Hebbian-Oscillatory Co-Learning

Hasi Hays

cs.NEFeb 21, 2026

Hebbian-Oscillatory Co-Learning

Authors: Hasi Hays

Organizations: Department of Chemical Engineering, University of Arkansas, Fayetteville, AR 72701, USA

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 & tables

Panel of Figure 1 — Figure 1: Overall HOC-L architecture. The left stream (blue) implements the RSGN [ 18 ] 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 [ 21 ] 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 III ), 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 3 . Contour lines at the base reveal the basin geometry projected onto the $(\|W\|_{F},\bar{\phi})$ plane.

Sparse Neural Networks

Sparse neural networks use a subset of weights, connections, activations, or experts to reduce model complexity and computation.

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Laboratory of Intelligent Systems, EPFL, Switzerland

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Open paper AskPDF arXiv

Biological neural networks continuously adapt and modify themselves in response to experiences throughout their lifetime - a capability largely absent in artificial neural networks. Hebbian plasticity offers a promising path toward rapid adaptation in changing environments. Here, we introduce Hebbian Attractor Networks (HAN), a class of plastic neural networks in which local weight update normalization induces emergent attractor dynamics. Unlike prior approaches, HANs employ dual-timescale plasticity and temporal averaging of pre- and postsynaptic activations to induce either co-dynamic limit cycles or fixed-point weight attractors. Using simulated locomotion benchmarks, we gain insight into how Hebbian update frequency and activation averaging influence weight dynamics and control performance. Our results show that slower updates, combined with averaged pre- and postsynaptic activations, promote convergence to stable weight configurations, while faster updates yield oscillatory co-dynamic systems. We further demonstrate that these findings generalize to high-dimensional quadrupedal locomotion with a simulated Unitree Go1 robot. These results highlight how the timing of plasticity shapes neural dynamics in embodied systems, providing a principled characterization of the attractor regimes that emerge in self-modifying networks.

Abstract

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