Cardiac arrhythmias are conventionally analyzed by locating phase singularities (PSs), the point vortices of the electrical activation phase. However, linear-core rotors (spiral-wave rotors whose tip traces a line-like rather than circular trajectory) and conduction block — the building blocks of fibrillation-like patterns — are organized by extended phase defect lines (PDLs, in 2D) and phase defect surfaces (PDSs, in 3D), across which the phase is nearly discontinuous. The phase-defect framework has classified these structures and their topology (conserved charges, quasiparticle endpoints, branes and strings — defect surfaces and their bounding curves), but no quantitative theory of their dynamics exists: there are no equations of motion, stability criteria, or interaction laws for PDLs/PDSs, and it is unknown whether the classical filament concepts of tension and rigidity possess surface analogues. The open problem, posed explicitly by Arno et al. (Frontiers in Physiology 12, 690453, 2021), is to derive such a theory — including defect response to external stimuli, boundaries, and genuinely 3D effects — which the source identifies as a key step toward predicting how conduction-block regions respond to defibrillation and pacing. Later literature (2022–2026) advances detection algorithms and topological classification but explicitly defers the dynamics.

Background

During a cardiac arrhythmia, the heart's electrical activation self-organizes into spatio-temporal patterns — rotating spiral waves (rotors) in two-dimensional views, scroll waves in the three-dimensional wall — whose organization underlies disorders such as atrial and ventricular fibrillation. A standard analysis tool is phase mapping: each tissue point is assigned a phase describing where it lies on the repeating excitation–recovery cycle, and one then locates phase singularities (PSs), the points around which every phase value occurs once. In three dimensions, PSs trace out curves called rotor filaments, and a highly successful classical theory exists for these filaments: filament velocity is driven by curvature (Keener, 1988), an effective 'filament tension' — the tendency of curved filaments to shorten or lengthen, whose sign depends on the medium's kinetics — governs the stability of scroll-wave turbulence (Biktashev, Holden and Zhang, 1994), and an effective bending rigidity explains the buckling of filaments carried by meandering scroll waves (Dierckx et al., 2012; Dierckx et al., 2017).

Two ubiquitous phenomena fall outside this PS/filament description. First, rotors with linear cores: for strongly meandering kinetics the rotor tip traces a line-like trajectory, and the center of the rotor is a line, not a point. Second, conduction block lines (CBLs): curves where a propagating wave front fails because the tissue ahead has not sufficiently recovered; such lines are routinely observed in experiments and clinical procedures. Arno et al. (2021) — with contemporaneous, independent numerical work by Tomii et al. (2021) — proposed the phase defect framework: in both cases the activation phase is nearly discontinuous across an extended curve (in 2D) or surface (in 3D), called a phase defect line (PDL) or phase defect surface (PDS). A PDL behaves like the branch cut of a complex logarithm: PSs detected by classical algorithms appear where the smeared phase field still winds, and in optical-mapping experiments (optical mapping: camera imaging of the heart's electrical activity via a voltage-sensitive fluorescent dye) nearly all detected PSs lie on PDLs, while PDLs persist considerably longer than the PSs on them. In 3D, linear-core scroll waves rotate around ribbon-shaped PDSs rather than filament curves. Later work built the topological layer of this framework: the endpoints of wave fronts, wave backs and defect lines act as charged quasiparticles ('heads', 'tails', 'pivots') whose interactions can be drawn as Feynman-like diagrams (Arno, Kabus and Dierckx, 2024), and in 3D phase defect surfaces ('branes') are bounded by topologically conserved curves ('strings'), with twistons at their crossings (Arno, Kabus and Dierckx, 2025).

What does not exist is any quantitative theory of how these defects move. For circular cores, adjoint 'response functions' (the linear sensitivity of a rotor to weak perturbations) localize stimulus sensitivity at the spiral-wave tip, and for linear cores near the turning points at the ends of the PDL (Dierckx, Verschelde and Panfilov, 2017); by contrast, for PDLs and PDSs there are no equations of motion, no stability criteria, no interaction laws, and no determination of whether filament tension and rigidity have surface analogues. Arno et al. (2021) posed this explicitly: 'How dynamical concepts such as filament tension and rigidity generalize to PDSs is still an open problem', and identified a quantitative theory of phase-defect dynamics — including motion under external stimuli, electrotonic currents from boundaries (electrotonic current: passive, diffusion-like spread of electric current through the coupled tissue), and 3D effects — as a key step toward predicting how conduction-block regions respond to control interventions such as defibrillation shocks and pacing. The problem below asks for this theory.

Problem Statement

Derive a quantitative dynamical theory of phase defect lines (PDLs, 2D) and phase defect surfaces (PDSs, 3D) in excitable media — at a level of rigor and predictive power comparable to what classical filament theory achieves for circular-core scroll waves. Within the regime where the local kinetics admit a single-loop attractor with a forbidden zone in state space (a single-loop attractor: the local dynamics settle onto a closed one-dimensional loop in state space, and its forbidden zone is the region the loop encircles but never enters — the feature that allows a phase variable to be defined; it is in this regime that linear-core rotors and conduction block occur), the research target comprises the following jointly posed components:

  1. Equations of motion for PDLs and PDSs: the velocity of a defect as a functional of its geometry and local state (such as curvature, defect width and orientation, incidence of wave fronts and wave backs, recovery state), including growth, shrinkage, splitting and merging of defects, and the motion of their topological endpoints (heads, tails, pivots).

  2. Stability criteria for isolated defects and their endpoint structures, analogous to the tension-based stability criteria for scroll-wave filaments.

  3. Interaction laws: how two nearby defects, and a defect and an approaching wave, attract, repel, merge, or annihilate.

  4. The determination of whether PDSs possess analogues of filament tension and filament rigidity — i.e. whether a defect surface has an effective surface tension and bending stiffness, with computable signs and magnitudes as functions of the medium's kinetic parameters — or a rigorous demonstration that no such local analogue exists.

  5. The response of defects to external stimuli and geometry: motion under applied stimuli (pacing, defibrillation-type shocks, impeding waves), electrotonic effects from medium boundaries and inhomogeneities, and genuinely 3D effects (such as twist, anisotropy, transmural structure), sufficient to predict how conduction-block regions respond to such interventions.

Within this objective, the intermediate flower-like meander regime (rotor-tip trajectories between circular and linear cores that trace flower-like petals) also requires classification — whether its central phase structure is a PS, a PDL, or a different object — a question the source explicitly leaves open. The derived laws must be validated against direct numerical simulations of standard reaction-diffusion models of excitable media (and, where possible, optical-mapping data) and must reduce to classical filament dynamics in the circular-core limit. Companion questions posed by the same source (multi-parameter phase descriptions for alternans regimes, i.e. beat-to-beat alternation of the excitation cycle; defect detection in noisy clinical data; clinical translation to local-activation-time maps; and the mechanism of incomplete epicardial rotor rotations) are separate applied directions and are not part of this problem.

References

  1. Louise Arno, Jan Quan, Nhan T. Nguyen, Maarten Vanmarcke, Elena G. Tolkacheva, Hans Dierckx, 'A phase defect framework for the analysis of cardiac arrhythmia patterns', Frontiers in Physiology 12, 690453 (2021). DOI: 10.3389/fphys.2021.690453. https://doi.org/10.3389/fphys.2021.690453 (preprint arXiv:2101.00315, https://arxiv.org/abs/2101.00315)
  2. Naoki Tomii, Masatoshi Yamazaki, Takashi Ashihara, Kazuo Nakazawa, Nitaro Shibata, Haruo Honjo, Ichiro Sakuma, 'Spatial phase discontinuity at the center of moving cardiac spiral waves', Computers in Biology and Medicine 130, 104217 (2021). DOI: 10.1016/j.compbiomed.2021.104217. https://doi.org/10.1016/j.compbiomed.2021.104217
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