Higher-Order Kuramoto Oscillator Network for Dense Associative Memory

TL;DR

This paper introduces a generalized Kuramoto oscillator network with higher-order coupling, demonstrating enhanced memory capacity and stability for dense associative memory applications through analytical and simulation results.

Contribution

It develops a new higher-order Kuramoto model inspired by dense Hopfield memory, revealing improved memory capacity and bistability for associative memory.

Findings

Bistable phase-locked states support stored memories.

Memory capacity scales superlinearly with network size.

Simulations confirm rapid, robust pattern retrieval.

Abstract

Networks of phase oscillators can serve as dense associative memories if they incorporate higher-order coupling beyond the classical Kuramoto model's pairwise interactions. Here we introduce a generalized Kuramoto model with combined second-harmonic (pairwise) and fourth-harmonic (quartic) coupling, inspired by dense Hopfield memory theory. Using mean-field theory and its dynamical approximation, we obtain a phase diagram for dense associative memory model that exhibits a tricritical point at which the continuous onset of memory retrieval is supplanted by a discontinuous, hysteretic transition. In the quartic-dominated regime, the system supports bistable phase-locked states corresponding to stored memory patterns, with a sizable energy barrier between memory and incoherent states. We analytically determine this bistable region and show that the escape time from a memory state (due to noise) grows exponentially with network size, indicating robust storage. Extending the theory to finite memory load, we show that higher-order couplings achieve superlinear scaling of memory capacity with system size, far exceeding the limit of pairwise-only oscillators. Large-scale simulations of the oscillator network confirm our theoretical predictions, demonstrating rapid pattern retrieval and robust storage of many phase patterns. These results bridge the Kuramoto synchronization with modern Hopfield memories, pointing toward experimental realization of high-capacity, analog associative memory in oscillator systems.