Understanding the transition from paroxysmal to persistent atrial fibrillation from micro-anatomical re-entry in a simple model

Alberto Ciacci; Kim Christensen; Kishan A. Manani; Max Falkenberg; Nicholas S. Peters; Tim S. Evans

REVIEW 4 major objections 6 minor 67 references

Understanding the transition from paroxysmal to persistent atrial fibrillation from micro-anatomical re-entry in a simple model

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that in a simple model of atrial fibrillation, persistent episodes arise from re-entrant circuits that are easier to activate than to deactivate, with no change in model parameters.

desk verdict A careful dissection of the CMP model that identifies asymmetric re-entrant circuits as the driver of persistent AF, with a real but fixable weakness in the cCMP control. read the letter →

arxiv 1908.01646 v2 pith:BFZAU5OC submitted 2019-08-05 q-bio.TO cond-mat.stat-mech

Alberto Ciacci , Max Falkenberg , Kishan A. Manani , Tim S. Evans , Nicholas S. Peters , Kim Christensen This is my paper · ORCID

classification q-bio.TOcond-mat.stat-mech

keywords atrialfibrillationparoxysmaltopersistenttransitionre-entrantcircuitsmicro-anatomicalre-entryconductionblockcellularautomatonmodelmean-fieldbirth-deathcriticalstructures

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 is a dissection of the Christensen-Manani-Peters (CMP) model of atrial fibrillation, a cellular automaton on a square lattice in which randomly placed conduction-blocking nodes occasionally fail to fire. The authors want to know why the model, at fixed parameter values, can show every clinical AF pattern: permanent sinus rhythm, short self-terminating paroxysmal episodes, occasional long episodes, and persistent AF lasting a simulated month. They establish that the difference is made by 'complex' re-entrant circuits whose activation probability exceeds their deactivation probability, because they require fewer consecutive conduction-block failures to start than to stop. A mean-field birth-death version and a controlled spatial version that contain only simple circuits both underestimate time in AF, and only the full model with complex circuits reproduces the observed spectrum. The clinical significance, if the claim transfers, is that a patient's paroxysmal versus persistent status could depend on the local arrangement of fibrosis-related conduction blocks rather than on total fibrosis burden or progressive remodelling.

What carries the argument

The load-bearing object is the CMP lattice model's 'critical structure', a local arrangement of a conduction-blocking node and a transverse-connection-free segment of length at least $\tau/2$ that can host a re-entrant circuit when the block fails. The paper classifies structures as simple or complex depending on how many successive conduction-block failures are needed to activate and to terminate them. Two auxiliary constructions carry the argument: a mean-field model in which each simple critical structure is a particle undergoing birth-death dynamics with rates set by $\epsilon/T$ and $\epsilon/\langle \ell \rangle$, and a controlled CMP model (cCMP) in which conduction-blocking nodes are placed only on isolated simple-critical segments so that complex circuits cannot form. The difference in AF probability and time in AF between the full model and these simplifications is what isolates the contribution of asymmetric complex circuits. The analytic risk formula $R=1-[1-(1-\nu_\perp)^\tau]^{\delta L^2}$ provides the baseline against which the model's excess AF is measured.

What would settle it

Run the cCMP construction but instead of deleting the 'extra' conduction-blocking nodes, keep them in place with failure probability set to zero (inert nodes), so the lattice geometry and connectivity are identical to the CMP model while no conduction block can occur outside simple segments. If time in AF in this inert-node control matches the cCMP value, the gap is due to the presence of blocking-prone nodes in complex configurations; if it rises toward the CMP value, the paper's attribution to complex circuits is not the whole story.

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Extended reading notes

Core claim

The paper's central claim is that persistent AF in the CMP model is caused by re-entrant circuits with an asymmetry in the probability of activation relative to deactivation. A simple circuit turns on when one susceptible node fails and turns off when one susceptible node fails, so its on and off rates are balanced, and a mean-field birth-death description captures its behaviour; the continuous mean-field solution shows that a finite set of such circuits cannot keep the system in AF almost all the time. Complex circuits break this balance: some need two successive cell failures to initiate but four to terminate, and others are coupled so that the termination of one circuit immediately re-ignites a neighbour. With these circuits present, the same lattice parameters can generate fibrillatory episodes from a few seconds to more than a month, and the gap between the full model and the controlled model collapses as the density of conduction-blocking nodes is reduced and complex circuits become rare.

Load-bearing premise

The argument that the cCMP-CMP gap is caused by complex asymmetric circuits assumes that removing all conduction-blocking nodes that are not part of an isolated simple critical segment changes nothing else about wavefront collisions, refractoriness, or suppression of neighbouring circuits; if deleting those nodes alters wave dynamics for reasons unrelated to circuit complexity, the measured gap could be produced without the proposed mechanism.

Editorial extensions

If this is right

If the claim is correct, the full spectrum of AF persistence in the model emerges without changing any parameter; a single fixed coupling value $\nu_\perp=0.11$ can yield sinus rhythm, paroxysmal AF, and persistent AF depending only on the local positions of vulnerable nodes.
Mean-field birth-death descriptions with a finite number of independent drivers cannot explain persistent AF, so any model that treats AF persistence as a simple on/off balance of fixed-rate drivers will miss the mechanism.
Reducing the fraction of conduction-blocking nodes $\delta$ suppresses complex circuits and makes the full model's behaviour coincide with the controlled model, identifying high local density of vulnerable nodes as proarrhythmic in a way that simple-circuit count alone does not capture.
Ablation strategies would preferentially need to target complex or coupled structures whose termination requires many simultaneous failures, rather than merely any region that can host a simple re-entrant circuit.

Reading between the lines

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

A testable extension is to build an 'inert-node control': in a cCMP lattice, keep the deleted conduction-blocking nodes in place but set their failure probability to zero; if the cCMP-CMP gap persists, connectivity changes, not circuit asymmetry, are responsible.
If the asymmetry mechanism generalizes, the distribution of AF episode durations in a fixed patient substrate should be heavy-tailed over many orders of magnitude, much wider than a single exponential; clinical event-duration histograms could be inspected for such spread before attributing persistent AF to remodelling.
The number of required simultaneous failures suggests a quantitative link between local fibrosis density and persistence: scanning a realistic fibre map for regions where multiple vulnerable nodes lie within one activation cycle of a re-entry path should predict where persistent drivers are anchored.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

What you should know: this is a genuinely useful paper. It takes the Christensen-Manani-Peters model and, instead of just running it, pulls it apart layer by layer. The mean-field model with analytic steady-state solution, the enhanced MF, and the controlled CMP are each introduced to isolate what makes persistent AF possible in the model. The central claim is that asymmetric critical structures - circuits that require fewer failures to activate than to terminate - explain why the same parameters can yield episodes from seconds to months. That claim is actually supported by the structure of the paper: the MF and cCMP models both get the phase boundary roughly right but miss the long tails, and only the full CMP has the long-lived asymmetric structures. The authors are honest that results are qualitative and that event durations need not map clinically. Good credit where due: the cMF formula in Eq. A7 is a real derivation, and the eMF model rules out the obvious alternative explanation that non-spatial details of simple circuits account for the gap.

The soft spots are real but not fatal. The cCMP control is the main one. The stress-test note is right: cCMP removes both the complex structures and most of the random block nodes, so the CMP-cCMP gap conflates total block density with structure complexity. The authors notice the density effect and show that at low delta the curves collapse, but that does not establish that the remaining gap at high delta is caused by the proposed asymmetric circuits rather than by generic density effects on wavefront collisions and refractoriness. They also identify the complex structures by inspection - hand-picked examples from figs 9 and 10 - and admit there is no systematic detection algorithm. The phase diagrams have no error bars or statistical tests, so the reader cannot judge whether the CMP-cCMP differences at intermediate delta are robust. None of this refutes the core idea, but it means the paper's central comparison is suggestive rather than conclusive.

Who this is for: anyone modeling AF with discrete cellular automata, or interested in how simple rules produce heterogeneous arrhythmia persistence. It deserves a serious referee and, with the control fixed (matching total block density, adding a detection algorithm, or at least variance estimates), it would be a solid contribution. I would cite it for the MF derivation and the asymmetric-structure hypothesis.

Recommendation: send it to review, with the clear expectation that the cCMP comparison needs a density-matched control before the mechanistic claim is accepted.

Referee Report

4 major / 6 minor

Summary. This paper analyzes the Christensen-Manani-Peters (CMP) cellular-automaton model of atrial fibrillation to explain why different simulations at identical parameters produce AF episodes ranging from seconds to months. The authors first derive a mean-field (MF) birth-death model in which the number of 'simple' re-entrant circuits, counted from the lattice, determines AF statistics, and they show analytically that this MF model underestimates time in AF and cannot produce persistent AF for realistic N. They then introduce a 'controlled' CMP (cCMP) model in which conduction-block-susceptible nodes are placed only on isolated segments that can support simple re-entrant circuits, and they compare this with a random-placement CMP copy. The gap in AF probability and time in AF between CMP and cCMP is attributed to complex critical structures whose activation requires fewer conduction-block failures than their deactivation, including self-contained asymmetric structures and coupled structures that re-initiate on termination. The paper concludes that persistent AF arises from these asymmetric re-entrant circuits, so that parameter variation or remodelling is not needed to explain diverse AF persistence.

Significance. If the central claim holds, the paper provides a mechanism for the paroxysmal-to-persistent transition in a minimal model, with potential implications for how local fibrosis architecture, rather than global burden, determines AF persistence. The work has genuine strengths: the continuous-time mean-field solution in Appendix A is an exact derivation; the enhanced MF model in Appendix B is a useful robustness check; the simulations extend to 10^9 timesteps, much longer than typical biophysical models; and the structural examples in Figs. 9 and 10 are clearly presented. The paper also explicitly acknowledges several limitations of the CMP model and the difficulty of verifying complete detection of critical structures. However, the central evidence for the mechanism rests on a controlled comparison that is confounded by changes in total block-node density, and the inference to higher-order structures is not quantitatively validated. These issues are load-bearing for the main claim, so the paper requires major revision rather than acceptance in its current form.

major comments (4)

[Section IV A] The cCMP control does not isolate complex critical structures because it changes the total density of conduction-blocking nodes. In cCMP, block-susceptible nodes are placed only on eligible nodes within isolated segments of length at least tau/2, at fraction delta of those eligible nodes, while the CMP comparison places block nodes at probability delta on all L^2 nodes. The two models therefore differ not only in the presence of multi-block complex structures but also in total block-node count, refractory landscape, and wavefront-collision effects. The conclusion in Section IV B that the CMP-cCMP gap is caused by complex asymmetric structures requires a control in which total block-node density and, ideally, the number of simple critical structures are matched while only the complex multi-block configurations are suppressed.
[Section IV B] The statement that 'by construction, the cCMP and CMP models contain the same number of simple critical structures' is not supported by the construction described in Section IV A. The cCMP model places block nodes only on selected segment nodes, whereas the CMP copy places them randomly across the whole lattice; these two procedures do not generally yield the same number of simple critical structures. If the counts differ, the subsequent inference that 'some higher order critical structures must exist' is a non sequitur, because the observed gap could be due to a different number of simple structures or to different total density rather than to complex structures.
[Section IV B and Fig. 12] The collapse of the CMP and cCMP curves at low delta in Fig. 12 is presented as evidence that complex structures dominate at high delta, but this collapse is also exactly what would be expected if the gap were driven by the total density of conduction-blocking nodes, since the absolute density difference between the two models shrinks as delta decreases. The figure therefore does not discriminate between the proposed asymmetric-structure mechanism and a generic density effect. A matched-density control, or a direct quantitative count of complex structures, is needed to make the mechanism claim.
[Section IV A and Section VI] The paper itself states in Section IV A that there is 'no easy method to verify that all circuits have been detected,' and the central mechanism is supported by hand-selected examples (Figs. 9-11) and by elimination through the cCMP comparison. Given the confound identified above, the existence and causal role of asymmetric complex structures is not quantitatively established. The authors should either provide a validated detection algorithm for complex structures or design a cleaner intervention that affects only multi-block complex configurations.

minor comments (6)

[Abstract] There is a typo in the first sentence: 'arrhytmia' should be 'arrhythmia', and later 'assymetry' should be 'asymmetry'.
[Figs. 7, 8, 12] The phase diagrams show averages over 200 simulations but no error bars or confidence intervals, making it difficult to assess whether the reported differences, especially near the sharp transitions, are statistically meaningful. Adding confidence bands would strengthen the quantitative claims.
[Section II C, Eq. (5)] The AF threshold is defined as 1.1 times L, which equals 220 for L=200, but the text states '1.1×L (220) nodes' without noting that this is specific to the chosen lattice size. Please clarify whether the threshold is 220 nodes or 1.1L generally.
[Section III] The statement that the MF model 'spends significantly less time in AF' than the CMP model is not accompanied by any statistical test or uncertainty estimate; a quantitative comparison of distributions or confidence intervals would be appropriate.
[Appendix B] The eMF model is described as 'perfectly compatible' with the MF model, but again no error bars or statistical measures are given; a quantitative statement of compatibility would be more informative.
[Appendix E, Fig. 19] The error bars in Fig. 19 are stated to be 95% confidence intervals over 50 simulations, but the main-text figures (e.g., Figs. 7 and 8) do not report the number of simulations beyond '200 simulations'; please make the simulation counts and uncertainty measures consistent throughout.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the MF/cMF calculations are genuine derivations from calibrated inputs, and the cCMP comparison is an external controlled simulation whose main weakness is a confound, not a definitional reduction.

full rationale

The paper's derivation chain is self-contained rather than circular. The mean-field model is calibrated from the CMP lattice (N counted from simple critical structures, p and q from epsilon, T, and <ell>), and the continuous-time formula Eq. (A7) follows from a genuine master-equation solution; the MF/cMF underprediction of CMP time-in-AF is an empirical comparison to simulation output, not a quantity defined to match it. The cCMP construction is a controlled intervention: block-susceptible nodes are restricted to isolated segments of length at least tau/2, and the CMP baseline is an independent copy with random block nodes at probability delta, so the CMP-cCMP gap is measured rather than postulated. The risk curve Eq. (4) and AF definition Eq. (5) are cited from prior work by the same group, but they are not load-bearing in a circular way: Eqs. (1)-(4) are re-derived geometrically in the paper and Eq. (5) is a threshold whose qualitative conclusions are robust and which is checked against simulated electrograms. The genuine weakness is a confound: CMP has approximately delta L^2 block nodes while cCMP has roughly delta M with M much smaller than L^2, so the gap conflates higher-order critical structures with total block-node density and wavefront-interaction effects; the 'therefore' in Section IV B overstates what the control establishes. That is a correctness risk, not a circular reduction of the conclusion to its inputs, so no circular step is recorded.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No new physical entities are introduced; the proposed structures are configurations of existing model elements.

free parameters (3)

AF threshold in active nodes = 220 active nodes (1.1L)
Eq. (5) defines AF as a(t) >= 220 for T consecutive steps; the threshold is set by inspection and is not derived. It affects the measured event durations and probabilities, though the authors argue it correlates with clinical AF.
Failure probability epsilon = 0.05
Set arbitrarily in Section II A; the paper says it mainly sets the time scale and has minimal effect on risk, but it directly sets MF activation/deactivation rates p and q.
Default fraction of conduction-block nodes delta = 0.01 (varied in Fig. 12)
Control parameter; the central claim about density dependence (Fig. 12) is tested by varying it, but the default is arbitrary.

assumptions (4)

ad hoc to paper Simple critical structures activate with rate epsilon/T in sinus rhythm and epsilon/<l> in AF, and deactivate with rate epsilon/<l> (Eq. 6).
This maps the CMP dynamics to a birth-death process; the rates are chosen by analogy, not derived from the CMP rules.
ad hoc to paper The number N of simple critical structures detected by inspecting the CMP lattice fully represents the system's potential re-entrant circuits.
Section III and IV; the paper states no easy method to verify that all circuits have been detected.
domain assumption The cCMP model differs from the CMP model only in the placement of conduction-blocking nodes.
Section IV A; used to attribute the CMP-cCMP difference to complex critical structures.
domain assumption One simulation time step equals approximately 3 ms of real time.
Section II A; converts event durations into seconds and months.

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Cite this review

Pith. "Pith review of Understanding the transition from paroxysmal to persistent atrial fibrillation from micro-anatomical re-entry in a simple model." pith.science (2026). https://pith.science/paper/BFZAU5OC

@misc{pith2026190801646,
  author       = {Pith},
  title        = {Pith review of: Understanding the transition from paroxysmal to persistent atrial fibrillation from micro-anatomical re-entry in a simple model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BFZAU5OC}},
  note         = {Machine review of arXiv:1908.01646}
}

read the original abstract

Atrial fibrillation (AF) is the most common cardiac arrhytmia, characterised by the chaotic motion of electrical wavefronts in the atria. In clinical practice, AF is classified under two primary categories: paroxysmal AF, short intermittent episodes separated by periods of normal electrical activity, and persistent AF, longer uninterrupted episodes of chaotic electrical activity. However, the precise reasons why AF in a given patient is paroxysmal or persistent is poorly understood. Recently, we have introduced the percolation based Christensen-Manani-Peters (CMP) model of AF which naturally exhibits both paroxysmal and persistent AF, but precisely how these differences emerge in the model is unclear. In this paper, we dissect the CMP model to identify the cause of these different AF classifications. Starting from a mean-field model where we describe AF as a simple birth-death process, we add layers of complexity to the model and show that persistent AF arises from re-entrant circuits which exhibit an asymmetry in their probability of activation relative to deactivation. As a result, different simulations generated at identical model parameters can exhibit fibrillatory episodes spanning several orders of magnitude from a few seconds to months. These findings demonstrate that diverse, complex fibrillatory dynamics can emerge from very simple dynamics in models of AF.

Figures

Figures reproduced from arXiv: 1908.01646 by the authors.

**Figure 2.** FIG. 2. The formation of a re-entrant circuit in the CMP [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Critical structures in the CMP model. The black [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (18 more)

**Figure 3.** Figure 3: To better understand the discrepancy between theory and experiment, we look at the trace of the number of active nodes in the model. AF is paroxysmal when this statistic exhibits large fluctuations which prevent it from stabilizing above the AF threshold, i.e., the num…

**Figure 4.** Figure 4: FIG. 4. Phase diagram of the probability of inducing AF as [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. The number of excited nodes per time step [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6. The CMP model is condensed into a mean-field (MF) [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: FIG. 7. Phase diagram of the time in AF as a function of the [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 8.** Figure 8: FIG. 8. Phase diagrams of (a) the probability of inducing [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

**Figure 3.** Figure 3: We now consider [PITH_FULL_IMAGE:figures/full_fig_p012_3.png]

**Figure 9.** Figure 9: FIG. 9. The formation of a complex re-entrant circuit with [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

**Figure 10.** Figure 10: FIG. 10. The coupling of multiple re-entrant circuits such that [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

**Figure 11.** Figure 11: FIG. 11. A simulation of the CMP model for 10 [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]

**Figure 13.** Figure 13: FIG. 13. Phase diagram of the time in AF as a function of the [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]

**Figure 14.** Figure 14: FIG. 14. Phase diagram of the time in AF as a function of the [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]

**Figure 7.** Figure 7: We find that the phase diagram of the time in [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]

**Figure 15.** Figure 15: FIG. 15. The classification of AF in the CMP model. (a) [PITH_FULL_IMAGE:figures/full_fig_p021_15.png]

**Figure 16.** Figure 16: FIG. 16. The number of active cells per time step, [PITH_FULL_IMAGE:figures/full_fig_p022_16.png]

**Figure 17.** Figure 17: FIG. 17. An example of persistent AF in the CMP model [PITH_FULL_IMAGE:figures/full_fig_p022_17.png]

**Figure 18.** Figure 18: FIG. 18. The average time in AF for the cCMP (red) and [PITH_FULL_IMAGE:figures/full_fig_p023_18.png]

**Figure 19.** Figure 19: FIG. 19. The average number of tracked simple critical struc [PITH_FULL_IMAGE:figures/full_fig_p023_19.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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