Distinguishing pairwise and higher-order interactions in coupled oscillators from time series

Abstract

Rhythmic phenomena, which are ubiquitous in biological systems, are typically modelled as systems of coupled limit cycle oscillators. Recently, there has been an increased interest in understanding the impact of higher-order interactions on the population dynamics of coupled oscillators. Meanwhile, the estimation of a mathematical model from experimental data is an essential step in understanding the dynamics of real-world complex systems. In coupled oscillator systems, identifying the type of interaction (e.g. pairwise or three-body) is challenging, because different interactions can exhibit similar dynamical states in experimental conditions. In this study, we have developed a method based on the adaptive LASSO (Least Absolute Shrinkage and Selection Operator) to infer the interactions among oscillators from time series data. The proposed method successfully identifies the type of interaction and estimates the probabilities of pairwise and three-body couplings. Through systematic analysis of synthetic datasets, we have demonstrated that our method outperforms two baseline methods, LASSO and OLS (Ordinary Least Squares), in accurately inferring the topology and strength of couplings between oscillators. Furthermore, the proposed method is applied to human brain network data, demonstrating its practical utility. Finally, we extend the method to more general oscillatory systems, including those exhibiting the deformation of limit cycles and those with four-body interactions. These results suggest that our method is a promising tool for identifying interaction mechanisms in oscillatory systems.

Figures (5)

• Figure 1: Synchronization dynamics of weakly coupled oscillators in the presence of white noise (Eq. \ref{['eq:Kuramoto']}). Left: Time series of the Kuramoto order parameter $r$ in a weakly synchronized state for three types of interactions; (a) Pairwise interaction only, (b) Three-body interaction only, and (c) Mixture interactions, where the system has both pairwise and three-body couplings. The average values of the order parameter were $r= 0.33, 0.25$, and $0.32$ for the pairwise, three-body, and mixture interactions, respectively. Right: Snapshots of the phase distribution of the oscillators when they are synchronized at similar levels ($r= 0.67, 0.61$ and $0.67$ for the pairwise, three-body, and mixture interaction, respectively). The weights ($K_2$, $K_3$) and the coupling probabilities ($q^{(2)}$, $q^{(3)}$) were set as follows: A. $K_2= 0.1$, $q^{(2)}= 0.1$and $q^{(3)}= 0.0$, B. $K_3= 0.3$, $q^{(2)}= 0.0$, and $q^{(3)}= 0.1$, and C. $K_2= 0.1$, $K_3= 0.3$, $q^{(2)}= 0.05$ and $q^{(3)}= 0.05$.
• Figure 2: Inferring the pairwise and three-body couplings from the phase time series. The couplings (i.e. individual interaction) inferred from synthetic data sets using the proposed method, LASSO, and OLS methods are compared with the true couplings. Three synthetic datasets are analysed: pairwise interaction (Top), three-body interaction (Middle), and mixture interaction (Bottom). The coupling probabilities were set to $q^{(2)}= 0.06$, $q^{(3)}= 0.00$ in the pairwise interaction, $q^{(2)}= 0.00$, $q^{(3)}= 0.01$ in the three-body interaction, and $q^{(2)}= 0.03$, $q^{(3)}= 0.005$ in the mixture interaction. Correctly inferred (true positive) pairwise and three-body couplings are shown in black and yellow, respectively, while incorrectly inferred (false positive) pairwise and three-body couplings are shown in red and blue, respectively.
• Figure 3: Effect of observation time on inference accuracy. Performance of coupling inference (MCC) was calculated for the three types of interaction: (a) Pairwise ($q^{(2)}= q$, $q^{(3)}= 0$), (b) Three-body ($q^{(2)}= 0$, $q^{(3)}= q$), and (c) Mixture of pairwise and three-body interactions ($q^{(2)}= q/2$, $q^{(3)}= q/2$). The parameter of the coupling probability $q$ was set to 0.05, 0.10 and 0.15, on the left, centre and right panel, respectively.
• Figure 4: Inferring coupling strength from time series. Inferred coupling strengths ($\hat{W}^{(2)}_{ij}$ and $\hat{W}^{(3)}_{ijl}$) obtained by the three methods (Left: proposed method, Centre: LASSO, and Right: OLS) are compared with the true strengths ($W^{(2)}_{ij}$ and $W^{(3)}_{ijl}$) for the three types of interaction: (a) Pairwise ($q^{(2)}= 0.1$, $q^{(3)}= 0$), (b) Three-body ($q^{(2)}= 0$, $q^{(3)}= 0.1$), and (c) Mixture of pairwise and three-body interactions ($q^{(2)}= 0.05$, $q^{(3)}= 0.05$). The points on the first quadrant represent true positives, while those on the non-zero $x$-axis and $y$-axis represent false negatives and false positives, respectively. The points above (below) the diagonal line indicate an overestimation (underestimation) of the coupling strength. The circle and triangle points represent the inferred results for pairwise and three-body interactions, respectively.
• Figure 5: Application to a human brain network (Subject ID: S038). (a) Inferred coupling network. The proposed method was applied to infer the couplings of the human brain network from time series data. The true network (left) is compared with the inferred network (right). Top panels display pairwise couplings; bottom panels show three-body couplings. Only strong couplings with magnitudes greater than 0.05 are visualized. Correctly inferred (true positive) couplings are shown in black (top: pairwise) and yellow (bottom: three-body). Incorrectly inferred (false positive) couplings are shown in red (top: pairwise) and blue (bottom: three-body). (b) Inferred coupling strengths. Scatter plots show inferred versus true coupling strengths for pairwise (top) and three-body (bottom) interactions. Points on the non-zero x-axis represent false negatives (missed true couplings), while those on the non-zero y-axis represent false positives (spurious couplings).