Threaded Rings that Swim in Excitable Media

Cardiac tissue and the Belousov-Zhabotinsky reaction provide two notable examples of excitable media that support scroll waves, in which a filament core is the source of spiral waves of excitation. Here we consider a novel topological configuration in which a closed filament loop, known as a scroll ring, is threaded by a pair of counterrotating filaments that are perpendicular to the plane of the ring and end on the boundary of a thin medium. We simulate the dynamics of this threaded ring (thring) in the photosensitive Belousov-Zhabotinsky excitable medium, using the modified Oregonator reaction-diffusion equations. These computations reveal that the threading topology induces an exotic motion in which the thring swims in the plane of the ring. We propose a light templating protocol to create a thring in the photosensitive Belousov-Zhabotinsky medium and provide experimental confirmation that this protocol indeed yields a thring.

Figure 1

Protocol for the initiation of a threaded ring: (a) and (b) A defect (black line) emits a cylindrical wave (green) from the center. (c) Light is used to cut the top half of the emitted wave. (d) The circular end of the wave forms a scroll ring (blue) as another cylindrical wave emanates from the defect. (e) Light is used to cut a segment of the cylindrical wave to yield (f) a pair of vertical scroll wave filaments (blue) that thread the ring. Note that the $z$ direction has been stretched to facilitate visualization.

Figure 2

A threaded ring: (a) Experimental image. (b)–(c) Numerical simulation. (b) A heat map of the average value of $v$. (c) The scroll ring and threaded vertical scroll wave filaments are shown in blue, together with a heat map of $v$ on the bottom surface. The black arrows indicate the direction of motion of the excitation waves.

Figure 3

A symmetric thring swimming along the $y$ axis. (a)–(b) The filaments (blue) and waves at increasing times. (c) A 3D view of the filaments. (d) The position along the $y$ axis of the thring as a function of time.

Figure 4

An asymmetric thring swimming in a circle. (a)–(b) The filaments (blue) and waves at increasing times. (c) A 3D view of the filaments. (d) The $x$ (black) and $y$ (red) coordinates of the thring as a function of time. (e) The trajectory in the ($x$, $y$) plane.