Low-Dimensional Behavior of a Kuramoto Model with Inertia and Hebbian Learning

Abstract

We study low-dimensional dynamics of a Kuramoto model with inertia with Hebbian learning.In this model, the coupling strength between oscillators depends on the phase differences between oscillators and changes according to a Hebbian learning rule. We analyze the special case of two coupled oscillators, which yields a three-dimensional dynamical system, and classify the stability of its equilibrium points using linear stability analysis. We show that the system is dissipative and that all of its trajectories are eventually confined to a bounded region. We approximate the limiting behavior and demarcate the parameter regions of three qualitatively different behaviors of the system. Using insights from our analysis of the low-dimensional dynamics, we study the original high-dimensional system when we draw the intrinsic frequencies of the oscillators from Gaussian distributions with different variances. Synchronization occurs ubiquitously in many systems[1], such as in electrical impulses of neurons and the flashing of fireflies. In response to changes in external environmental conditions, many systems of coupled oscillators adapt to enhance collective behavior. One example of adaptation is plasticity in networks of neurons, in which the synaptic strengths between neurons change based on their relative spike times or other features [2]. Another example of adaptation occurs in groups of fireflies [3], which have been modeled using coupled phase oscillators with inertia, which can reduce local synchronization speed while simultaneously facilitating global synchronization. We consider both plasticity and inertia by examining a modified version of the ubiquitous Kuramoto model of coupled oscillators [4]. To examine the interplay between oscillator plasticityand inertia, we mathematically analyze a small system of these coupled oscillators. We then use the results of this analysis to gain some insights into larger systems of these oscillators.