The 2013 theory of Dierckx and Verschelde reduces the dynamics of curved and twisted scroll waves in isotropic reaction-diffusion excitable media to an effective law of motion for a virtual filament, computed perturbatively to third order in the small parameter λ=max⁡(kd,∣w∣d)\lambda=\max(kd,|w|d), where kk is filament curvature, ww the twist rate, and dd the spiral-core diameter. The source itself delimits what remains open: extension beyond the circular-core regime to meandering and post-bifurcation regimes, explicit extension to strongly anisotropic or spatially varying diffusivity, quantification of the omitted higher-order corrections, the nonlinear (super- versus sub-critical) nature of the twist-driven sproing instability, the global dynamics of the resulting inhomogeneous Burgers-type phase equation, and numerical/experimental validation of the predicted coefficients. Since then, only the leading-order meandering facet has been resolved (Dierckx et al., Phys. Rev. Lett. 2017); no solution is known to date for the higher-order, anisotropic, criticality, phase-dynamics, and validation questions. The objective is a mathematically controlled extension of the virtual-filament framework, with explicit coefficient sets, to these regimes, together with numerical and experimental validation.

Background

Excitable media are systems whose local kinetics have a stable rest state with a threshold-like excitation and a refractory recovery; together with diffusion they are modeled by reaction-diffusion equations and support rotating spiral waves in two spatial dimensions, for example in the Belousov-Zhabotinsky chemical reaction and in cardiac electrical tissue, where such waves underlie reentrant arrhythmias. In three spatial dimensions the rotating wave generalizes to a scroll wave, organized around a one-dimensional curve — the filament — traced out by the spiral-wave cores in the planes transverse to it. Biktashev, Holden, and Zhang (1994) showed that a slowly curved filament moves approximately like a thin elastic string driven by its own tension: normal velocity vn=γ1kv_n = \gamma_1 k with filament tension coefficient γ1\gamma_1, so that positive tension shrinks rings and scrolls while negative tension inflates them, destabilizing the wave and leading to scroll-wave turbulence and breakup. When the rotation phase varies along the filament — a twisted scroll — twist couples to curvature, and the filament can undergo the twist or sproing instability, a buckling into a helical shape above a critical twist (Dierckx, Verschelde, Selsil, and Biktashev, 2012); such helical scroll waves are observed experimentally. In cardiac muscle the diffusion tensor is strongly anisotropic (fiber architecture), for which Verschelde, Dierckx, and Bernus (2007) derived leading-order covariant filament laws by viewing anisotropic propagation as effectively isotropic dynamics on a curved (Riemannian) space. Building on the response-function method — the use of adjoint eigenmodes of the linearization about the rotating spiral wave to extract drift coefficients via solvability conditions — Dierckx and Verschelde (2013) derived effective equations of motion to third order in the small parameter λ=max⁡(kd,∣w∣d)\lambda=\max(kd,|w|d) (with kk the filament curvature, ww the twist rate, and dd the spiral-core diameter) for a virtual filament: a corrected, phase-averaged filament coordinate whose dynamics decouple from the fast internal rotation. Their theory yields the tension and twist-curvature coupling terms with explicit coefficient formulas, a phase equation of inhomogeneous Burgers type (a nonlinear first-order PDE for the rotation phase along the filament, known to admit shock and rarefaction waves), and an analysis of sproing that assumes the bifurcation is supercritical in deriving small-amplitude helical scaling. The derivation is restricted to isotropic media and to the circular-core regime, i.e. rigidly rotating spiral tips; spiral tips can also meander, executing quasiperiodic flower-like trajectories arising from a Hopf bifurcation of the rotating wave, and this meandering regime is the one actually observed over wide parameter ranges in chemistry and cardiology. The source paper explicitly acknowledges that extending the virtual-filament reduced laws and their coefficient formulas beyond the small-λ\lambda, circular-core, isotropic, leading-order regime — to meandering cores, strongly anisotropic or inhomogeneous media, higher perturbative orders, the nonlinear character of the sproing bifurcation, and the global behavior of the phase equation — remains unresolved, and that some predicted coefficients (notably a2a_2) could not be validated at reachable numerical resolution.

Problem Statement

Establish the range of validity of the virtual-filament effective description of scroll-wave dynamics derived by Dierckx and Verschelde (2013) and extend it beyond the regime in which it was derived — small perturbation parameter λ=max⁡(kd,∣w∣d)\lambda=\max(kd,|w|d), circular (rigidly rotating, non-meandering) spiral cores, isotropic diffusion, and leading computed perturbative orders. The single scientific objective is a quantitatively validated effective theory of twisted and curved scroll-wave filaments in the regimes actually encountered in applications, with explicit, testable coefficient formulas; its linked components are: (i) effective filament equations of motion, including twist-curvature coupling and the full higher-order coefficient set, for meandering (quasiperiodic-core) spirals and for strongly nonlinear post-bifurcation regimes, beyond the leading-order averaging of Dierckx et al. (2017); (ii) extension of the reduced laws and coefficient formulas to strongly anisotropic and spatially varying diffusivity, going beyond the leading-order covariant laws of Verschelde, Dierckx, and Bernus (2007); (iii) quantification of the effect of the omitted higher-order corrections (field corrections beyond the computed orders and terms of order O(λ5)O(\lambda^5) and beyond) on long-time filament dynamics and on thresholds near marginal stability; (iv) determination of the nonlinear character — supercritical or subcritical — of the twist-driven sproing bifurcation for given excitable kinetics and parameter ranges, replacing the assumption of supercriticality used in the 2013 small-amplitude helical scaling; (v) the global dynamics of the derived inhomogeneous Burgers-type phase equation for the filament rotation phase, including whether and how shocks, rarefaction waves, or singularities form along the filament; and (vi) numerical and experimental validation of the resulting laws and coefficients (including the still-unvalidated coefficient a2a_2 and the reconstruction of virtual filaments from measured tip-line data). These components jointly delimit the validity and extend the reach of one effective theory.

References

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