Metastability induced by non-reciprocal adaptive couplings in Kuramoto models

TL;DR

The paper investigates metastability in generalized Kuramoto models with non-reciprocal, adaptive couplings arising from Hebbian and anti-Hebbian rules acting on distinct time scales. By combining fast Hebbian upper-triangular and slow anti-Hebbian lower-triangular adaptations, and using both all-to-all and sparse topologies, the authors demonstrate sustained metastable switching between two anti-phase clusters, governed by the slow adaptation of couplings and the system size. Through stability analysis and a snapshot-based predictor of switching events, they show that a small set of oscillators repeatedly switch clusters, while the rest remain in their respective clusters, producing quasi-deterministic yet effectively stochastic dynamics. The findings offer a potential mechanistic link to metastable brain activity, and point to future work extending to other limit-cycle models and topologies to further understand adaptive non-reciprocal synchronization. $R$ and $R_2$ quantify coherence and cluster structure, while $\dot{\theta}_i$, $\dot{k}_{ij}$ equations with $\beta_1=-\frac{\pi}{2}$ and $\beta_2=\frac{\pi}{2}$ capture the essential slow-fast coupling dynamics underlying the observed phenomena.

Abstract

Non-reciprocal couplings are frequently found in systems out-of-equilibrium such as neuronal networks. We consider generalized Kuramoto models with non-reciprocal adaptive couplings. The non-reciprocity refers to the type of couplings according to Hebbian or anti-Hebbian rules and to different time scales, on which the couplings evolve. The main effect of this specific combination of deterministic dynamics is an induced metastability of anti-phase synchronized clusters of oscillators. Metastable switching is typical for neuronal networks and a characteristic of brain dynamics. We analyze the metatstability as a function of the system parameters, in particular of the size and the network connectivity. The mechanism behind sudden changes in the order parameters are individual oscillators which change their cluster affiliation from time to time, providing ``weak ties" between clusters of synchronized oscillators. The time series have random features, but derive from deterministic dynamics.

Figures (14)

• Figure 1: Categorical heatmap of collective states as a function of $(\varepsilon_1, \varepsilon_2)$ for (a) $N=10$ oscillators and (b) $N=20$ oscillators, averaged over $5$ independent realizations. Colors indicate two anti-phase clusters (orange), and incoherent (gray) states; $\epsilon_1$ is varying within [0, 0.1] and $\epsilon_2$ is varying within [0, 0.003] with a step length of 0.0001. The red dashed curve should guide the eyes and roughly marks the transition boundary between two anti-phase clusters and incoherent states via a quadratic least square fit.
• Figure 2: (a) Time series of the first- and second-order Kuramoto order parameters $R(t)$ (dashed line) and $R_2(t)$ (solid line) for $\varepsilon_1=0.01$ and $\varepsilon_2=0.0001$ with $N = 10$. After initial convergence to $R_2 \approx 1$, occasional deviations occur, visible in spikes, which subsequently return to the same state. (b-c) Upper part of the couplings $k_{ij}$ (for $i < j$) and lower part of $k_{ij}$ (for $i > j$) . As soon as $R_2$ deviates from 1, a few oscillators switch clusters, which leads to changes in the slopes of the corresponding $k_{ij}$ values. Since $\varepsilon_1 = 0.01$ is significantly larger than $\varepsilon_2 = 0.0001$, the upper $k_{ij}$ values quickly reach their equilibrium values of $\pm 1$, whereas the lower $k_{ij}$ values adjust more slowly. (d) Extended time view of (c) to see some structure in the evolution of the lower couplings.
• Figure 3: Snapshots at $t=2000$ of the phases(a),the order parametres $R$ and $R_2$ (b) and the cluster sizes (c) before a switching event for $N=20$.
• Figure 4: Same as Fig. \ref{['fig3']}, but snapshot at $t=4000$ after a switching event. Note that the cluster sizes have changed to a different partition before and after the switching of oscillators. In both panels (a) and (b), two anti-phase clusters are visible, each perfectly synchronized.
• Figure 5: Snapshot-based prediction performance across $1,000$ random realizations for (a) $N=6$ and (b) $N=10$. The bars represent the total count of switching events categorized by prediction accuracy. The percentage shown above each bar indicates the frequency of that outcome relative to the total number of detected events. The overall success rate (combined Perfect and Good categories) remains remarkably stable as the system size increases.