Two-Population Kuramoto-Sakaguchi Model

• The two-population Kuramoto-Sakaguchi model is a framework for studying dual-ensemble synchronization where distinct coupling parameters, phase lags, and noise yield states like incoherence, zero-lag, and traveling waves.
• It employs reduced ODE systems and bifurcation analysis to capture the evolution of order parameters and phase differences, clarifying transitions between synchronized and discordant states.
• Its applications span neural, chemical, and engineered systems, offering insights into bistability, chimera states, and noise-induced order restoration in complex networks.

The two-population Kuramoto–Sakaguchi model describes the collective dynamics of two interacting ensembles of phase oscillators, each of which can exhibit internal synchronization while also competing or cooperating to achieve a preferred phase relationship with the other population. This model generalizes the classic Kuramoto and Kuramoto–Sakaguchi formulations by introducing population-specific coupling parameters (both in- and out-couplings), phase lags ("frustration"), and often stochastic influences, thereby producing a wide array of dynamical phenomena including discordant synchronization, bistability, hysteresis, traveling waves, chimera states, and chaotic dynamics.

1. Mathematical Formulation and Model Setup

The dynamical equations for a prototypical two-population Kuramoto–Sakaguchi system take the form

$$\dot{\phi}_i = \omega_0 + \frac{K_i}{N} \sum_{j=1}^N G_j \sin(\phi_j - \phi_i) + \xi_i(t),$$

where $i$ indexes oscillators across both populations; $K_i$ ("in-coupling") determines the sensitivity of oscillator $i$ to the population mean field and $G_j$ ("out-coupling") governs the contribution of oscillator $j$ to the mean field. For two equally sized populations, coupling strengths are parameterized as

$$K_{1,2} = K_0 \pm \frac{\Delta K}{2}, \quad G_{1,2} = G_0 \pm \frac{\Delta G}{2}.$$

Noise is typically modeled as Gaussian white noise with intensity $D$, but extensions include tempered stable Lévy noise with tunable heaviness of the tails and exponential tempering, enabling the paper of rare, large stochastic events (Kalloniatis et al., 2021).

Collective states are characterized by order parameters $r_{1,2} e^{i\Theta_{1,2}}$ for each subpopulation, and their phase difference $\delta = \Theta_1 - \Theta_2$. The evolution of these order parameters and the phase lag between populations encapsulates the macroscopic behavior.

2. Collective States: Synchronization, Discordance, and Traveling Waves

This model supports several qualitatively distinct states:

• Incoherent state: Both $r_{1,2}=0$; oscillators are desynchronized.
• Zero-lag synchronization: $r_{1,2}>0$ and $\delta=0$.
• $\pi$-state (discordant synchronization): $r_{1,2}>0$, $\delta=\pi$. Each cluster is synchronized but precisely anti-aligned, yielding no net drift of the global phase.
• Traveling wave state: $r_{1,2}>0$, $0<\delta<\pi$. The ensemble collectively rotates with net frequency $\Omega$ differing from the intrinsic frequency, induced by the nonzero phase lag: $$\lim_{t\to\infty} \dot{\Theta}_1 = \lim_{t\to\infty} \dot{\Theta}_2 = \sin\delta \; \frac{r_2^{-1}+r_2^3}{4} K_2 G_1 r_1,$$ with analogous expressions for other parameter regimes.

These dynamical behaviors arise from fixed points and limit cycles of the reduced system (Sonnenschein et al., 2015).

3. Reduced ODE Description and Bifurcation Structure

The continuum-limit Fokker–Planck equations for each subpopulation can, under suitable approximations (such as a Gaussian phase approximation), be reduced to a three-dimensional ODE system for the order parameters and their phase lag: $$\begin{aligned} &\dot{r}_1 = -Dr_1 + \frac{1 - r_1^4}{4} K_1 [ r_1 G_1 + r_2 G_2 \cos\delta ],\ &\dot{r}_2 = -Dr_2 + \frac{1 - r_2^4}{4} K_2 [ r_2 G_2 + r_1 G_1 \cos\delta ],\ &\dot{\delta} = -\frac{\sin\delta}{4}\left[ (r_1^{-1} + r_1^3) K_1 r_2 G_2 + (r_2^{-1} + r_2^3) K_2 r_1 G_1 \right]. \end{aligned}$$ Fixed points of these equations correspond to synchronized states (zero-lag, $\pi$-state, or intermediate/traveling wave), delineated by bifurcation curves in the $(K_0, G_0, \Delta K, \Delta G, D)$ parameter space. For instance, the incoherent state loses stability when

$\Delta K\,\Delta G = 8D - 4K_0G_0.$

Transitions between states, including onset of traveling waves and hysteresis/bistability, are revealed in bifurcation diagrams derived analytically and via numerical continuation (Sonnenschein et al., 2015).

4. Discordance, Phase Lags, and Bistability

A key feature is "discordant synchronization," in which the phase lag $\delta$ between the two subpopulations is nontrivial (i.e., neither $0$ nor $\pi$), splitting the system into two distinct synchronized clusters. For $\delta = \pi$, the clusters exactly cancel each other's mean fields, inhibiting net drift. When $\delta \neq \pi$, as in the traveling wave regime, collective motion emerges.

Bistability arises when parameter regions support multiple attractors (incoherence, zero-lag, $\pi$-state, or traveling wave), resulting in multistability and hysteresis. The outcome can depend heavily on initial conditions or noise: perturbations may trigger abrupt transitions between different collective behaviors.

5. Stochastic Effects and Order Restoration Mechanisms

Noise, particularly of the non-Gaussian tempered stable Lévy variety, exerts nuanced control over synchronization. As the power-law index $\alpha$ departs from the Gaussian limit ($\alpha=2$), the system first loses phase locking due to rare but large stochastic excursions, but sufficient tempering (exponential cutoff parameter $\lambda$) suppresses these events and can restore or enhance phase locking—even for very low $\alpha$ (Kalloniatis et al., 2021). This yields a non-monotonic dependence of synchronization on noise parameters, traceable to metastability in the "tilted ratchet potential" governing the centroid phase difference. Analytic reduction to an effective 1D SDE for this phase difference demonstrates how noise-induced hops between metastable wells can be tuned, counterintuitively, to either disrupt or stabilize global ordering in the face of competition.

6. Broader Implications: Competition, Chimera States, and Application Domains

The two-population model is foundational to the paper of competitive dynamics, exemplified in contexts ranging from neural assemblies (competition for phase advantage, leading and lagging) to chemical, ecological, and engineered systems. The emergence of bistable or multistable regimes, as well as chimera states (coexistence of coherent and incoherent subpopulations), is typical when network topology, frustration, or noise is appropriately engineered. Mechanisms discovered in this model—such as order restoration via tuning noise properties—suggest novel control paradigms for collective dynamics in complex networks and provide explanatory paradigms for diverse phenomena including neuronal competition, chemical wave propagation, and social or ecological synchronization.

7. Analytical Methods and Parametric Tuning

The mathematical architecture of the two-population Kuramoto–Sakaguchi model encompasses Fokker–Planck/PDE descriptions, mean-field order parameter reductions, and the use of ratchet (tilted periodic) potentials in effective SDE analyses. Bifurcation structure is often mapped via numerical continuation (e.g., MATCONT), and the impact of noise is treated both with direct numerical simulation and via analytical solutions to fractional/tempered Fokker–Planck equations.

Crucially, the system's dynamic regime can be selectively tuned via coupling asymmetries ($\Delta K$, $\Delta G$), noise parameters ($\alpha$, $\lambda$), or frustration/phase lag parameters, yielding a high degree of controllability over transitions between incoherence, synchronization, discordance, bistability, and traveling waves. This control framework provides a toolkit for theoretical, experimental, and applied investigations into collective dynamics and complex synchronization.