Distinguishing Direct from Indirect Interactions in Oscillatory Networks with Multiple Time Scales

We propose a method to infer the coupling structure in networks of nonlinear oscillatory systems with multiple time scales. The method of partial phase synchronization allows us to infer the coupling structure for coupled nonlinear oscillators with one well-defined time scale. The case of oscillators with multiple time scales has remained a challenge until now. Here, we introduce partial recurrence based synchronization analysis to tackle this challenge. We successfully apply the proposed method to model systems and experimental data from coupled electrochemical oscillators. The statistical significance of the results is evaluated based on a surrogate hypothesis test.

Figure 1

Networks of coupled oscillators. Dashed arrows indicate indirect interactions.

Figure 2

(a)–(c) Bivariate synchronization indices and (d)–(f) multivariate PARS indices depending on the coupling strength $\mu$ for three interdependent funnel Rössler systems exhibiting multiple time scales. The significance levels (gray) are based on the method of twin surrogates. The curves have been smoothed over 8 data points. (e) Only the PARS indices correctly determine the indirect coupling between oscillators ${\mathit{\xi }}_1(t)$ and ${\mathit{\xi }}_3(t)$.

Figure 3

Multivariate PARS analysis with significance levels (gray) of four interdependent funnel Rössler systems with coupling strengths ${\mu }_{1,2}={\mu }_{2,4}={\mu }_{1,4}=0$ and $\mu ={\mu }_{2,3}={\mu }_{1,3}={\mu }_{3,4}>0$. All oscillators are contaminated by 50% of auto- and cross-correlated observational noise ${\eta }_i(t)=a{\eta }_i(t-1)+{\chi }_i(t)+b\sum _{j\ne i}{\chi }_j(t)$ where $i=1,\dots ,4$, $a=0.1$, $b=0.01$ and ${\chi }_i$ is standard Gaussian white noise. The PARS indices correctly determine both the indirect couplings (a)–(c) and the direct couplings (d)–(f).

Figure 4

(a) Segment of the time series and (b) power spectrum of current oscillations exhibiting multiple time scales of one of the three electrochemical oscillators.

Figure 5

(a)–(c) Bivariate analysis and (d)–(f) multivariate PARS analysis with significance levels (gray) of three electrochemical oscillators with varied local interactions depending on the resistance $R$. (e) PARS analysis correctly determines the indirect coupling between the oscillators $\{1,3\}$.