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Spiral defect chaos with intermittency increases mean termination time

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Lengthening one recovery-time parameter converts cardiac spiral chaos into long-lived intermittent activity, raising mean termination time by nearly three orders of magnitude.

desk verdict Genuine new observation of intermittent non-SDC states in the FK model with a sharp Tterm increase, but one missing domain-size control keeps the central claim from being fully settled. read the letter →

arxiv 2505.06427 v1 pith:3EZGRJMN submitted 2025-05-09 nlin.CD physics.bio-phphysics.comp-ph

Mahesh Kumar Mulimani , Wouter-Jan Rappel This is my paper · ORCID
classification nlin.CDphysics.bio-phphysics.comp-ph
keywords cardiacarrhythmiasspiraldefectchaosintermittencyterminationtimeFenton-Karmamodelexcitablemediaphasesingularityatrialfibrillation
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

This paper argues that spiral defect chaos, the disordered self-renewing wave state thought to underlie atrial fibrillation, is not inevitably a short-lived state in cardiac tissue models. Increasing the recovery time constant $\tau_w^-$ in the Fenton-Karma model carries the system from continuous chaos through an intermittent regime in which chaotic intervals alternate with intervals holding a small, stable number of spiral waves. The authors quantify this intermittency and show that the mean time until all electrical activity stops rises sharply, by almost three orders of magnitude between $\tau_w^-=150$ ms and $200$ ms. The result matters because it identifies a route, other than tissue size, by which arrhythmic activity can become long-lived.

What carries the argument

The paper's central quantitative tool is the intermittency parameter $f$, defined by averaging, over successive non-SDC/SDC interval pairs, the fraction $f_i=t_{\text{non-SDC},i}/(t_{\text{non-SDC},i}+t_{\text{SDC},i})$ of each pair spent in the non-SDC state; $f=0$ is continuous SDC and $f=1$ is a single stable spiral. A non-SDC interval is operationally defined as an interval longer than 1.5 s (at least 19 rotations) in which the number of spiral wave tips stays constant. The companion quantity is the probability distribution $p(N_{\rm PS})$ of the number of phase singularities: in continuous SDC the mean termination time is known to scale as $1/p(N_{\rm PS}=1)$, and the intermittent regime introduces a bump at small $N_{\rm PS}$ that eventually dominates the distribution. These measures are what link the visually distinct dynamical regimes to the reported thousand-fold increase in survival time.

What would settle it

Repeat the same protocol at $\tau_w^-=200$ ms in larger domains, for example $N=360$ and $N=720$, keeping all other parameters fixed. If the intermittency parameter $f$ drops toward zero or the sharp increase in $T_{\rm term}$ moves to different parameter values, the reported effect is a finite-size artifact; if the non-SDC intervals and the increase persist, the claim stands.

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Extended reading notes

Core claim

Varying the single recovery-time parameter $\tau_w^-$ moves the Fenton-Karma model from continuous spiral defect chaos (SDC) to an intermittent regime and finally, for $\tau_w^-\ge 330$ ms, to a single stable spiral wave. In the intermittent regime, SDC intervals are separated by non-SDC intervals during which the number of phase singularities remains small and constant for many rotations; the intermittency parameter $f$, the average fraction of time spent in the non-SDC state, grows from 0 toward 1 as $\tau_w^-$ increases. The mean termination time $T_{\rm term}$ grows linearly with $\tau_w^-$ in the continuous SDC regime and then sharply, by nearly three orders of magnitude, between $\tau_w^-=150$ ms and $200$ ms. Simulations with more non-SDC intervals terminate later, and individual spiral wave lifetimes grow to more than $10^3$ s. The authors also report quasi-stable spiral waves coexisting with SDC in different parts of the same domain, increasingly for $\tau_w^-\ge 200$ ms, and state that similar intermittency occurs for other parameter choices and in a generic two-variable model.

Load-bearing premise

The long-lived orderly intervals are treated as a genuine dynamical state, but every simulation uses the same $N=180$ tissue size, and survival times in these models are known to grow with domain size, so the intervals and the thousand-fold increase could be artifacts of the fixed box rather than a property of the model.

Editorial extensions

If this is right

  • In the continuous SDC regime the mean termination time increases linearly with $\tau_w^-$, but entering the intermittent regime changes the growth to a sharp, nearly exponential rise.
  • Termination-time probability distributions switch from exponential in continuous SDC to long-tailed in the intermittent regime, reflecting runs with one or several long non-SDC intervals.
  • Recovery time acts as a control axis alongside domain size: shorter recovery gives short-lived chaos, longer recovery near the single-spiral boundary gives activity that can outlive the simulation budget.
  • The same intermittency appearing under other parameter choices and in a generic two-variable model indicates that the mechanism is not locked to one parameter set of the Fenton-Karma model.
  • If the mapping to atrial fibrillation holds, the results imply that interventions altering recovery dynamics could change whether fibrillation self-terminates or persists.

Reading between the lines

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

  • Editorial inference: the sharp rise in $T_{\rm term}$ resembles a critical slowing down at the SDC-to-spiral transition; one testable consequence is that non-SDC interval durations near the transition should obey a power-law distribution, a quantity the paper does not report.
  • Editorial inference: the coexistence of quasi-stable spirals with SDC in separate subregions suggests that ordered regions act as local stabilizers; perturbing recovery parameters only in a subdomain could create or destroy such coexistence in a controlled way.
  • Editorial inference: because survival times are known to depend exponentially on domain size, the reported thousand-fold increase might be amplified or suppressed by system size; comparing $f$ across domain sizes would separate the intrinsic intermittent regime from a finite-size effect.
  • Editorial inference: clinically, if non-SDC intervals correspond to organized activity, the model predicts a counterintuitive hazard: drugs that flatten the action-potential-duration restitution curve (effectively raising $\tau_w^-$) could lengthen fibrillatory episodes even as they make the waves more orderly.
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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. The paper studies the Fenton-Karma (FK) cardiac model in a 2D domain and reports that varying a single time constant, τ−w, takes the system from continuous spiral defect chaos (SDC) through a regime of intermittent SDC, in which intervals of chaos are punctuated by quasi-stable intervals with a small, constant number of spiral waves, to a final state with a single stable spiral wave. The authors define an intermittency parameter f, measure the probability distribution of spiral tip numbers, and compute the mean termination time Tterm. They find that as τ−w approaches the stable-spiral regime, Tterm increases sharply, by almost three orders of magnitude between τ−w = 150 ms and 200 ms, and that the number of intermittent non-SDC intervals per simulation correlates positively with termination time. They also report local coexistence of quasi-stable spiral waves with SDC and suggest clinical relevance for paroxysmal vs. persistent atrial fibrillation.

Significance. If the central claim survives scrutiny, this is a valuable contribution to the dynamics of excitable media and cardiac arrhythmia modeling: it identifies a single-parameter route from short-lived SDC to much longer-lived intermittent activity, and it provides quantitative observables (the fraction f of time in the non-SDC state, tip-number distributions, and lifetime distributions) that could be compared with clinical data. The paper also reproduces and extends earlier master-equation results by relating termination-time distributions to the probability of small tip numbers. The authors are explicit about the computational limitations, and the qualitative phenomenology is well illustrated by the time series and snapshots. However, the quantitative claim of a three-orders-of-magnitude increase in Tterm currently rests on a single system size and on a small number of simulations for the most interesting parameter values, which limits the strength of the conclusions.

major comments (4)
  1. [II.B, Fig. 5e] The central quantitative claim is not yet supported against a finite-size artifact. All simulations use a single N=180 domain with no-flux boundaries. The non-SDC intervals consist of a small, fixed number of spiral waves in the entire box. Since previous work, cited in the Introduction (Refs. 28-30), shows that Tterm grows exponentially with domain size, the observed collapse from many-tip SDC to a few quasi-stable tips could be a metastable, finite-size effect that would disappear or become negligible in larger domains. The authors should vary N (e.g., 120, 180, 240, 360) and show how f, the tip-number distribution, and Tterm scale with system size, or provide a theoretical argument that the fluctuation to small tip numbers is not controlled by area. Without such a test, the extrapolation to clinical atrial fibrillation in the Concluding Remarks is not justified.
  2. [Fig. 5e, Section III.C] The quantitative claim of a sharp, three-orders-of-magnitude increase in Tterm between τ−w = 150 ms and 200 ms is not accompanied by error bars for Tterm, and several of the data points are based on very few simulations: only 7 simulations for τ−w = 215 ms and 250 ms, and 8 for τ−w = 300 ms. Because the intermittent-regime termination-time distributions are heavy-tailed (Fig. 5b-d), the sample mean is highly sensitive to rare long-lived realizations and the uncertainty is large. The authors should provide bootstrap confidence intervals or standard errors for every point in Fig. 5e, report the number of simulations for each point, and discuss whether any simulations were terminated by a maximum-time cutoff (censoring), which would bias the mean. The qualitative trend may be robust, but the specific magnitude and the transition point need statistical support.
  3. [Section III.A, Eq. (8)] The definition of a non-SDC interval is arbitrary: any interval longer than 1.5 s (≥19 rotations) in which the number of spiral tips remains stable. The intermittency parameter f and, consequently, the location of regime boundaries in Fig. 3 depend on this threshold and on the precise meaning of "stable" (is zero change in tip number required, or is a small tolerance allowed?). The authors should test the sensitivity of f to the threshold (e.g., 1.0, 2.0, 3.0 s) and state the tip-counting tolerance. Without this, the quantitative transition curve f(τ−w) is not well defined.
  4. [Section III.C, Fig. 5e] The quantity TSDC, the average time spent in the SDC state before termination, is not precisely defined. It is unclear whether TSDC is the sum of all SDC intervals in a simulation, or the average length of individual SDC intervals, and how the coexistence case of Fig. 7, in which quasi-stable spirals appear locally while the rest of the domain is in SDC, is treated. The exponential fit to TSDC (blue line in Fig. 5e) is also not described: which points are included, what is the fitting form, and what is the goodness of fit? These details are needed to support the claim that the growth in Tterm is not simply due to longer non-SDC intervals.
minor comments (5)
  1. [Throughout] There are several typographical issues: "Fig. 1(a))" has an extra parenthesis; "liftetimes" should be "lifetimes"; the superscripted "a" in "For this, we compute... a As shown in Fig. 6a&b" appears to be a LaTeX artifact; and the notation τ−w would read more cleanly as τ_w^- .
  2. [Fig. 5e caption] The caption uses τSDC while the text uses TSDC for the same quantity; please unify the notation.
  3. [Section III.A, Fig. 2] The authors use the term "quasi-stable" for spiral waves in non-SDC intervals but do not define it precisely; a short definition (e.g., a tip that survives for at least some multiple of the dominant period and does not change its distance to the boundary) would improve clarity.
  4. [Concluding Remarks] The statement that the intermittent behavior is not restricted to the particular parameter, parameter set, or model is not supported by any displayed data, figure, or reference to a supplementary file. Either add a supporting figure or weaken the claim to a qualitative observation.
  5. [Table I] The range for τ−w is given as "50-300 ms" with a hyphen; using an en dash and explicitly stating the units for all parameters (e.g., usi_c is dimensionless) would be clearer.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central termination-time result is measured directly rather than derived from fitted inputs or self-referential definitions.

full rationale

The paper's central claim is that varying tau_w drives the FK model from spiral defect chaos through an intermittent SDC/non-SDC regime to a single spiral wave, and that the appearance of non-SDC intervals sharply increases the measured mean termination time. This claim is supported by direct numerical simulations: Tterm is computed from termination times of many independent runs, f is a separately defined descriptive measure of the fraction of time spent in the non-SDC state, and p(NPS) is a measured histogram. None of these quantities is fitted to a subset of data and then presented as a prediction; the exponential and linear fits in Fig. 5(e) are descriptive summaries of the measured curves, not inputs to the claim. The observed correlation between the number of non-SDC intervals and termination time is presented as a statistical relation between two independently measured observables, not as an identity or a derived consequence of a fitted parameter. The paper's use of prior work, including self-citations such as refs. [13], [27], and [34], is contextual: these supply background on restitution-mediated instability, the stochastic birth-death description of spiral tips, and the exponential termination-time distribution in continuous SDC. The new intermittent regime is not defined in terms of those results, and the main termination-time increase does not reduce to any of them. The absence of a domain-size study is a legitimate robustness concern about finite-size artifacts, but it is a correctness or generalization risk, not evidence that the derivation is circular. Overall, the derivation chain is self-contained: the phenomenon is observed and quantified directly rather than manufactured from its own assumptions.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new free model parameters or entities. It uses the standard FK model with a varied recovery time constant, a hand-chosen threshold for the intermittency definition, and implicit domain assumptions about the representativeness of the simulation domain and initial conditions.

free parameters (1)
  • non_SDC_interval_threshold = 1.5 s
    Chosen by hand to classify stable non-SDC intervals (>=19 rotations). Used to compute f, but the central claim about Tterm does not depend on this threshold.
assumptions (4)
  • domain assumption The Fenton-Karma model is a valid representation of the relevant cardiac tissue dynamics.
    All results are obtained in this model; clinical extrapolation relies on its validity. Invoked in Section II.A.
  • domain assumption Phase singularity tracking correctly identifies spiral wave tips.
    Used to count NPS and compute lifetimes. Invoked in Section II.B with the standard method from ref. 32.
  • domain assumption The N=180 domain is sufficiently large that the observed intermittency is not a finite-size artifact.
    No system-size variation is reported; the known exponential domain-size dependence of Tterm makes this an untested premise. Underlies the central claim and the clinical discussion.
  • domain assumption The independent noise-perturbed initial conditions sample the SDC attractor.
    Termination times are averaged over these samples; if the samples miss the relevant basin, the mean could be biased. Invoked in Section II.B.

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Pith. "Pith review of Spiral defect chaos with intermittency increases mean termination time." pith.science (2026). https://pith.science/paper/3EZGRJMN

@misc{pith2026250506427,
  author       = {Pith},
  title        = {Pith review of: Spiral defect chaos with intermittency increases mean termination time},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3EZGRJMN}},
  note         = {Machine review of arXiv:2505.06427}
}
read the original abstract

Cardiac models are examples of excitable systems and can support stable spiral waves. For certain parameter values, however, these spiral waves can become unstable, resulting in spiral defect chaos (SDC), characterized by the continuous creation and annihilation of spiral waves and thought to underlie atrial fibrillation. During SDC, the number of spiral waves fluctuates and eventually drops to zero, marking the termination of activity. In this work, we demonstrate that varying a single parameter allows the system to transition from SDC to a single spiral wave, passing through an intermediate regime of intermittency. In this intermittent dynamics, intervals of SDC are sandwiched between non-SDC intervals during which the number of spiral waves remains small and constant. We quantify this intermittency and show that the mean termination time increases significantly as the control parameter approaches values for which a single spiral wave is stable. In addition, we find that it is also possible to have intermittently present quasi-stable spiral waves in part of the computational domain while the remainder of the domain exhibits SDC. Our results may have implications for clinical atrial fibrillation, which often shows intermittency, switching back-and-forth between fibrillation and normal sinus rhythm.

Figures

Figures reproduced from arXiv: 2505.06427 by the authors.

Figure 1
Figure 1. FIG. 1. The number of phase singularities [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Number of phase singularities (upper panel) and dis [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Intermittency parameter [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a-d): Probability distribution of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a)-(d) Semi-logarithmic plot of the probability distribution of termination times for increasing values of [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (a-b) Mean termination time as a function of the number of non-SDC intervals for [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Number of spiral wave tips (upper panel) and distance [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

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