Asymptotic dynamics of reflecting spiral waves

Resonantly forced spiral waves in excitable media drift in straight-line paths, their rotation centers behaving as pointlike objects moving along trajectories with a constant velocity. Interaction with medium boundaries alters this velocity and may often result in a reflection of the drift trajectory. Such reflections have diverse characteristics and are known to be highly nonspecular in general. In this context we apply the theory of response functions, which via numerically computable integrals, reduces the reaction-diffusion equations governing the whole excitable medium to the dynamics of just the rotation center and rotation phase of a spiral wave. Spiral reflection trajectories are computed by this method for both small- and large-core spiral waves in the Barkley model. Such calculations provide insight into the process of reflection as well as explanations for differences in trajectories across parameters, including the effects of incidence angle and forcing amplitude. Qualitative aspects of these results are preserved far beyond the asymptotic limit of weak boundary effects and slow resonant drift.

Figure 1

Two examples of resonantly drifting spirals reflecting in the Barkley model of a generic excitable medium. The trajectories of the wave tips are drawn in black. Arrows indicate the overall direction of drift. The spiral waves at the final point in the plotted trajectory are visualized by the $u$ field of the Barkley model. Both plots use the same length scale. The boundaries are generated by a step change in medium properties, indicated by gray shading at the left-hand edges. (a) A “small-core” spiral wave approaches a boundary and doubles back on itself; its reflection angle lies on the same side of the boundary normal as its incidence angle. (b) A “large-core” spiral wave speeds up close to the boundary and travels alongside it for a short while before reflecting sharply away. (The plots were cropped to $25\times 40$ space units from simulations performed on a $50\times 50$ square domain, discretized in space with grid spacing $h=1/12$ and in time with time step $▵t=2.09\times {10}^{-3}$. The step change was located 12 space units from the left-hand domain wall. Parameters: (a) $a=0.8,b=0.05,c=0.02,{\epsilon }_s=0.035,{\epsilon }_f=1.44\times {10}^{-3},{\omega }_f=1.7893$; (b) $a=0.6,b=0.07,c=0.02,{\epsilon }_s=0.035,{\epsilon }_f=4.4\times {10}^{-4},{\omega }_f=0.9504$. Details concerning these parameters and the methods used are given in Sec. 3.)

Figure 2

$S_X,S_Y$, and $S_{\Phi }$ for a small-core spiral with $a=0.8,b=0.05$, and $c=0.02$. Also plotted in dotted gray are the vertical lines $x=\pm 2.9$, which enclose the effective boundary region. [For $\left|x\right|>2.9,S_X\left(x\right)$ and $S_Y\left(x\right)$ are less than 0.1% of $S_X\left(0\right)$ and $S_Y\left(0\right)$ respectively.]

Figure 3

Theoretical trajectory of a small-core spiral-wave reflection with ${\theta }_i=0^{\circ }$ and ${\epsilon }_f/{\epsilon }_s=1/25$. Initial conditions: $X_0=6,Y_0=0,{\Phi }_0=\pi$. Each horizontal row of vectors plots the velocity field at the instant at which the spiral center attained the given $Y$. These vectors depend on $X$ and the phase $\Phi$. The value of $\Phi$ at each horizontal slice is indicated on the right-hand side. Vector magnitudes have been scaled nonlinearly for visual clarity. The ratio of the $X$:$Y$ scales is 1:4.

Figure 4

Two theoretical trajectories in the small-core regime, initiated at $X_0=6,Y_0=0$. ${\epsilon }_f/{\epsilon }_s=1/25$. The filled points plotted along the trajectories are equally spaced in time to indicate drift speed. Incident angles are (a) ${\theta }_i\approx -{70}^{\circ }$ and (b) ${\theta }_i\approx +{70}^{\circ }$. Both spirals reflect with angle ${\theta }_r\approx +{88}^{\circ }$. The ratio of the $X$:$Y$ scales is 1:1.

Figure 5

$S_X,S_Y$, and $S_{\Phi }$ curves, together with representative theoretical reflection trajectories for three different small-core spiral waves. ${\epsilon }_f/{\epsilon }_s=1/50$. Each pair of reflection trajectories is plotted below the corresponding boundary curves. The left- and right-hand trajectories are ${\theta }_i\approx 0^{\circ }$ and ${\theta }_i\approx -{70}^{\circ }$, respectively, and include filled points, matched to the time step of the corresponding points in Fig. 4, indicating drift speed. Model parameters: in (a) and (b) $a=0.7,b=0.01$; in (c) and (d) $a=0.95,b=0.01$; in (e) and (f) $a=0.95,b=0.08$. In all cases $c=0.02$. These span a substantial extent of the small-core regime.

Figure 6

$S_X,S_Y$ and $S_{\Phi }$ for a large-core spiral with $a=0.6,b=0.07,c=0.02$. Also plotted in dotted gray are the vertical lines $x=\pm 5.0$, which enclose the effective boundary region. [For $\left|x\right|>5.0,|S_X\left(x\right)|$ and $|S_Y\left(x\right)|$ are less than 0.1% of $S_X\left(0\right)$ and $S_Y\left(0\right)$, respectively.]

Figure 7

Theoretical trajectory of a large-core spiral reflection with ${\theta }_i=0^{\circ }$ and ${\epsilon }_f/{\epsilon }_s=1/87.5$. Initial conditions: $X_0=10,Y_0=0,{\Phi }_0=\pi$. Each horizontal row of vectors plots the velocity field at the instant at which the spiral center attained the given $Y$. These vectors depend on $X$ and the phase $\Phi$. The value of $\Phi$ at each horizontal slice is indicated on the right-hand side. Vector magnitudes have been scaled nonlinearly for visual clarity. The ratio of the $X$:$Y$ axes is 1:1.

Figure 8

Effect of incident angle ${\theta }_i$ for a large-core spiral. Various theoretical trajectories are shown with different initial ${\Phi }_0$ and ${\epsilon }_f/{\epsilon }_s=1/87.5$. The filled points plotted along the trajectories are equally spaced in time to indicate drift speed. Incident angles: (a) ${\theta }_i={60}^{\circ }$, (b) ${\theta }_i=67.5^{\circ }$, (c) ${\theta }_i={75}^{\circ }$.

Figure 9

Phase dynamics for large-core spirals approaching the boundary with different incident angles. ${\epsilon }_f/{\epsilon }_s=1/87.5$. The top plot shows the curve $S_X$, for reference. The bottom plot shows the theoretical “trajectory” of the spiral phase as the spiral moves in and out of the boundary region for various incident angles. Incoming trajectories have $\Phi \in (\pi /2,3\pi /2)$ and outgoing trajectories have $\Phi \in (-\pi /2,\pi /2)$. The solid black trajectory corresponds to the reflection in Fig. 7 and the dotted and dashed trajectories correspond to the reflections in Fig. 8.

Figure 10

Effect of forcing amplitude on large-core spiral waves. Three theoretical trajectories are shown in order of increasing amplitude and include filled points, matched to the time step of the corresponding points in Fig. 8, indicating drift speed. The perturbation ratio ${\epsilon }_f/{\epsilon }_s$ in each case equals (a) $1/75$, (b) $1/50$, and (c) $1/25$.

Figure 11

Reflected angle ${\theta }_r$ versus incident angle ${\theta }_i$ for large-core spirals at different forcing amplitudes. The angles were measured from the theoretical response function trajectories at the given ${\epsilon }_f/{\epsilon }_s$ ratios.

Figure 12

Comparison between theory and direct numerical simulation (DNS) of the Barkley model for a variety of parameter values and incident angles in the small-core regime. In each case the rotation center of the spiral wave in the DNS is plotted (open circles) every 30th rotation period. The theoretical trajectories (curves with solid dots) use an initial condition selected such that they agree with the DNS trajectory at a point close to the boundary. Solid dots are separated by a time corresponding to 30 rotation periods. Also shown are the rotating spiral tip trajectories, dotted in gray, and the step boundary at $x=0$, dashed in gray. Each of the three columns corresponds to a different choice of model parameters broadly spanning the small-core parameter regime. Within each column two incident angles are shown: one normal and one oblique to the boundary. Parameters in (a) and (b): $a=0.8,b=0.05,c=0.02,{\omega }_f=1.850564,{\epsilon }_f/{\epsilon }_s=4\times {10}^{-5}/3.5\times {10}^{-3}=1/87.5$; in (c) and (d): $a=0.7,b=0.01,c=0.02,{\omega }_f=2.043489,{\epsilon }_f/{\epsilon }_s=4\times {10}^{-5}/2\times {10}^{-3}=1/50$; in (e) and (f): $a=0.95,b=0.08,c=0.02,{\omega }_f=1.768359,{\epsilon }_f/{\epsilon }_s=4\times {10}^{-5}/2\times {10}^{-3}=1/50$.

Figure 13

Comparison between theory and direct numerical simulation (DNS) of the Barkley model in the large-core regime verifying the theoretical predictions for the role of incident angle and forcing amplitude. Plot (a) shows a simulation of the case explained theoretically in Fig. 7, in which ${\theta }_i\approx 0^{\circ }$. Among (a), (b), and (c), the incident angle was varied from $0^{\circ }$ to approximately ${60}^{\circ }$ and ${70}^{\circ }$, respectively, keeping all other parameters fixed. In (d), the resonant forcing perturbation used was twice that of (a), while the incident angle and all other parameters remained fixed. Thus the effects of incident angle and forcing amplitude are seen to agree with those predicted in Figs. 8 and 10 and explained theoretically in Sec. 4b. In each case the rotation center of the spiral in the DNS is plotted (open circles) every 10th rotation period. The theoretical trajectories (curves with solid dots) use an initial condition selected such that they agree with the DNS trajectory at a point close to the boundary and are plotted with a time step (time between successive points) corresponding to 10 rotation periods of the simulation. Also shown are the rotating spiral tip trajectories, dotted in gray, and the step boundary at $x=0$, dashed in gray. Parameters: $a=0.6,b=0.07,c=0.02$, and ${\omega }_f=0.9164372$; in (a)–(c) ${\epsilon }_f/{\epsilon }_s=4\times {10}^{-5}/3.5\times {10}^{-3}=1/87.5$; in (d) ${\epsilon }_f/{\epsilon }_s=8\times {10}^{-5}/3.5\times {10}^{-3}=1/43.75$.