Density wave propagation of a wave train in a closed excitable medium

A wave train in an excitable reaction-diffusion medium shows a variety of spatiotemporal patterns as a result of interactions between the individual waves. In this paper, we report a novel spatiotemporal pattern in a wave train in a closed excitable medium. We carried out experiments using a photosensitive Belousov-Zhabotinsky reaction with Ru(bpy)${}_3$${}^{2+}$ as a catalyst and a numerical calculation using the FitzHugh-Nagumo equation. A wave train was locally distributed as an initial condition and the number of waves was systematically varied. In both the experiment and numerical calculation, density wave propagation was formed in a wave train during relaxation with a large number of waves. Our results suggest that density wave propagation originates from inhibitory interaction between the waves.

Figure 1

(a) Schematic illustration of the experimental setup based on the photosensitive BZ reaction. BZ film was placed on a glass plate and covered with silicone oil to prevent evaporation of the solvent. The reaction solution was maintained at a reaction temperature of 298 ± 1 K using a Peltier device. The condition of the reaction field was controlled by light field projected with a liquid-crystal projector. (b) Schematic illustration of the reaction field on the BZ film. There were three different regions [i.e., (I) oscillatory (<160 lx), (II) suppressed (17500 lx), and (III) excitable (160 lx) regions].

Figure 2

(a) Snapshots and (b) time series of the traveling velocity of chemical waves. A regular periodic distribution and an inhomogeneous distribution were observed for (i) monotonic relaxation ($N_w=8$) and (ii) density wave propagation ($N_w=13$), respectively. The chemical waves travelled with a constant velocity of 0.10 ± 0.02 mm s${}^{-1}$ during monotonic relaxation ($N_w=8$), while the chemical waves repeatedly alternated between acceleration and deceleration during density wave propagation ($N_w=13$).

Figure 3

Space-time diagrams for (a) monotonic relaxation ($N_w$ = 8) and (b) density wave propagation ($N_w$ = 13). (c) Time series of snapshots during density wave propagation. The chemical waves are numbered from the head to tail of a wave train at the initial state ($t=0$). The circled numbers indicate that the wave was in a high-density region, and the filled circles show that the wave was in a low-density region.

Figure 4

Average velocity of waves $v_{\mathrm{avg}}$ depending on the number of waves, $N_w$. The error bar denotes the standard deviation of the time mean velocity of individual waves, and we performed at least three experiments for each $N_w$, except for $N_w$ = 13. The background color indicates the relaxation process (white: monotonic relaxation, gray: density wave propagation).

Figure 5

Illustration of (a) a wave train in a closed reaction field and (b) profile of the inhibitor concentration. We defined the interval between the waves, $d_i$, and the decay length of the inhibitor, ${\lambda }_i$, for the $i$th wave as shown in the profile (b).

Figure 6

Average velocity of traveling waves $v_{\mathrm{avg}}$ depending on the number of waves $N_w$ as the result of numerical calculation (black filled circles). When the velocity oscillated, both the local maximum and local minimum velocities are shown as black bars.

Figure 7

Traveling velocity $v_i$ depending on the interval between the waves $d_i$ for a 1D system of numerical calculation. The empty circles denote the velocities for the steady state. We set ten waves with a regular interval and determined the traveling velocity in a steady state by a numerical calculation [16]. The broken and solid lines indicate the trajectories for $N_w$ = 7 and 13, respectively. The arrows exhibit the direction of time evolution.

Figure 8

Decay length λ of the inhibitor concentration depending on the traveling velocity $v$ obtained by numerical calculation ($N_w$ = 13).