Weakly Coupled Oscillators in a Slowly Varying World

TL;DR

This paper extends weakly coupled oscillator theory to include slow variations in inputs and parameters, deriving phase equations and demonstrating effects on neuronal synchrony during modulation.

Contribution

It introduces a combined perturbation and adiabatic approach to analyze coupled oscillators with slow parameter changes, including coupling effects.

Findings

Derived phase equations for oscillators with slow modulation.

Applied method to neuronal models showing waxing and waning synchrony.

Extended previous work to include coupling effects.

Abstract

We extend the theory of weakly coupled oscillators to incorporate slowly varying inputs and parameters. We employ a combination of regular perturbation and an adiabatic approximation to derive equations for the phase-difference between a pair of oscillators. We apply this to the simple Hopf oscillator and then to a biophysical model. The latter represents the behavior of a neuron that is subject to slow modulation of a muscarinic current such as would occur during transient attention through cholinergic activation. Our method extends and simplifies the recent work of Kurebayashi (Physical Review Letters, 111, 214101, 2013) to include coupling. We apply the method to an all-to-all network and show that there is a waxing and waning of synchrony of modulated neurons.