Chaotic tip trajectories of a single spiral wave in the presence of heterogeneities

Spiral waves have been observed in a variety of physical, chemical, and biological systems. They play a major role in cardiac arrhythmias, including fibrillation, where the observed irregular activation patterns are generally thought to arise from the continuous breakup of multiple unstable spiral waves. Using spatially extended simulations of different electrophysiological models of cardiac tissue, we show that a single spiral wave in the presence of heterogeneities can display chaotic tip trajectories, consistent with fibrillation. We also show that the simulated spiral tip dynamics, including chaotic trajectories, can be captured by a simple particle model which only describes the dynamics of the spiral tip. This shows that spiral wave breakup, or interactions with other waves, are not necessary to initiate chaos in spiral waves.

Figure 1

(a), (c) Snapshot of a counterclockwise rotating spiral wave for two different parameter sets of the FK model [voltage color-coded ranging from high (red) to low (blue) values, tip trajectory shown in white; (a) set I, (c) set II]. (b), (d) Phase diagrams in the spacing-size space for the FK model. In this, and all other figures, blue area indicates regular trajectories that wrap around both heterogeneities, purple region represents tip trajectories that circle either one of the heterogeneities, and yellow region corresponds to chaotic trajectories. Displayed spiral tip trajectories correspond to the white dots in the phase diagram. Red X's mark the locations of the circular regions with decreased excitability. Lyapunov exponents ($\lambda$) for the chaotic trajectories are given in units of bits/second [(b) set I, (d) set II].

Figure 2

(A) Power spectrum of the $x$ component of the tip trajectory for the different electrophysiological models and parameter sets used in the main text. The spectra have either one or two peaks, corresponding to the frequencies ${\omega }_1=2\pi /T_1$ and ${\omega }_2=2\pi /T_2$ (left panel, FK set I; middle panel, FK set II; right panel, KKT). (B) The spiral tip trajectory for set II. The maximum and minimum value of the spatial extent of the tip trajectory are denoted by $A_1$ and $A_2$, respectively.

Figure 3

(A) Snapshot of a counterclockwise rotating spiral wave in the homogeneous KKT model. (B) Sample tip trajectories of the KKT model, for different spacings of heterogeneities with radius 0.5 cm. (C) Phase diagrams of tip trajectories for the SP model. All scale bars are 1 cm. See also Fig. 1.

Figure 4

A: Comparison between the full models (FK set I right panel, FK set II middle panel, and KKT right panel) and SP model for homogeneous media. Scale bars are 0.5 cm. (B, C) Cumulative error (measured in cm, and plotted using a color scheme) in the tip trajectory from the SP model for set I (B) and set II (C) as compared to the trajectory of the full model. The error was computed for every ms and for an entire rotation of the full model. (D) Example of a trajectory of the SP model for the nonoptimal parameters corresponding to the white symbol in C ($A_1=1.075\mathrm{cm}$, $A_2=0.2\mathrm{cm}$, ${\omega }_1=0.0058{\text{ms}}^{-1}$, and ${\omega }_2=0.0439{\text{ms}}^{-1}$).

Figure 5

Schematic representation of a particle moving in a potential landscape with two wells, representing tissue heterogeneities.

Figure 6

A, B: Phase diagrams of tip trajectories for set I (A) and set II (B) in the SP model, with tip trajectories and Lyapunov exponents. See also Fig. 1. (C) Phase diagram for set I, illustrating the effect of different well depths $g$. The circles correspond to the parameter values of the circles in (A).

Figure 7

Spiral trajectories for the FK (left column) and SP models (right column) in the presence of six randomly placed heterogeneities. Regular (A, B) and chaotic (C, D) trajectories with matching placement of the heterogeneities. The radius of the heterogeneities in the FK model was 0.3 cm, and the $s$ parameter in the SP model was 0.275 cm. All scale bars are 1 cm.

Figure 8

Example trajectories for the homogeneous (left column) and heterogeneous (right column) case with added anisotropy. In the FK model, the diffusion constant in the $y$ (vertical) direction was chosen to be 50% of the value for the homogeneous case (i.e., $D_y=0.0005{\text{cm}}^2/\text{ms}$). In the SP model, the forcing amplitude in the $x$ and $y$ equation can be directly determined from the tip trajectory in the homogeneous FK model. The tip trajectories in the SP model capture the FK trajectories for both the homogeneous (left column) and heterogeneous case (right column). Specifically, the chaotic regime is still present in both models. All scale bars are 1 cm.