Hebbian-Oscillatory Co-Learning

TL;DR

Hebbian-Oscillatory Co-Learning (HOC-L), a unified two-timescale dynamical framework for joint structural plasticity and phase synchronization in bio-inspired sparse neural architectures, is introduced and emergent cluster-aligned connectivity and monotonic Lyapunov decrease are demonstrated.

Abstract

We introduce Hebbian-Oscillatory Co-Learning (HOC-L), a unified two-timescale dynamical framework for joint structural plasticity and phase synchronization in bio-inspired sparse neural architectures. HOC-L couples two recent frameworks: the hyperbolic sparse geometry of Resonant Sparse Geometry Networks (RSGN), which employs Poincaré ball embeddings with Hebbian-driven dynamic sparsity, and the oscillator-based attention of Selective Synchronization Attention (SSA), which replaces dot-product attention with Kuramoto-type phase-locking dynamics. The key mechanism is synchronization-gated plasticity: the macroscopic order parameter $r(t)$ of the oscillator ensemble gates Hebbian structural updates, so that connectivity consolidation occurs only when sufficient phase coherence signals a meaningful computational pattern. We prove convergence of the joint system to a stable equilibrium via a composite Lyapunov function and derive explicit timescale separation bounds. The resulting architecture achieves $O(n \cdot k)$ complexity with $k \ll n$, preserving the sparsity of both parent frameworks. Numerical simulations confirm the theoretical predictions, demonstrating emergent cluster-aligned connectivity and monotonic Lyapunov decrease.

Figures (4)

• Figure 1: Overall HOC-L architecture. The left stream (blue) implements the RSGN rsgn2025 structural pathway: inputs are embedded in the Poincaré ball, sparse hyperbolic neighborhoods are constructed, and Hebbian plasticity modifies structural connectivity. The right stream (orange) implements the SSA ssa2026 oscillatory pathway: tokens receive oscillator encodings, phase-locking attention computes synchronization-based weights, and the macroscopic order parameter $r(t)$ is extracted. The central coupling bridge (green) implements synchronization-gated plasticity: the order parameter gates Hebbian updates, and structural changes modulate future synchronization.
• Figure 2: Simulation results: Two-timescale dynamics in HOC-L. A Kuramoto oscillator system ($N=50$, $K=2.0$) is simulated jointly with synchronization-gated Hebbian weight updates for 1000 time steps ($dt=0.05$), using the smooth sigmoid gate $G(r) = \sigma(\beta(r - r_c))$ with $\beta = 20$. The fast curve (red) shows the running average of SSA phase update magnitudes $|d\theta_i/dt|$, exhibiting rapid oscillations as oscillators seek synchronization. The slow curve (blue) shows the running average of Hebbian weight update magnitudes $|\Delta W_{ij}|$, which are continuously modulated by $G(r)$: negligible when $r \ll r_c$ and substantial once $r$ exceeds $r_c$. The vertical dashed line (green) marks the moment when $G(r)$ crosses $0.5$ (i.e., $r$ crosses $r_c = 0.5$). The orange curve (right axis) traces the macroscopic order parameter $r(t)$. The timescale separation $\tau_{\text{fast}} \ll \tau_{\text{slow}}$ (Section \ref{['sec:framework']}), where $\tau_{\text{fast}}$ is the characteristic timescale of phase updates and $\tau_{\text{slow}}$ is that of Hebbian structural updates, is clearly visible in the distinct frequency content of the two signals.
• Figure 3: Simulation results: Phase synchronization to structural plasticity coupling mechanism in a Kuramoto system with $N=8$ oscillators ($K=3.0$, 2000 time steps). (A) Final oscillator phases on the unit circle: orange markers denote phase-locked (synchronized) oscillators that have formed a coherent cluster, while grey markers denote desynchronized oscillators with distinct intrinsic frequencies. (B) Order parameter $r(t)$ trajectory over time, with the critical threshold $r_c = 0.5$ shown as a dashed line; the shaded region indicates intervals where the plasticity gate is open ($r > r_c$). (C) Emergent weight matrix $W_{ij}$ after simulation: strong positive weights (warm colors) develop between synchronized oscillators, while connections involving desynchronized oscillators remain weak or near zero, demonstrating that synchronization-gated Hebbian plasticity produces sparse, cluster-aligned connectivity.
• Figure 4: Simulation results: Convergence basin of the HOC-L Lyapunov function. The 3D surface shows the composite Lyapunov function $V(\mathbf{W}, \bm{\theta}) = -\frac{K}{2N}\sum_{i,j}\cos(\theta_i - \theta_j) + \frac{\lambda}{2}\|\mathbf{W}\|_F^2$ evaluated on a grid over the projected weight norm ($\|W\|_F$-axis) and mean phase deviation ($\bar{\phi}$-axis). Colored trajectories trace the joint dynamics of 8 random initializations through the $(\|W\|_F, \bar{\phi}, V)$ space, each computed from actual numerical integration of the coupled Kuramoto--Hebbian ODE system ($N=20$ oscillators, 600 steps per trajectory). All trajectories converge toward the basin minimum (red star) where oscillators are synchronized ($\bar{\phi} \approx 0$) and weights are regularized, confirming the monotonic decrease of $V$ along system trajectories as guaranteed by Theorem \ref{['thm:convergence']}. Contour lines at the base reveal the basin geometry projected onto the $(\|W\|_F, \bar{\phi})$ plane.

Theorems & Definitions (11)

• Lemma 1: Lyapunov decrease along fast dynamics
• proof
• Lemma 2: Ultimate boundedness of slow dynamics
• proof
• Theorem 3: Local convergence of HOC-L
• proof
• Proposition 4: Required timescale separation
• proof
• Remark 1
• Proposition 5: Complexity of HOC-L