{"id":"5150978c-6d4d-4787-b949-6aa2bebddbed","arxiv_id":"2412.06802","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A coarse grid spacing of about 1.6 cm in cardiac tissue models reproduces action potential duration profiles within 5% during discordant alternans.","lead":"This paper tests how coarsely a computer heart model can sample cell-to-cell differences in electrical properties while still accurately simulating an irregular rhythm called discordant alternans. The finding that roughly 1.6 centimeter spacing is enough could make personalized heart simulations far less expensive to run.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 1.0–1.6 cm spacing result is only demonstrated for interpolation of exactly known parameter values at coarse grid points; the leap to experimental matching skips the inverse problem of estimating those values from noisy optical-mapping data.","rationale":"The reader's weakest_assumption is the load-bearing gap: the benchmark validates interpolation of parameters that are exact at coarse grid points, while the paper's practical claim requires estimating those parameters from data. The computational result itself is internally consistent and does not involve fitting parameters to the predicted output, so I see no flaw in the interpolation benchmark. However, the abstract overreaches by suggesting that efficient matching to experimental data follows directly. The paper explicitly discloses the exact-value assumption, which is good, but the assumption is not tested and is known to be hard: optical mapping does not directly provide ionic-model parameter values, and parameter-estimation errors can induce alternans phase flips that produce APD errors above the 5% threshold even at very fine grids. The proposed noise-sensitivity study would settle whether the 1.6 cm spacing survives realistic parameter uncertainty. Because the reader already issued a CONDITIONAL verdict and identified the same central assumption, my stress-test does not change the verdict; it sharpens the condition that must be met before the practical conclusion can be accepted.","tokens_in":4455,"tokens_out":5755,"duration_ms":59439,"concrete_test":"Add controlled noise to the coarse-grid parameters and repeat the Tables 1-2 analysis. For each parameter, gradient function, cable length, and grid spacing, perturb the 'known' value at each coarse grid point by zero-mean Gaussian noise with sigma = 1%, 5%, and 10% of the parameter interval (or, more realistically, by errors from an independent parameter-estimation pipeline applied to synthetic APD maps). Then recompute the maximum spacing that keeps the mean relative APD error at or below 5%. If the 1.6 cm spacing no longer meets the threshold under modest noise, the abstract's experimental-data conclusion requires re-qualification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central computational benchmark supports the interpolation claim, but the abstract's practical conclusion ('matching the output of models ... to heterogeneous experimental data can be done efficiently') depends on the assumption, stated in Section 2, that exact parameter values are known at the coarse grid points. Optical mapping measures voltage and APD, not ionic parameters such as tau_v^+, tau_w^-, tau_d, or k. To use a 1.6 cm grid in practice, one must solve a parameter-estimation inverse problem at each coarse node, with noise, non-identifiability, and model mismatch. The paper provides no test of that step. The problem is not only missing precision: under discordant alternans, small parameter differences can shift nodal lines or flip alternans phase. The paper itself reports a case (tau_w^+ reflected sinusoid, long cable) where even a 0.05 cm grid fails the 5% threshold because an alternans flipped. Such sensitivity means the 5% mean-relative-error criterion may not be robust to the small parameter perturbations introduced by any realistic estimation procedure. Thus the strongest defensible claim is conditional: 1.6 cm spacing is sufficient when exact parameter values are available, but there is no evidence yet that it is sufficient for experimentally derived parameter values.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":4644,"tokens_out":4869,"duration_ms":49474,"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":[{"comment":"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.","section":"Section 2 and Abstract"},{"comment":"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.","section":"Results and Tables 1-2"},{"comment":"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.","section":"Methods and Figure 2.3"},{"comment":"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.","section":"Section 2 (Heterogeneity model)"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"The text contains typos: 'al functions' should be 'all functions', and 'we sticked to' should be 'we stuck to' or 'we continued to use'.","section":"Results"},{"comment":"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.","section":"Figure 2.2 caption"},{"comment":"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.","section":"Tables 1-2"},{"comment":"Reference [2] has garbled page formatting: \"20 '¨A ` ı47\" should be cleaned up to the correct page range.","section":"References"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper earns its headline for the narrow question it actually answers. For Fenton-Karma cables with smooth, single-parameter heterogeneity, a parameter grid of roughly 1.6 cm (often coarser) reproduces APD profiles within 5% during discordant alternans, provided the parameter values at grid points are exact. That conditional is doing more work than the abstract acknowledges.\n\nWhat's genuinely new is the systematic spacing threshold: eight parameters, four gradient shapes, two cable lengths, two interpolation schemes, with tables showing the largest spacing that meets the 5% criterion. The interpolation methods are standard, but the quantitative guidance is not in the prior literature. The authors also report the ugly parts: error is not always monotonic in spacing, and the tau_w+ case on the long cable fails even at 0.05 cm because an alternans flips. Reporting that is a point in their favor.\n\nThe soft spots are the usual ones for a benchmark. The assumption that exact parameter values are known at the coarse points skips the inverse problem; optical mapping gives voltage and APD, not tau_v^+, tau_w^-, or k. The stress-test note is on target: small parameter errors from any realistic estimation procedure could shift nodal lines or flip alternans phase, and the tau_w+ case shows the 5% criterion is not robust to that kind of sensitivity. The scope is also narrow—one ionic model, 1D cables, smooth gradients, a hand-picked 5% threshold—and no code or data are released. These are limitations, not fatal flaws; the computational finding stands for the cases tested.\n\nWho should read this: anyone doing efficient parameterization of cardiac tissue models in simulation, where you control the parameters. It's less useful for matching experimental data until the estimation step is addressed. I'd accept it for peer review with a request to temper the experimental conclusion and add a sensitivity test with noisy or perturbed parameter values at the coarse nodes. Sharing code and data would also help.","headline":"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.","tokens_in":5189,"tokens_out":2262,"would_cite":true,"duration_ms":21366,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["92C30","65M06"],"pacs":[],"model":"deepseek-v4-flash","headline":"A parameter grid spacing of about 1.6 cm reproduces cardiac action-potential-duration profiles within 5 percent error during discordant alternans.","keywords":["cardiac electrophysiology","spatial heterogeneity","action potential duration","discordant alternans","coarse-grid parameterization","piecewise linear interpolation","piecewise constant interpolation","Fenton-Karma model"],"falsifier":"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.","tokens_in":1321,"feed_emoji":"💓","tokens_out":3617,"duration_ms":72354,"temperature":0.7,"pith_summary":"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.","feed_headline":"Coarse 1.6-cm grids capture cardiac alternation patterns","feed_subtitle":"Sparse parameter points reproduce action potential profiles within 5 percent error, easing data fitting.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the parameterization approach for cardiac action potential models that motivates representing heterogeneous tissue with sparse parameter grids.","marker":"[1]"},{"why":"Supplies the Fenton-Karma model equations used for all simulations, including the eight parameters varied in space.","marker":"[2]"},{"why":"Supplies the mechanisms for discordant alternans, the dynamical state used as the test bed for the coarse-grid representations.","marker":"[3]"}],"fun_headline_variants":["1.6-cm grid captures cardiac alternans patterns","Sparse 1.6-cm spacing reproduces heart dynamics","Coarse cardiac grid matches complex alternans","Efficient representation of heart tissue heterogeneity","1.6-cm steps suffice for cardiac model fitting"],"cache_read_input_tokens":7296,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["1.6-cm grid captures cardiac alternans patterns","Sparse 1.6-cm spacing reproduces heart dynamics","Coarse cardiac grid matches complex alternans","Efficient representation of heart tissue heterogeneity","1.6-cm steps suffice for cardiac model fitting"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000235,"raw_usage":{"total_tokens":1514,"prompt_tokens":976,"completion_tokens":538,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":592,"completion_tokens_details":{"reasoning_tokens":462}},"tokens_in":592,"tokens_out":538,"duration_ms":5138,"temperature":1.0,"reasoning_tokens":462,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:00:38.886557+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Fenton, and E.M","cited_arxiv_id":null,"evidence_quote":"Supplies the parameterization approach for cardiac action potential models that motivates representing heterogeneous tissue with sparse parameter grids."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Fenton-Karma model equations used for all simulations, including the eight parameters varied in space."},{"cited_title":"Fenton, S.J","cited_arxiv_id":null,"evidence_quote":"Supplies the mechanisms for discordant alternans, the dynamical state used as the test bed for the coarse-grid representations."}],"review_version":1}