Propagation failure dynamics of wave trains in excitable systems

We report experimental and numerical results on temporal patterns of propagation failures in reaction-diffusion systems. Experiments employ the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction. The propagation failures occur in the frontier region of the wave train and can profoundly affect its expansion speed. The specific rhythms observed vary from simple periodic to highly complex and possibly chaotic sequences. All but the period-1 sequences are found in the transition region between “merging” and “tracking” dynamics, which correspond to wave behavior caused by two qualitatively different types of anomalous dispersion relations.

Figure 1

Examples of stacking (a, b) and merging (c, d) dynamics. Space-time plots of wave propagation in quasi-one-dimensional systems (a, c) and snapshots of waves in two-dimensional systems (b, c). Initial concentrations: $\left[{\mathrm{H}}_2{\mathrm{SO}}_4\right]=2.00M$ (a, b) and $0.60M$ (c, d), $\left[\mathrm{CHD}\right]=0.11M$, $\left[{\mathrm{NaBrO}}_3\right]=0.07M$ (a, b) and $0.10M$ (c, d), and $\left[\mathrm{ferroin}\right]=0.5\mathrm{m}M$. The horizontal and vertical axes span $11.7\mathrm{mm}$ and $224\mathrm{s}$ (a) and $9.5\mathrm{mm}$ and $500\mathrm{s}$ (c), respectively. Field of view: $(20.1\times 16.7){\mathrm{mm}}^2$ (b) and $(25.0\times 20.3){\mathrm{mm}}^2$ (d).

Figure 2

Wave dynamics in the tracking regime. Space-time plots of wave propagation in quasi-one-dimensional systems (a,b), a dispersion relation (c) evaluated from (a), and a snap-shot of waves in a two-dimensional system (d). Initial concentrations: $\left[{\mathrm{H}}_2{\mathrm{SO}}_4\right]=0.6M$, $\left[\mathrm{CHD}\right]=0.20M$ (a, b) and $0.11M$ (d), $\left[{\mathrm{NaBrO}}_3\right]=0.20M$ (a, b), and $0.10M$ (d), and $\left[\mathrm{Fe}\right[\mathrm{batho}\left({\mathrm{SO}}_3{)}_2{]}_3\right]=0.5\mathrm{m}M$. The horizontal and vertical axes span $8.1\mathrm{mm}$ and $600\mathrm{s}$ (a) and $13.1\mathrm{mm}$ and $570\mathrm{s}$ (b). Field of view in (d): $(21.2\times 13.0){\mathrm{mm}}^2$.

Figure 3

Dependence of the tracking velocity $c_t$ on $\left[{\mathrm{NaBrO}}_3\right]$. At higher concentrations of ${\mathrm{NaBrO}}_3$, i.e., in the merging regime, $c_t$ becomes the velocity of the leading front. Other initial concentrations are the same as in Figs. 2a, 2b.

Figure 4

Experimental space-time plots of wave dynamics in the transition region between tracking and merging and corresponding diagrams showing the time $\mathrm{\Delta }{T}_n$ elapsed between the $(n-1)$st and the $n$th pulse death. Initial concentrations as in Figs. 2a, 2b, except $\left[{\mathrm{NaBrO}}_3\right]=0.21M$ (a), $0.22M$ (b), $0.23M$ (c), and $0.24M$ (d). The horizontal and vertical axes span $6.0\mathrm{mm}$ and $250\mathrm{s}$, respectively.

Figure 5

Experimental space-time plots showing complex wave dynamics and corresponding analyses of $\mathrm{\Delta }{T}_n$. Initial concentrations as in Fig. 2a, 2b except $\left[{\mathrm{NaBrO}}_3\right]=0.20M$ (a) and $0.22M$ (b). The horizontal and vertical axes span $5.4\mathrm{mm}$ and $600\mathrm{s}$ (a) and $8.6\mathrm{mm}$ and $450\mathrm{s}$ (b).

Figure 6

Numerically obtained space-time plots and their corresponding $\mathrm{\Delta }{T}_n$ diagrams for the following $\delta$-values: 0.2700 (a), 0.2670 (b), 0.2666 (c), 0.2660 (d), 0.2652 (e), and 0.2640 (f). The horizontal space and vertical time axes span 150 and 100 units, respectively. Wave trains are created at a constant period of 5.0. The other numerical parameters are $a=0.7$, $\beta =0.3$, $ϵ={10}^{-2}$, and $\gamma =5.0$.

Figure 7

Numerical results of the $\mathrm{\Delta }{T}_n$ dependence on the parameter $\delta$ revealing a variety of dynamical states in the transition between the merging (M) and the tracking (T) regime. All numerical parameters are the same as in Fig. 6.

Figure 8

$\mathrm{\Delta }{T}_n$ dependence on the parameter $T_{\mathrm{ini}}$ at $\delta =0.2660$. All other numerical parameters are the same as in Fig. 6. The dashed line indicates $\mathrm{\Delta }{T}_n=T_{\mathrm{ini}}$.

Figure 9

Space-time plot of wave dynamics in the transition region between merging and stacking. Initial concentrations: $\left[{\mathrm{H}}_2{\mathrm{SO}}_4\right]=0.6M$, $\left[\mathrm{CHD}\right]=0.11M$, $\left[{\mathrm{NaBrO}}_3\right]=0.15M$, and $\left[\mathrm{ferroin}\right]=0.5\mathrm{m}M$. The horizontal and vertical axes span $13\mathrm{mm}$ and $300\mathrm{s}$.