Revival of oscillation from mean-field-induced death: Theory and experiment

The revival of oscillation and maintaining rhythmicity in a network of coupled oscillators offer an open challenge to researchers as the cessation of oscillation often leads to a fatal system degradation and an irrecoverable malfunctioning in many physical, biological, and physiological systems. Recently a general technique of restoration of rhythmicity in diffusively coupled networks of nonlinear oscillators has been proposed in Zou et al. [Nat. Commun. 6, 7709 (2015)], where it is shown that a proper feedback parameter that controls the rate of diffusion can effectively revive oscillation from an oscillation suppressed state. In this paper we show that the mean-field diffusive coupling, which can suppress oscillation even in a network of identical oscillators, can be modified in order to revoke the cessation of oscillation induced by it. Using a rigorous bifurcation analysis we show that, unlike other diffusive coupling schemes, here one has two control parameters, namely the density of the mean-field and the feedback parameter that can be controlled to revive oscillation from a death state. We demonstrate that an appropriate choice of density of the mean field is capable of inducing rhythmicity even in the presence of complete diffusion, which is a unique feature of this mean-field coupling that is not available in other coupling schemes. Finally, we report the experimental observation of revival of oscillation from the mean-field-induced oscillation suppression state that supports our theoretical results.

Figure 1

Two ($N=2$) mean-field coupled van der Pol oscillators [Eqs. (1)]: Bifurcation diagram with $\epsilon$ for (a) $\alpha =1$, (b) $\alpha =0.8$, and (c) $\alpha =0.701$ at $Q=0.7$. The red (gray) line is for the stable fixed point, black lines are for unstable fixed points, the green solid circle represents amplitude of stable limit cycle that emerges through Hopf bifurcation, and the blue open circle represents that of an unstable limit cycle. (d) Two parameter bifurcation diagram in the $\epsilon -Q$ space for $\alpha =1$. (e) Shrinking of death region in $\epsilon -Q$ space for decreasing $\alpha$. Area below each of the curves for a particular $\alpha$ represents the oscillation suppression region and above that shows the oscillating zone. (f) The variation of death and limit cycle (LC) region are shown in the $\alpha -Q$ space for different $\epsilon$: The upper (lower) portion of each curve represents the stable LC (death) zone. Other parameter: $a=0.4$.

Figure 2

Network of mean-field coupled van der Pol oscillators [Eqs. (1)] with $Q=0.7, a=0.4, N=100$: Spatiotemporal plots showing [(a) and (b)] AD and its revival, respectively, with decreasing $\alpha$ at $\epsilon =2$. [(c) and (d)] The OD and its revival, respectively, with a decreasing $\alpha$ at $\epsilon =5$. The upper rows [i.e., (a) and (c)] have $\alpha =1$ and the lower rows [i.e., (b) and (d)] have $\alpha =0.75$. The first $t=5000$ time is excluded and then $x_{i}s$ for the next $t=100$ are plotted.

Figure 3

Stuart-Landau oscillators, $\omega =2$: Bifurcation diagram with $\epsilon$ (a) $\alpha =1$ ($Q=0.5$), (b) $\alpha =0.5$ ($Q=0.5$). (c) Two parameter bifurcation diagram in $\epsilon -Q$ space for different $\alpha$. (d) Spatiotemporal plots of $N=100$ Stuart-Landau oscillators showing OD at $\alpha =1$ (upper panel) and the revival of oscillation at $\alpha =0.6$ (lower panel): $Q=0.5$ and $\epsilon =6$. The time scale is the same as in Fig. 2.

Figure 4

Experimental circuit diagram of the modified mean-field coupled VdP oscillators. A, A1-A4, and AQ are realized with TL074 op-amps. All the unlabeled resistors have value $R=10 \mathrm{k}\mathrm{\Omega }$. $C=10$ nF, $R_a=250\mathrm{\Omega }, V_{\delta }=0.1$ V. Box denoted by “B” is op-amp based buffers; inverters are realized with the unity gain inverting op-amps. $\otimes$ sign indicates squarer using AD633.

Figure 5

[(a) and (c)] Experimental real time traces of $V_{x1}$ and $V_{x2}$ along with the [(b) and (d)] numerical time series plots of $x_1$ and $x_2$. [(a) and (b)] With $R_{\epsilon }=5 \mathrm{k}\mathrm{\Omega }$ (i.e., $\epsilon =2$) a decreasing $R_{\alpha }$ ($\alpha$) restores oscillation (LC) from AD: AD at $R_{\alpha }=9.7 \mathrm{k}\mathrm{\Omega }$ ($\alpha =0.97$) and LC at $R_{\alpha }=7.07 \mathrm{k}\mathrm{\Omega }$ ($\alpha =0.707$). [(c) and (d)] With $R_{\epsilon }=2$ $\mathrm{k}\mathrm{\Omega }$ (i.e., $\epsilon =5$) a decreasing $R_{\alpha }$ ($\alpha$) restores oscillation (LC) from OD: OD at $R_{\alpha }=9.42 \mathrm{k}\mathrm{\Omega }$ ($\alpha =0.942$) and LC at $R_{\alpha }=7.31 \mathrm{k}\mathrm{\Omega }$ ($\alpha =0.731$). Others parameters are $R_Q=3.5 \mathrm{k}\mathrm{\Omega }$ ($Q=0.7$) and $R_a=250$ $\mathrm{\Omega }$ ($a=0.4$). $y$ axis: (a) 200 mv/div and (c) 100 mv/div; $x$ axis: 380 $\mu \mathrm{s/div}$.