Reduced-order dynamics of coupled self-induced stochastic resonance quasiperiodic oscillators driven by varying noise intensities

Abstract

Noise can induce nontrivial behaviors in systems with multiple timescales. The dynamical reduction of fast-slow systems remains a challenging and intriguing problem. Self-induced stochastic resonance is a noise-induced coherent phenomenon that noise is introduced to the fast dynamics. By applying the embedding phase reduction approach, the synchronization behaviors of two non-identically coupled self-induced stochastic resonance oscillators are investigated. The first passage time distribution on the slow manifolds, which is crucial in the reduced phase equation, is well approximated by the Weibull distribution and is continuously estimated regarding the noise intensity by the polynomial interpolation. The effectiveness of the reduced phase equation is validated by the Monte Carlo simulations. The methods and results may motivate data-based investigations on other multiple timescale systems and experimental findings.

1 Introduction

Noise can play nontrivial roles in multifarious dynamical systems in nature and science. Dating back to the 1980s [1, 2], the discovery of stochastic resonance (SR) violated our intuition in that appropriate amount of noise can enhance the system’s signal-to-noise ratio [3]. Numerous theoretical and experimental investigations have been inspired by the counterintuitive feature of SR, e.g., inverse stochastic resonance [4,5,6,7], double stochastic resonance [8], vibrational resonance [9, 10], coherence resonance [11, 12], chaotic resonance [13, 14], self-induced stochastic resonance [15,16,17,18,19], rotational stochastic resonance [20], to name but a few. Especially in neuroscience, extensive theoretical and experimental researches have shown that noise can have important influences on the neuronal information processing [21].

Coupled oscillators, constructing various typologies in complex networks, exhibit fruitful dynamics (e.g. in signal propagation [22]). How to reduce the dimensionality of these systems remains a big challenge [23, 24]. Reduced-order dynamics for noisy oscillators (i.e., deterministic oscillations perturbed by noise) and noise-induced oscillators (i.e., stochastic oscillations which do not exist without noise) have long been a hot topic. The former based on the phase reduction approach has aroused a warm debate around two decade ago [25,26,27,28,29,30]. The later can be more complex in that there is no deterministic limit cycle in the noise-free system as the reference orbit, but also very attractive as noise-induced oscillations are omnipresent in excitable systems [11, 31]. Phase description of noise-induced stochastic oscillators is one of the promising directions on this issue. Based on the mean first return time, Schwabedal and Pikovsky [32] introduced a phase description by generalizing the standard isophases. Thomas, Lindner, and Pérez-Cervera established the phase-amplitude description of the stochastic oscillators in terms of the slowest decaying complex and real eigenfunctions of the backward Kolmogorov operator [33, 34], which was proved to be in the same path via defining the stochastic Koopman operator by Kato et al. [35] due to the connection between the infinitesimal generator of the stochastic Koopman operator and the backward Fokker-Planck operator.

As a special case, self-induced stochastic resonance (SISR) is an interesting phenomenon that noise acting on the fast dynamics can induce very coherent oscillations that cannot exist in the absence of noise [15, 36, 37]. Compared with coherence resonance (CR) [11, 38, 39], another well-known noise-induced rhythmic behavior, SISR does not require the system to be close to bifurcation and depends critically on the large timescale separation. The large timescale separation between the fast and slow subsystems renders the transition among the slow manifolds to be instantaneous and thus, allows for the analysis of the marginal probability density [40] and enables the consideration of the slow variable as a control parameter based on the large deviation theory [15, 17, 36, 41]. SISR exhibits better periodicity of the stochastic oscillations than that of CR, despite the stronger noisy trajectories in the phase portrait [37]. Thus, SISR proves to be a better candidate for the application of the phase reduction approach [41,42,43]. In Ref. [43], the embedding phase reduction approach was proposed, which heavily relies on the calculation of the first passage time distribution (FPTD) of the jump positions on the slow manifolds. However, in practice, we may only have the raw data of the FPTD. In this work, we proved that the FPTD can be well approximated by the Weibull form and utilized the maximal likelihood estimation to obtain the scale and shape parameters. Accordingly, the reduced phase equation can be significantly simplified for various noise intensities. The proposed method is verified by the example of two non-identically coupled SISR oscillators.

2 Embedding phase reduction of the SISR oscillator

We consider the classical FitzHugh-Nagumo neuron model [44, 45] for the SISR oscillator, which is governed by

$$\begin{aligned} \begin{aligned} \begin{array}{lcl} \varepsilon {\dot{v}}&{}=&{} (v-\frac{v^3}{3}-u) + \sqrt{\sigma } \xi (t),\\ {\dot{u}}&{}=&{} v+a, \end{array} \end{aligned} \end{aligned}$$

(1)

where v and u represent the fast (membrane potential) and slow (recovery) variables, respectively. The Gaussian white noise $\xi (t)$ satisfies $\left<\xi (t)\right>=0$ and $\left<\xi (t)\xi (\tau )\right>=\delta (t-\tau )$, where $\sigma$ is the noise intensity. The small parameter $\epsilon =0.0001$ indicates the large timescale separation between the fast and slow dynamics. The bifurcation parameter is set as $a=0.9$ so that the deterministic system (i.e., $\sigma =0$) has a stable limit cycle ($\gamma _d$ in Fig. 1).

In the absence of noise, we can apply the traditional phase reduction approach since the system (1) without noise has a deterministic stable limit cycle $\gamma _d$. For the simplicity of notation, we rewrite it as: $\dot{\textbf{X}}=\textbf{F}(\textbf{X})$, where $\textbf{X}=[v,u]^\textsf{T}$ and $\textbf{F}(\textbf{X})$ is the deterministic vector field of the system (1). When subjected to the weak perturbation $\textbf{P}(\textbf{X},t)$ (e.g., in Sec. 4, $\textbf{P}(\textbf{X},t)=[0, \mu (u_j-u_i)]^\textsf{T}$), by applying the traditional phase reduction approach [46, 47], it can be reduced to the phase equation: ${{\dot{\theta }}} = \frac{\textrm{d}\Theta (\textbf{X}(t))}{\textrm{d}t} = \omega +\frac{\partial \Theta (\textbf{X})}{\partial \textbf{X}}\cdot \textbf{P}(\textbf{X},t)\approx \omega +\textbf{Z}(\theta )\cdot \textbf{P}(\theta ,t),$ where $\theta =\Theta (\textbf{X}(t))$ is the phase of the reduced system. $\textbf{Z}(\theta )=[Z_v(\theta ),Z_u(\theta )]=\frac{\partial \Theta (\textbf{X})}{\partial \textbf{X}}\big |_{\textbf{X}(\theta )=\textbf{X}_0(\theta )}$ and $\textbf{P}(\theta ,t)$ are the phase sensitivity function and perturbation evaluated at $\textbf{X}_0(\theta )$ on $\gamma _d$ [46, 47], respectively. $\omega =\frac{\partial \Theta (\textbf{X})}{\partial \textbf{X}} \cdot \textbf{F}(\textbf{X})$ denotes the frequency of the reduced system without perturbations. The phase sensitivity function $\textbf{Z}(\theta )$ can be numerically obtained by solving the adjoint equation $\omega \frac{\textrm{d}\textbf{Z}(\theta )}{\textrm{d}\theta }=-\textbf{J}(\theta )^\dagger \textbf{Z}(\theta )$ [47, 48] (see Fig. 1b). The effect of the large timescale separation can be revealed by the small values of the v component and the fast variation of the u component near the tip points (i.e., (−1, −2/3) and (1, 2/3), respectively) of the v nullcline.

Fig. 1

Phase portrait, phase sensitivity function, jump strength, and FPTD of the SISR oscillator (1). a The stable limit cycle $\gamma _d$ of the deterministic system (bold black curve) and the noisy trajectories (magenta for $\sigma =0.05$ and blue for $\sigma =0.01$). Dot-dashed lines are the nullclines. b The phase sensitivity function ${\textbf{Z}}(\theta )=[Z_v(\theta ), Z_u(\theta )]$. c The calculation of the jump strength function $K(\theta )=\theta (t+0)-\theta (t)$. The inset shows the jump function from the left (solid) and right (dashed) branches of the v nullcline. d The Monte Carlo simulation results (bars) and theoretical predictions (dashed curves, Eq.(3)) of the jump position distributions on the left (blue) and right (yellow) branches for the noise intensity $\sigma =0.03$. Other parameters: $\varepsilon =0.0001, a=0.9$

When noise is added to the fast subsystem as in the system (1), the stochastic trajectories can exhibit coherent noise-induced oscillations (for $a>1$) [15, 37, 49] or non-mean field quasiperiodic orbits (for $a<1$) [43, 50]. Shown in Fig. 1a, for $a=0.9$ in our case, it can be noticed that different noise intensities can lead to stochastic quasiperiodic orbits with different sizes [15, 17, 49](larger noise can induce earlier transitions on both branches resulting in smaller stochastic orbits). These noise-induced stochastic quasiperiodic orbits (hereafter we call them SISR oscillators for simplicity) are markedly different from the deterministic limit cycle $\gamma _d$. Therefore, traditional phase reduction approach on limit-cycle oscillators cannot be applied. To tackle with this issue, in Ref. [43], the embedding phase reduction method was proposed, wherein the SISR oscillators under various noise intensities can be uniformly embedded into one single limit cycle $\gamma _d$, sharing the same phase sensitivity function ${\textbf{Z}(\theta )}$ while having different jump position distributions. According to [43], the system (1) can be reduced as

$$\begin{aligned} {\dot{\theta }}= & {} \omega +\textbf{Z}(\theta )\cdot \textbf{P}(\theta ,t) \nonumber \\{} & {} + K(\theta _c)\delta (\theta -\theta _c), \theta _c \sim \rho _\sigma (\theta _c), \end{aligned}$$

(2)

where $K(\theta _c)$ denotes the jump strength at the jump phase $\theta _c$ as is shown in Fig. 1c ($K(\theta _c(t))=\theta _c(t+0)-\theta _c(t)$). The jump phase $\theta _c$ satisfies the FPTD $\rho _\sigma (\theta _c)$ on the left and right branches of the v nullcline. Figure 1d illustrates the FPTD $\rho _\sigma (u)$ on each branch for the noise intensity $\sigma =0.03$ ($\rho _\sigma (\theta _c)$ can be accordingly obtained via the variable transformation based on the phase reduction on $\gamma _d$). The narrow distributions indicate the coherence of the SISR oscillator. Due to the large timescale separation in the system (1), the jump process from the left and right stable branches of the cubic v nullcline can be approximated as a one-dimensional escape problem [41] with a slow varying parameter (i.e., the slow variable u). The survival probability $G(t)=\exp \left( {-\int ^{t}\frac{1}{T_\textrm{e}(t^\prime )}\textrm{d}t^\prime }\right)$ [41] where $T_{\textrm{e}}(t^\prime )$ is the mean first passage time (MFPT) from the stable (left or right) to the unstable (middle) branch and $\rho _\sigma (t)=-{\dot{G(t)}}$. After the variable transformation, the FPTD $\rho _\sigma (u)$ can be analytically calculated as [41, 43, 51,52,53]

$$\begin{aligned} \rho _\sigma ^i(u)= & {} \frac{1}{\left| \varepsilon (v^i(u)+a) T_{\textrm{e}}^i(u;\sigma )\right| }\nonumber \\{} & {} \exp \left( {-\int ^{u}\frac{1}{\varepsilon (v^i(u^\prime )+a) T_{\textrm{e}}^i(u^\prime ;\sigma )}\textrm{d}u^\prime }\right) , i=l,r,\nonumber \\ \end{aligned}$$

(3)

where l, r represent the corresponding quantity on the left and right branches of the v nullcline. $T_{\textrm{e}}^i(u;\sigma )$ is the MFPT which can be obtained as [51]

$$\begin{aligned} T_{\textrm{e}}^i(u;\sigma )= & {} \frac{2 \pi }{\sqrt{|U''(v^m) |U''(v^i)}}\textrm{exp}\nonumber \\{} & {} \left( \frac{2\left( U(v^m)-U(v^i)\right) }{\sigma }\right) , i=l,r, \end{aligned}$$

(4)

where $U(v)=-\frac{v^2}{2}+\frac{v^4}{12}+vu+C$ is the potential function of the fast subsystem for each fixed u, and C is a constant. Eq.(3) can be numerically solved since both v and $T_{\textrm{e}}^i(u;\sigma )$ are functions of u on each of the three branches of the cubic v nullcline. The accuracy of Eq.(3) can be validated by the Monte Carlo simulations (see Fig. 1d).

With the FPTD $\rho _\sigma (\theta _c)$, i.e., Eq.(3), we can obtain the dynamical evolution of the reduced system (2). However, for different noise intensities, the calculation of Eqs.(3)–(4) is cumbersome. Besides, in practical applications, the precise value of the noise strength may not be achieved. In the following section, we first show that the FPTD of the system (1) on the left and right branches of the v nullcline can be approximated by the Weibull distribution. Later, we apply the maximal likelihood estimation to obtain a continuous approximation for the FPTD against the noise intensity.

3 Maximal likelihood estimation of FPTD

3.1 Weibull distribution approximation

For each fixed $u \in (-2/3,2/3)$, the fast variables on the three branches (left, middle, and right) are $v_l = 2 \cos (\varphi +\frac{2}{3}\pi ), v_m = 2 \cos (\varphi +\frac{4}{3}\pi ),$ and $v_r = 2 \cos (\varphi )$, respectively, where $\varphi =\frac{1}{3} \arccos (-\frac{3}{2} u)$ with $\varphi \in [0,\pi /3]$. Thus, the potential function U(v) for the fixed u can be calculated on three branches. After some manipulation, the MFPT can be simplified as

$$\begin{aligned}{} & {} T_{\textrm{e}}^l(\varphi ;\sigma )=\frac{\pi }{\sin (\varphi )\sqrt{2\cos (2\varphi )+1}} \nonumber \\{} & {} \quad \exp {\frac{8\sqrt{3}}{\sigma } \sin (\varphi )^3 \cos (\varphi )}, \end{aligned}$$

(5)

$$\begin{aligned}{} & {} \quad T_{\textrm{e}}^r(\varphi ;\sigma )=\frac{\sqrt{2}\pi }{\sqrt{\sin (4\varphi +\frac{\pi }{6})-\cos (2\varphi +\frac{\pi }{3})}}\nonumber \\{} & {} \quad \exp {\frac{\sqrt{3}}{\sigma } \left[ \cos (4\varphi +\frac{\pi }{6})+2\sin (2\varphi +\frac{\pi }{3})\right] }. \end{aligned}$$

(6)

Fig. 2

Jump position distributions for various noise intensities. The dashed and solid curves represent the accurate results derived from the survival probability (Eq.(3)) and the approximation results (Eqs.(11)–(12)), respectively. Other parameters: $\varepsilon =0.0001, a=0.9$

Table 1 Maximal likelihood estimation of FPTD. $\hat{\lambda }_L$, $\hat{k}_L$, $\hat{\lambda }_R$, and $\hat{k}_R$ are the scale and shape parameters on the left and right branches, respectively

Substituting Eqs.(5)–(6) back to the survival probability, we have

$$\begin{aligned}{} & {} G_\sigma ^l(\varphi )=\exp {\int _{\frac{\pi }{3}}^{\varphi }\frac{2\sin (3\varphi ^\prime )\sin (\varphi ^\prime )\sqrt{2\cos (2\varphi ^\prime )+1}}{\varepsilon \pi (2\sin (\varphi ^\prime +\frac{\pi }{6})-a)\textrm{e}^{\frac{8\sqrt{3}\sin (\varphi ^\prime )^3\cos (\varphi ^\prime )}{\sigma }}}\textrm{d}\varphi ^\prime }, \end{aligned}$$

(7)

$$\begin{aligned}{} & {} G_\sigma ^r(\varphi )=\exp {\int _{0}^{\varphi }\frac{\sqrt{2}\sin (3\varphi ^\prime )\sqrt{\sin (4\varphi ^\prime +\frac{\pi }{6})-\cos (2\varphi ^\prime +\frac{\pi }{3})}}{\varepsilon \pi (2\cos (\varphi ^\prime )+a)\textrm{e}^{\frac{\sqrt{3}}{\sigma }\left[ \cos (4\varphi ^\prime +\frac{\pi }{6})+2\sin (2\varphi ^\prime +\frac{\pi }{3})\right] }}\textrm{d}\varphi ^\prime }.\nonumber \\ \end{aligned}$$

(8)

Eqs.(7)–(8) are still complex to handle. We note that for small noise intensities, the jump positions can be close to the tip points of the cubic v nullcline, implying that $\varphi$ is close to 0 on the left branch and $\frac{\pi }{3}$ on the right branch, respectively. From this, after having omitted high order terms, we have

$$\begin{aligned} G_\sigma ^l(\varphi ) \approx \exp (-\frac{\sigma \textrm{e}^{-\frac{8\sqrt{3}\varphi ^3}{\sigma }}}{4 \varepsilon \pi (1-a)}), \text {for}\, \varphi \rightarrow 0, \end{aligned}$$

(9)

$$\begin{aligned} G_\sigma ^r(\varphi ) \approx \exp (-\frac{\sigma \textrm{e}^{-\frac{8\sqrt{3}(\frac{\pi }{3}-\varphi )^3}{\sigma }}}{4 \varepsilon \pi (1+a)}), \text {for}\, \varphi \rightarrow \frac{\pi }{3}. \end{aligned}$$

(10)

For the jump positions to be close to the tip points of the v nullcline, to the lowest order approximation, $\varphi _l\approx \frac{\sqrt{3u+2}}{3}$ and $\varphi _r\approx \frac{\pi }{3}-\frac{\sqrt{6-3u}}{3}$. Replacing $\varphi$ with u in Eqs.(9)-(10) and using the relation between $G_\sigma ^i$ and $\rho _\sigma ^i$, we have

$$\begin{aligned} \rho _\sigma ^l(u) \approx \frac{\sqrt{9u+6}}{3\varepsilon \pi (1-a)}\exp (L(u,\sigma )-\frac{\sigma \textrm{e}^{L(u,\sigma )}}{4\varepsilon \pi (1-a)}), \end{aligned}$$

(11)

$$\begin{aligned} \rho _\sigma ^r(u) \approx \frac{\sqrt{6-9u}}{3\varepsilon \pi (1+a)}\exp (R(u,\sigma )-\frac{\sigma \textrm{e}^{R(u,\sigma )}}{4\varepsilon \pi (1+a)}),\nonumber \\ \end{aligned}$$

(12)

where $L(u,\sigma )=-\frac{8}{81}\frac{(9u+6)^{\frac{3}{2}}}{\sigma }$ and $R(u,\sigma )=-\frac{8}{81}\frac{(6-9u)^{\frac{3}{2}}}{\sigma }$. Figure 2 compared the results of the FPTD from Eqs.(11)–(12) with those from Eq.(3), showing reasonable agreement. It can be seen that Eq.(12) performs better as the jump positions on the right branch are closer to the tip point. More accurate approximations can be achieved by considering higher order terms. We note that the form of Eqs.(11)–(12) is still more complex than the traditional Weibull distribution. We can only approximate Eqs.(11)–(12) as the Weibull distribution in the sense that the first term and the second term in the first exponential term of Eqs.(11)–(12) determine the shape and magnitude of the approximated Weibull distribution, respectively. The maximal likelihood estimation in the next subsection further verifies the Weibull distribution approximation.

3.2 Maximal likelihood estimation

The traditional Weibull distribution takes the following form:

$$\begin{aligned} \rho (u)=\frac{k}{\lambda }\left( \frac{u}{\lambda }\right) ^{k-1} \exp [-\left( \frac{u}{\lambda }\right) ^{k}], \end{aligned}$$

(13)

where $\lambda$ and k are the scale and shape parameters, respectively. As the noise intensity is our only concern, we hope to figure out the dependence of these parameters on the noise. In practice, we may have the data of the FPTD. By employing the maximal likelihood estimation, the estimators of $\lambda$ and k are:

$$\begin{aligned} \begin{aligned} \begin{array}{lcl} {{\hat{\lambda }}}&{}=&{}\left[ \frac{1}{n}\sum \limits _{i=1}^{n}u_i^{{\hat{k}}}\right] ^{\frac{1}{{\hat{k}}}},\\ {{\hat{k}}}&{}=&{}\frac{\sum \limits _{i=1}^{n}u_i^{{\hat{k}}}}{\sum \limits _{i=1}^{n}u_i^{{\hat{k}}} \log (u_i)-\frac{1}{n}\sum \limits _{i=1}^{n}u_i^{{\hat{k}}} \sum \limits _{i=1}^{n}\log (u_i)}. \end{array} \end{aligned} \end{aligned}$$

(14)

Based on the jump position data via the Monte Carlo simulations (for the u values on the left branch of the v nullcline, we use $\left| u\right|$ due to the requirement of the Weibull distribution for $u\ge 0$), the scale and shape parameters in the Weibull distribution for various noise intensities can be estimated as in Table 1. Both $\lambda$ and k decrease with the increase of the noise strength. Fig. 3 plots FPTD of both the maximal likelihood estimation (13) with scale and shape parameters in Table 1 and Monte Carlo simulations, showing excellent agreement.

Fig. 3

Maximal likelihood estimation of FPTD in compared with Monte Carlo simulation results. a Left branch. b Right branch. The colored bars and black curves are obtained by Monte Carlo simulations and the Weibull distribution (13), respectively. The scale and shape parameters of Eq.(13) are those in Table 1. Other parameters: $\varepsilon =0.0001, a=0.9$

In order to generalize the results to other noise intensities, we apply the first-order and third-order polynomial interpolations on $\lambda$ and k, respectively (see Fig. 4a–b). By substituting these interpolations back to the Weibull distribution (i.e., Eq.(13)), a continuous (w.r.t $\sigma$) estimation of the FPTD can be achieved as shown in Fig. 4c–d. Good agreement can be observed which, on the one hand, validates the effectiveness of the Weibull distribution approximation, on the other hand, ensures the reliability of later applications in the reduced phase equation (e.g., Eq.(2)).

Fig. 4

Maximal likelihood estimation for FPTD approximated by the Weibull distribution (continuous estimation against the noise intensity). a–b Polynomial interpolations of the scale and shape parameters on the left ($\hat{\lambda }_L$, $\hat{k}_L$) and right ($\hat{\lambda }_R$, $\hat{k}_R$) branches. Symbols denote the Monte Carlo simulation results as those in Table 1. Lines are polynomial interpolations with $\hat{\lambda }_L=-3.3324\sigma +0.6448$, $\hat{k}_L=-9.8474\textrm{E}5\sigma ^3+1.1936\textrm{E}5\sigma ^2-5.1635\textrm{E}3\sigma +100.5536$, $\hat{\lambda }_R=-2.7606\sigma +0.6659$, and $\hat{k}_R=-5.7189\textrm{E}5\sigma ^3+7.1609\textrm{E}4\sigma ^2-3.2690\textrm{E}3\sigma +72.1296$. c–d Continuous (w.r.t $\sigma$) estimation of the FPTD (gray surfaces) on the left (c) and right (d) branches, respectively. Colored curves are those via Monte Carlo simulations. Parameters: $\varepsilon =0.0001, a=0.9$

4 Phase dynamics of non-identically coupled SISR oscillators

To validate the effectiveness of the reduced phase equation (2) based on the Weibull approximation of the FPTD (13), we consider the following two non-identically coupled SISR oscillators:

$$\begin{aligned} \begin{aligned} \begin{array}{lcl} \varepsilon {\dot{v}_i}&{} = &{} (v_i-\frac{v_i^3}{3}-u_i) + \sqrt{\sigma _i} \xi _i(t),\\ {\dot{u}_i}&{} = &{} v_i+a+\mu (u_j-u_i), \end{array} \end{aligned} \end{aligned}$$

(15)

where $i=1,2$ and $j=3-i$ indicate the label of the SISR oscillator. The independent Gaussian white noises satisfy $\left<\xi _i(t)\right>=0$ and $\left<\xi _i(t)\xi _j(\tau )\right>=\delta _{ij}\delta (t-\tau )$. The noise intensities are $\sigma _1=0.01$ and $\sigma _2=0.012$, which is the only different parameter between the two oscillators. The other parameters are the same as previous. The diffusive coupling is $G(u_i,u_j)=\mu (u_j-u_i)$, where $\mu \ll 1$ represents the coupling strength.

Fig. 5

Prediction of the phase difference dynamics by the phase coupling function and the reduced equation. a Coupling function $\Gamma _d(\phi )$. Dashed and dot-dashed lines illustrate the positions of $-\Delta \omega /\mu$ for the cases of $\mu =0.15$ and $\mu =0.02$, respectively. The red dot and circle indicate the stable and unstable synchronized states for $\mu =0.15$. b Dynamics of the phase difference with different initial conditions. Solid and dashed curves represent the cases of $\mu =0.15$ and $\mu =0.02$, respectively. Other Parameters: $\varepsilon =0.0001, a=0.9$

Fig. 6

Dynamical evolution of the phase difference of the two non-identically coupled SISR oscillator. Blue, purple and black curves represent the simulations of the original system (15), the reduced phase equation before averaging (16), and the reduced phase equation after averaging (18). The coupling strength: a $\mu =0.15$ and b $\mu =0.02$. The simulation results of Eq.(15) and (16) are averaged over 100 samples. Other parameters: $\varepsilon =0.0001, a=0.9$

According to the embedding phase reduction method, the reduced phase equation reads

$$\begin{aligned} \begin{aligned} \begin{array}{lcl} {{{\dot{\theta }}}_1}&{} = &{} \omega +\mu Z_u(\theta _1) G(\theta _1,\theta _2) + K(\theta _{c1})\delta (\theta _1-\theta _{c1}),\\ &{}&{} \theta _{c1} \sim \rho _{\sigma _1}(\theta ),\\ {{{\dot{\theta }}}_2}&{} = &{} \omega +\mu Z_u(\theta _2) G(\theta _2,\theta _1) + K(\theta _{c2})\delta (\theta _2-\theta _{c2}),\\ &{}&{} \theta _{c2} \sim \rho _{\sigma _2}(\theta ),\\ \end{array} \end{aligned}\nonumber \\ \end{aligned}$$

(16)

where $\omega$ and $Z_u(\theta )$ are the frequency of $\gamma _d$ and the u component of the phase sensitivity function evaluated on $\gamma _d$. The FPTD $\rho _{\sigma }(\theta )$ is our main focus, which can be directly obtained through the Weibull distribution approximation (13) with the parameters depending on the noise intensity as in Fig. 4. Due to the jump feature of SISR, the real frequency of the oscillator is larger than $\omega$. In fact, the average frequency can be calculated as [43] $\bar{\omega }=\omega /[1-\left<K(\theta _c)\right>/2\pi ]$, where $\left<K(\theta _c)\right>=\sum _{i=l,r}\int _{0}^{2\pi }K^i(\theta )\rho ^i_\sigma (\theta )\textrm{d}\theta$, from which, $\bar{\omega }_1=2.9248$, and $\bar{\omega }_2=2.9686$. Note that $\theta _1$ and $\theta _2$ are not continuous in time due to the jumps. However, we can define a new phase variable $\theta _*$ by re-parameterization [43] to ensure the time continuity. In terms of $\theta _*$, $Z_u(\theta )$ should be also modified as [43] $Z_u^{(\theta _l,\theta _r)}(\theta _*)=M(\theta _l,\theta _r)Z_u(\theta )$, where $M(\theta _l,\theta _r)=2\pi /(2\pi -K(\theta _l)-K(\theta _r))$ is the magnification factor. Then, the system (16) can be simplified as

$$\begin{aligned} {{{\dot{\theta }}}_{*i}} = \bar{\omega }_i+\mu {{\bar{Z}}}_{ui}(\theta _{*i}) G(\theta _{*i},\theta _{*j}), i=1,2, j=3-i,\nonumber \\ \end{aligned}$$

(17)

where we have omitted the superscript of the phase sensitivity function by considering the averaging jump phases on the two branches of the v nullcline.

To investigate the synchronization behaviors of the coupled SISR oscillators, we define the phase difference $\phi =\theta _{*1}-\theta _{*2}$. From Eq.(17) and by averaging, we have

$$\begin{aligned} {{{\dot{\phi }}}} = \Delta \omega +\mu \Gamma _d(\phi ), \end{aligned}$$

(18)

where $\Delta \omega =\bar{\omega }_1-\bar{\omega }_2$ is the frequency difference. $\Gamma _d(\phi )=\Gamma _1(\phi )-\Gamma _2(-\phi )$, and $\Gamma _i(\phi )=\frac{1}{2\pi }\int _{0}^{2\pi }{\bar{Z}}_{ui}(\phi +\psi )G(\phi +\psi ,\psi )\textrm{d}\psi$ ($i=1,2$) is the phase coupling function. It should be noted that different noise intensities can induce stochastic quasiperiodic orbits with different sizes [15, 17, 41, 54], which leads to the difference in positions in the configuration space for $\theta _{*1}=\theta _{*2}$. This causes the symmetry breaking in computing $\Gamma _d(\phi )$ (i.e., $\Gamma _1(\phi ) \ne \Gamma _2(\phi )$). Figure 5a shows the curve of the coupling function $\Gamma _d(\phi )$. From the right hand side of Eq.(18), the stationary state of the phase difference $\phi$ is by calculating $\Delta \omega +\mu \Gamma _d(\phi _s)=0$, i.e., $\Gamma _d(\phi _s)=-\Delta \omega /\mu$. The stability depends on $\Gamma _d'(\phi _s)$ (stable for $\Gamma _d'(\phi _s)<0$ and unstable for $\Gamma _d'(\phi _s)>0$). We consider two cases for the coupling strength: $\mu =0.02$ and $\mu =0.15$. It can be inferred from Fig. 5a that when $\mu =0.02$, two SISR oscillators cannot synchronize and the phase difference keeps decreasing (see the dashed curves in Fig. 5b). While for $\mu =0.15$, there is a stable fixed points near $\phi =0$ (or $2\pi$). The phase difference will converge to the equilibrium point in the long term (see the solid curves in Fig. 5b).

To verify our prediction based on the averaged phase equation (18), we perform Monte Carlo simulations on the original stochastic system (15), together with the reduced phase equation before averaging (i.e., Eq.(16)). The good agreement shows the effectiveness of the reduced phase equation. The reduced equation before averaging can even reproduce the jiggling of the phase difference due to the phase jumps.

5 Conclusion and discussion

In conclusion, two non-identically coupled SISR oscillators are investigated in this paper through the embedding phase reduction approach. The FPTD is approximated by the Weibull distribution, wherein the scale and shape parameters are estimated via the maximal likelihood estimation. Through the polynomial interpolation for these parameters with respect to the noise intensity, a continuous estimation can be achieved, which significantly simplifies the calculation of the FPTD applied in the reduced phase equation. The accuracy and effectiveness are verified by comparing the results of the reduced phase equation and the Monte Carlo simulations of the original system.

In practical applications, with the aid of the experimentally measured phase sensitivity function [55], the SISR oscillator can be completely modeled through data. Thus, the method and analysis employed in this paper may contribute to the investigation of related realistic noisy biological oscillators. Besides, the continuous estimation of the FPTD for the noise intensity can be easily extended to large coupling networks of SISR oscillators with dispersing noise strengths. As other parameters including the timescale separation parameter are fixed in this work, it would be interesting to investigate whether the Weibull distribution approximation and the corresponding phase reduction equation are still valid when the other parameters are changed.