Hebbian-Oscillatory Co-Learning

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.

💡 Research Summary

The paper introduces Hebbian‑Oscillatory Co‑Learning (HOC‑L), a unified two‑timescale dynamical framework that simultaneously addresses structural plasticity and fast phase synchronization in sparse, hyperbolic neural architectures. HOC‑L builds on two previously proposed systems: Resonant Sparse Geometry Networks (RSGN), which embed neurons in the Poincaré ball and enforce input‑dependent sparsity via a distance threshold, and Selective Synchronization Attention (SSA), which replaces the dot‑product attention of transformers with a Kuramoto‑type oscillator model. The core idea is “synchronization‑gated Hebbian plasticity”: the macroscopic order parameter (r(t)) of the oscillator ensemble acts as a gate for Hebbian weight updates. When (r(t)) exceeds a critical threshold (r_c), indicating that a coherent computational pattern has emerged, the Hebbian rule (\Delta W_{ij}= \eta,x_i x_j,\mathbf{1}