Hebbian-Oscillatory Co-Learning

TL;DR

This paper presents a unified framework combining Hebbian plasticity and oscillatory phase synchronization to enhance sparse neural architectures, with proven convergence and demonstrated emergent connectivity patterns.

Contribution

It introduces Hebbian-Oscillatory Co-Learning (HOC-L), integrating RSGN and SSA frameworks, with a novel synchronization-gated plasticity mechanism and theoretical convergence guarantees.

Findings

Achieves $O(n imes k)$ complexity with $k \

Demonstrates emergent cluster-aligned connectivity in simulations.

Proves convergence to a stable equilibrium via Lyapunov analysis.

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 Poincar\'{e} 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.