  Signals from interacting brain regions display transient synchronization of
phases and amplitudes in different frequencies. Commonly, the interaction
between regions of the brain is quantitatively described by either analyzing
the correlations of amplitudes of the measured signals or by calculating
phase-synchronization measures. However, for a complete picture of the
interactions it is important to analyze the dynamics of both the amplitude and
the phase. In this work, we present a new method for finding the coupling
between brain regions by reconstructing the phase-amplitude dynamics directly
from the measured electrophysiological signals. For this purpose, we use the
recent advances in the field of phase-amplitude reduction of oscillatory
systems, which allow the representation of an uncoupled oscillatory system as a
phase-amplitude oscillator in a unique form using transformations
(parametrizations) related to the eigenfunctions of the Koopman operator. By
combining the parametrization method and the Fourier-Laplace averaging method
of finding the eigenfunctions of the Koopman operator, we developed a novel
method of assessing the transformation functions from the signals of the
interacting oscillatory systems. The resulting reconstructed dynamical system
is a network of phase-amplitude oscillators with the interactions between them
represented as coupling functions in phase and amplitude coordinates. Using
synthetic signals generated from several models with known and unknown
theoretical phase-amplitude reduced forms, we demonstrate that our method is
capable of finding the proper unique dynamic form of these oscillatory systems
in the reduced phase-amplitude space. Our method can be applied to describe any
network of interacting oscillators as a dynamical system using signals of the
network elements.
