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Edited by: Norbert Marwan, Potsdam-Institut für Klimafolgenforschung (PIK), Germany

Reviewed by: Alessandro Giuliani, Istituto Superiore di Sanità (ISS), Italy; Chen Cheng-Bang, Pennsylvania State University, United States

This article was submitted to Dynamical Systems, a section of the journal Frontiers in Applied Mathematics and Statistics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Complex dynamical systems arise in many fields of science, e.g., in biology, physics, climatology, and engineering. Those are systems exhibiting nonlinear, non-stationary, and possibly even chaotic behavior which is generally difficult to model or analyze. Several dynamical systems are also characterized by a well defined spatial organization (like the human brain for instance), and to properly describe their dynamics, one must take into account both their spatial and temporal properties. A way to achieve this is by analyzing the spatio-temporal data that can be observed from the complex system. In this respect, an important observation is that dynamical systems tend to return to former states, and this recurrence can be recognized as a fundamental characteristic of many dynamical systems [

Originally, RPs were developed for the study of phase-space trajectories, or for uni-variate time series through time-delay embeddings. Several extensions have been proposed to deal with multi-variate time series data and with spatio-temporal processes which can also exhibit typical recurrent structures of interest (the interested reader is invited to read section 2 for more background information about multi-variate and spatial recurrence plots). Those approaches for multi-variate RPs and their accompanying RQA generally exploit the whole record of spatio-temporal data observed from the complex dynamical system under investigation; they do not aim to find a subset on which to perform a more focused and informative analysis. However, for a given system with an arbitrary spatial or geometric structure, recurrence (perceived as quasi-periodic repetitive patterns) may occur in localized regions of this structure only (limited over space), and may last for just a limited time (relative to the overall time span of the data). Not focusing on such most informative spatio-temporal regions of interest in the data, may hinder the identification and characterization of complex dynamics with RQA, as relevant information becomes obscured by getting drowned in uninformative data and noise. In addition, from a computational point of view, it is also inefficient to build RPs and perform RQA on complete multi-variate datasets if one is predominantly interested in spatio-temporally localized recurrences. We therefore argue that, before carrying out multi-variate RPs and RQA, it is important to first detect if, where, and when recurrence patterns are present and recognizable. This will allow one to select the spatial region(s) and time interval(s) of interest for performing a more efficient and accurate analysis of recurrence, leaving out much of the unwanted disinformation.

In this study, we propose a novel such framework for the detection of spatio-temporal recurrence in complex dynamical systems, This framework focuses on a specific type of recurrence, namely, recurrences from repetitive (relatively short lived) quasi-periodic oscillatory patterns, with a limited frequency bandwidth. This kind of recurrence is important in signal processing applications, especially in the field of medicine from which we shall present an example with real data. The framework (see

Schematic of the framework. Briefly, the time series associated with each point in the geometric structure are collected into a matrix

As mentioned in the Introduction, several extensions of RP and RQA have been proposed to deal with multi-variate time series data and with spatial data. Romano et al. [

When the focus is on spatial data structures characterized by high-dimension and geometric correlation patterns (different from time series data), traditional RP is limited in analyzing it, since recurrene may occur in any spatial direction (and not simply over time). In this respect, Vasconcelos et al. [

One underlying assumption of the methods listed above seems to be that all the data (multi-variate data or spatial structures) is relevant to recurrence behavior analysis. For instance, recurrence may characterize the entire distributed network of sensors in Nichols et al. [

This section is divided in two main parts. The first part focuses on introducing an algorithm for detection of spatio-temporal recurrence. The second part focuses on an approach for generating recurrence plots from multi-variate time series.

Given a dynamical system with ^{n}, which is known as the state-space or as the phase-space of the system. The evolution of the system over time traces a path through this space, referred to as the phase-space trajectory of the system. A recurrence in the phase-space can then be defined as a time instant in which the state returns close to a location it has visited before (

For the purpose of detecting recurring geometric patterns, and given that we have a geometrical structure of

where (

Example of spatio-temporally localized recurrence. A 2D sinusoidal wave of limited time duration appears in a specific region of a random noise image, and propagates in the direction indicated by the white arrow. The sequence consists of

The first step is to identify the most relevant points in the geometric structure associated with a specific recurrent pattern. This is achieved by first generating a matrix ^{T}, with

The second step is to identify how many PCs of _{j}(

of each PC _{j} (with _{j}(

where _{j} for the computation of the SC of component _{j}.

Normalized power spectrum of the first principal component of the matrix _{j} are also shown. The horizontal axis shows the frequency indices; the vertical axis shows the normalized power spectrum (n.u.: normalized units).

Once all relevant PCs of a recurrent pattern have been selected, the third step is to identify all points in the original geometric structure associated with those PCs, i.e., the points on which the selected PCs are reflected the most. This is achieved as follows: for each selected PC _{j}, the entries in the (_{j}, the _{j} to each point in the original structure. The most relevant points in the geometric structure associated with _{j} can therefore be defined as those points with the largest contribution from _{j}. The following criterion is used on _{j} to select all relevant points associated with a selected _{j}. Given each entry _{j}(_{j}, the vector _{j}: ã_{j}(_{j}(_{j}(

To be able to estimate the time span of a region of recurrence in a geometric structure, the multi-variate signal can be split up into (overlapping) time windows, and the algorithm for spatial detection of recurrence introduced in the previous section can be applied to each window individually. All clusters identified in a window need then to be compared with those found in the previous window, to be able to say for each cluster whether it is a new cluster or not, or whether a cluster from the previous window has ceased. Experience shows that the window size

Example of application of the framework for spatio-temporal detection of recurrence introduced in section 3.1 to the example of

For the estimate of the time interval of a region of recurrence, we used the first and last window where the region was detected. The mid points of those windows are used to estimate the start and the end of the time interval of recurrence, respectively. In the example, the estimated interval (from frame 35 to 65, as displayed in

As mentioned before, RPs provide a way to visualize and quantify recurrent behavior of the phase-space trajectory of a dynamical system [

with time-delay τ and embedding dimension _{ij} between two points

Given a multi-variate process, the uni-variate framework in (5) needs to be extended to handle multi-variate (spatio-temporal) data. This would allow to process the signals from all points of a recurrent region simultaneously, and take into account the contribution of space and time together. A _{i}(_{i}(

As in the multi-variate case of a dynamical system, the embedding dimension of all _{i} = 1, _{1}(_{2}(_{ℓ}(

As a distance measure, we propose to use the cosine of the angle between two vectors instead of the Euclidean distance, namely:

The rationale for this choice is that this distance provides a normalized correlation which is not affected by the magnitude of the vectors (for two signals, that means that the shapes of the signals are compared without being influenced by their respective amplitudes). This approach for generating a multi-variate RP was used in Meste et al. [

_{i∈S; 1:N} of size _{i∈S; j∈T} of size

Multi-variate RP for the case presented in

Two numerical simulations are presented in this section. The first one shows the ability of the proposed approach to handle multiple recurrent patterns simultaneously occurring in a geometric structure. The second one shows the ability of the proposed approach to detect recurrent patterns from spiraling waves, which may occur in several real world problems, among which cardiac arrhythmia like AF. All analysis and computations were performed in MATLAB (MATLAB and Statistics Toolbox Release 2018a, The MathWorks, Inc., Natick, MA, USA). Code can be requested from the authors.

The first simulation concerns a sequence of random noise images (each frame of size 256 × 256 points), with two different 2D sinusoidal waves of limited time duration appearing in two different regions of the random images. The sequence is composed of

Simulation characterized by a sequence of random images with two different 2D sinusoidal waves of limited time duration, appearing in two different regions of the random images. The two sinusoidal waves are simultaneously present over 20 frames, and they have different direction of propagation (indicated by the white arrows), different frequency, and different speed of propagation (with the smaller wave having a higher speed than the larger wave).

Result of applying the framework for spatio-temporal detection of recurrence introduced in section 3.1 to the example of

Multi-variate RPs computed on the regions of interest and time intervals reported in

The second simulation concerns a sequence of random noise images (each frame with a size of 300 × 300 points). The sequence is composed of

where

Simulation characterized by a sequence of random images with a spiraling wave of limited time duration added to it. The spiraling pattern was obtained by means of a complex Ginzburg-Landau equation, with parameters

Result of applying the framework for spatio-temporal detection of recurrence introduced in section 3.1 to the example of

Multi-variate RP computed on the region of interest and time interval reported in

The animal study performed to acquire the data used in this study was carried out in accordance with the principles of the Basel Declaration and regulations of European directive 2010/63/EU. The local ethical board for animal experimentation of Maastricht University approved the protocol. Epicardial (outer surface of the heart) high-density direct contact mapping was performed in an ovine model of acute atrial fibrillation (AF). Uni-polar electrograms were measured on the left atrial free wall during an open-thorax procedure with a regular grid of electrodes (16x16 electrodes, 1.5mm inter-electrode distance, 1 KHz sampling frequency, see

Application of the spatio-temporal recurrence detection framework to high-density contact mapping recordings in an ovine model of AF.

Recurrence of wave patterns was determined in a selected segment of the mapping, which comprised 10 s of AF, with an average cycle length of 67 ms (~15 Hz). Recurrence analysis was based on automated annotation of local atrial deflections and activation-based phase map similarity as described in previous work [

The proposed framework for spatio-temporal detection of recurrence was applied to the uni-polar electrograms. The frequency content around the fundamental frequency of AF was emphasized by filtering the electrograms with a band-pass filter (40–250 Hz, 3rd order, zero-phase Chebyshev), followed by rectification and a low-pass filter (20 Hz, 3rd order, zero-phase Chebyshev). Window size was set to

This study proposed a novel framework for spatio-temporal detection of recurrence in complex dynamical systems characterized by repetitive (quasi-periodic) spatio-temporal patterns. This framework allows to address the following questions: whether a recurrent pattern is present in the geometric structure, where it is located (in which region of the geometric structure), and when it occurs (at what time instants it appears and then disappears). This is relevant since several dynamical systems are characterized by a well defined spatial organization, and accurate localization of recurrence both in space and time becomes important to be able to properly describe the recurrent behavior of such systems, and filter out any noise or unwanted information. This can be seen in

The two numerical simulations presented in section 4 suggest that the proposed framework is able to achieve a detailed identification of the spatial and temporal locations of different types of spatio-temporal repetitive patterns in a dynamical system. The first simulation shows that an accurate identification in both space and time is possible even when several recurrent patterns are simultaneously present in the geometric structure of a dynamical system. The second simulation shows that accuracy is also maintained with more complicated types of recurrent patterns like spiral waves from reaction diffusion models, simulated by means of the complex Ginzburg-Landau equation, whose recurrent patterns become sparser both in space and time, and thus more difficult to be detected. This scenario is also encountered in cardiac arrhythmia like AF, in which abnormal electrical activity may generate and propagate in specific regions of the atrial tissue, which may also offer the physiological conditions for this activity to self-perpetuate over a certain time span. When applied to actual invasive recordings of atrial activity during AF in an animal model, the proposed framework was able to unveil regions on the atrial walls characterized by recurrent behaviors, which were not visible from the RP generated from the full available spatio-temporal data. This is relevant, as an accurate spatial and temporal identification of those regions may help characterize the progression of the disease, the level of impairment of the cardiac tissue, and improve patients stratification and personalize therapy [

The proposed framework was also tested on simulations similar to those of

Several extensions of RP techniques for spatially distributed data have been introduced. In this respect, Riedl et al. [

The task of detecting the start and end points of a recurrent pattern may appear very similar to the one of change point detection (CPD) in time series data [

In this study, we did not analyze the influence of the noise level on the accuracy in the detection of spatio-temporal recurrence, which should be addressed by a future study. One limitation of the proposed framework is that it only considers information about proximity when clustering neighboring points to build a region of recurrence. Additional information could be taken into account, such as distances between the points, which may become relevant when clustering points in arbitrary irregular geometric structures.

In this study, we presented a novel framework for spatio-temporal detection of recurrence in complex dynamical systems characterized by a well defined spatial structure. This framework focuses on spatio-temporally localized repetitive patterns and is able to retrieve the correct recurrence plots associated with known traveling waves in a geometric structure, by focusing on the spatial regions and time intervals involved by a recurrence behavior, and leaving out all unwanted information. This framework may be integrated with state of the art methods for multi-scale and multi-variate recurrence plots, to help improve recurrence quantification analysis of spatio-temporal data.

The datasets generated for this study are available on request to the corresponding author.

The animal study was reviewed and approved by Animal Ethics Committee Maastricht University Secretariat DEC, UNS 50, box 48 6200MD Maastricht, The Netherlands.

PB, RP, OM, and JK developed the theoretical formalism. PB designed the numerical experiments and analyzed the data. SZ and AvH designed and performed the animal experiments, and analyzed the corresponding data. PB wrote the first draft of the manuscript. All authors contributed to the final version and contributed in interpreting the results.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.