Strings, branes and twistons: topological analysis of phase defects in excitable media such as the heart

Abstract

Several excitable systems, such as the heart, self-organize into complex spatio-temporal patterns that involve wave collisions, wave breaks, and rotating vortices, of which the dynamics are incompletely understood. Recently, conduction block lines in two-dimensional media were recognized as phase defects, on which quasi-particles can be defined. These particles also form bound states, one of which corresponds to the classical phase singularity. Here, we relate the quasi-particles to the structure of the dynamical attractor in state space and extend the framework to three spatial dimensions. We reveal that 3D excitable media are governed by phase defect surfaces, i.e. branes, and three flavors of topologically preserved curves, i.e. strings: heads, tails, and pivot curves. We identify previously coined twistons as points of co-dimension three at the crossing of a head curve and a pivot curve. Our framework predicts splitting and branching phase defect surfaces that can connect multiple classical filaments, thereby proposing a new mechanism for the origin, perpetuation, and control of complex excitation patterns, including cardiac fibrillation.

Figures (13)

• Figure 1: Key excitable structures in this paper: from simple to complex excitation patterns. Simulated patterns are (a) circular-core spiral wave, using the Aliev-Panfilov (AP) cardiac excitation model Aliev:1996 in an isotropic medium, (b) biperiodically rotating (meandering) spiral wave with linear core in the Bueno-Orovio-Cherry-Fenton (BOCF) modelBuenoOrovio:2008, (c) 3D rotating scroll wave in the BOCF model BuenoOrovio:2008 for the anisotropic case with $D_1= 1$ and $D_2=1/5$ for which the fibre angle varies from $-60 ^{\circ}$ (bottom) to $60 ^{\circ}$ (top).
• Figure 2: Overview of classical phase analysis and recently introduced phase defects. (a) Plotting two state variables with respect to each other allows us to define a phase angle in this plane. (b) The circular-core spiral wave from Fig. \ref{['fig:spirals']}a colored according to the phase, showing a point where all phases meet, the phase singularity. (c) The linear-core spiral wave from Fig. \ref{['fig:spirals']}b colored using the activation phase $\phi^{act}$. (d) Same spiral wave, using a phase $\phi^{\rm LAT}$ based on local activation time, such that the wave front is no longer emphasized Kabus:2022. Instead, a phase defect line or conduction block line is observed at the rotor core. (e) The phase defect density $\rho = ||\vec{\nabla} \phi ||$ allows to localize the phase defect Kabus:2022. The data were down-sampled to make the otherwise few pixels thin line more visible.
• Figure 3: Origin of phase defects from the geometric structure of state space in excitable media. (a) state space view in 2D and 3D. (b) Point cloud of states observed during the numerical simulation of a spiral wave with circular core (c) with Aliev-Panfilov kinetics Aliev:1996, in an isotropic medium. Few points located at the rotor core lie in the forbidden zone of state space in panel b and are brought their by electrotonic effects, i.e. diffusion; they form the phase defect in physical space. (d-e) similar analysis for the three-variable Fenton-Karma (FK) model in an isotropic medium, supporting a linear-core rotor. Colors indicate the presence of fronts, backs, heads, tails and pivots and are used consistently as in Tab. \ref{['tab:structures']}.
• Figure 4: Overview of spatial transitions and temporal processes involving heads and tails in 2D. (a) A topological charge $Q=\pm \frac{1}{2}$ is associated to heads (h) and tails (t). (b) Illustration of the only four processes that can occur at the PD boundary changing the number of heads. Heads are created or annihilated in a pair of opposite charge. (c) Same as b. for wave back. (d) PD creation by collision of a wave front and back, introducing a quadruple $(\text{h}^+,\text{t}^+,\text{h}^-,\text{t}^-)$. (e) Example of the classical tip, indicated in purple, with charge $Q=\pm 1$, appearing as a bound state of heads and tails with equal charge $Q=\pm \frac{1}{2}$.
• Figure 5: Mutual relation of head, tails and pivots during growth and shrinking of a phase defect. (a) initial PD formation via a wave block since an impeding wave (top left) touches an incompletely recovered region (right). A growth point is formed, consisting of a head, tail, and pivot of the same chirality and topological charge. (b) During further evolution, the pivot will generally not coincide with head and tail. (c) Retraction of the oldest PD end (pivot), as a bound state with a tail point. In 3D, similar stages of development occur. Only case (a) is covered by the classical theory of phase singularities and filaments.