Averaging approach to phase coherence of uncoupled limit-cycle oscillators receiving common random impulses

Populations of uncoupled limit-cycle oscillators receiving common random impulses show various types of phase-coherent states, which are characterized by the distribution of phase differences between pairs of oscillators. We develop a theory to predict the stationary distribution of pairwise phase differences from the phase response curve, which quantitatively encapsulates the oscillator dynamics, via averaging of the Frobenius-Perron equation describing the impulse-driven oscillators. The validity of our theory is confirmed by direct numerical simulations using the FitzHugh-Nagumo neural oscillator receiving common Poisson impulses as an example.

Figure 1

(Color online) (a) PRC $G\left(\varphi \right)$ for various values of additive impulse intensity $c$ for the FHN oscillator with $I=0.34$, with the PRCs of smaller amplitudes shown enlarged in the inset. (b) Averaged phase difference transition probability $X(\xi ,{\xi }^{\prime })$ for additive impulses with $c=-0.2$, $D=2.5\times {10}^{-5}$, corresponding to the case shown in Fig. 2. (c), (d) PRCs for $I=0.875$ with multiplicative impulses, and corresponding transition probability for $c=0.5$, $D=2.5\times {10}^{-5}$, corresponding to the case shown in Fig. 5. The PRC of FHN gains additional symmetry $G\left(\varphi \right)=G(\varphi +0.5)$ [as does the transition probability $X(\xi ,{\xi }^{\prime })=X(\xi \pm 0.5,{\xi }^{\prime }\pm 0.5)$] with application of balanced, multiplicative noise, $\sigma (v,c)=cv$.

Figure 2

(Color online) Comparison of $U\left(\xi \right)$ for the case of $\Lambda <0$ calculated using the averaged Frobenius-Perron equation (FPE) and measured via simulation (Sim). (a) shows the global distribution on a semilogarithmic scale, and (b) the distribution near $\xi =0$ on a log-log scale for $\xi >0$. The intensity of independent, additive noise (diffusion) is varied ($D=9\times {10}^{-8}$, $1\times {10}^{-6}$, $2.5\times {10}^{-5}$) while the intensity of the common impulse $(c=0.5)$ is kept constant for FHN oscillators with $I_0=0.875$. It can be seen that lowering the independent noise narrows and increases the height of the peaks of the distribution near $\xi =0$. Because the Lyapunov exponent remains constant, the slope is preserved for various diffusion strengths.

Figure 3

(Color online) Comparison of $U\left(\xi \right)$ for the case of $\Lambda >0$ calculated using the averaged Frobenius-Perron equation (FPE) and measured via simulation (Sim). (a) shows the global distribution on a semilogarithmic scale (note the $y$-axis range in comparison with Figs. 2, 5), and (b) the distribution near $\xi =0$ on a log-log scale. The intensity of independent, additive noise is varied ($D=9\times {10}^{-8}$, $1\times {10}^{-6}$, $2.5\times {10}^{-5}$) while the intensity of the common impulse $(c=-0.2)$ is kept constant for FHN oscillators with $I_0=0.34$. Due to the inherent instability of the $\xi =0$ state, the distribution of $\xi$ reaches a limiting value as the independent, additive noise is lowered.

Figure 4

(Color online) Power-law distributions of phase difference $U\left(\xi \right)$ near $\xi =0$ on a log-log scales for the FHN oscillator with $I=0.34$. The intensity of independent, additive noise is kept constant $(D=1\times {10}^{-6})$ while the intensity of the common impulse is varied $(c=-0.2,0.05,0.1)$. As the Lyapunov exponent of the system is changed, the slope of the power law changes correspondingly.

Figure 5

(Color online) Comparison of two-clustered $\xi$ distribution for the case of $\Lambda <0$ calculated using the averaged Frobenius-Perron equation (FPE) and measured via simulation (Sim) for impulses with $c=0.5$, FHN bifurcation parameter $I_0=0.875$, and independent additive noise ($D=9\times {10}^{-8}$, $1\times {10}^{-6}$, $2.5\times {10}^{-5}$). (a) shows the global distribution on a semilogarithmic scale and (b) the distribution near $\xi =0$ on a log-log scale.