the coarse-parameter representation and compare them with a full-resolution reference simulation, using a 5% mean-relative-error threshold. The main reported finding is that a parameter grid spacing of about 1.0-1.6 cm, and usually about 1.6 cm, is sufficient for both cable lengths and most parameters, with a documented exception for tau_w^+ on the longer cable, where even a 0.05 cm grid fails because an alternans flips. The paper concludes that matching models to heterogeneous experimental data can be done efficiently.

Significance. The computational benchmark is useful and not circular: the coarse approximations are compared against an independent full-resolution simulation, and no model parameter is fitted to the target APD profiles. If the idealized result holds, it provides a practical guideline for reducing the number of parameter sample points needed to represent smooth spatial heterogeneity in cardiac tissue models. The main limitations are the exact-value idealization, single-parameter variation, smooth-only gradients, one chosen pacing period per parameter, and the hand-selected 5% threshold. These limitations do not invalidate the interpolation benchmark itself, but they do restrict how strongly the practical, experiment-facing conclusion can be stated.

major comments (4)
  1. [Section 2 and Abstract] The abstract's final sentence, 'matching the output of models of cardiac tissue to heterogeneous experimental data can be done efficiently,' goes beyond what the simulations show. Section 2 explicitly assumes that 'the exact parameter values are known at the points of the coarser grid' and that these values 'could be obtained from experimental data,' but optical mapping provides voltage and APD measurements, not Fenton-Karma parameters such as tau_v^+, tau_w^-, tau_d, or k. The paper contains no test of the parameter-estimation inverse problem at the coarse nodes, including noise, identifiability, or model mismatch. This is load-bearing for the practical claim: if the coarse-point values are not exact, the reported 1.0-1.6 cm spacings need not preserve the 5% APD accuracy. I recommend either adding a propagation-of-error test in which the coarse-point values are perturbed by realistic noise (and reporting how the maximum allowable spacing changes), or explicitly limiting the conclusions to the idealized benchmark and revising the abstract accordingly.
  2. [Results and Tables 1-2] The quantitative claim 'spacing of about 1.6 cm produces profiles within 5% of the true profiles' is only established for one selected pacing period per parameter and one selected 5% threshold. The authors acknowledge in the Results that convergence is not always monotonic and that for tau_w^+ on the longer cable even a 0.05 cm grid fails the threshold because an alternans flips. This means the 'maximum spacing' values in Tables 1 and 2 are threshold-crossing points for particular choices, not robust convergence properties. I ask for a sensitivity analysis: vary the threshold (e.g., 2% and 10%) and vary pacing periods within the alternans range for at least a few parameters, and report whether the 1.0-1.6 cm spacing rule remains stable. Without this, the central number may be an artifact of the chosen criterion.
  3. [Methods and Figure 2.3] The mean relative error used throughout is not defined precisely. The text refers to 'the average relative error over space' and to values 'across all functions,' but it is unclear whether the tables report the mean over functions, the maximum over functions, or some other aggregation, and whether the error is averaged over the last two beats. The exact formula and aggregation rule are needed to reproduce the tables. In addition, the reference APD is computed on a 0.025 cm computational grid with threshold detection from discrete voltage values; the contribution of that finite-resolution/APD-detection error to the 5% criterion is not quantified, and this matters because the reported maximum spacings are close to the threshold in several cases.
  4. [Section 2 (Heterogeneity model)] The study varies only one parameter at a time, and all heterogeneity shapes are smooth deterministic functions (sigmoid, quadratic, cubic, sinusoid and their reflections). Real cardiac tissue exhibits simultaneous multi-parameter variation, noise, and potentially sharper or discontinuous spatial transitions. The abstract's phrase 'generally results in spatial profiles that agree well' is therefore too broad. At minimum, I would like to see a multi-parameter test case and a noise-corrupted gradient test, or the conclusions explicitly restricted to single-parameter smooth gradients. As it stands, the practical relevance of the claimed spacing for experimentally mapped tissue is not established.
minor comments (5)
  1. [Throughout] The parameter notation is inconsistent in places (for example, 't−v1' instead of tau^-_{v1}, and missing superscripts in 'τ+v', 'τ+w'). Please normalize the notation.
  2. [Results] The text contains typos: 'al functions' should be 'all functions', and 'we sticked to' should be 'we stuck to' or 'we continued to use'.
  3. [Figure 2.2 caption] The caption says the sinusoidal function is 'defined on the interval [29, 29.2]', but this interval appears to be the parameter range for tau_si, not the spatial definition of the function. Please clarify.
  4. [Tables 1-2] The 'NA' entry in Table 2 for tau_w^+ under PW Linear is not explained in the caption or text; please state explicitly that no tested spacing achieved the 5% threshold for that case.
  5. [References] Reference [2] has garbled page formatting: "20 '¨A ` ı47" should be cleaned up to the correct page range.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the coarse-grid benchmark is evaluated against an independent full-resolution simulation, and no parameter is fitted to the target APD profiles.

full rationale

The paper's derivation chain is self-contained as a computational benchmark. It defines true APD profiles from full-resolution Fenton-Karma simulations, then approximates spatial parameter heterogeneity by piecewise-constant or piecewise-linear interpolation from exact coarse-grid parameter values, and compares the resulting APD profiles to the true profiles. No model parameter is inferred from the APD data being matched; the 5% mean-relative-error threshold is an externally chosen evaluation criterion, not a fitted constant. The assumption that exact parameter values are known at coarse points is explicitly stated in the abstract and in Section 2, and it is a limitation about applicability to experimental data, not a circular step: the paper does not claim to solve the inverse parameter-estimation problem, and the main quantitative claim is conditional on exact coarse-point values. The few references are to the standard Fenton-Karma model and prior electrophysiology work; they are not invoked to justify the numerical result. The anomalous case (tau_w^+ reflected sinusoid on the long cable) is reported as a failure, which is evidence against curve-fitting circularity. Therefore no self-definitional, fitted-input, or self-citation circularity is present.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central result rests on a benchmark comparison rather than a new physical entity. No free model parameters are fit to the data being predicted; the only hand-set numbers are the evaluation threshold and the pacing/parameter ranges. The main idealized assumptions are exact knowledge of coarse-grid parameter values and smooth single-parameter gradients.

free parameters (3)
  • mean relative error threshold = 5%
    Chosen by the authors as the criterion for 'good agreement'; all reported maximum grid spacings are conditional on this threshold. Not fitted to data, but a hand-set evaluation cutoff.
  • pacing period per parameter = 300, 305, 310, 325, 330, 335 ms
    Periods are selected by hand to produce discordant alternans and avoid block; the maximum spacing result is specific to these periods and may not hold at other cycle lengths.
  • parameter value intervals = k 9-10; tau_d 0.05-0.25; tau_r 32.8-33.4; tau_si 29-29.2; tau_v1- 20-25; tau_v+ 2.9-5; tau_w- 35-110; tau_w+ 750-870
    Ranges are chosen by hand to produce significant dynamical differences and alternans; they are inputs rather than results, but the threshold spacings depend on them.
assumptions (3)
  • domain assumption The Fenton-Karma model with the given current equations is an adequate representation of cardiac electrical behavior for studying discordant alternans.
    The entire study uses this model as a proxy for cardiac tissue; the coarse-grid thresholds are not tested in other ionic models. Invoked in Section 2.
  • ad hoc to paper Exact parameter values are known at the coarse grid points and could be obtained from experimental data.
    Optical mapping does not directly provide ionic model parameters; this idealized assumption skips parameter estimation and noise. Stated in Section 2.
  • domain assumption Heterogeneity is smooth and can be represented by one of four functional shapes: sigmoid, quadratic, cubic, or sinusoid.
    Only smooth single-parameter gradients are tested; realistic tissue may have sharper or more complex gradients, limiting the generality of the 1.6 cm guideline. Stated in Section 2.

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Cite this review

Pith. "Pith review of Efficient Representations of Cardiac Spatial Heterogeneity in Computational Models." pith.science (2026). https://pith.science/paper/XPTMFBA6

@misc{pith2026241206802,
  author       = {Pith},
  title        = {Pith review of: Efficient Representations of Cardiac Spatial Heterogeneity in Computational Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XPTMFBA6}},
  note         = {Machine review of arXiv:2412.06802}
}
read the original abstract

It is generally assumed that all cells in models of the electrical behavior of cardiac tissue have the same properties. However, there are differences in cardiac cells that are not well characterized but cause spatial heterogeneity of the electrical properties in tissue. Optical mapping can be used to obtain experimental data from cardiac surfaces at high spatial resolution. Variations in model parameters can be defined on a coarser grid than considering each single pixel, which would allow a representation of heterogeneous tissue to be obtained more efficiently. Here, we address how coarse the parameterization grid can be while still obtaining accurate results for complicated dynamical states of spatially discordant alternans. We use the Fenton-Karma model with heterogeneity included as a smooth nonlinear gradient over space for more model parameters. To obtain the more efficient representations, we set parameter values everywhere in space based on the assumption that the exact parameter values are known at the points of the coarser grid; we assume the parameter values could be obtained from experimental data. We assign parameter values in space by fitting either a piecewise-constant or piecewise-linear function to the spatially coarse known data. We wish to identify the maximal grid spacing of such points to obtain good agreement with spatial profiles of action potential duration during complex states. We find that coarse grid spacing of about 1.0-1.6 cm generally results in spatial profiles that agree well with the true profiles for a range of different model parameters and different functions of those parameters over space. In addition, the piecewise-constant and piecewise-linear functions perform similarly. Our results to date suggest that matching the output of models of cardiac tissue to heterogeneous experimental data can be done efficiently, even during complex dynamical states.

Figures

Figures reproduced from arXiv: 2412.06802 by the authors.

Figure 2.1
Figure 2.1. Example of the two kinds of the approximation using a quadratic original function for the [PITH_FULL_IMAGE:figures/full_fig_p003_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Example of discordant alternans. These profiles were obtained for the [PITH_FULL_IMAGE:figures/full_fig_p004_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Example of the average relative error over space for the parameter [PITH_FULL_IMAGE:figures/full_fig_p005_2_3.png] view at source ↗
Figures from the paper (6 more)
Figure 2.4
Figure 2.4. Figure 2.4: Example of different discordant alternans profiles for the parameter [PITH_FULL_IMAGE:figures/full_fig_p006_2_4.png]
Figure 2.5
Figure 2.5. Figure 2.5: Parameter τd, with an original quadratic gradient function and a maximum spacing of 12.8 (left) and 25.6 cm (right). In general, the number of nodes increased for the longe cable. (a) Piecewise approximations. (b) Relative error [PITH_FULL_IMAGE:figures/full_fig_p00…
Figure 2.6
Figure 2.6. Figure 2.6: Parameter τ − v1 , with an original cubic gradient function and a maximum spacing of 12.8 cm. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_2_6.png]
Figure 2.7
Figure 2.7. Figure 2.7: Parameter τ + v , with a reflected sigmoid gradient function and a maximum spacing of 25.6 cm. (a) Piecewise approximations. (b) Relative error [PITH_FULL_IMAGE:figures/full_fig_p008_2_7.png]
Figure 2.8
Figure 2.8. Figure 2.8: Parameter τr, with a reflected sinusoidal gradient function and a maximum spacing of 12.8 cm. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_2_8.png]
Figure 2.9
Figure 2.9. Figure 2.9: Comparison of the average relative error for the two different lengths (12.8 and 25.6 cm) [PITH_FULL_IMAGE:figures/full_fig_p011_2_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

  1. [1]

    Fenton, and E.M

    Cairns, D.I., F.H. Fenton, and E.M. Cherry. 2017. Efficient parame- terization of cardiac action potential models using a genetic algorithm. Chaos. 27: 093922

    work page 2017

  2. [2]

    Fenton, F., and A. Karma. 1998. Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation. Chaos. 8: 20 ’ ¨A ` ı47. 11

    work page 1998

  3. [3]

    Fenton, S.J

    Watanabe, M.A., F.H. Fenton, S.J. Evans, H.M. Hastings, and A. Karma. 2001. Mechanisms for discordant alternans. J. Cardiovasc. Elec- trophysiol. 12: 196 ’ ¨A ` ı206. 12

    work page 2001

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