Wave-train-induced termination of weakly anchored vortices in excitable media

A free vortex in excitable media can be displaced and removed by a wave train. However, simple physical arguments suggest that vortices anchored to large inexcitable obstacles cannot be removed similarly. We show that unpinning of vortices attached to obstacles smaller than the core radius of the free vortex is possible through pacing. The wave-train frequency necessary for unpinning increases with the obstacle size and we present a geometric explanation of this dependence. Our model-independent results suggest that decreasing excitability of the medium can facilitate pacing-induced removal of vortices in cardiac tissue.

Figure 1

(Color online) (a) Wave $S_0$, pinned to an obstacle (shaded), rotates counterclockwise; wave 1 is the first pacing wave. (b) Wave 1 hits the obstacle and separates into a wave rotating counterclockwise $\left(1a\right)$ and a wave rotating clockwise $\left(1b\right)$. (c) Waves $S_0$ and $1b$ collide and merge leaving only one rotating wave $1a$ denoted $S_1$ hereafter. (d) The wave resulting from the merging of $S_0$ and $1b$ leaves the system. The interaction between the following pacing wave, 2 and $S_1$, is similar to that shown in (a)–(c). Thus, the pinned vortex persists. Numerical simulation of the Barkley model with parameters: $a=0.9$, $b=0.17$; the pacing period is $T_p=6.7$ and the radius of the obstacle is $R=6.5$.

Figure 2

(Color online) Lowering excitability results in successful detachment of pinned vortex by pacing. $S_0$ is a rotating wave whose core (dashed line) is larger than the pinning center (shaded). (a)–(c) are topologically as in Fig. 1. (d) A wavelet $1c$ is produced after collision of waves $S_0$ and $1b$ in contrast with Fig. 1d. (e) The wavelet $1c$ collides with $1a$ and the resulting wave $S_1$ is displaced away from the obstacle. (f) Subsequent pacing induces drift of the spiral wave $S_1$ to the boundary eventually removing it from the medium. The parameters are as in Fig. 1, except for $a=0.895$ and $b=0.1725$, resulting in increasing the vortex core size.

Figure 3

(Color online) (a) Parameter space of the Barkley model. Unpinning is possible in the shaded portion of the SW region, which exhibits persistent spiral waves. The thick line indicates the boundary with the SE region, where spirals cannot form. The domain where unpinning is possible shrinks with increasing size of the pinning center, the three dashed lines corresponding to $R_{obst}=0$, i.e., no obstacle (square), $R_{obst}=1.25$ (plus), and $R_{obst}=6.5$ (diamond). (b) Radius $R_{FS}$ of the free spiral and the maximum obstacle radius $R_{obst}^{\text{max}}$ from which wave trains can unpin vortices, as a function of the distance $d$ from the SE-SW boundary, along the dot-dashed line indicated in (a). Note that $R_{FS}>R_{obst}^{\text{max}}$ and both increase with decreasing $d$. [In (a), NW (BI) indicates the parameters for which steady waves are absent (the medium is bistable).]

Figure 4

(Color online) (a) The maximum pacing period $T_p^{\text{max}}$ at which unpinning is possible as a function of the obstacle radius $R_{obst}$. For the parameters $a=1.1323,b=0.2459$ that we have used, the maximum radius of obstacle from which depinning can occur is $R_{obst}^{\text{max}}=4$. $T_{FS}$ is the period of a free spiral wave and $T_{ref}$ is the refractory period. The dashed line indicates the prediction from Eq. (2). (b) The wavelet formation mechanism leading to the detachment of the pinned vortex (schematic). (c)–(f) Numerical simulation of the Barkley model. $S$ collides with wave 1 at point $C$ at an angle $\theta$. The part $1b$ of the pacing wave merges with $S$, moving out of the system. The remaining part of the pacing wave collides with the obstacle (shaded) separating into $1a$ and a small wavelet $1c$. When the length $l$ of wavelet $1c$ is larger than the critical nucleation length, $1c$ survives and collides with $S$. This results in unpinning of $S$.