Detection of wandering spatio-temporal recurrent patterns in dynamical systems

For systems characterized by a well-defined spatial structure, it may be relevant to detect if repetitive (quasi-periodic) patterns are present. However, this may be difficult if these repetitive patterns are confined to specific local regions of the spatial structure, and they only last for limited time. This can decrease the efficacy of recurrence plots in showing such patterns, if all points of the structure and all time instants available are used concurrently. In such a situation, the information of interest may be buried by less repetitive components or noise. Another complicating factor is that these local recurrent patterns may not be always confined to the same region, but they may wander across space. It then becomes important to first detect whether such wandering repetitive spatio-temporal patterns are present, and if so, localize them both in space and time, so that one can zoom-in on the relevant spatio-temporal region, by only using the information from those regions and time intervals. In this way, more meaningful recurrence plots can be generated. In a previous study, we proposed a method to detect static spatio-temporal repetitive patters. In this study, we extended this framework to also be able to detect and follow wandering spatio-temporal patterns across time and space.
In terms of the method, the spatio-temporal data observed from the system are decomposed using principal component analysis to identify the points in the spatial structure exhibiting quasi-periodic recurrent patterns. The frequency content (and spectral concentration) of the principal components is used to determine whether such patterns are present, and the corresponding eigenvectors are used to identify the points associated with those components. Geometric information indicating proximity of these points is used to cluster them into local regions of recurrence. The steps above are then repeated on sliding temporal windows to detect recurrent regions over time. Regions of recurrence from consecutive time windows are checked for overlaps and linked together if overlap occurs. In this way, we can describe how regions of recurrence traverse the spatial structure over time.

Such an approach could for instance be useful in the detection of abnormal sources of electrical activity in the heart during pathologies like atrial fibrillation, where such sources may not be necessarily anchored to specific regions of the heart, but they wander around in a more or less organized manner.