Phase Coupling Estimation from Multivariate Phase Statistics

Abstract

Coupled oscillators are prevalent throughout the physical world. Dynamical system formulations of weakly coupled oscillator systems have proven effective at capturing the properties of real-world systems. However, these formulations usually deal with the 'forward problem': simulating a system from known coupling parameters. Here we provide a solution to the 'inverse problem': determining the coupling parameters from measurements. Starting from the dynamic equations of a system of coupled phase oscillators, given by a nonlinear Langevin equation, we derive the corresponding equilibrium distribution. This formulation leads us to the maximum entropy distribution that captures pair-wise phase relationships. To solve the inverse problem for this distribution, we derive a closed form solution for estimating the phase coupling parameters from observed phase statistics. Through simulations, we show that the algorithm performs well in high dimensions (d=100) and in cases with limited data (as few as 100 samples per dimension). Because the distribution serves as the unique maximum entropy solution for pairwise phase statistics, the distribution and estimation technique can be broadly applied to phase coupling estimation in any system of phase oscillators.

Oscillatory phase coupling coordinates anatomically dispersed functional cell assemblies

“…Two different monkeys engaged in a working memory task and had acute recordings made from multiple prefrontal areas. The pairwise phase distribution of LFP measurements was modeled using a probabilistic model (30). For each neuron, two models were fitted using either all LFP data or LFP phases occurring at spike times alone and then related using Bayes' rule.…”

Learning Intermediate-Level Representations of Form and Motion from Natural Movies

“…How to model the structure in absolute phase is an important and difficult open problem. Some recent progress has been made on this issue through the development of multivariate models of phase dependencies (Cadieu & Koepsell, 2010a, 2010b, and incorporating these ideas into the model is a goal of ongoing work.…”

“…In separate work, we have proposed statistical models that may be relevant for capturing these unmodeled dependencies (Cadieu & Koepsell, 2010a, 2010b, and it will be important to include these forms of structure in order to develop a complete model of higher-order form and motion.…”

Multivariate Phase–Amplitude Cross-Frequency Coupling in Neurophysiological Signals

“…In fact, such a multivariate phase distribution was derived recently [11]. Given an N -dimensional vector of phases, i.e., θ = ( θ 1 , θ 2 , …, θ N ), extracted from N distinct signals, the maximum entropy distribution corresponding to the observed pairwise phase statistics is given as $$pfalse(\theta \mid \mathrm{boldK}false)=\frac{1}{Zfalse(\mathrm{boldK}false)}\exp \left[\frac{1}{2}\sum _{m,n=1}^N{\kappa }_{\mathrm{italicmn}}\cos ({\theta }_m-{\theta }_n-{\mu }_{\mathrm{italicmn}})\right]$$where the terms κ mn and μ mn are the coupling parameters between channels m and n .…”

“…The term κ representing coupling strength in the multivariate phase distribution is analogous to the concentration parameter γ in the von Mises distribution. The normalization constant Z ( K ) is a function of the coupling matrix and a general analytic formula for Z ( K ) has not yet been found [11]. However, an efficient technique based on score-matching can nonetheless be used to estimate distribution parameters from observed phase data [11].…”