Phase Diffusion in Unequally Noisy Coupled Oscillators

We consider the dynamics of two directionally coupled unequally noisy oscillators, the first oscillator being noisier than the second oscillator. We derive analytically the phase diffusion coefficient of both oscillators in a heterogeneous setup (different frequencies, coupling coefficients, and intrinsic noise intensities) and show that the phase coherence of the second oscillator depends in a nonmonotonic fashion on the noise intensity of the first oscillator: as the first oscillator becomes less coherent, i.e., worse, the second one becomes more coherent, i.e., better. This surprising effect is related to the statistics of the first oscillator which provides a source of noise for the second oscillator, that is non-Gaussian, bounded, and possesses a finite bandwidth. We verify that the effect is robust by numerical simulations of two coupled FitzHugh-Nagumo models.

Figure 1

Effective diffusion coefficients of two oscillators ${\mathcal{D}}_{\text{eff},1,2}$ versus noise intensity in the first oscillator $D_1$. Symbols show the result of direct numerical simulation of an ensemble of $2^{20}$ pairs of stochastic oscillators, Eq. (1). Solid lines correspond to the theory. (a),(b) Effect of the frequency detuning $\nu$, with $G_{2\to 1}=0.05$, $G_{1\to 2}=0.1$, $D_2={10}^{-3}$, ${\omega }_{1,2}=1\pm \nu /2$. (c),(d) Effect of the coupling strength $G_{2\to 1}$, with ${\omega }_{1,2}=1$, $G_{1\to 2}=0.1$, $D_2={10}^{-3}$.

Figure 2

Numerical simulations of coupled FitzHugh-Nagumo models. Rows 1, 2, and 3 correspond to the values of noise intensity in the first oscillator, $D_1=0.01$, 0.1, and 1.0, respectively. (a) Time traces of oscillators’ voltage variables. Upper trace (red line) shows $v_1(t)$, lower trace (black line) shows $v_2(t)$. (b) Power spectral densities (PSDs), corresponding to time traces in (a). (c) Normalized autocorrelation functions $C(\tau )$ corresponding to PSDs in (b). In panels (b) and (c), dotted (red) and solid (black) lines correspond to the first and second oscillators, respectively. Other parameters are $G_{2\to 1}=0$, $G_{1\to 2}=0.1$, $I_{\mathrm{ext}}=0.4$, $D_2={10}^{-3}$.

Figure 3

Direct numerical simulation of an ensemble of $2^{15}$ pairs of stochastic FHN oscillators Eq. (10). Effective phase diffusion coefficient of the first and second oscillators ${\mathcal{D}}_{\text{eff},1,2}$ versus noise intensity of the first oscillator $D_1$ for the indicated values of the coupling strength $G_{2\to 1}$. Other parameters are $G_{1\to 2}=0.1$, $D_2={10}^{-3}$.