Weakly Coupled Oscillators in a Slowly Varying World

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.

arXiv comment28 pages, 11 figures

References in corpus (3)

• Robustness of the noise-induced phase synchronization in a general class of limit cycle oscillators
• Phase Reduction Method for Strongly Perturbed Limit Cycle Oscillators
• A criterion for timescale decomposition of external inputs for generalized phase reduction of limit-cycle oscillators

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• On the Concept of Dynamical Reduction : The Case of Coupled Oscillators
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• Nonautonomous driving induces stability in network of identical oscillators
• Nonlinear phase-amplitude reduction of delay-induced oscillations
• A Phase Model with Large Time Delayed Coupling
• High-Order Accuracy Computation of Coupling Functions for Strongly Coupled Oscillators
• Optimal Control of Oscillation Timing and Entrainment Using Large Magnitude Inputs: An Adaptive Phase-Amplitude-Coordinate-Based Approach
• Hysteresis bifurcation and application to delayed Fitzhugh-Nagumo neural systems