A 2019 Physical Review Letters study (Otani, Wheeler, Krinsky, and Luther) showed in simulations of the Barkley model — a standard generic two-variable description of an excitable medium — that a single ~5 ms electric-field pulse with a large component perpendicular to the tissue boundary can simultaneously terminate multiple three-dimensional scroll waves: the pulse depolarizes a thin surface layer, detaches the scroll-wave filaments from the boundary, and the detached filaments reattach to a parallel surface as C- or U-shaped filaments that shrink and vanish under positive filament tension, with a 50% termination-success threshold near 1 V/cm independent of the number of vortices — roughly a 25-fold energy reduction relative to standard ~5 V/cm defibrillation shocks. The authors themselves stated what remains unresolved: the result was obtained only in a highly idealized model and simple geometries; the required field (~1 V/cm) is surprisingly about five times the experimentally measured threshold for mere surface excitation (~0.2 V/cm), indicating that mechanisms beyond mere surface depolarization — plausibly the filament-reattachment processes the authors labeled Mechanisms 4 and 5, whose quantitative roles remain unestablished — set the termination threshold; and the critical depolarized-layer thickness below which the new filament segment fails to survive was introduced only qualitatively and never quantified. This problem asks for the physical basis and robustness of surface-impact scroll-wave termination to be established: (i) whether the detachment-and-collapse mechanism operates in more realistic models of cardiac tissue and at what field strengths; (ii) identification of the mechanisms responsible for the gap between the ~0.2 V/cm excitation threshold and the ~1 V/cm termination threshold, and whether the required field can be reduced below 1 V/cm; and (iii) the quantitative value of the critical depolarized-layer thickness and its dependence on medium properties and geometry. No study known to date has addressed these three questions, so the problem remains open.

Background

Rotating excitation waves underlie many self-organizing phenomena: spiral waves on catalytic surfaces, waves that direct cell movement in morphogenesis, and, most consequentially, the electrical vortices that drive cardiac fibrillation. In three dimensions a rotating wave becomes a scroll wave, whose organizing axis is the scroll filament. Standard clinical defibrillation terminates fibrillation by applying a high-energy shock (roughly 360 J, creating fields around 5 V/cm in tissue) that extinguishes all wave activity, at the cost of significant side effects. Low-energy alternatives have therefore been pursued. One line of work exploits the vulnerable window (the narrow phase interval in which a stimulus extinguishes a rotor by emitting a wave of opposite topological charge), but with many vortices of unknown phase one must scan all phases with multiple pulses, as demonstrated experimentally by Fenton et al. (Circulation 120, 467, 2009). A second line, the LEAP approach (low-energy anti-fibrillation pacing) of Luther et al. (Nature 475, 235, 2011), terminates pinned vortices with trains of far-field pulses near ~0.2 V/cm, exploiting wave emission from small heterogeneities. Both lines either need phase information or rely on heterogeneity-induced wave emission. The surface-impact mechanism of Otani, Wheeler, Krinsky, and Luther (Phys. Rev. Lett. 123, 068102, 2019) is different: it targets the filament itself rather than the rotating wave. The filament dynamics of scroll waves are governed by filament tension: when the tension is positive, curved filaments shrink (their radius obeys dR/dt=−α/RdR/dt=-\alpha/R), a regime studied systematically since Barkley's linear stability analysis of rotating spiral waves (Phys. Rev. Lett. 68, 2090, 1992) and contrasted with the negative-tension turbulence tamed by Alonso, Sagués, and Mikhailov (Science 299, 1722, 2003). In the Barkley two-variable model of an excitable medium (∂u/∂t=D∇2u+ϵ−1u(1−u)(u−(v+b)/a)\partial u/\partial t = D\nabla^2 u + \epsilon^{-1}u(1-u)(u-(v+b)/a), ∂v/∂t=u−v\partial v/\partial t = u-v, where uu is the membrane potential and vv its slow recovery variable), with the boundary condition n^⋅∇(u−E0⋅x)=0\hat n\cdot\nabla(u-\mathbf{E}_0\cdot\mathbf{x})=0 that follows from the bidomain description of cardiac tissue (the standard two-compartment description of intra- and extracellular current), an electric-field pulse perpendicular to a boundary depolarizes a thin layer near that surface. If this layer is thicker than some critical distance, a filament ending on that surface detaches and reattaches to a parallel surface as a C-shaped filament, which then shrinks and disappears under positive tension. The authors verified this in boxes, cylinders, elliptical cylinders, and a hemispherical shell: a radially directed field (perpendicular to the shell surfaces) terminated ensembles of one to five scroll waves with 50% success at E50≈1E_{50}\approx1 V/cm, independent of vortex number and insensitive to the scroll phase, whereas a uniform field perpendicular to only one pair of surfaces fails whenever the number of filaments is odd, because unpaired filament ends reattach to the same wall. The success of the method thus hinges on avoiding the processes the authors enumerated: filament-end reattachment (their Mechanism 4) and filament-center reattachment (their Mechanism 5), which occur when the depolarized layer is too thin — that is, thinner than the critical thickness. Two quantitative facts frame the open questions. First, mere surface excitation of cardiac tissue requires only about 0.2 V/cm in experiments (Luther et al. 2011), yet the simulations needed about 1 V/cm to terminate scrolls, so processes beyond simple surface depolarization — presumably the reattachment mechanisms — set the threshold, and their investigation might allow the field to be reduced below 1 V/cm. Second, the critical depolarized-layer thickness was introduced only as a qualitative existence threshold whose numerical value and dependence on medium parameters, geometry, and boundary curvature were never established. The simulations themselves used a two-variable generic excitable medium with isotropic diffusion, simple geometric domains, and an idealized field profile, whereas real cardiac tissue has detailed ionic kinetics, anisotropic and heterogeneous conductivity, variable thickness, and realistic anatomical shape.

Problem Statement

Establish the physical basis and robustness of surface-impact termination of scroll waves beyond the idealized setting in which it was discovered, at the generality the source poses. This comprises three jointly linked questions. Their common objective is to determine whether the detachment-and-collapse mechanism is a physically viable low-energy route to terminating three-dimensional reentry, and what field it actually requires: (1) Realistic-media performance — determine how the surface-impact mechanism performs when the highly idealized Barkley-medium simulations are replaced by a more realistic description of cardiac tissue. Any single established cardiac ionic model with anisotropic conductivity in one realistic, anatomically shaped geometry is a sufficient qualifying setting; report that setting's termination success rate and required field strengths (E50 and the success rate as a function of field amplitude), without requiring a survey of additional models or geometries. (2) Threshold mechanisms — identify the mechanisms, beyond mere surface depolarization, that set the termination threshold: the source's numerical threshold (~1 V/cm) is about five times the experimental surface-excitation threshold (~0.2 V/cm), indicating that additional processes are involved; the authors pointed to the filament-end and filament-center reattachment mechanisms they labeled Mechanisms 4 and 5 as likely candidates, and their quantitative roles remain to be established and determine whether the field required for termination can thereby be reduced below 1 V/cm. (3) Critical depolarized-layer thickness — determine the quantitative value of the critical thickness lcritl_{\text{crit}} such that filament detachment produces a surviving C-shaped filament only when the depolarized layer is thicker. Report that value both for the Barkley medium of the source and for the single realistic setting of part (1), and determine its dependence on the medium's excitability and refractoriness parameters and on the boundary geometry class (e.g. flat and spherically curved boundaries), without requiring an exhaustive parameter or geometry sweep.

References

  1. Niels F. Otani, Kayleigh Wheeler, Valentin Krinsky, Stefan Luther, 'Termination of Scroll Waves by Surface Impacts', Physical Review Letters 123, 068102 (2019). DOI: 10.1103/PhysRevLett.123.068102, https://doi.org/10.1103/PhysRevLett.123.068102
  2. Stefan Luther, Flavio H. Fenton, Bruce G. Kornreich, Amgad Squires, Philip Bittihn, Daniel Hornung, Markus Zabel, James Flanders, Andrea Gladuli, Luis Campoy, Elizabeth M. Cherry, Gisa Luther, Gerd Hasenfuss, Valentin I. Krinsky, Alain Pumir, Robert F. Gilmour, Eberhard Bodenschatz, 'Low-energy control of electrical turbulence in the heart', Nature 475, 235–239 (2011). DOI: 10.1038/nature10216, https://doi.org/10.1038/nature10216
  3. Flavio H. Fenton, Stefan Luther, Elizabeth M. Cherry, Niels F. Otani, Valentin Krinsky, Alain Pumir, Eberhard Bodenschatz, Robert F. Gilmour, 'Termination of Atrial Fibrillation Using Pulsed Low-Energy Far-Field Stimulation', Circulation 120, 467–476 (2009). DOI: 10.1161/CIRCULATIONAHA.108.825091, https://doi.org/10.1161/CIRCULATIONAHA.108.825091
  4. Dwight Barkley, 'Linear stability analysis of rotating spiral waves in excitable media', Physical Review Letters 68, 2090 (1992). DOI: 10.1103/PhysRevLett.68.2090, https://doi.org/10.1103/PhysRevLett.68.2090
  5. Sergio Alonso, Francesc Sagués, Alexander S. Mikhailov, 'Taming Winfree Turbulence of Scroll Waves in Excitable Media', Science 299, 1722–1725 (2003). DOI: 10.1126/science.1080207, https://doi.org/10.1126/science.1080207
  6. Teng-Chao Li, Bao-quan Ai, Yi Chen, Bing-Wei Li, 'Stabilizing meandering scroll waves and ordering scroll wave turbulence by a circularly polarized electric field in highly excitable media', Physics Letters A 388, 127058 (2021). DOI: 10.1016/j.physleta.2020.127058, https://doi.org/10.1016/j.physleta.2020.127058
  7. Thomas Lilienkamp, Ulrich Parlitz, Stefan Luther, 'Non-monotonous dose response function of the termination of spiral wave chaos', Scientific Reports 12, 12043 (2022). DOI: 10.1038/s41598-022-16068-8, https://doi.org/10.1038/s41598-022-16068-8
  8. Thomas Lilienkamp, Ulrich Parlitz, Stefan Luther, 'Taming cardiac arrhythmias: Terminating spiral wave chaos by adaptive deceleration pacing', Chaos 32, 121105 (2022). DOI: 10.1063/5.0126682, https://doi.org/10.1063/5.0126682
  9. S. V. Amrutha, Anupama Sebastian, Puthiyapurayil Sibeesh, Shreyas Punacha, T. K. Shajahan, 'Mechanism of Spiral Wave Unpinning in the Belousov–Zhabotinsky Reaction with a DC Electric Field', The Journal of Physical Chemistry C 126, 19618–19626 (2022). DOI: 10.1021/acs.jpcc.2c01452, https://doi.org/10.1021/acs.jpcc.2c01452
  10. Marcel Aron, Thomas Lilienkamp, Stefan Luther, Ulrich Parlitz, 'Optimising low-energy defibrillation in 2D cardiac tissue with a genetic algorithm', Frontiers in Network Physiology 3, 1172454 (2023). DOI: 10.3389/fnetp.2023.1172454, https://doi.org/10.3389/fnetp.2023.1172454
  11. Daniel Suth, Stefan Luther, Thomas Lilienkamp, 'Chaos control in cardiac dynamics: terminating chaotic states with local minima pacing', Frontiers in Network Physiology 4, 1401661 (2024). DOI: 10.3389/fnetp.2024.1401661, https://doi.org/10.3389/fnetp.2024.1401661
  12. Qianming Ding, Yipeng Hu, Yong Wu, Xueyan Hu, Ya Jia, Lijian Yang, 'Elimination of reentry spiral waves using adaptive optogenetical illumination based on dynamic learning techniques', Chaos, Solitons & Fractals 191, 115846 (2025). DOI: 10.1016/j.chaos.2024.115846, https://doi.org/10.1016/j.chaos.2024.115846
  13. Marcel Aron, Stefan Luther, Ulrich Parlitz, 'Success rates of simulated multi-pulse defibrillation protocols are sensitive to application timing with individual, protocol-specific optimal timings', Frontiers in Network Physiology 5, 1572834 (2025). DOI: 10.3389/fnetp.2025.1572834, https://doi.org/10.3389/fnetp.2025.1572834
  14. Daniel Frühwald, Thomas Lilienkamp, 'Leading mechanisms of defibrillation: a computational approach to study differences between monophasic and biphasic waveforms', Nonlinear Dynamics 114, 162 (2026). DOI: 10.1007/s11071-025-11992-6, https://doi.org/10.1007/s11071-025-11992-6