Oscillatory activity regulation in an ensemble of autonomous mercury beating heart oscillators

Collective behavior of an ensemble of directly or indirectly coupled oscillators can be a function of population density. Experiments using autonomous mercury beating heart (MBH) oscillators coupled through their surroundings are employed, to study the existence of quorum-like (population dependent) phenomena. Two coupling mechanisms are used, namely, static and dynamic coupling. For the static coupling scheme, the transitions of a subset of the coupled oscillators occur from active (oscillatory) to inactive (quiescent) state and vice versa. A continuous variation of collective dynamics was observed as the population of the oscillators increased. For the dynamic coupling scheme, the time for which the coupled oscillators are active changes sharply as the population increases beyond a certain threshold.

Figure 1

Schematic circuit diagram to implement the static coupling for the ($i=1$) autonomous MBH oscillator with its surroundings. Configurations for (a) the emergence of oscillations and (b) the extinction of oscillations. (c) Surrounding dynamics for the $i\mathrm{th}$ ($i=1$) MBH oscillator.

Figure 2

Normalized area time series ($A$) of the dynamical state of a single MBH oscillator for the emergence of oscillations case showing (a) the inactive state (before coupling) and (b) the active state of the oscillator (after coupling).

Figure 3

${\mathrm{N}}_{\mathrm{dead}}$/N vs N plot for the inactive to active state transition. The coupling resistance $R_C=10\mathrm{\Omega }$ is used. A solution of 0.5 M ${\mathrm{H}}_2{\mathrm{SO}}_4$ and 2 mM ${\mathrm{K}}_2{\mathrm{Cr}}_2{\mathrm{O}}_7$ oxidizing agent is used as the common electrolyte.

Figure 4

Normalized area time series ($A$) of the dynamical state of a single MBH oscillator for the extinction of oscillation case showing (a) the active state (before coupling) and (b) the inactive state of the oscillator (after coupling).

Figure 5

${\mathrm{N}}_{\mathrm{alive}}$/N vs N plot for the active to inactive state transition. The coupling resistance is $R_C=2\mathrm{\Omega }$. A solution of 2 M ${\mathrm{H}}_2{\mathrm{SO}}_4$ and 8 mM ${\mathrm{K}}_2{\mathrm{Cr}}_2{\mathrm{O}}_7$ is used as the common electrolyte.

Figure 6

Schematic circuit diagram showing the oscillators are coupled to the common medium ($\tilde{S}$). $R_c\approx 2\mathrm{\Omega }$ is the coupling resistance. All the iron nails are connected to a common ground. The unequal sizes of mercury shown in the schematic diagram indicate the different quantities of mercury for the oscillators and the common medium. Each oscillator contains an equal volume of mercury drop of 0.6 ml, and the common medium $\tilde{S}$ contains 13 ml of mercury drop.

Figure 7

Normalized area time series ($A$ and $A_i$) of the surrounding and $N=4$ quasi-identical MBH oscillators showing (a) the inactive state of the common medium ($\tilde{S}$) before coupling and (b) the active state of the surrounding after coupling. (c) Unsynchronized state of the oscillators before coupled through the common medium and (d) synchronized activity of the coupled oscillators.

Figure 8

Semilogarithmic plot between $T$ vs $N$, whre $T$ represents the oscillation persistence time of the coupled oscillators and $N$ is the oscillator number. The exponent $\beta$ of the linear fit of the plot is $0.90\pm 0.02$. The error bars are from the five independent experimental trials.