J. Kurths

The application of methods of non-linear dynamics for the improved and predictive recognition of patients threatened by sudden cardiac death

OBJECTIVES This study introduces new methods of non-linear dynamics (NLD) and compares these with traditional methods of heart rate variability (HRV) and high resolution ECG (HRECG) analysis in order to improve the reliability of high risk stratification. METHODS Simultaneous 30 min high resolution ECGs and long-term ECGs were recorded from 26 cardiac patients after myocardial infarction (MI). They were divided into two groups depending upon the electrical risk, a low risk group (group 2, n = 10) and a high risk group (group 3, n = 16). The control group consisted of 35 healthy persons (group 1). From these electrocardiograms we extracted standard measures in time and frequency domain as well as measures from the new non-linear methods of symbolic dynamics and renormalized entropy. RESULTS Applying discriminant function techniques on HRV analysis the parameters of non-linear dynamics led to an acceptable differentiation between healthy persons and high risk patients of 96%. The time domain and frequency domain parameters were successful in less than 90%. The combination of parameters from all domains and a stepwise discriminant function separated these groups completely (100%). Use of this discriminant function classified three patients with apparently low (no) risk into the same cluster as high risk patients. The combination of the HRECG and HRV analysis showed the same individual clustering but increased the positive value of separation. CONCLUSIONS The methods of NLD describe complex rhythm fluctuations and separate structures of non-linear behavior in the heart rate time series more successfully than classical methods of time and frequency domains. This leads to an improved discrimination between a normal (healthy persons) and an abnormal (high risk patients) type of heart beat generation. Some patients with an unknown risk exhibit similar patterns to high risk patients and this suggests a hidden high risk. The methods of symbolic dynamics and renormalized entropy were particularly useful measures for classifying the dynamics of HRV.

Quantitative analysis of heart rate variability

In the modern industrialized countries every year several hundred thousands of people die due to sudden cardiac death. The individual risk for this sudden cardiac death cannot be defined precisely by common available, noninvasive diagnostic tools like Holter monitoring, highly amplified ECG and traditional linear analysis of heart rate variability (HRV). Therefore, we apply some rather unconventional methods of nonlinear dynamics to analyze the HRV. Especially, some complexity measures that are based on symbolic dynamics as well as a new measure, the renormalized entropy, detect some abnormalities in the HRV of several patients who have been classified in the low risk group by traditional methods. A combination of these complexity measures with the parameters in the frequency domain seems to be a promising way to get a more precise definition of the individual risk. These findings have to be validated by a representative number of patients. (c) 1995 American Institute of Physics.

Network synchronization, diffusion, and the paradox of heterogeneity

Many complex networks display strong heterogeneity in the degree (connectivity) distribution. Heterogeneity in the degree distribution often reduces the average distance between nodes but, paradoxically, may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength. Here we offer a solution to this apparent paradox. Our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states. We show that, for a given degree distribution, the maximum synchronizability is achieved when the network of couplings is weighted and directed and the overall cost involved in the couplings is minimum. This enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size. Numerical verification of the main results is provided for representative classes of small-world and scale-free networks.

Enhancing complex-network synchronization

Heterogeneity in the degree (connectivity) distribution has been shown to suppress synchronization in networks of symmetrically coupled oscillators with uniform coupling strength (unweighted coupling). Here we uncover a condition for enhanced synchronization in weighted networks with asymmetric coupling. We show that, in the optimum regime, synchronizability is solely determined by the average degree and does not depend on the system size and the details of the degree distribution. In scale-free networks, where the average degree may increase with heterogeneity, synchronizability is drastically enhanced and may become positively correlated with heterogeneity, while the overall cost involved in the network coupling is significantly reduced as compared to the case of unweighted coupling.

The backbone of the climate network

We propose a method to reconstruct and analyze a complex network from data generated by a spatio-temporal dynamical system, relying on the nonlinear mutual information of time series analysis and betweenness centrality of the complex network theory. We show that this approach reveals a rich internal structure in complex climate networks constructed from reanalysis and model surface air temperature data. Our novel method uncovers peculiar wave-like structures of high-energy flow, that we relate to global surface ocean currents. This points to a major role of the oceanic surface circulation in coupling and stabilizing the global temperature field in the long-term mean (140 years for the model run and 60 years for reanalysis data). We find that these results cannot be obtained using classical linear methods of multivariate data analysis, and have ensured their robustness by intensive significance testing.

The Kuramoto model in complex networks

Synchronization of an ensemble of oscillators is an emergent phenomenon present in several complex systems, ranging from social and physical to biological and technological systems. The most successful approach to describe how coherent behavior emerges in these complex systems is given by the paradigmatic Kuramoto model. This model has been traditionally studied in complete graphs. However, besides being intrinsically dynamical, complex systems present very heterogeneous structure, which can be represented as complex networks. This report is dedicated to review main contributions in the field of synchronization in networks of Kuramoto oscillators. In particular, we provide an overview of the impact of network patterns on the local and global dynamics of coupled phase oscillators. We cover many relevant topics, which encompass a description of the most used analytical approaches and the analysis of several numerical results. Furthermore, we discuss recent developments on variations of the Kuramoto model in networks, including the presence of noise and inertia. The rich potential for applications is discussed for special fields in engineering, neuroscience, physics and Earth science. Finally, we conclude by discussing problems that remain open after the last decade of intensive research on the Kuramoto model and point out some promising directions for future research.

Multiparametric Analysis of Heart Rate Variability Used for Risk Stratification Among Survivors of Acute Myocardial Infarction

A multiparametric heart rate variability analysis was performed to prove if combined heart rate variability (HRV) measures of different domains improve the result of risk stratification in patients after myocardial infarction. In this study, standard time domain, frequency domain and non‐linear dynamics measures of HRV assessment were applied to 572 survivors of acute myocardial infarction. Three parameter sets each consisting of 4 parameters were applied and compared with the standard measurement of global heart rate variability HRVi. Discriminant analysis technique and t‐test were performed to separate the high risk groups from the survivors. The predictive value of this approach was evaluated with receiver operator (ROC) and positive predictive accuracy (PPA) curves. Results ‐ The discriminant analysis shows a separation of patients suffered by all cause mortality in 80% (best single parameter 74%) and sudden arrhythmic death in 86% (73%). All parameters of set I show a high significant difference (p<0.001) between survivors and non‐survivors based on two‐tailed t‐test. The specificity level of the multivariate parameter sets is at the 70% sensitivity level (ROC) about 85–90%, whereas HRVi shows maximum levels of 70%. The PPA in the all cause mortality group is at the 70% sensitivity level twice as high as the univarihate HRV measure and increases to more than fourfold as high within the VT/VF group. In conclusion, in this population, the multiparametric approach with the combination of four parameters from all domains especially from NLD seems to be a better predictor of high arrhythmia risk than the standard measurement of global heart rate variability.

Chapter 9 Phase synchronization: From theory to data analysis

Publisher Summary The chapter describes particular experiments and searching for phase synchronization. The phase synchronization of chaotic system is the appearance of a certain relation between the phases of interacting systems (or between the phase of a system and that of an external force), while the amplitudes can remain chaotic and are noncorrelated. The properties of phase synchronization in chaotic systems are similar to those of synchronization in periodic noisy oscillators. Synchronization plays an important role in several neurological diseases such as epilepsies and pathological tremors. The chapter reviews the ideas and results of theoretical studies of the synchronization phenomena that are used to time series analysis. The chapter presents techniques of the bivariate data analysis and illustrates them by examples of physiological data. These examples are given in the ascending order of the signal analysis complexity.

A Constrained Evolutionary Computation Method for Detecting Controlling Regions of Cortical Networks

Controlling regions in cortical networks, which serve as key nodes to control the dynamics of networks to a desired state, can be detected by minimizing the eigenratio R and the maximum imaginary part σ of an extended connection matrix. Until now, optimal selection of the set of controlling regions is still an open problem and this paper represents the first attempt to include two measures of controllability into one unified framework. The detection problem of controlling regions in cortical networks is converted into a constrained optimization problem (COP), where the objective function R is minimized and σ is regarded as a constraint. Then, the detection of controlling regions of a weighted and directed complex network (e.g., a cortical network of a cat), is thoroughly investigated. The controlling regions of cortical networks are successfully detected by means of an improved dynamic hybrid framework (IDyHF). Our experiments verify that the proposed IDyHF outperforms two recently developed evolutionary computation methods in constrained optimization field and some traditional methods in control theory as well as graph theory. Based on the IDyHF, the controlling regions are detected in a microscopic and macroscopic way. Our results unveil the dependence of controlling regions on the number of driver nodes I and the constraint r. The controlling regions are largely selected from the regions with a large in-degree and a small out-degree. When r = + ∞, there exists a concave shape of the mean degrees of the driver nodes, i.e., the regions with a large degree are of great importance to the control of the networks when I is small and the regions with a small degree are helpful to control the networks when I increases. When r = 0, the mean degrees of the driver nodes increase as a function of I. We find that controlling σ is becoming more important in controlling a cortical network with increasing I. The methods and results of detecting controlling regions in this paper would promote the coordination and information consensus of various kinds of real-world complex networks including transportation networks, genetic regulatory networks, and social networks, etc.

Line structures in recurrence plots

Recurrence plots exhibit line structures which represent typical behaviour of the investigated system. The local slope of these line structures is connected with a specific transformation of the time scales of different segments of the phase-space trajectory. This provides us a better understanding of the structures occurring in recurrence plots. The relationship between the time-scales and line structures are of practical importance in cross recurrence plots. Using this relationship within cross recurrence plots, the time-scales of differently sampled or time-transformed measurements can be adjusted. An application to geophysical measurements illustrates the capability of this method for the adjustment of time-scales in different measurements.