Frontiers in Applied Mathematics and Statistics (Aug 2019)
Detection of Spatio-Temporal Recurrent Patterns in Dynamical Systems
Abstract
Background: The study of repetitive (quasi-periodic) spatio-temporal patterns in complex dynamical systems with a well defined spatial structure may be complicated if the recurrent behavior is confined to specific local regions, where it lasts for a limited time. This can decrease the efficacy of recurrence plots (RPs) in recognizing such patterns. It then becomes important to first detect whether repetitive spatio-temporal patterns are present, and if so, where they are located (both in space and time), to facilitate a focused RP analysis approach. This study proposes a novel framework for spatio-temporal detection of local recurrence of a quasi-periodic nature in complex dynamical systems. A motivating application for this framework is the analysis of atrial fibrillation to better understand the heart tissue involved.Methods: The spatio-temporal data observed from the system are decomposed by means of principal component analysis to identify the points in the spatial structure exhibiting quasi-periodic recurrent patterns. The frequency content of the principal components is used to determine if such patterns are present, and the corresponding eigenvectors are used to identify the points associated with those components. Geometric information about proximity of these points is used to cluster them into local regions. A sliding temporal window is used to detect the start and end of each pattern.Results: A first simulation shows how the proposed framework can handle multiple recurrent patterns simultaneously occurring in a spatial structure of a dynamical system. A second simulation shows how the method can handle more complicated patterns like 2D nonlinear spiraling waves, typical of many diffusion processes. The framework is then applied to real data to detect recurrent patterns in wave fronts propagating inside the heart during atrial fibrillation. This analysis can unveil regions of recurrence in the atria that were not visible with standard RP analysis.Conclusion: A novel framework for detecting spatio-temporal repetitive patterns in complex dynamical systems is introduced. It allows retrieve the correct recurrences associated with known 2D traveling waves, while the same information is not visible with standard RP analysis. This framework can be effectively used to investigate recurrence in real dynamical systems as cardiac arrhythmia.
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