Phase-amplitude reduction and optimal phase locking of collectively oscillating networks

Area of Science:

• Dynamical systems theory
• Network science
• Computational neuroscience

Background:

• Collective oscillations are fundamental in biological and artificial networks.
• Existing phase reduction methods often simplify dynamics by focusing solely on limit cycles.
• Understanding deviations from limit cycle dynamics is crucial for complex network behavior.

Purpose of the Study:

• To develop a novel phase-amplitude reduction framework for analyzing collective oscillations in networked dynamical systems.
• To extend the capabilities of traditional phase reduction by incorporating amplitude dynamics.
• To provide a tool for studying network responses to external inputs and coupling, including synchronization and phase locking.

Main Methods:

• Developed a phase-amplitude reduction framework building upon established phase reduction techniques.
• Incorporated amplitude variables alongside phase variables to capture deviations from limit cycle dynamics.
• Applied the framework to networks of FitzHugh-Nagumo elements using numerical simulations.

Main Results:

• The phase-amplitude reduction framework effectively analyzes collective oscillations and deviations from unperturbed dynamics.
• Demonstrated the framework's efficacy on networks composed of FitzHugh-Nagumo elements.
• The derived phase-amplitude equations offer insights into network synchronization and stability.

Conclusions:

• The phase-amplitude reduction framework offers a more comprehensive analysis of networked dynamical systems than phase-only methods.
• This framework facilitates the study of synchronization, phase locking, and stability in complex networks.
• The derived equations can inform the design of optimal control strategies for network behavior.