In bounded three-dimensional (3D) excitable media whose scroll-wave filaments have negative line tension, turbulence is generated and sustained by unstable filaments breeding in the bulk, and it manifests on the medium surface as repeatedly created and annihilated spiral-wave defects. Davidsen, Zhan and Kapral (Phys. Rev. Lett. 101, 208302, 2008) showed, for the Barkley model in a weakly excitable regime, that the population of these surface defects is well described by Markov birth-death rate equations with essentially affine (linear-plus-constant-offset) rates, yielding a stationary defect-number distribution distinguishable from two-dimensional defect turbulence. Several quantitative issues were left open by that study and remain unsettled. The subsequent literature characterizes the Markov rate-equation description as an unfinished, mostly empirical attempt (Biktashev 2014); the reviews cite its linear-plus-constant-offset result without completing the constant annihilation term, its validity domain, or its regime generality (Alonso, Bär and Panfilov 2012; Alonso, Bär and Echebarria 2016); and Alonso, Bär and Echebarria (2016) note the resulting possibility of determining the essential mechanism of electrical turbulence from experimental phase-singularity data. The open issues are: (i) whether the inferred population-independent constant contribution to the annihilation rate, hypothesized in the source to arise from transient triple-stranded filament bound states, genuinely exists and what its value is; (ii) the domain of validity of the affine-rate description at small system sizes and small defect numbers, where finite-size correlations break the independence assumption; (iii) how the surface-rate forms, the stationary distribution, and the role of bound states change across excitability regimes and different excitation kinetics, including regimes where filaments fragment and produce scroll rings; and (iv) whether the surface-statistics diagnostic transfers to realistic cardiac-tissue models and to experimental surface measurements, where the relevance of the 3D filament mechanism to ventricular fibrillation is debated. These questions remain open, with related but non-overlapping progress in cardiac modeling and clinical phase-singularity statistics.

Background

An excitable medium is a spatially extended system whose elements respond to super-threshold perturbations with a pulse of activity followed by a refractory period; reaction-diffusion equations of this type describe oxidation waves in the Belousov-Zhabotinsky reaction and electrical excitation waves in cardiac tissue. In two dimensions (2D), such media support rotating spiral waves, and when the spirals lose stability the result is defect-mediated turbulence: a gas of phase singularities (the spiral cores, points where the phase of the oscillation is undefined) that are continually created and annihilated in pairs. The statistics of such defect populations in 2D — approximately Poisson-distributed defect numbers and exponentially distributed lifetimes under the assumption of effectively independent creation and annihilation events — have been characterized theoretically and observed across fluid, chemical, quantum, and biological systems (Coullet, Gil and Lega 1989; Wang 2004; Tan et al. 2020). In 2D cardiology-relevant models, spiral breakup provides one route to turbulent electrical activity (Fenton et al. 2002). In three dimensions, the rotation axis of a spiral generalizes to a vortex filament, and the rotating object is a scroll wave. A filament behaves, to leading order, like a curve moving by curvature with a coefficient called the filament line tension (Biktashev, Holden and Zhang 1994). When this tension is negative, any curvature grows: filaments lengthen, bend, and breed new scroll waves, producing a distinct, purely 3D turbulence mechanism ('negative-tension' or Winfree turbulence) that has been reproduced in models and in chemical experiments (Alonso et al. 2004; Luengviriya et al. 2008; Bánsági and Steinbock 2007) and was early on proposed as a mechanism of ventricular fibrillation in heart muscle — ventricular fibrillation being the chaotically disorganized electrical activation of the heart's pumping chambers that underlies sudden cardiac arrest (Winfree 1994; Zaritski, Mironov and Pertsov 2004). The Barkley model is a standard two-variable reaction-diffusion excitable system whose parameters allow both stable spiral waves and negative line tension, so it is a common testbed for these dynamics. Davidsen, Zhan and Kapral (2008) studied negative-tension turbulence of the Barkley model in a bounded 3D domain and asked how the bulk filament turbulence is seen from outside: filaments intersecting the boundary nucleate and remove surface phase singularities, so the surface carries its own defect-mediated turbulence whose stochastic dynamics the authors modeled with Markov birth-death rate equations for the surface defect number nn, with creation rate b(n)b(n) and annihilation rate a(n)a(n). They found the rates to be well approximated by affine laws, b(n)=b1n+b0b(n) = b_1 n + b_0 and a(n)=a1n+a0a(n) = a_1 n + a_0, linear in nn with constant offsets — a form that differs from the 2D analog and lets surface statistics discriminate the 3D filament mechanism — and obtained the corresponding stationary distribution of defect numbers (their Eq. (3)). The constant (population-independent) part of the annihilation rate was attributed to transient triple-stranded filament bound states reaching the surface, but with statistical fluctuations too large to confirm or parameterize it; the fitted linear laws broke down at small system sizes LL and small nn, where correlations violate the independence assumption behind the rate equations; and all results were obtained for one weakly excitable parameter set with negative line tension, leaving other excitability regimes, other kinetics (including those where filaments fragment and give scroll rings), realistic cardiac tissue models, and actual experimental surface measurements unexplored. Whether this surface-statistics diagnostic can identify 3D filament-induced turbulence in real tissue bears directly on the long-standing controversy over whether scroll-wave filaments sustain ventricular fibrillation (Winfree 1994; Clayton 2008).

Problem Statement

Complete and validate the Markov rate-equation description of surface spiral-defect statistics for filament-induced (negative-line-tension) turbulence in three-dimensional excitable media, as initiated by Davidsen, Zhan and Kapral (2008) for the Barkley model. The objective is to establish, quantitatively, when and how the statistics of defects observed on the surface of a bounded 3D excitable medium carry the signature of bulk filament turbulence. The following four open questions each contribute to that objective and remain unresolved. They are independent. (1) Rate structure: writing the birth and death rates of the surface defect number nn in the affine form b(n)=b1n+b0b(n) = b_1 n + b_0 and a(n)=a1n+a0a(n) = a_1 n + a_0, determine whether the annihilation rate contains a population-independent constant contribution a0a_0 — hypothesized in the source to arise from transient triple-stranded filament bound states reaching the boundary — and measure the constants b0,b1,a0,a1b_0, b_1, a_0, a_1 with sufficient statistics to settle the existence and magnitudes of the constant terms. The affine form is the parameterization found in the source. (2) Validity domain: map out where the affine-rate, effectively independent-event description holds — specifically its breakdown and crossover behavior at small system sizes LL and small defect numbers nn, where finite-size correlations violate the independence assumption — characterizing how the stationary defect-number distribution is modified in that regime. (3) Robustness: establish how the rate structure established in (1), the stationary distribution characterized in (2), and the role of transient bound states change across excitability regimes and across excitation kinetics, including parameter regimes and models in which filaments fragment in the interior and produce scroll rings, rather than only the single weakly excitable Barkley parameter set with negative line tension used originally. (4) Diagnostic transfer: determine whether the surface-statistics diagnostic identifies 3D filament-induced turbulence in realistic cardiac-tissue models and in experimental surface measurements (surface phase-singularity data from existing or new optical, catheter, or ultrasound recordings) — i.e., whether surface-observed creation/annihilation-rate structure distinguishes the 3D filament mechanism from 2D spiral breakup in settings relevant to the controversy over the role of scroll-wave filaments in ventricular fibrillation.

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