Search

Discretizing a nonlinear time series enables us to calculate its statistics fast and rigorously. Before the turn of the century, the approach using partitions was dominant. In the last two decades, discretization via permutations has been developed to a powerful methodology, while recurrence plots have recently begun to be recognized as a method of discretization. In the meantime, horizontal visibility graphs have also been proposed to discretize time series. In this review, we summarize these methods and compare them from the viewpoint of symbolic dynamics, which is the right framework to study the symbolic representation of nonlinear time series and the inverse process: the symbolic reconstruction of dynamical systems. As we will show, symbolic dynamics is currently a very active research field with interesting applications.

The connectome is a wiring diagram mapping all the neural connections in the brain. At the cellular level, it provides a map of the neurons and synapses within a part or all of the brain of an organism. In recent years, significant advances have been made in the study of the connectome via network science and graph theory. This analysis is fundamental to understand neurotransmission (fast synaptic transmission) networks. However, neurons use other forms of communication as neuromodulation that, instead of conveying excitation or inhibition, change neuronal and synaptic properties. This additional neuromodulatory layers condition and reconfigure the connectome. In this paper, we propose that multilayer adaptive networks, in which different synaptic and neurochemical layers interact, are the appropriate framework to explain neuronal processing. Then, we describe a simplified multilayer adaptive network model that accounts for these extra-layers of interaction and analyse the emergence of interesting computational capabilities., 11 pages, 2 fugures

The non-homogenous flow of the cerebrospinal fluid within the ventricular catheter is one of the causative factors in shunt obstructions during the treatment of hydrocephalus. Previously, we studied the flow in ventricular catheters under the steady and pulsatile boundary conditions by means of computational fluid dynamics (CFD) in three-dimensional paradigms. Subsequently, several catheter designs with homogeneous flow patterns were developed out of which one prototype was chosen after a validation study. To test the effectiveness of the flow ventricular catheter in a prospective, multicenter, comparative study. Eligible centers were three pediatric hospitals: two with sole adult practice and one a mixed pediatric-adult. Standard silicone material was used to develop a parametric catheter model with homogenous flow characteristics. The flow catheters were inserted in pediatric (n = 30) and adult (n = 10) patients with all types of hydrocephalus. Simultaneously, regular ventricular catheters were inserted in another 43 control patients in the participating centers. Catheter positioning was standardized according to the Schaumann and Thomale classification. All ventricular catheters had a cephalad grade I or II positioning, and caudally, its extension had a peritoneal location. Programmable valves were utilized in 70% and antisiphon devices in 20% of the cases. Regular differential pressure valves were utilized in the remaining. No case of flow catheter obstruction was identified during a mean follow-up period of 2 years at the time of this writing. There were four catheter obstructions in the control cohort, all pediatric cases, during the first year. Shunt infections occurred in two cases in the control group, while there was one recurrent case of adult ventriculitis in the flow catheter group. This prototype model represents the next generation of ventricular catheters with a homogeneous flow pattern. The flow catheter can be inserted safely in hydrocephalic patients, and this preliminary prospective comparative study showed a possible obstruction-free functionality.

This is a review of group entropy and its application to permutation complexity. Specifically, we revisit a new approach to the notion of complexity in the time series analysis based on both permutation entropy and group entropy. As a result, the permutation entropy rate can be extended from deterministic dynamics to random processes. More generally, our approach provides a unified framework to discuss chaotic and random behaviors.

Causal relationships are important to understand the dynamics of coupled processes and, moreover, to influence or control the effects by acting on the causes. Among the different approaches to determine cause-effect relationships and, in particular, coupling directions in interacting random or deterministic processes, we focus in this paper on information-theoretic measures. So, we study in the theoretical part the difference between directionality indicators based on transfer entropy as well as on its dimensional reduction via transcripts in algebraic time series representations. In the applications we consider specifically the lowest dimensional case, i.e., 3-dimensional transfer entropy, which is currently one of the most popular causality indicators, and the (2-dimensional) mutual information of transcripts. Needless to say, the lower dimensionality of the transcript-based indicator can make a difference in practice, where datasets are usually small. To compare numerically the performance of both directionality indicators, synthetic data (obtained with random processes) and real world data (in the form of biomedical recordings) are used. As happened in previous related work, we found again that the transcript mutual information performs as good as, and in some cases even better than, the lowest dimensional binned and symbolic transfer entropy, the symbols being ordinal patterns.

Permutation entropy measures the complexity of a deterministic time series via a data symbolic quantization consisting of rank vectors called ordinal patterns or simply permutations. Reasons for the increasing popularity of this entropy in time series analysis include that (i) it converges to the Kolmogorov–Sinai entropy of the underlying dynamics in the limit of ever longer permutations and (ii) its computation dispenses with generating and ad hoc partitions. However, permutation entropy diverges when the number of allowed permutations grows super-exponentially with their length, as happens when time series are output by dynamical systems with observational or dynamical noise or purely random processes. In this paper, we propose a generalized permutation entropy, belonging to the class of group entropies, that is finite in that situation, which is actually the one found in practice. The theoretical results are illustrated numerically by random processes with short- and long-term dependencies, as well as by noisy deterministic signals.

Neuroscience and machine learning are two interrelated fields with obvious synergies. In recent years we have seen profound advances in these fields and in the integration between them. However, there remain challenges for a greater integration between the two disciplines. In this chapter we describe two areas, different but related, that can serve as a guide in the future for this integration. On the one hand, it is necessary to have more explanatory algorithms in order to understand information processing in the brain. The combination of multilayer and adaptive networks can be the right framework to understand this processing and analyse the interesting computational capabilities that occur in the brain. On the other hand, machine learning algorithms should have, similar to brain processing, more innate structure. This prior structure could make the process of learning more efficient and intuitive, and support artificial intelligence.

The main result of this paper is a proof using real analysis of the monotonicity of the topological entropy for the family of quadratic maps, sometimes called Milnor's Monotonicity Conjecture. In contrast, the existing proofs rely in one way or another on complex analysis. Our proof is based on tools and algorithms previously developed by the authors and collaborators to compute the topological entropy of multimodal maps. Specifically, we use the number of transverse intersections of the map iterates with the so-called critical line. The approach is technically simple and geometrical. The same approach is also used to briefly revisit the superstable cycles of the quadratic maps, since both topics are closely related., Comment: 19 pages, 10 figures

Since its inception, control of data congestion on the Internet has been based on stochastic models. One of the first such models was Random Early Detection. Later, this model was reformulated as a dynamical system, with the average queue sizes at a router’s buffer being the states. Recently, the dynamical model has been generalized to improve global stability. In this paper we review the original stochastic model and both nonlinear models of Random Early Detection with a two-fold objective: (i) illustrate how a random model can be “smoothed out” to a deterministic one through data aggregation and (ii) how this translation can shed light into complex processes such as the Internet data traffic. Furthermore, this paper contains new materials concerning the occurrence of chaos, bifurcation diagrams, Lyapunov exponents and global stability robustness with respect to control parameters. The results reviewed and reported here are expected to help design an active queue management algorithm in real conditions, that is, when system parameters such as the number of users and the round-trip time of the data packets change over time. The topic also illustrates the much-needed synergy of a theoretical approach, practical intuition and numerical simulations in engineering.

This is a paper in the intersection of time series analysis and complexity theory that presents new results on permutation complexity in general and permutation entropy in particular. In this context, permutation complexity refers to the characterization of time series by means of ordinal patterns (permutations), entropic measures, decay rates of missing ordinal patterns, and more. Since the inception of this \textquotedblleft ordinal\textquotedblright\ methodology, its practical application to any type of scalar time series and real-valued processes have proven to be simple and useful. However, the theoretical aspects have remained limited to noiseless deterministic series and dynamical systems, the main obstacle being the super-exponential growth of visible permutations with length when randomness (also in form of observational noise) is present in the data. To overcome this difficulty, we take a new approach through complexity classes, which are precisely defined by the growth of visible permutations with length, regardless of the deterministic or noisy nature of the data. We consider three major classes: exponential, sub-factorial and factorial. The next step is to adapt the concept of Z-entropy to each of those classes, which we call permutation entropy because it coincides with the conventional permutation entropy on the exponential class. Z-entropies are a family of group entropies, each of them extensive on a given complexity class. The result is a unified approach to the ordinal analysis of deterministic and random processes, from dynamical systems to white noise, with new concepts and tools. Numerical simulations show that permutation entropy discriminates time series from all complexity classes., 22 pages, 4 figures

To the best of our knowledge, the method of prediction coordinates is the only forecasting method in nonlinear time series analysis that explicitly uses the stochastic characteristics of a system with dynamical noise. Specifically, it generates multiple predictions to jointly infer the current states and dynamical noises. Recent findings based on hypothesis testing show that weather is nonlinear and stochastic and, therefore, so are renewable energy power outputs. This being the case, in this paper, we apply the method of prediction coordinates to forecast wind power ramps, which are rapid transitions in the wind power output that can deteriorate the quality of the electricity supply. First, the method of prediction coordinates is tested using numerical simulations. Then, we present an example of wind power ramp forecasting with empirical data. The results show that the method of prediction coordinates compares favorably with other methods, validating it as a reliable tool for forecasting transitions in nonlinear stochastic dynamics, particularly in the field of renewable energies.

The flow pattern of the cerebrospinal fluid is probably the most important factor related to obstruction of ventricular catheters during the normal treatment of hydrocephalus. To better comprehend the flow pattern, we have carried out a parametric study via numerical models of ventricular catheters. In previous studies, the flow was studied under steady and, recently, in pulsatile boundary conditions by means of computational fluid dynamics (CFD) in three-dimensional catheter models. This study aimed to bring in prototype models of catheter CFD flow solutions as well to introduce the theory behind parametric development of ventricular catheters. A preceding study allowed deriving basic principles which lead to designs with improved flow patterns of ventricular catheters. The parameters chosen were the number of drainage segments, the distances between them, the number and diameter of the holes on each segment, as well as their relative angular position. CFD results of previously unreleased models of ventricular catheter flow solutions are presented in this study. Parametric development guided new designs with better flow distribution while lowering the shear stress of the catheters holes. High-resolution 3D printed catheter solutions of three models and basic benchmark testing are introduced as well. The next generation of catheter with homogeneous flow patterns based on parametric designs may represent a step forward for the treatment of hydrocephalus, by possibly broadening their lifespan.

We propose a method for generating surrogate data that preserves all the properties of ordinal patterns up to a certain length, such as the numbers of allowed/forbidden ordinal patterns and transition likelihoods from ordinal patterns into others. The null hypothesis is that the details of the underlying dynamics do not matter beyond the refinements of ordinal patterns finer than a predefined length. The proposed surrogate data help construct a test of determinism that is free from the common linearity assumption for a null-hypothesis.

The bifurcation and chaotic behavior of TCP may cause heavy oscillation of average queue length at routers and induce network instability. In this context, congestion control in the Internet is a challenging problem of key importance. This work presents a new discrete dynamical model of Random Early Detection (RED) using beta distribution for controlling bifurcations and chaos in the internet congestion control. The numerical analysis and the simulation experiments show that this new Active Queue Management (AQM) model can obtain the stable average queue length to the desired fix point. The model programmed with Python and Mathematica incorporates new parameters (α, β) that make it possible to stabilize oscillations of averaged router queue length and to be close to stationary state.