Phase coherence in an ensemble of uncoupled limit-cycle oscillators receiving common Poisson impulses

An ensemble of uncoupled limit-cycle oscillators receiving common Poisson impulses shows a range of nontrivial behavior, from synchronization, desynchronization, to clustering. The group behavior that arises in the ensemble can be predicted from the phase response of a single oscillator to a given impulsive perturbation. We present a theory based on phase reduction of a jump stochastic process describing a Poisson-driven limit-cycle oscillator, and verify the results through numerical simulations and electric circuit experiments. We also give a geometrical interpretation of the synchronizing mechanism, a perturbative expansion to the stationary phase distribution, and the diffusion limit of our jump stochastic model.

Figure 1

(Color online) Asymptotic phase for FHN oscillator with limit cycle in black. $I_0=0.8$ and 0.34 in (a), (b), respectively. The center of the spiral in (b) occurs at the intersection of the nullclines, and is the remnant of a destabilized fixed point as the oscillator passes through a subcritical Hopf bifurcation where $I_0$ is the bifurcation parameter.

Figure 2

Raster plots of FHN oscillators showing synchronization and desynchronization due to common impulses with rate $\lambda =1∕4T$ for both. Time axis normalized by natural frequency of oscillators. Each tick indicates the time oscillator passed through $\varphi =0$ near the limit cycle. (a) Synchronization of FHN ($I_0=0.8$, $c=0.3$, $D=5\times {10}^{-6}$) and (b) desynchronization of FHN ($I_0=0.34$, $c=0.2$, $D=5\times {10}^{-8}$). The wide blank without ticks in (b) corresponds to the situation where the orbit is transiently trapped around the unstable focus.

Figure 3

(Color online) Comparison of Ito vs Stratonovich interpretations of Eq. (1) on the PRC $G(\varphi ,c)$ of FHN for an impulse whose jump size is $c=0.2$. (a) Additive impulse $[\sigma (v,c)=c]$ and (b) linear multiplicative impulse $[\sigma (v,c)=cv]$. The curve “Ito” is calculated by affecting a discontinuous jump, i.e., impulse duration is 0. The curve “Stratonovich” and “Strat.” is calculated by continuously changing $v$ using a narrow rectangular wave form of temporal width 0.0002. The curve “Exp.” is calculated using the Wang-Zakai-Marcus approximation for the continuous narrow impulse, namely, discontinuously changing the orbit by an amount $g(v,c)=(e^c-1)v$.

Figure 4

(Color online) (a) PRCs for FHN with $I_0=0.8$ and additive impulse intensities $c∊[-0.4,0.4]$, (b) PRCs for FHN with $I_0=0.34$ and additive impulse intensities $c=-0.20,0.03,0.20$.

Figure 5

(Color online) Various PRCs and coherent states of FHN oscillators. (a), (c), (e) show PRCs $G(\varphi ,c)$, and (b), (d), (f) show corresponding phase-space diagrams of 200 oscillators a sufficient time after the initial condition shown in the insets. Poisson rate is $\lambda =1∕4T$ for (a), (b), and (f), and independent noise is $D=5\times {10}^{-6}$ for (b) and (f), and $D=9\times {10}^{-9}$ for (d). The control parameter is $I_0=0.875$ for (a), (b), (e), (f), and $I_0=0.34$ for (c) and (d). For weak additive impulse, the PRC $G(\varphi ,c)$ is a periodic function, (a), and the one-cluster (synchronized) state, (b), appears. At $I_0=0.34$, the PRC $G(\varphi ,c)$ becomes jagged when the additive impulse intensity is in a certain range as shown in (c), which often leads to common-impulse induced desynchronization, (d). For multiplicative impulse at $I_0=0.875$, a doubly periodic PRC $G(\varphi ,c)$, (e), leads to the two-cluster state, (f).

Figure 6

(Color online) Pairwise growth times of perturbations to FHN at $I_0=0.34$ for several values of the impulse intensity ($A$: $c=0.03$, $B$: $c=-0.6$, $C$: $c=0.4$, $D$: $c=0.2$, $E$: $c=-0.35$, and $F$: $c=-0.2$). Data was taken for 100–200 trials, each with an ensemble of 20 oscillators, and with Poisson impulse rate $\lambda =1∕4T$. Slope of linear least-square fit gives the Lyapunov exponent.

Figure 7

(Color online) Comparison of the Lyapunov exponents $\Lambda$ between the direct measurement from the raster plot and the theoretical prediction from the PRC for FHN with parameter $I_0=0.8$ driven by additive impulses of intensity $c$, and Poisson impulse rate $\lambda =1∕4T$.

Figure 8

(Color online) Comparison of the Lyapunov exponents $\Lambda$ between the direct measurement from the raster plot and the theoretical prediction from the PRC for FHN with parameter $I_0=0.34$ and additive impulses with rate $\lambda =1∕4T$. Labels $A,B,\dots ,F$ correspond to those in Fig. 6.

Figure 9

(Color online) On-off intermittency exhibited by two oscillators in synchronized ($I_0=0.8$, additive impulses with $c=0.1$), and clustered ($I_0=0.875$, linear multiplicative impulses with $c=0.1$) states. Independent noise with $D=9\times {10}^{-9}$, and impulses with $\lambda =1∕4T$ were used. Phase difference $\Delta \varphi$ between the oscillators from stable configuration is small (laminar region) much of the time, but large occasional bursts occur. (a) Long-time evolution of $\Delta \varphi \left(t\right)$, which shows excursions away from the synchronized state. (b) Distribution of laminar duration corresponding to (a) (arbitrary normalization). (c) Long-time evolution of $\Delta \varphi \left(t\right)$ from the $1∕2$-out-of-phase clustered state, and (d) distribution of laminar duration corresponding to (c) (arbitrary normalization). Oscillators are considered to be in the laminar state when $\Delta \varphi <0.0013$ away from synchronized or clustered states. Laminar distributions exhibit power laws with exponent $-1.5$. At this weak independent noise intensity, the phase difference between the oscillators takes only either 0 or $1∕2$ depending on the initial condition, and switching between the clustered state occurs is a very rare event.

Figure 10

(Color online) Transition between clustered states for two oscillators. $\Delta \varphi$ is the phase difference between the two oscillators. Poisson impulses with rate $\lambda =1∕4T$ and $c=0.1$ was used. A larger independent noise with $D=3\times {10}^{-4}$ is added in order to facilitate the transitions between single- and two-cluster states.

Figure 11

(a) Diagram of electrical circuit with limit-cycle behavior. Computer-generated impulses control $V_g$, which turns MOSFET $M_1$ current source on and/or off. Switch $S_1$ allows us to send common impulse to either $C{h}_1$ or $C{h}_2$. (b) Limit cycle of electrical circuit produce by measuring voltages at $C{h}_1$ or $C{h}_2$ as given in (a).

Figure 12

(Color online) PRCs $G_1(\varphi ,V_g)$ and $G_2(\varphi ,V_g)$ of electrical oscillator obtained by stimulating (a) $C{h}_1$, and (b) $C{h}_2$, which show responses of oscillators that desynchronize and synchronize, respectively, upon receiving common impulses. Each curve is labeled with a letter $(A,B,C,\dots )$ that corresponds to a location where impulse was applied ($C{h}_1$ or $C{h}_2$), and the MOSFET gate voltage creating the impulse, which corresponds to the Poisson mark $c$.

Figure 13

(Color online) Representative wave forms of electrical oscillators undergoing common-impulse-induced synchronization ($V_g=-7.79\mathrm{V}$ added to $C{h}_2$) (a) and desynchronization ($V_g=-7.69\mathrm{V}$ added to $C{h}_1$) (b) measured at $C{h}_1$. The Poisson impulse rate is $\lambda =1∕4T$. Voltages traces in (b) actually extend below $0.2\mathrm{V}$, but have been clipped to show detail.

Figure 14

Raster plots of electrical oscillators showing synchronization and desynchronization. The Poisson impulse rate is $\lambda =1∕4T$. Each tick indicates the time oscillator passed through $\varphi =0$. (a) Synchronization of electrical oscillators (impulse added to $C{h}_2$, $V_g=-7.79\mathrm{V}$) and (b) desynchronization of electrical oscillators (impulse added to $C{h}_1$, $V_g=-7.64\mathrm{V}$).

Figure 15

(Color online) Growth times of perturbations to electrical oscillator, where $(A,B,C,\dots ,F)$ correspond to that introduced in Fig. 12. Data was taken for 80–150 trials, each with an ensemble of 20 oscillators, with Poisson impulse rate $\lambda =1∕4T$. A comparison of the Lyapunov exponent with theory is shown in the inset.

Figure 16

(Color online) Schematic of evolution two nearby orbits. $a={\mathbf{X}}_0\left(\varphi \right)$ and $b={\mathbf{X}}_0\left(\varphi \right)+\mathbf{z}\left(0\right)$ represent the spatial points at which two orbits receive a common additive impulse $\mathbf{c}$ on the limit cycle. The oscillators jump to $\tilde{a}={\mathbf{X}}_0\left(\varphi \right)+\mathbf{c}$ and $\tilde{b}={\mathbf{X}}_0\left(\varphi \right)+\mathbf{z}\left(0\right)+\mathbf{c}$. After the oscillator completes one period of unperturbed motion, through analysis of the Floquet eigenvectors and eigenvalues, we see that the difference vector has shrunk, i.e., $\mid \mathbf{z}\left(T\right)\mid <\mid \mathbf{z}\left(0\right)\mid$.

Figure 17

(Color online) The stationary phase distribution of a FitzHugh-Nagumo oscillator receiving random Poisson impulses, $I_0=0.8$, $c=0.5$, and $\lambda =1∕4T$, calculated using a perturbation expansion of the forward Kolmogorov equation and by direct numerical simulation.

Figure 18

(Color online) (a) PRCs for SL model $(c_0=-c_2=12)$ for various additive impulse intensities. (b) Comparison of the Lyapunov exponents $\Lambda$ between the direct measurement from the raster plot and the theoretical prediction from the PRC for SL oscillators driven by additive impulses of intensity $c$, and Poisson impulse rate $\lambda =1∕380T$. We chose a large inter-impulse interval $\left(380T\right)$ because the SL oscillators have a very slow relaxation back to the limit-cycle orbit following a perturbation.