Echo Behavior in Large Populations of Chemical Oscillators

Experimental and theoretical studies are reported, for the first time, on the observation and characterization of echo phenomena in oscillatory chemical reactions. Populations of uncoupled and coupled oscillators are globally perturbed. The macroscopic response to this perturbation dies out with time: At some time $\tau$ after the perturbation (where $\tau$ is long enough that the response has died out), the system is again perturbed, and the initial response to this second perturbation again dies out. Echoes can potentially appear as responses that arise at $2\tau ,3\tau ,\mathrm{...}$ after the first perturbation. The phase-resetting character of the chemical oscillators allows a detailed analysis, offering insights into the origin of the echo in terms of an intricate structure of phase relationships. Groups of oscillators experiencing different perturbations are analyzed with a geometric approach and in an analytical theory. The characterization of echo phenomena in populations of chemical oscillators reinforces recent theoretical studies of the behavior in populations of phase oscillators [E. Ott et al., Chaos 18, 037115 (2008)]. This indicates the generality of the behavior, including its likely occurrence in biological systems.

Popular Summary

Periodically flashing fireflies and rhythmically beating heart cells are just two examples of oscillatory behavior, which is ubiquitous in the natural world. Studies of oscillatory behavior are often carried out on populations of oscillators, theoretically using mathematical models or experimentally using physical model systems (e.g., collections of metronomes or chemical oscillators). Here, we present a study of large populations of chemical oscillators that are successively perturbed and subsequently exhibit a macroscopic response called an echo. We characterize the echo behavior with experiments, simulations, and analytical theory.

The echo is a macroscopically observable response that arises after a population of heterogeneous oscillators has been perturbed successively twice. After all macroscopic evidence of the two perturbations has disappeared, an echo occurs at time $2\tau$, where $\tau$ is the time between the perturbations. The manifestation of the echo implies that the system retains memory of the past application of the two pulses, even though macroscopically this information appears to have been lost. The retention of the information resides in the microscopic state through an intricate structure of oscillator phase relationships that is not apparent macroscopically. We consider ensembles of both uncoupled and coupled oscillators, and we study, for the first time, echo behavior in populations of photosensitive chemical oscillators. These oscillators are catalyst-loaded beads, roughly 200 microns in diameter, placed in a catalyst-free oscillator solution, and they are coupled via photochemical feedback. We find that echo is reliably present and that it damps out as the system is perturbed. We also conduct simulations of over 100,000 oscillators, which reveal that the echo decreases as noise increases.

We expect that our findings will pave the way for future studies examining the appearance of echo behavior in biological systems such as populations of yeast or bacteria.

Figure 1

Experimental measurements illustrating echo behavior in populations of Belousov-Zhabotinsky (BZ) oscillators. The order parameter $R$ is calculated from the phases of the oscillators and plotted as a function of time. The phases of the oscillators are determined using linear interpolation between consecutive peak times. An echo is exhibited in the magnitude of the order parameter $R$ at time $t_p+2\tau$ for a system perturbed at times $t_p$ and $t_p+\tau$. (a) Uncoupled system, $k=0$, with $t_p=376\mathrm{s}$, $\tau =378\mathrm{s}$, and $N=1295$. (b) Coupled system, $k=0.25$, with $t_p=373\mathrm{s}$, $\tau =336\mathrm{s}$, and $N=1001$. In these experiments, the critical coupling strength for the onset of synchronization is $k_c=0.70$. Average natural period and standard deviation: (a) $T_0=36.5\pm 3.2\mathrm{s}$ and (b) $T_0=42.6\pm 8.2\mathrm{s}$.

Figure 2

Simulations of the modified ZBKE model [19, 20, 21, 26] illustrating echo behavior in populations of uncoupled photosensitive BZ oscillators. The order parameter $R$ is calculated from the phases of the oscillators and plotted as a function of time. An echo is exhibited in the magnitude of $R$ at time $t_p+2\tau$ for a system perturbed at times $t_p$ and $t_p+\tau$. (a) System with $k=0$, $t_p=500$, $\tau =650$, ${\varphi }_p=0.147$, and $N=5000$. Half width at half maximum for first and second perturbations ($\mathbf{+}$) and width at half maximum for first and second echoes ($\mathbf{*}$). Inset: Maximum value of $R$ at the time of the echo as a function of the noise intensity $D$. (b) System with $k=0$, $t_p=5000$, $\tau =40000$, ${\varphi }_p=0.147$, and $N=5000$. Average natural period and standard deviation $T_0=42.5\pm 2.4$ (dimensionless time units).

Figure 3

(a) Numerical simulation of a coupled system with $k=3.35\times {10}^{-4}$, $\tau =650$, ${\varphi }_p=0.147$, and $N=5000$. Note the slower phase dispersion following each perturbation and the slightly larger magnitude of the echo. (b) Maximum value of the order parameter $R$ at the time of the echo as a function of coupling strength from simulations. The critical coupling strength for the onset of synchronization is $k_c=3.0\times {10}^{-3}$ for the simulations in (a) and (b).

Figure 4

(a) Magnitude of the order parameter $R$ at the time of the echo as a function of the size of the phase-resetting region $X$, which increases monotonically with the size of the perturbation. We show ZBKE simulation values (dark blue line) and values calculated with the geometric approach using groups I and II (light blue line). The red dashed line shows values calculated with the geometric approach using groups I–IV or with Eq. (16) from the analytical theory. (b) Phase response curves for stimuli ${\varphi }_p=0.147$ (blue line) and ${\varphi }_p=0.0218$ (red line) from ZBKE simulations. The sizes of the associated phase-resetting regions $X$ are shown by the red and blue horizontal arrows, respectively. Simulation parameters are as in Fig. 2.

Figure 5

Plots of phase as a function of frequency of oscillators in the ZBKE simulation shown in Fig. 2 at the following times: (a) immediately prior to the first perturbation at $t_p$, (b) immediately following the first perturbation, in which oscillators in the resetting region of their phase are phase reset to 0, (c) halfway between the two perturbations, $t_p+\tau /2$, (d) immediately prior to the second perturbation, (e) immediately following the second perturbation, in which oscillators in the resetting region of their phase are phase reset to 0, and (f) at the time of the echo, when a series of narrow bands is observed, each with a single branch segment. The phase-resetting region at each perturbation is given by the phase range ($1-X$) to 1, where $X$ is the size of the region as defined in Fig. 4. Here, $X=0.6$ and the phase-resetting region spans the phase interval [0.4, 1). Parameters: $\tau =650$. Other parameters as in Fig. 2. See text for color coding of the oscillators.

Figure 6

Plots of phase as a function of frequency of oscillators from the experiment shown in Fig. 1 at the following times: (a) immediately prior to the first perturbation at $t_p$, (b) immediately following the first perturbation, in which oscillators in the resetting region of their phase are phase reset to 0, (c) halfway between the two perturbations, $t_p+\tau /2$, (d) immediately prior to the second perturbation, (e) immediately following the second perturbation, in which oscillators in the phase range ($1-X$) to 1 are phase reset to 0, and (f) at the time of the echo. Note that there is no color coding, as in Fig. 5, of the oscillator groups I–IV.

Figure 7

Phase as a function of frequency of oscillators from a ZBKE simulation for (a) group I oscillators and (b) group II oscillators at the following times: (i) immediately prior to the first perturbation, (ii) immediately following the second perturbation, and (iii) at the time of the echo. All parameters are as in Fig. 2.

Figure 8

(a) Geometrical prediction of phase distribution of oscillators (solid lines) and ZBKE simulation values (symbols) at the time of the echo. Green (open circle), red (plus), black (diamond), and blue (star) symbols and lines correspond to groups I, II, III, and IV, respectively. The violet symbols (times) and line correspond to the sum of the distributions of the four groups. The ZBKE simulations were carried out with a large number of oscillators, $N=100$ 000. (b) Detail of structure from Fig. 5, color coded according to the four different groups in (a) at the time of the echo. The frequencies indicated by the left and right filled red circles correspond to $14/\tau$ and $15/\tau$, respectively. Only half of the population of group III oscillators (black) are plotted in order to not obscure the group I oscillators (green).

Figure 9

Schematic representation for the geometric analysis of the phase distribution of group III oscillators. (a) At time $t_p$, the oscillators are uniformly distributed through the refractory region. (b) Just prior to the second perturbation, the same oscillators are now uniformly distributed through the resetting region. (c) The oscillators are then phase reset by the perturbation at time $t_p+\tau$. The red, light blue, and green regions indicate the oscillators in one of three narrow phase ranges at time $t_p$. The order of their vertical stacking in (b) and (c) is arbitrary. Each horizontal band contributes equally to the total number of oscillators at a particular phase. (d) The phase range of each set (red, blue, and green) of oscillators at the time of the echo. Since the choice of phase ${\theta }_0$ is arbitrary, the phase spread of the other oscillators is bound by the sides of the parallelogram (dotted black line).

Figure 10

Values ($\xi$, ${\theta }_0$) in region $A_K$ ($K=\mathrm{I}$, II, III, IV) corresponding to oscillators in group $K$.