Unpinning of scroll waves under the influence of a thermal gradient

Three-dimensional scroll waves of electrical activity are among the mechanisms believed to be responsible for the rapid, unsynchronized contraction of cardiac ventricles, thereby reducing the heart's ability to pump blood. Scroll waves can attach themselves to unexcitable obstacles, and this sometimes highly elongates their life span. Hence, the unpinning and annihilation of these vortices has attracted much attention in recent decades. In this work, we study the influence of a thermal gradient on scroll waves pinned to inert obstacles, in the Belousov-Zhabotinsky reaction. Under a temperature gradient, scroll rings were seen to unpin from these obstacles, thus strikingly reducing their lifetimes. These results were also reproduced by numerical simulations using the Barkley model.

Figure 1

Schematic diagram of experimental set-up.

Figure 2

Unpinning of scroll waves under a constant temperature gradient. Snapshots (a, b, d, e, g, h, j, k) and time-space plots (c, f, i, l) of four different experiments. A pinned, stable vortex is shown in (a) and (b). The shining round-shaped objects are glass beads of 1-mm radius that act as our inert obstacles. In the next three experiments, the angle between the interbead axis and gradient vector ($\theta$) varies as 0°  (d, e, f), 90°  (g, h, i), and 45°  (j, k, l). In each of the snapshots the left side is the hot end and the temperature gradient is 2° C ${\mathrm{cm}}^{-1}$. The star on the time-space plot marks the time at which the thermal gradient has been applied on the system. The two arrows mark the position of the snapshots, one taken before the application of thermal gradient and the other after at least one end of the scrolls has detached itself from the pinning obstacles. The time-space plots (time going left to right) span time intervals of 220 min (c), 130 min (f), 95 min (i), and 90 min (l). Each snapshot has an area of $16\times 16$ ${\mathrm{mm}}^2$. Straight, black lines on the snapshot show the stripe along which the corresponding time-space plot has been generated. Movies available [25].

Figure 3

Variation of gradient strength and its effect on unpinning. (a, b), (c, d), and (e, f) are snapshots (area $16\times 8$ ${\mathrm{mm}}^2$) and time-space plots of three different experiments with thermal gradients of 1° C ${\mathrm{cm}}^{-1}$, 2° C ${\mathrm{cm}}^{-1}$, and 3° C ${\mathrm{cm}}^{-1}$, respectively. The time-space plots (time going left to right) span time intervals of 240 min (b), 133 min (d), and 82 min (f). Each frame has been chosen at the instant when one end of the scroll wave has detached itself from the heterogeneous obstacles for the first time, denoted by the arrow in the corresponding time-space plot. The star on the time-space plot denotes the advent of thermal gradient. (g) The time required for unpinning ($t_{\text{unpin}}$) as a function of the temperature gradient. Squares (black line) stand for $\theta$ = 0°, circles (red curve) for $\theta$ = 90°, and triangles (blue line) for $\theta$ = 45°.

Figure 4

Simulation studies of unpinning of a scroll vortex under the influence of a thermal gradient. Two-dimensional snapshots (a–h) of area 65 space units $\times$ 65 space units and a representative time-space plot (i) generated from simulation of the three-dimensional Barkley model. (a, b) show the stabilization of a scroll that is attached to two inert obstacles (black spheres) in the absence of a temperature gradient. All other sets of simulations have been carried out for $\nabla T$ = 10. Each set shows one snapshot for the stable scroll before the introduction of thermal gradient (a, c, e, g) and one snapshot after the advent of unpinning (b, d, f, h). The time-space plot (i) (time increasing from top to bottom) corresponds to the case where $\theta$ = 0°  (c, d), along the black line shown in (c). The star on the time-space plot marks the time at which the thermal gradient has been applied on the system. The two arrows mark the position of the snapshots, one taken before the application of thermal gradient and the other after one end of the scroll has detached itself from the pinning obstacle.

Figure 5

Filament dynamics of unpinning for different orientation of inert obstacles. Results of two different sets of simulation (a, b, c) and (d, e, f) for the three-dimensional Barkley model. The curves (red) in the center of the cube are the filaments, the spheres (blue) are the pinning obstacles, and the curves (blue) along with the circles (purple) at the bottom of the box are the two-dimensional projections of the filament and the obstacle. In (a, b, c) the interbead axis is parallel to the gradient vector, while in (d, e, f) it is perpendicular. In (a) and (d) the vortex filament is shown attached to the two unexcitable obstacles, before the application of thermal gradient. (b) and (e) show the distortion of the filament under the influence of thermal gradient, finally leading to unpinning in (c) and (f).