Universality of the spectral exponents and the possibility of a Kolmogorov-like scaling theory for spiral- and scroll-wave turbulence in cardiac excitable media

Spiral- and scroll-wave turbulence underlies ventricular fibrillation, the disorganized electrical activity that leads to sudden cardiac death. A 2017 GPU-based direct numerical simulation study by Rajany, Gupta, Panfilov, and Pandit characterized this turbulence in two- and three-dimensional Panfilov and Aliev-Panfilov models of cardiac tissue using the tools of classical fluid turbulence: Fourier spectra $E_V(k)$ and $E_g(k)$ of the transmembrane potential and recovery variable, probability distribution functions, and structure functions with extended self-similarity. The spectra show approximate power laws, but their exponents are not demonstrably universal across the two models, and the limited power-law ranges leave genuine multiscaling unsettled. The open problem is threefold and linked: determine whether the spectral exponents and multiscaling exponent ratios are universal across additional models and parameter regimes; settle whether the structure functions display genuine multiscaling; and on that basis decide whether a phenomenological scaling theory analogous to Kolmogorov's 1941 theory of fluid turbulence can be formulated for this turbulence, and what parameters govern it. No later work has addressed these questions.

Background

Cardiac tissue is an excitable medium: a nonlinear reaction-diffusion system in which a local disturbance above threshold launches a regenerative electrical pulse, the action potential, that propagates without damping. Two-variable models capture this with a fast transmembrane potential $V$ and a slow recovery variable $g$; the study at issue used the Panfilov model and the Aliev-Panfilov model, two widely used, minimally parametrized models of this type. In such media, rotating waves in two dimensions (spiral waves) and their three-dimensional generalizations (scroll waves) can destabilize and break up into a disordered, statistically steady state called spiral- or scroll-wave turbulence. In heart muscle this state is ventricular fibrillation (VF), often described as electrical turbulence, and it causes sudden cardiac death.

Classical fluid turbulence offers a mature statistical language that this problem borrows. In statistically homogeneous, isotropic fluid turbulence the energy spectrum $E(k)$ follows an approximate power law $E(k)\sim k^{-\alpha}$ over an inertial range of wavenumbers $k$; Kolmogorov's 1941 phenomenological theory (K41) predicts the universal value $\alpha=5/3$ and structure-function exponents $\zeta_p=p/3$, where the order-$p$ structure function $S_p(r)=\langle[\delta u(r)]^p\rangle$ is built from velocity differences over separation $r$. When direct power-law ranges are short, the extended-self-similarity (ESS) procedure — plotting $S_p(r)$ against $S_2(r)$ — extends the scaling range and yields exponent ratios $\zeta_p/\zeta_2$; curvature of $\zeta_p/\zeta_2$ versus $p$ signals multiscaling (intermittency). Electrical turbulence in excitable media differs fundamentally: it is neither forced nor decaying — once initiated, a large spiral or scroll breaks into smaller ones (a forward-cascade analog of Richardson's energy cascade), and diffusive terms do not fully dissipate the activity because the medium is excitable. The dimensionless ratio $L/\lambda$, with $L$ the domain size and $\lambda$ the plane-wave wavelength, plays the role of a Reynolds-number-like control parameter.

Earlier work by the same group had characterized spiral turbulence and its control in excitable-media models (Pandit, Pande, Sinha, and Sen 2002; Shajahan, Nayak, and Pandit 2009). The 2017 study (Rajany, Gupta, Panfilov, and Pandit, arXiv:1705.09935) was the first to apply the full fluid-turbulence statistical toolkit to this turbulence, via large-scale direct numerical simulations (DNS) on graphics processing units: $4096\times4096$ domains in 2D and $640^3$ domains in 3D, for both models. It found the analog spectra $E_V(k)$ and $E_g(k)$ spread over several decades of $k$ with approximate power laws — in 2D, $m_V\simeq-2.7$ (Panfilov) versus $-3.2$ (Aliev-Panfilov) for $E_V$, and $m_g\simeq-1.9$ versus $-3.1$ for $E_g$; in 3D, $m_V\simeq-3.8$ versus $-3.9$ and $m_g\simeq-1.8$ versus $-4.4$ — comparable between models but not equal at the achievable resolution, and hence not shown to be universal in the way their fluid-turbulence counterparts are. The ESS exponent ratios $\zeta_p/\zeta_2$ showed curvature suggestive of multiscaling, but over limited ranges and as ratios of small numbers, so the authors state that whether genuine multiscaling exists can be settled only by DNSs with much larger domains and longer runs. Scroll-wave turbulence with negative filament tension showed no significant statistical difference from the positive-tension case at their resolution. The authors note that if the exponents are not universal — possibly because of the large number of model parameters — then no spiral-turbulence analog of the K41 phenomenological theory can be developed in any obvious way.

Problem Statement

For statistically steady, statistically homogeneous and isotropic spiral- and scroll-wave turbulence in excitable media of the type used for cardiac tissue, settle the following linked questions, which the source study leaves open:

1. Universality of the statistical exponents. Determine whether the approximate power-law exponents of the spectra $E_V(k)$ and $E_g(k)$ and the extended-self-similarity exponent ratios $\zeta_p/\zeta_2$ of the structure functions are universal, or whether they depend systematically on the model and its parameters — and, if they are not universal, characterize that dependence.

2. Genuine multiscaling. Establish whether the structure functions of $V$ and $g$ display genuine multiscaling (i.e., nonlinear $\zeta_p$ versus $p$ beyond what finite ESS ranges or ratios of small numbers could produce).

3. Phenomenological scaling theory. On the basis of (1) and (2), decide whether a phenomenological scaling theory for spiral- and scroll-wave turbulence, analogous to Kolmogorov's 1941 theory for fluid turbulence, can be formulated; if so, derive its predictions (spectral exponents and structure-function scaling) and identify the fundamental parameters that govern it (the source's candidate control parameter is $L/\lambda$, the ratio of the domain size to the plane-wave wavelength); if not, delimit the classes of models or parameter regimes within which such a theory can hold.

These three components jointly constitute one scientific objective: a K41-like statistical characterization of electrical turbulence in excitable media.

References

1. K. V. Rajany, Anupam Gupta, Alexander V. Panfilov, Rahul Pandit, "The statistical properties of spiral- and scroll-wave turbulence in cardiac tissue", arXiv:1705.09935 (2017), DOI 10.48550/arXiv.1705.09935, https://arxiv.org/abs/1705.09935
2. A. N. Kolmogorov, "The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers" (1941; English translation), Proceedings of the Royal Society of London. Series A 434, 9-13 (1991), DOI 10.1098/rspa.1991.0075, https://doi.org/10.1098/rspa.1991.0075
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