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arxiv: 2604.13391 · v3 · pith:3BZLIVTYnew · submitted 2026-04-15 · ❄️ cond-mat.soft · nlin.AO

Dynamical Theory of Elastic Synchronization of Cardiomyocytes

Akinari Tomiie , Nariya Uchida This is my paper

Pith reviewed 2026-05-15 06:57 UTC · model grok-4.3

classification ❄️ cond-mat.soft nlin.AO
keywords cardiomyocyteselastic synchronizationphase reductionforce dipolesubstrate elasticityin-phase anti-phasegeometry dependencemechanical coupling
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The pith

Elastic interactions through the substrate make two cardiomyocytes synchronize to in-phase or anti-phase beating depending on their mutual orientation.

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

The paper models each cardiomyocyte as an oscillating force dipole whose motion follows a Rayleigh-type equation. From the linear elastic response of the surrounding medium it derives an effective coupling between the two cells. Phase reduction theory then produces a dynamical equation for the phase difference whose stable fixed points are either in-phase or anti-phase synchronization, with the choice determined by the angle between the cells. Direct simulations confirm that the cells converge robustly to the predicted state and that the time to synchronize varies strongly with both distance and orientation. This supplies a dynamical bridge between earlier static energy calculations for two-body systems and single-cell dynamical models.

Core claim

By treating each cardiomyocyte as a force dipole oscillating according to a Rayleigh equation and computing the elastic interaction from the substrate's linear response, phase reduction yields a dynamical equation whose fixed points select in-phase synchronization for some orientations and anti-phase for others, with a nontrivial boundary in the orientation map.

What carries the argument

The phase-reduced dynamical description of two interacting force dipoles, where the coupling function is obtained from the elastic Green's function of the medium and selects the stable synchronized states.

If this is right

  • The time required to reach synchronization depends strongly on both separation and mutual angle.
  • An orientation-dependent state map exists with a nontrivial boundary separating regions of in-phase and anti-phase locking.
  • The cells converge robustly to the selected state from a wide range of initial phase differences.
  • The framework connects static energetic two-body theories to dynamical single-cell descriptions.

Where Pith is reading between the lines

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

  • In extended tissues, local cell orientations could create spatial domains of differing synchronization type that influence contraction wave speed.
  • The predicted state boundary can be tested by fabricating micropatterned substrates that fix cell angles while varying substrate stiffness.
  • The same phase-reduction approach may apply to other mechanically coupled biological oscillators such as synthetic cell sheets or engineered muscle tissues.

Load-bearing premise

Each cardiomyocyte can be modeled as an independent oscillating force dipole governed by a Rayleigh-type equation whose only interaction is the linear elastic response of the substrate.

What would settle it

Place two cardiomyocytes on a soft elastic gel at a series of controlled angles and distances, measure their beating phases over time, and check whether the observed transition between in-phase and anti-phase states matches the predicted orientation boundary.

Figures

Figures reproduced from arXiv: 2604.13391 by Akinari Tomiie, Nariya Uchida.

Figure 1
Figure 1. Figure 1: FIG. 1. Geometry of a pair of cardiomyocytes modeled as force [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Synchronization patterns for the representative configurations [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Orientation-dependent dynamical state maps for (a) [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

We study synchronization of two cardiomyocytes mediated by elastic interactions through the substrate. Modeling each cell as an oscillating force dipole governed by a Rayleigh-type equation, we derive an effective mechanical coupling from the elastic response of the surrounding medium. Using phase reduction theory, supported by direct numerical simulations, we obtain a dynamical phase description for two cardiomyocytes that predicts geometry-dependent selection of synchronized states. Depending on the mutual orientation, the cells robustly converge to either in-phase or anti-phase beating, yielding an orientation-dependent state map with a nontrivial state boundary. The synchronization time also depends strongly on the distance and mutual orientation of the cells. These results bridge earlier energetic two-body theory and dynamical single-cell theory, and provide a dynamical framework for elastic synchronization of cardiomyocytes.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a dynamical theory for elastic synchronization of two cardiomyocytes modeled as oscillating force dipoles governed by Rayleigh-type equations. Effective mechanical coupling is derived from the substrate's linear elastic Green's function. Phase reduction yields a phase equation predicting geometry-dependent convergence to in-phase or anti-phase states, producing an orientation-dependent state map with a nontrivial boundary. Synchronization times depend strongly on distance and orientation. The analytic results are supported by direct numerical simulations, bridging energetic two-body and dynamical single-cell approaches.

Significance. If the central claims hold, the work supplies a useful dynamical framework for substrate-mediated cardiomyocyte synchronization with clear, testable predictions for orientation-dependent states and distance-dependent timescales. The combination of phase reduction and direct simulations is a strength, offering a bridge between prior energetic and single-cell models that could inform mechanobiology experiments on cardiac cell pairs.

major comments (2)
  1. [Phase reduction and effective coupling derivation] The phase reduction step assumes weak coupling so that trajectories remain close to the unperturbed limit cycle, yet the effective coupling is obtained from the elastic Green's function and therefore increases as inter-cell distance decreases. The reported strong distance dependence of synchronization time implies that the smallest separations may lie outside the weak-coupling regime, where amplitude fluctuations or higher-order terms could shift the predicted state boundary. Direct simulations are invoked for support, but no explicit check (e.g., comparison of reduced-model attractor versus full-model trajectories at small separations) is described.
  2. [Model setup and phase equation] The Rayleigh-type equation for each force dipole contains free parameters whose values are not constrained by the elastic coupling derivation. Because the phase equation inherits these parameters, the location of the nontrivial state boundary in the orientation map is not parameter-free; a brief sensitivity analysis or explicit statement of the parameter regime used for the state map would be required to substantiate the geometry-dependent selection claim.
minor comments (2)
  1. [Abstract] The abstract states that the state boundary is 'nontrivial' without indicating what feature (e.g., curvature, dependence on dipole orientation angles) makes it so; a single clarifying clause would improve readability.
  2. [Notation and equations] Notation for the dipole orientation angles and the Green's function components should be introduced once and used consistently; occasional redefinition of symbols across sections reduces clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment point by point below. Where the comments identify areas needing clarification or additional verification, we have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: The phase reduction step assumes weak coupling so that trajectories remain close to the unperturbed limit cycle, yet the effective coupling is obtained from the elastic Green's function and therefore increases as inter-cell distance decreases. The reported strong distance dependence of synchronization time implies that the smallest separations may lie outside the weak-coupling regime, where amplitude fluctuations or higher-order terms could shift the predicted state boundary. Direct simulations are invoked for support, but no explicit check (e.g., comparison of reduced-model attractor versus full-model trajectories at small separations) is described.

    Authors: We agree that an explicit verification of the phase reduction at small separations is warranted to confirm the robustness of the predicted state boundary. In the revised manuscript we have added a direct comparison between the phase-reduced model and full numerical integration of the two-dipole system at the smallest distances considered. The comparison shows that the attractors (in-phase versus anti-phase) remain identical and that quantitative differences in synchronization time are modest, indicating that higher-order effects do not alter the geometry-dependent selection within the parameter range explored. We have also added a brief discussion of the distance range over which the weak-coupling assumption remains quantitatively accurate. revision: yes

  2. Referee: The Rayleigh-type equation for each force dipole contains free parameters whose values are not constrained by the elastic coupling derivation. Because the phase equation inherits these parameters, the location of the nontrivial state boundary in the orientation map is not parameter-free; a brief sensitivity analysis or explicit statement of the parameter regime used for the state map would be required to substantiate the geometry-dependent selection claim.

    Authors: The parameters of the Rayleigh equation are indeed phenomenological and chosen to match typical single-cell beating amplitudes and frequencies reported in the literature. In the revised manuscript we now state the specific parameter values used for the orientation map and include a short sensitivity analysis. This analysis demonstrates that the location of the nontrivial boundary in the orientation map remains qualitatively unchanged when the nonlinear damping coefficients are varied over a range consistent with experimental observations. The geometry-dependent selection of in-phase versus anti-phase states is therefore robust within the biologically relevant regime. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses external elastic Green's function and phase reduction without self-referential reduction

full rationale

The paper models each cardiomyocyte as a Rayleigh oscillator and derives the effective coupling directly from the linear elastic Green's function of the substrate (an independent physical input). This coupling is then inserted into the standard phase-reduction formula to obtain the phase equation. No equation reduces by construction to a fitted parameter, self-citation, or ansatz that was defined using the target result. The orientation-dependent state map and synchronization times are obtained from this reduced dynamics and cross-checked against direct numerical simulations of the original coupled system, confirming the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the force-dipole representation of a cardiomyocyte and the applicability of phase reduction to the resulting weakly coupled system; both are standard modeling choices rather than new postulates.

free parameters (1)
  • Rayleigh-equation parameters
    Amplitude and frequency parameters in the single-cell Rayleigh oscillator are required to close the model but are not derived from first principles.
axioms (1)
  • domain assumption Phase reduction theory applies to the weakly coupled cardiomyocyte system
    Invoked to reduce the full dynamical equations to a phase-difference equation.

pith-pipeline@v0.9.0 · 5418 in / 1281 out tokens · 42033 ms · 2026-05-15T06:57:12.353312+00:00 · methodology

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