Oscillatory Network Based on Kuramoto Model for Image Segmentation

A global synchronization theorem for oscillators on a random graph

Consider n identical Kuramoto oscillators on a random graph. Specifically, consider Erdős-Rényi random graphs in which any two oscillators are bidirectionally coupled with unit strength, independently and at random, with probability 0 ≤ p ≤ 1. We say that a network is globally synchronizing if the oscillators converge to the all-in-phase synchronous state for almost all initial conditions. Is there a critical threshold for p above which global synchrony is extremely likely but below which it is extremely rare? It is suspected that a critical threshold exists and is close to the so-called connectivity threshold, namely, p ∼ log ⁡ ( n ) / n for n ≫ 1. Ling, Xu, and Bandeira made the first progress toward proving a result in this direction: they showed that if p ≫ log ⁡ ( n ) / ⁿ, then Erdős-Rényi networks of Kuramoto oscillators are globally synchronizing with high probability as n → ∞. Here, we improve that result by showing that p ≫ ^log ⁡ ( n ) / n suffices. Our estimates are explicit: for example, we can say that there is more than a 99.9996 % chance that a random network with n = ¹⁰ and p > 0.011 17 is globally synchronizing.

Two-dimensional mutually synchronized spin Hall nano-oscillator arrays for neuromorphic computing

In spin Hall nano-oscillators (SHNOs), pure spin currents drive local regions of magnetic films and nanostructures into auto-oscillating precession. If such regions are placed in close proximity to each other they can interact and may mutually synchronize. Here, we demonstrate robust mutual synchronization of two-dimensional SHNO arrays ranging from 2 × 2 to 8 × 8 nano-constrictions, observed both electrically and using micro-Brillouin light scattering microscopy. On short time scales, where the auto-oscillation linewidth [Formula: see text] is governed by white noise, the signal quality factor, [Formula: see text], increases linearly with the number of mutually synchronized nano-constrictions (N), reaching 170,000 in the largest arrays. We also show that SHNO arrays exposed to two independently tuned microwave frequencies exhibit the same synchronization maps as can be used for neuromorphic vowel recognition. Our demonstrations may hence enable the use of SHNO arrays in two-dimensional oscillator networks for high-quality microwave signal generation and ultra-fast neuromorphic computing.