Colonies of threaded rings in excitable media

A thring is a recent addition to the zoo of spiral wave phenomena found in excitable media and consists of a scroll ring that is threaded by a pair of counter-rotating scroll waves. This arrangement behaves as a particle that swims through the medium. Here, we present results on the dynamics, interaction, and collective behavior of several thrings via numerical simulation of the reaction-diffusion equations that model thrings created in chemical experiments. We reveal an attraction between two thrings that leads to a stable bound pair that thwarts their individual locomotion. Furthermore, such a pair emits waves at a higher frequency than a single thring, which protects the pair from the advances of any other thring and rules out the formation of a triplet bound state. As a result, the long-term evolution of a colony of thrings ultimately yields an unusual frozen nonequilibrium state consisting of a collection of pairs accompanied by isolated thrings that are inhibited from further motion by the waves emanating from the pairs.

Figure 1

The interaction of a pair of thrings, with chemical waves displayed using a heat map of $v$ and filaments shown in blue. (a) The situation shortly after simulated optical initialization ($t/T=2.5$), (b), (c) the collision ($t/T=15.8,23.5$), and (d) the long-term dynamics, which involves reorientation and the subsequent formation of a stable bound pair ($t/T=92.4$).

Figure 2

The interaction of three thrings. (a) The situation shortly after simulated optical initialization ($t/T=2.8$), (b), (c) the formation of a bound pair that is protected from invasion by an isolated thring ($t/T=15.8,61.5$), and (d) the long-term dynamics is an isolated thring trapped by the waves emanating from the bound pair ($t/T=92.7$).

Figure 3

The evolution of a colony containing 36 thrings. (a) The initial array of $6\times 6$ thrings initialized with a regular spacing but random swimming directions ($t/T=2.5$). (b) The early stages of the motion before the pairs are formed ($t/T=20.4$). (c) Collisions take place and pairs begin to form (including with mirror images at the no-flux boundary). A thring is destroyed by a reconnection event ($t/T=34$). (d) Ultimately, a frozen state is reached, consisting of stable bound pairs together with single thrings trapped by the waves emitted from the pairs ($t/T=213$).