Higher-Order Kuramoto Oscillator Network for Dense Associative Memory

Abstract

Networks of phase oscillators can serve as dense associative memories when they incorporate genuine many-body coupling beyond the classical Kuramoto model's pairwise interaction. Here we introduce a generalized Hebbian Kuramoto model that combines a conventional two-body, first-Fourier-harmonic coupling with a genuine four-body phase interaction, inspired by dense Hopfield memory theory. For identical oscillators with Langevin noise, equilibrium mean-field theory yields a phase diagram with a tricritical point when the four-body coupling is three times the pairwise coupling, where continuous retrieval onset gives way to a discontinuous, hysteretic transition. We separately analyze a deterministic model with Lorentzian frequency disorder using the Ott--Antonsen ansatz. In that model the onset changes from supercritical to subcritical when the four-body coupling is six times the pairwise coupling, and the two descriptions agree only at the linear instability threshold. In the four-body-dominated equilibrium regime, stored-pattern and incoherent states coexist. We determine the bistable region and the free-energy barriers for transitions in both directions, and show that the noise-induced memory-loss time grows exponentially with the number of oscillators, with a rate set by the relevant barrier. At finite memory load, a phenomenological cavity signal--to--noise closure models pattern crosstalk, while simulations measure finite-size strong-cue recovery and retention thresholds. Nominal power-law fits over the simulated size range have exponents above one in several four-body-dominated cases, with the largest fitted exponent in the purely four-body zero-temperature data; these fits are not asymptotic capacity laws.