Adaptive, locally-linear models of complex dynamics

TL;DR

This paper introduces an adaptive local linear modeling approach to analyze complex, high-dimensional, and non-stationary dynamics, enabling detailed characterization of systems like chaotic Lorenz systems, nematode behavior, and brain activity.

Contribution

It presents a novel adaptive local linear modeling framework combined with hierarchical clustering and eigenvalue analysis for complex dynamic systems.

Findings

Successfully characterizes Lorenz system dynamics in different regimes.

Identifies behavioral states and bifurcations in C. elegans posture data.

Detects changes in brain state stability with oxygen levels.

Abstract

The dynamics of complex systems generally include high-dimensional, non-stationary and non-linear behavior, all of which pose fundamental challenges to quantitative understanding. To address these difficulties we detail a new approach based on local linear models within windows determined adaptively from the data. While the dynamics within each window are simple, consisting of exponential decay, growth and oscillations, the collection of local parameters across all windows provides a principled characterization of the full time series. To explore the resulting model space, we develop a novel likelihood-based hierarchical clustering and we examine the eigenvalues of the linear dynamics. We demonstrate our analysis with the Lorenz system undergoing stable spiral dynamics and in the standard chaotic regime. Applied to the posture dynamics of the nematode $C. elegans$ our approach identifies fine-grained behavioral states and model dynamics which fluctuate close to an instability boundary, and we detail a bifurcation in a transition from forward to backward crawling. Finally, we analyze whole-brain imaging in $C. elegans$ and show that the stability of global brain states changes with oxygen concentration.