Phase coherence is not related to topology.

Phase coherence is an important measure in nonlinear science. Whereas there is no generally accepted definition for phase and therefore for phase coherence, many works associate this feature with topological aspects of the systems, such as having a well-defined rotating center. Given the relevance of this concept for synchronization problems, one aim of this paper is to argue by means of a couple of counterexamples that phase coherence is not related to the topology of the attractor. A second aim is to introduce a phase-coherence measure based on recurrence plots, for which probabilities of recurrences for two different trajectories are similar for a phase-coherent system and dissimilar for non-phase-coherent systems. The measure does not require a phase variable defined a priori.

Navigating differential structures in complex networks.

Structural changes in a network representation of a system, due to different experimental conditions, different connectivity across layers, or to its time evolution, can provide insight on its organization, function, and on how it responds to external perturbations. The deeper understanding of how gene networks cope with diseases and treatments is maybe the most incisive demonstration of the gains obtained through this differential network analysis point of view, which led to an explosion of new numeric techniques in the last decade. However, where to focus one's attention, or how to navigate through the differential structures in the context of large networks, can be overwhelming even for a few experimental conditions. In this paper, we propose a theory and a methodological implementation for the characterization of shared "structural roles" of nodes simultaneously within and between networks. Inspired by recent methodological advances in chaotic phase synchronization analysis, we show how the information about the shared structures of a set of networks can be split and organized in an automatic fashion, in scenarios with very different (i) community sizes, (ii) total number of communities, and (iii) even for a large number of 100 networks compared using numerical benchmarks generated by a stochastic block model. Then, we investigate how the network size, number of networks, and mean size of communities influence the method performance in a series of Monte Carlo experiments. To illustrate its potential use in a more challenging scenario with real-world data, we show evidence that the method can still split and organize the structural information of a set of four gene coexpression networks obtained from two cell types × two treatments (interferon-ß stimulated or control). Aside from its potential use as for automatic feature extraction and preprocessing tool, we discuss that another strength of the method is its "story-telling"-like characterization of the information encoded in a set of networks, which can be used to pinpoint unexpected shared structure, leading to further investigations and providing new insights. Finally, the method is flexible to address different research-field-specific questions, by not restricting what scientific-meaningful characteristic (or relevant feature) of a node shall be used.

Faint phase synchronization detection through structured orthomax rotations in singular spectrum analysis.

Multivariate singular spectrum analysis (M-SSA), with a structured varimax rotation, is a method that allows a deep characterization of phase synchronization (PS) phenomena in an almost automatic fashion. It has been increasingly used in the study of PS in networks of nonlinear, real-world, and numeric systems. This paper investigates the impact of the other recently developed structured orthomax rotations on the M-SSA ability to characterize PS. The results show that by using the structured quartimax rotation, a very faint and intermittent PS regime can be detected, in contrast with the structured varimax (which demands a stronger, more consolidated PS regime). This is due to the fact that the different rotations do not have the same efficiency in achieving a simple structure of the M-SSA eigenvectors. Nevertheless, for well-established PS regimes, the same robustness of the original M-SSA approach against high levels of additive Gaussian noise was found for the structured quartimax and biquartimax rotations. However, for all approaches we found an overshoot of the qualitative range for the PS onset due to noise.

The reliability of recurrence network analysis is influenced by the observability properties of the recorded time series.

Recurrence network analysis (RNA) is a remarkable technique for the detection of dynamical transitions in experimental applications. However, in practical experiments, often only a scalar time series is recorded. This requires the state-space reconstruction from this single time series which, as established by embedding and observability theory, is shown to be hampered if the recorded variable conveys poor observability. In this work, we investigate how RNA metrics are impacted by the observability properties of the recorded time series. Following the framework of Zou et al. [Chaos 20, 043130 (2010)], we use the Rössler and Duffing-Ueda systems as benchmark models for our study. It is shown that usually RNA metrics perform badly with variables of poor observability as for recurrence quantification analysis. An exception is the clustering coefficient, which is rather robust to observability issues. Along with its efficacy to detect dynamical transitions, it is shown to be an efficient tool for RNA-especially when no prior information of the variable observability is available.

Structural, dynamical and symbolic observability: From dynamical systems to networks.

Classical definitions of observability classify a system as either being observable or not. Observability has been recognized as an important feature to study complex networks, and as for dynamical systems the focus has been on determining conditions for a network to be observable. About twenty years ago continuous measures of observability for nonlinear dynamical systems started to be used. In this paper various aspects of observability that are established for dynamical systems will be investigated in the context of networks. In particular it will be discussed in which ways simple networks can be ranked in terms of observability using continuous measures of such a property. Also it is pointed out that the analysis of the network topology is typically not sufficient for observability purposes, since both the dynamics and the coupling of such nodes play a vital role. Some of the main ideas are illustrated by means of numerical simulations.

Recurrence plots for the assessment of patient-ventilator interactions quality during invasive mechanical ventilation.

Inappropriate patient-ventilator interactions' (PVI) quality is associated with adverse clinical consequences, such as patient anxiety/fear and increased need of sedative and paralytic agents. Thus, technological devices/tools to support the recognition and monitoring of different PVI quality are of great interest. In the present study, we investigate two tools based on a recent landmark study which applied recurrence plots (RPs) and recurrence quantification analysis (RQA) techniques in non-invasive mechanical ventilation. Our interest is in how this approach could be a daily part of critical care professionals' routine (which are not familiar with dynamical systems theory methods and concepts). Two representative time series of three typical PVI "scenarios" were selected from 6 critically ill patients subjected to invasive mechanical ventilation. First, both the (i) main signatures in RPs and the (ii) respective signals that provide the most (visually) discriminant RPs were identified. This allows one to propose a visual identification protocol for PVIs' quality through the RPs' overall aspect. Support for the effectiveness of this visual based assessment tool is given by a RQA-based assessment tool. A statistical analysis shows that both the recurrence rate and the Shannon entropy are able to identify the selected PVI scenarios. It is then expected that the development of an objective method can reliably identify PVI quality, where the results corroborate the potential of RPs/RQA in the field of respiratory pattern analysis.

Observability and synchronization of neuron models.

Observability is the property that enables recovering the state of a dynamical system from a reduced number of measured variables. In high-dimensional systems, it is therefore important to make sure that the variable recorded to perform the analysis conveys good observability of the system dynamics. The observability of a network of neuron models depends nontrivially on the observability of the node dynamics and on the topology of the network. The aim of this paper is twofold. First, to perform a study of observability using four well-known neuron models by computing three different observability coefficients. This not only clarifies observability properties of the models but also shows the limitations of applicability of each type of coefficients in the context of such models. Second, to study the emergence of phase synchronization in networks composed of neuron models. This is done performing multivariate singular spectrum analysis which, to the best of the authors' knowledge, has not been used in the context of networks of neuron models. It is shown that it is possible to detect phase synchronization: (i) without having to measure all the state variables, but only one (that provides greatest observability) from each node and (ii) without having to estimate the phase.

Enhancing multivariate singular spectrum analysis for phase synchronization: The role of observability.

Multivariate singular spectrum analysis (M-SSA) was recently adapted to study systems of coupled oscillators. It does not require an a priori definition for phase nor detailed knowledge of the individual oscillators, but it uses all the variables of each system. This aspect could be restrictive for practical applications, since usually just a few (sometimes only one) variables are measured. Based on dynamical systems and observability theories, we first show how to apply the M-SSA with only one variable and show the conditions to achieve good performance. Next, we provide numerical evidence that this single-variable approach enhances the explanatory power compared to the original M-SSA when computed with all the system variables. This could have important practical implications, as pointed out using benchmark oscillators.

Matrix formulation and singular-value decomposition algorithm for structured varimax rotation in multivariate singular spectrum analysis.

Groth and Ghil [Phys. Rev. E 84, 036206 (2011)PLEEE81539-375510.1103/PhysRevE.84.036206] developed a modified varimax rotation aimed at enhancing the ability of the multivariate singular spectrum analysis (M-SSA) to characterize phase synchronization in systems of coupled chaotic oscillators. Due to the special structure of the M-SSA eigenvectors, the modification proposed by Groth and Ghil imposes a constraint in the rotation of blocks of components associated with the different subsystems. Accordingly, here we call it a structured varimax rotation (SVR). The SVR was presented as successive pairwise rotations of the eigenvectors. The aim of this paper is threefold. First, we develop a closed matrix formulation for the entire family of structured orthomax rotation criteria, for which the SVR is a special case. Second, this matrix approach is used to enable the use of known singular value algorithms for fast computation, allowing a simultaneous rotation of the M-SSA eigenvectors (a Python code is provided in the Appendix). This could be critical in the characterization of phase synchronization phenomena in large real systems of coupled oscillators. Furthermore, the closed algebraic matrix formulation could be used in theoretical studies of the (modified) M-SSA approach. Third, we illustrate the use of the proposed singular value algorithm for the SVR in the context of the two benchmark examples of Groth and Ghil: the Rössler system in the chaotic (i) phase-coherent and (ii) funnel regimes. Comparison with the results obtained with Kaiser's original (unstructured) varimax rotation (UVR) reveals that both SVR and UVR give the same result for the phase-coherent scenario, but for the more complex behavior (ii) only the SVR improves on the M-SSA.