Spiral wave dynamics under feedback via an equilateral triangular sensory domain

We perform a numerical study of the trajectories of spiral wave cores in excitable systems whose excitability is modulated in proportion to the integral of the activity in a sensory domain in the shape of an equilateral triangle. As a result of this domain shape having vertices opposite sides, unusual forms of lobed limit cycles occur, which are destroyed and then re-form as the domain size is varied. Some key results are also demonstrated experimentally using the light-sensitive Belousov-Zhabotinsky reaction. To characterize the observed behavior, we introduce the concept of express and stagnation zones, which are regions where the trajectory moves particularly rapidly or slowly. The location and strength of the zones far from the domain are accounted for by approximating the parts of the spiral wave crossing the domain by a series of plane waves.

Figure 1

Drift vector plots. In this and subsequent plots of this type, the black triangle is the domain, the white curves are the spiral tip paths, and the arrows indicate the drift velocity. The length of all but the small, narrow-headed arrows is proportional to the drift speed (although the proportionality constant differs for each plot). Black (white) arrows are in regions where the field divergence is positive (negative). The background is shaded according to the inverse-magnitude scaled divergence. Regions of negative (positive) divergence have a dark (light) background. $d=\left(\mathrm{a}\right)$ 1 and (b) 1.5. In (a) and (b) the tip trajectory starts near the central unstable focus and ends up on the innermost stable limit cycle. (c) $d=1.7$—tip trajectories are shown following the first two stable limit cycles.

Figure 2

Drift velocity plot for $d=1.85$. Tip trajectories follow (parts of) the first three stable limit cycles. The second and third are examples of vertex and lateral lobed limit cycles, respectively. The isolated circle marks the position of a stable focus.

Figure 3

Drift velocity plot for $d=2.70$. Selected fixed points referred to in the text are inside white circles.

Figure 4

Experimental results for various $(d,k_{\mathrm{fb}})$: (a) (1.0, 0.6); (b) (1.3, 0.6); (c) (2.3, 1.0); (d) (2.5, 0.9); (e) (1.8, 0.6); (f) (1.8, 1.4). The spiral wave image is taken at the start of the tip trajectory. Feedback is initiated after a few revolutions of the tip.

Figure 5

Plane wave approximation. (a) Construction used to find $L\left(p\right)$. The spiral core is in the direction of the dashed line. Solid lines perpendicular to the dashed line are wave fronts. In (b) the small circle represents the spiral core and the arrow the current direction of motion of the tip.

Figure 6

Distances of centers of attracting express zones in the directions of (a) the vertices and (b) the midpoints of the sides. Solid lines, from computational results; dashed line, from PWA. The dotted lines show the edge of the domain.

Figure 7

$(1∕d,A)$ plot of maximum and minimum drift vector amplitudes lying in the directions $\theta =30°$ (open circles) and 90° (filled circles) for the express zones at a distance of about 6 from the centroid. The dotted line passes through the value with the smallest range of amplitudes.

Figure 8

Inverse-magnitude scaled divergence plots for $d=\left(\mathrm{a}\right)$ 3.0 and (b) 4.0.