Adaptive Kuramoto–Sakaguchi Models

• Adaptive Kuramoto–Sakaguchi models are generalized oscillator networks with time-varying couplings that adapt according to activity-dependent plasticity rules.
• They employ non-reciprocal Hebbian and anti-Hebbian adaptation mechanisms to induce bistability, clustering, and rich bifurcation behaviors in phase dynamics.
• Higher-order extensions, including hypergraph interactions, reveal complex multistability and synchronization transitions applicable to neuroscience and complex network studies.

Adaptive Kuramoto–Sakaguchi models are a generalization of the classical Kuramoto and Kuramoto–Sakaguchi oscillator networks in which interaction strengths (couplings) evolve dynamically according to activity-dependent rules, often motivated by neurobiological plasticity mechanisms such as Hebbian or spike-timing-dependent plasticity. These models combine the complexity of nonlinearly coupled phase oscillators with adaptive, possibly non-reciprocal, and higher-order time-dependent network architectures.

1. Model Formulation and Mathematical Structure

Adaptive Kuramoto–Sakaguchi networks generalize the conventional phase oscillator system by allowing the coupling matrix or functions to adapt dynamically in response to the evolving oscillator phases. The foundational model for $N$ oscillators with phase variables $\theta_i$ and time-dependent couplings $k_{ij}(t)$ incorporates both classic Kuramoto–Sakaguchi phase interactions and adaptation:

$$\dot\theta_i = \omega_i + \frac{1}{N}\sum_{j=1}^{N} k_{ij}(t)\,\sin[\theta_j - \theta_i - \alpha] + \xi_i(t),$$

$$\dot k_{ij} = -\varepsilon_{(1,2)}[k_{ij} + \sin(\theta_i - \theta_j - \alpha + \beta_{(1,2)})],$$

where $\omega_i$ is the natural frequency, $\xi_i(t)$ is noise, $\alpha$ is the Sakaguchi phase-lag, $\varepsilon_{(1,2)}$ are adaptation rates, and $\beta_{(1,2)}$ encode Hebbian/anti-Hebbian learning phase offsets. More generally, plasticity rules may involve more complex functions $\theta_i$0, such as

$\theta_i$1

with $\theta_i$2, $\theta_i$3 the adaptivity strength, and $\theta_i$4 an STDP-like shift (Jüttner et al., 2022).

Higher-order and global adaptation mechanisms extend this framework by coupling strengths to global or clustered order parameters, or by including triadic/hypergraph interactions with adaptive weighting (Dutta et al., 2024, Biswas et al., 10 Nov 2025).

2. Non-Reciprocal and Hebbian/Anti-Hebbian Adaptation

A distinctive feature of recent work is the explicit modeling of non-reciprocal adaptive couplings, wherein $\theta_i$5 and adaptation rules employ different rates and phase offsets. For example, upper triangular elements $\theta_i$6 adapt rapidly under Hebbian rules ($\theta_i$7, $\theta_i$8 fast), while lower triangular $\theta_i$9 adapt slowly under anti-Hebbian rules ($k_{ij}(t)$0, $k_{ij}(t)$1 slow). This non-reciprocity structurally embeds causality and asymmetry found in neural substrates (Chowdhury et al., 23 Dec 2025).

The resulting couplings, after transients, exhibit bistability ($k_{ij}(t)$2), creating robust clusters with anti-phase relationships. This mechanism gives rise to high-index saddle equilibrium states: locally metastable, but with many unstable directions, enabling stochastic switching of oscillator cluster memberships—an important motif in models of brain state transitions and heteroclinic switching.

3. Bifurcation Structure and Critical Adaptivity

Adaptivity introduces novel bifurcation scenarios absent from static-coupling Kuramoto–Sakaguchi models. For the minimal $k_{ij}(t)$3 adaptive case, the critical adaptivity threshold is $k_{ij}(t)$4. Below threshold, only phase drift or single locking regimes exist; above, bistability, saddle–node curves, cusp points, and Bogdanov–Takens bifurcation points emerge. Asymmetric adaptivity ($k_{ij}(t)$5) introduces true Hopf bifurcations, homoclinic connections, period doubling, and chaos, as revealed by comprehensive bifurcation analyses (Jüttner et al., 2022).

These bifurcation structures become richer with higher-order and global adaptation: super- and subcritical pitchforks, multiple saddle-node bifurcations, and tiered synchronization transitions (multiple jumps in the order parameter as control parameters are tuned) are observed, allowing both continuous (second order), explosive (first order), and two-step synchronization transitions depending on adaptation exponent and phase-lag values (Dutta et al., 2024).

4. Metastability, Cluster Dynamics, and Synchronization Regimes

Non-reciprocal adaptive Kuramoto–Sakaguchi models robustly generate deterministic metastability: systems organize into two or more internally coherent clusters (often anti-phase), but finite-size instabilities or high-index saddle structure lead to intermittent switching events. The dominant order parameters, such as $k_{ij}(t)$6 (second harmonic of phase distribution), display pseudo-irregular bursts as groups of oscillators temporarily escape or change cluster affiliation, interpreted as "weak ties" linking otherwise distinct phase clusters (Chowdhury et al., 23 Dec 2025).

System size and connectivity modulate these dynamics: increasing $k_{ij}(t)$7 raises the number of escape channels (unstable directions), raising the frequency but reducing the amplitude of cluster-switching events. On random graphs, increased connectivity prolongs dwell-times in cluster states. Small noise further accelerates switching; however, noise-induced complete synchronization is not observed under these adaptive rules (Chowdhury et al., 23 Dec 2025).

In large adaptive networks, the macroscopic states observed include:

• Antipodal (AP): Two phase-locked clusters at $k_{ij}(t)$8
• Locked (L): Global phase coherence
• Partially synchronous (PS): A locked core with a set of drifting oscillators
• Drift (D): All oscillators incoherent

Adaptivity enhances synchronizability by expanding the locking region in parameter space and supporting robust multi-stable and partially synchronous states (Jüttner et al., 2022).

5. Order Parameters and Reduced Modeling

Order parameter techniques—most notably the Ott–Antonsen ansatz—enable dimension reduction for analysis of collective adaptive Kuramoto–Sakaguchi dynamics. The total (first harmonic) complex order parameter

$k_{ij}(t)$9

evolves according to ODEs derived from the continuum limit. For adaptive higher-order interactions, the ODEs for $$\dot\theta_i = \omega_i + \frac{1}{N}\sum_{j=1}^{N} k_{ij}(t)\,\sin[\theta_j - \theta_i - \alpha] + \xi_i(t),$$0 and $$\dot\theta_i = \omega_i + \frac{1}{N}\sum_{j=1}^{N} k_{ij}(t)\,\sin[\theta_j - \theta_i - \alpha] + \xi_i(t),$$1 capture the full macroscopic bifurcation structure, enabling explicit calculation of critical points for synchronization and the nature of the transitions (super- vs subcritical, saddle-node, tiered, etc.) as functions of adaption exponents and phase-lag (Dutta et al., 2024, Biswas et al., 10 Nov 2025).

A selection of the reduced model structure is as follows: $$\dot\theta_i = \omega_i + \frac{1}{N}\sum_{j=1}^{N} k_{ij}(t)\,\sin[\theta_j - \theta_i - \alpha] + \xi_i(t),$$2

$$\dot\theta_i = \omega_i + \frac{1}{N}\sum_{j=1}^{N} k_{ij}(t)\,\sin[\theta_j - \theta_i - \alpha] + \xi_i(t),$$3

where $$\dot\theta_i = \omega_i + \frac{1}{N}\sum_{j=1}^{N} k_{ij}(t)\,\sin[\theta_j - \theta_i - \alpha] + \xi_i(t),$$4, $$\dot\theta_i = \omega_i + \frac{1}{N}\sum_{j=1}^{N} k_{ij}(t)\,\sin[\theta_j - \theta_i - \alpha] + \xi_i(t),$$5 are the coupling strengths for pairwise and triadic interactions, and $$\dot\theta_i = \omega_i + \frac{1}{N}\sum_{j=1}^{N} k_{ij}(t)\,\sin[\theta_j - \theta_i - \alpha] + \xi_i(t),$$6 the respective adaptation exponents. This structure remains analytically tractable for hypergraph (arbitrary $$\dot\theta_i = \omega_i + \frac{1}{N}\sum_{j=1}^{N} k_{ij}(t)\,\sin[\theta_j - \theta_i - \alpha] + \xi_i(t),$$7-body) interaction systems (Biswas et al., 10 Nov 2025).

6. Adaptive Higher-Order and Hypergraph Extensions

Adaptive Kuramoto–Sakaguchi models with higher-order (triadic and above) and hypergraph connectivity reflect the reality of complex networks—biological and technological—with non-pairwise interactions. The generalization introduces adaptation rules for hyperedge strengths, with feedback on global or order-parameter-dependent functions: $$\dot\theta_i = \omega_i + \frac{1}{N}\sum_{j=1}^{N} k_{ij}(t)\,\sin[\theta_j - \theta_i - \alpha] + \xi_i(t),$$8 where $$\dot\theta_i = \omega_i + \frac{1}{N}\sum_{j=1}^{N} k_{ij}(t)\,\sin[\theta_j - \theta_i - \alpha] + \xi_i(t),$$9 are adaptive feedback functions of the order parameter, $$\dot k_{ij} = -\varepsilon_{(1,2)}[k_{ij} + \sin(\theta_i - \theta_j - \alpha + \beta_{(1,2)})],$$0 coupling strengths, and $$\dot k_{ij} = -\varepsilon_{(1,2)}[k_{ij} + \sin(\theta_i - \theta_j - \alpha + \beta_{(1,2)})],$$1 phase-lags (Biswas et al., 10 Nov 2025).

This architecture produces new dynamical behaviors: multistability, hybrid synchronization, coexisting macrostates, and discontinuous transitions. Applications include models of epilepsy surgery (triadic motifs as "hidden" epileptogenic foci) and the rapid propagation of rumors via "hyper-seeds" in social networks (Biswas et al., 10 Nov 2025).

7. Applications and Implications

Adaptive Kuramoto–Sakaguchi models are directly relevant for systems exhibiting dynamic plasticity, multimodal interactions, and metastable switching, notably:

• Neuroscience: Modelling of metastable neuronal assemblies, critical dynamics resembling brain state transitions, synaptic plasticity, and the effect of higher-order motifs in pathological and healthy brain states.
• Complex Networks: Analysis of hierarchical or modular synchronization, resilience due to adaptive connectivity, and propagation of bursts or transitions in power-grid and social hypergraphs.
• Bifurcation Engineering: Systematic design of networks with tunable synchronization properties (locking window, hysteresis, explosive/tiered transitions) through adaption strength, phase-lag, and higher-order coupling features (Dutta et al., 2024, Biswas et al., 10 Nov 2025).

The emergence of deterministic yet seemingly stochastic macro-dynamics due to high-dimensional metastability and cluster re-affiliation suggests directions for exploiting and controlling coherence and flexibility in synthetic and biological oscillator networks.

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Key References:

• Metastability and non-reciprocal adaptation: (Chowdhury et al., 23 Dec 2025)
• General adaptive Kuramoto–Sakaguchi and bifurcation analysis: (Jüttner et al., 2022)
• Higher-order global adaptive interactions: (Dutta et al., 2024)
• Hypergraph adaptive dynamics and finite-size effects: (Biswas et al., 10 Nov 2025)