In cardiac excitable media, spiral defect chaos (SDC) — the continual creation and annihilation of spiral waves — can be captured at a fraction of the cost of full reaction-diffusion simulations by representing spiral-wave tips as diffusing particles subject to a short-range attraction and, after pair creation, a short-lived repulsion. This particle description has so far been validated only under idealized conditions: periodic domains, angle-averaged annihilation rules, and fixed ionic parameters. The open problem is to extend it to the regimes required for physiological and pharmacological use: (i) domains with non-conducting boundaries, where single spirals can annihilate at a wall and the pairwise-annihilation constraint no longer holds; (ii) annihilation conditioned on the relative activation-front angles of the colliding tips; and (iii) a quantitative characterization of how the effective tip-attraction coefficient depends on transmembrane (ionic) currents, so that termination statistics — and possible drug targets — can be predicted from ionic parameters.

Background

Cardiac tissue is an excitable medium: a local depolarization triggers a propagating electrical activation wave. In spatially extended models of such tissue, a wave front that curls around its own refractory tail forms a spiral wave whose core tip is a phase singularity. For certain parameters a single spiral is unstable and breaks up into many spirals, a regime called spiral defect chaos (SDC), in which spiral waves are continuously created and destroyed; SDC is the standard theoretical picture of the disordered activation underlying atrial and ventricular fibrillation. Because the number of coexisting spirals fluctuates and eventually reaches zero, SDC is a transient whose lifetime — the termination time — is a key statistical observable, known to be exponentially distributed in both detailed simulations and clinical recordings.

Direct simulation of SDC requires solving reaction-diffusion equations with ionic-current terms over large domains, which is computationally expensive. Tyree, Murphy, and Rappel (2024) therefore introduced a reduced, particle-level description: each spiral-wave tip is a particle obeying a stochastic Langevin equation with diffusion (coefficient DD) and an apparent attraction toward the tip of opposite chirality that it will annihilate with, of strength aa (in cm2^2/s). In this 'linear particle model' (LPM) the inter-tip distance obeys dR=−2a/R dt+8D dWdR = -2a/R\,dt + \sqrt{8D}\,dW, and the pair annihilates when RR reaches zero. The model parameters are fitted by comparing the mean squared inter-tip separation (MSR) before annihilation events in the full cardiac models — the Fenton-Karma (FK) model, a three-current simplification of ventricular electrophysiology, and the Luo-Rudy (LR) model, a more detailed ionic model — against the corresponding particle statistics. The LPM reproduces the power-law dependence of the annihilation rate on tip density, w−=M−nν−w_{-}=M_{-}n^{\nu_{-}}, and, combined with measured creation rates w+=M+nν+w_{+}=M_{+}n^{\nu_{+}} and birth-death process theory, reproduces the exponential termination-time distributions and their means. The authors also formulated an 'oscillatory particle model' (OPM) in which the attraction depends on the oscillatory activation-front angle (AAF) — the orientation of the activation front at each tip, which oscillates as the spiral rotates — but they linearized it to the LPM so that the AAFs never had to be computed. A central finding is that increasing aa sharply decreases the mean termination time, suggesting aa as a target for pharmacological intervention.

The 2024 paper explicitly listed what its framework had not yet achieved. The first item — particle-level creation events, possibly requiring the short-range tip repulsion reported in some earlier cardiac models — was subsequently resolved by Tyree, Reiss, and Rappel (2025), who added a short-lived repulsive interaction between newly created tip pairs and showed the extended model reproduces the termination statistics, including mean termination time, of the FK and LR models. Three limitations remain open and together define the surviving problem: all validations use periodic boundary conditions, which enforce pair-wise annihilation by conservation of topological charge (vorticity) and exclude the single-spiral annihilation that occurs when a spiral collides with a non-conducting tissue boundary; the annihilation rule is angle-averaged, so the role of the relative AAF at encounters is unknown; and the dependence of the attraction coefficient aa on specific transmembrane currents and ionic parameters was demonstrated only for two isolated parameter changes (an excitability time constant in FK and extracellular potassium in LR), not characterized as a mapping.

Problem Statement

Extend the particle-level (tip-based) description of spiral defect chaos in cardiac excitable media beyond its currently validated regime, so that its birth-death statistics faithfully reproduce those of full cardiac reaction-diffusion simulations under physiologically relevant conditions. The common objective uniting the requested results is a computationally cheap particle model whose predicted creation/annihilation rates and termination-time statistics remain valid where the existing framework is not. Concretely, the source literature leaves open:

  1. Boundary annihilation. Formulate and validate a particle-level description of the annihilation of a single spiral wave at a non-conducting (no-flux) boundary, in domains without periodic boundary conditions, where conservation of topological charge no longer restricts annihilation to oppositely rotating pairs. The extended model should be shown to reproduce the annihilation rates and the termination-time statistics of full cardiac-model simulations in such bounded domains.

  2. Angle-resolved annihilation. Incorporate the dependence of annihilation on the relative activation-front angles (AAFs) of the two tips at an encounter — currently averaged over in the linear particle model — and quantify how this changes encounter-to-annihilation probabilities and the predicted termination statistics relative to the angle-averaged model.

  3. Ionic-to-attraction mapping. Characterize how the effective attraction coefficient aa depends on the transmembrane (ionic) currents of the underlying cardiac model — e.g., on individual ionic conductances and ionic concentrations — providing a quantitative mapping that predicts aa, and hence termination statistics, from ionic parameters, as needed to identify pharmacological interventions that modify tip attraction.

References

  1. Timothy J. Tyree, Patrick Murphy, Wouter-Jan Rappel, 'Annihilation dynamics during spiral defect chaos revealed by particle models', Chaos 34, 053131 (2024), DOI: 10.1063/5.0203319, https://doi.org/10.1063/5.0203319 (preprint arXiv:2402.10308, https://arxiv.org/abs/2402.10308)
  2. Timothy J. Tyree, Michael Reiss, Wouter-Jan Rappel, 'Particle model of creation and annihilation captures termination dynamics of spiral defect chaos', Chaos 35, 093139 (2025), DOI: 10.1063/5.0277113, https://doi.org/10.1063/5.0277113
  3. Mahesh Kumar Mulimani, Wouter-Jan Rappel, 'Spiral defect chaos with intermittency increases mean termination time', Physical Review E 112, 034203 (2025), DOI: 10.1103/bwpl-qqyl, https://doi.org/10.1103/bwpl-qqyl (preprint arXiv:2505.06427, https://arxiv.org/abs/2505.06427)
  4. Sebastián Echeverría-Alar, Wouter-Jan Rappel, 'Boundary-Driven Delayed-Feedback Control of Spatiotemporal Dynamics in Excitable Media', Physical Review Letters 135, 267201 (2025), DOI: 10.1103/vlnh-3xzf, https://doi.org/10.1103/vlnh-3xzf
  5. Navneet Roshan, Rupamanjari Majumder, 'Termination of figure-of-eight reentry via resonant feedback pacing', Frontiers in Network Physiology 5, 1692372 (2025), DOI: 10.3389/fnetp.2025.1692372, https://doi.org/10.3389/fnetp.2025.1692372
  6. Ying-Qi Liu, Yi-Peng Hu, Qian-Ming Ding, Ying Xie, Ya Jia, 'Mutual annihilation of counter-rotating spiral waves induced by electric fields', Chinese Physics B 34, 120505 (2025), DOI: 10.1088/1674-1056/ae0561, https://doi.org/10.1088/1674-1056/ae0561