Signal amplification enhanced by large phase disorder in coupled bistable units

We study the maximum response of network-coupled bistable units to subthreshold signals focusing on the effect of phase disorder. We find that for signals with large levels of phase disorder, the network exhibits an enhanced response for intermediate coupling strength, while generating a damped response for low levels of phase disorder. We observe that the large phase-disorder-enhanced response depends mainly on the signal intensity but not on the signal frequency or the network topology. We show that a zero average activity of the units caused by large phase disorder plays a key role in the enhancement of the maximum response. With a detailed analysis, we demonstrate that large phase disorder can suppress the synchronization of the units, leading to the observed resonancelike response. Finally, we examine the robustness of this phenomenon to the unit bistability, the initial phase distribution, and various signal waveform. Our result demonstrates a potential benefit of phase disorder on signal amplification in complex systems.

Figure 1

(a) The maximum response $G$ of Eq. (3) versus coupling strength $\lambda$ for $\gamma =0$, 0.5, 1. (b) The normalized maximum response $\tilde{G}=G\left(\gamma \right)/G\left(0\right)$ versus coupling strength $\lambda$ and level of phase disorder $\gamma$.

Figure 2

(a) The degree of spatial synchronization $\rho$ of Eq. (3) versus coupling strength $\lambda$ for $\gamma =0$, 0.5, 1. Snapshots of the spatial positions of the units are shown in panel (b) for $\lambda =5\times {10}^{-5}$, in panel (c) for $\lambda =2\times {10}^{-4}$, and in panel (d) for $\lambda =5\times {10}^{-4}$, where the indices of the units are rearranged based on the rank of the initial phases of driven signals.

Figure 3

The maximum response $G$ and the degree of spatial synchronization $\rho$ versus signal intensity $A$ for $\gamma =0$ and $\gamma =1$. The period of external signals is $T=100$ for panels (a, b) and $T=10$ for panels (c, d). The coupling strength $\lambda =5\times {10}^{-4}$ is used in Eq. (3).

Figure 4

The maximum response $G$ and the degree of spatial synchronization $\rho$ versus coupling strength $\lambda$ for $\gamma =0$ and $\gamma =1$ in SW and SF networks. (a, b) SW network with $T=100$ and $A=0.35$, (c, d) SW network with $T=10$ and $A=0.65$, (e, f) SF network with $T=100$ and $A=0.35$, and (g, h) SF network with $T=10$ and $A=0.65$.

Figure 5

The potential $V_i$ of Eq. (9) for various combinations of parameters. (a) ${\varphi }_i=0, X=0, \lambda =1.4\times {10}^{-4}$, (b) $t=0, {\varphi }_i=\pm \pi /2, X=0, \lambda =1.4\times {10}^{-4}$, (d) $t=0, {\varphi }_i=\pm \pi /2, X=0.1, \lambda =1.4\times {10}^{-4}$, (f) $t=0, {\varphi }_i=\pm \pi /2, X=0.1, \lambda =1.8\times {10}^{-4}$, and (h) $t=0, {\varphi }_i=\pm \pi /2, X=0.1, \lambda =1\times {10}^{-3}$. Panels (c, e, g) are the enlarged views of panels (b, d, f), respectively.

Figure 6

(a) The analytical maximum response $G$ for $\gamma =0$ and $\gamma =1$, obtained from Eqs. (15) and (22) for a global network with $T=100$ and $A=0.35$. (b) The analytical $M_x$ of Eq. (25) versus coupling strength $\lambda$, where the dashed line denotes $M_x=1/3$.

Figure 7

(a) The maximum response $G$ of Eq. (28) versus coupling strength $\lambda$ for $\gamma =0$ and $\gamma =1$. Phase portraits of one representative oscillator $i$ from Eq. (28) for $\gamma =1$: (b) two possible intrawell rotations at $\lambda =1\times {10}^{-4}$ depending on the initial condition of the Duffing oscillator, (c) interwell oscillation at $\lambda =1\times {10}^{-4}$. The intensity $A=0.35$ and the period $T=100$ are used.

Figure 8

(a) The maximum response $G$ of Eq. (29) versus coupling strength $\lambda$ for $\gamma =0$ and $\gamma =1$. Phase portraits of one representative oscillator $i$ from Eq. (29) for $\gamma =1$: (b) two possible limit cycles about one of two fixed points at $\lambda =5\times {10}^{-5}$ depending on the initial condition of the Lorenz oscillator, (c) irregular switches between two fixed points at $\lambda =1\times {10}^{-3}$. The intensity $A=16$ and the period $T=100$ are used.

Figure 9

(a) Density of the von Mises distribution (30) for $\kappa =0.1$, 0.2, 0.5. (b) The maximum response $G$ of Eq. (3) versus coupling strength $\lambda$ for $\kappa =0.1$, 0.2, 0.5. The intensity $A=0.35$ and the period $T=100$ are used.

Figure 10

(a) Signal $S\left(t\right)$ of Eq. (31) versus $t$ for $m=0$, 0.72, 0.99. (b) The maximum response $G$ of Eq. (3) versus coupling strength $\lambda$ for $m=0.72$ and $m=0.99$. The parameters $A=0.35$ and $T=100$ are used.