Global phase-amplitude description of oscillatory dynamics via the parameterization method

Area of Science:

• Dynamical Systems Theory
• Computational Neuroscience
• Nonlinear Dynamics

Background:

• Classical phase reduction simplifies oscillator dynamics but is limited to asymptotic states.
• Understanding nonlinear dynamics beyond asymptotic states is crucial for complex systems like neural networks.
• Accurate characterization of invariant manifolds is essential for predicting system behavior.

Purpose of the Study:

• To present a parameterization method for a complete description of n-dimensional oscillator dynamics.
• To extend phase reduction by analyzing dynamics on the attracting invariant manifold.
• To provide accurate tools for characterizing oscillatory dynamics and response functions.

Main Methods:

• Utilizing a parameterization method with efficient algorithms to parameterize the attracting invariant manifold.
• Employing Fourier-Taylor expansions for analytical approximations.
• Developing numerical methods for manifold globalization and dimension reduction.

Main Results:

• Obtained accurate parameterizations of the attracting invariant manifold in phase-amplitude variables.
• Derived local and global isochrons and isostables, providing a geometrical portrait of dynamics.
• Generated infinitesimal phase and amplitude response functions for detailed perturbation analysis.

Conclusions:

• The parameterization method offers a comprehensive description of oscillator dynamics beyond classical phase reduction.
• The methodology accurately captures phase and amplitude shifts on the attracting invariant manifold.
• Applied successfully to neuroscience models, demonstrating its utility in analyzing neural dynamics.