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We apply the recently developed symbolic resonance analysis to electroencephalographic measurements of event- related brain potentials (ERPs) in a language processing experiment by using a three-symbol static encoding with varying thresholds for analyzing the ERP epochs, followed by a spin-flip transformation as a nonlinear filter. We compute an estimator of the signal-to-noise ratio (SNR) for the symbolic dynamics measuring the coherence of threshold-crossing events. Hence, we utilize the inherent noise of the EEG for sweeping the underlying ERP components beyond the encoding thresholds. Plotting the SNR computed within the time window of a particular ERP component (the N400) against the encoding thresholds, we find different resonance curves for the experimental conditions. The maximal differences of the SNR lead to the estimation of optimal encoding thresholds. We show that topographic brain maps of the optimal threshold voltages and of their associated coherence differences are able to dissociate the underlying physiological processes, while corresponding maps gained from the customary voltage averaging technique are unable to do so
We study Hamiltonian chaos generated by the dynamics of passive tracers moving in a two-dimensional fluid flow and describe the complex structure formed in a chaotic layer that separates a vortex region from the shear flow. The stable and unstable manifolds of unstable periodic orbits are computed. It is shown that their intersections in the Poincare map as an invariant set of homoclinic points constitute the backbone of the chaotic layer. Special attention is paid to the finite time properties of the chaotic layer. In particular, finite time Lyapunov exponents are computed and a scaling law of the variance of their distribution is derived. Additionally, the box counting dimension as an effective dimension to characterize the fractal properties of the layer is estimated for different duration times of simulation. Its behavior in the asymptotic time limit is discussed. By computing the Lyapunov exponents and by applying methods of symbolic dynamics, the formation of the layer as a function of the external forcing strength, which in turn represents the perturbation of the originally integrable system, is characterized. In particular, it is shown that the capture of KAM tori by the layer has a remarkable influence on the averaged Lyapunov exponents. (C) 2004 Elsevier Ltd. All rights reserved
We investigate the influence of noise on synchronization between the spiking activities of neurons with external impulsive forces. We first analyze the dependence of the synchronized firing on the amplitude and the angular frequency of the impulsive force in the noise-free system. Three cases (regular spiking, traveling wave, and chaotic spiking) with low synchronized firing are chosen to study effects due to noise. In each case we find that small noise can be a promoter of synchronization phenomena in neural activities, by choosing an appropriate noise intensity acting on some of the neurons. (C) 2005 American Institute of Physics
In integrated medical considerations of the biological human system, both intellectual and motor performances in a similar manner are considered as a result of the function of the nervous system. Consequently, universal minimal dysfunctions of the central nervous system may lead to both intellectual and physical anomalies. Therefore, this study tests the hypothesis that there is a connection between the balance ability as a motor parameter and school success as an intellectual parameter. A postural measuring system based on the force-moment sensor technique was used to record the postural balance regulation of 773 children (circle divide 11 +/- 1 years). The school achievement of each child was determined by school grades. Data analysis was performed by linear as well as by nonlinear time series analyses. There are highly significant differences in balance regulation between good and poor pupils recognized by several linear and nonlinear parameters. Good pupils could be discriminated from pupils with bad results in learning to 80 %. The results support the hypothesis mentioned above. One possible explanation for the poor regulation of balance in bad learners could be a deficit in the neural maturity. In future, further developments will be targeted on higher discrimination levels, possibly in order to predict school success. On the other hand, the effects of special movement exercises on the neural development in childhood will be the focus in our further work
Concepts from Ergodic Theory are used to describe the existence of special non-transitive maps in attractors of phase synchronous chaotic oscillators. In particular, it is shown that, for a class of phase-coherent oscillators, these special maps imply phase synchronization. We illustrate these ideas in the sinusoidally forced Chua's circuit and two coupled Rossler oscillators. Furthermore, these results are extended to other coupled chaotic systems. In addition, a phase for a chaotic attractor is defined from the tangent vector of the flow. Finally, it is discussed how these maps can be used for the real-time detection of phase synchronization in experimental systems. (c) 2005 Elsevier B.V. All rights reserved
Chaotic channel
(2005)
This work combines the theory of chaotic synchronization with the theory of information in order to introduce the chaotic channel, an active medium formed by connected chaotic systems. This subset of a large chaotic net represents the path along which information flows. We show that the possible amount of information exchange between the transmitter, where information enters the net, and the receiver, the destination of the information, is proportional to the level of synchronization between these two special subsystems
We present an automatic control method for phase locking of regular and chaotic non-identical oscillations, when all subsystems interact via feedback. This method is based on the well known principle of feedback control which takes place in nature and is successfully used in engineering. In contrast to unidirectional and bidirectional coupling, the approach presented here supposes the existence of a special controller, which allows to change the parameters of the controlled systems. First we discuss general principles of automatic phase synchronization (PS) for arbitrary coupled systems with a controller whose input is given by a special quadratic form of coordinates of the individual systems and its output is a result of the application of a linear differential operator. We demonstrate the effectiveness of our approach for controlled PS on several examples: (i) two coupled regular oscillators, (ii) coupled regular and chaotic oscillators, (iii) two coupled chaotic R"ossler oscillators, (iv) two coupled foodweb models, (v) coupled chaotic R"ossler and Lorenz oscillators, (vi) ensembles of locally coupled regular oscillators, (vii) ensembles of locally coupled chaotic oscillators, and (viii) ensembles of globally coupled chaotic oscillators.
An approach is presented for coupled chaotic systems with weak coherent motion, from which we estimate the upper bound value for the absolute phase difference in phase synchronous states. This approach shows that synchronicity in phase implies synchronicity in the time of events, a characteristic explored to derive an equation to detect phase synchronization, based on the absolute difference between the time of these events. We demonstrate the potential use of this approach for the phase coherent and the funnel attractor of the Rossler system, as well as for the spiking/bursting Rulkov map.
Understanding the functional dynamics of the mammalian brain is one of the central aims of modern neuroscience. Mathematical modeling and computational simulations of neural networks can help in this quest. In recent publications, a multilevel model has been presented to simulate the resting-state dynamics of the cortico-cortical connectivity of the mammalian brain. In the present work we investigate how much of the dynamical behavior of the multilevel model can be reproduced by a strongly simplified model. We find that replacing each cortical area by a single Rulkov map recreates the patterns of dynamical correlation of the multilevel model, while the outcome of other models and setups mainly depends on the local network properties, e. g. the input degree of each vertex. In general, we find that a simple simulation whose dynamics depends on the global topology of the whole network is far from trivial. A systematic analysis of different dynamical models and coupling setups is required.
In the present paper, two kinds of dynamical complex networks are considered. The first is that elements of every node have different time delays but all nodes in Such networks have the same time-delay vector. The second is that different nodes have different time-delay vectors, and the elements of each node also have different time delays. Corresponding synchronization theorems are established. Numerical examples show the efficiency of the derived theorems.
In this paper we show that delay embedding produces spurious structures in a recurrence plot (RP) that are not present in the real attractor. We analyze typical sets of simulated data, such as white noise and data from the chaotic Rossler system to show the relevance of this effect. In the second part of the paper we show that the second order Renyi entropy and the correlation dimension are dynamical invariants that can be estimated from Recurrence Plots with arbitrary embedding dimension and delay
Starting from an initial wiring of connections, we show that the synchronizability of a network can be significantly improved by evolving the graph along a time dependent connectivity matrix. We consider the case of connectivity matrices that commute at all times, and compare several approaches to engineer the corresponding commutative graphs. In particular, we show that synchronization in a dynamical network can be achieved even in the case in which each individual commutative graphs does not give rise to synchronized behavior
We propose a new autonomous dynamical system of dimension N=4 that demonstrates the regime of stable two- frequency motions and period-doubling bifurcations of a two-dimensional torus. It is shown that the period-doubling bifurcation of the two-dimensional torus is not followed by the resonance phenomenon, and the two-dimensional ergodic torus undergoes a period-doubling bifurcation. The interaction of two generators is also analyzed. The phenomenon of external and mutual synchronization of two-frequency oscillations is observed, for which winding number locking on a two- dimensional torus takes place
In this work, we reanalyze the heart rate variability (HRV) data from the 2002 Computers in Cardiology (CiC) Challenge using the concept of large-scale dimension densities and additionally apply this technique to data of healthy persons and of patients with cardiac diseases. The large-scale dimension density (LASDID) is estimated from the time series using a normalized Grassberger-Procaccia algorithm, which leads to a suitable correction of systematic errors produced by boundary effects in the rather large scales of a system. This way, it is possible to analyze rather short, nonstationary, and unfiltered data, such as HRV. Moreover, this method allows us to analyze short parts of the data and to look for differences between day and night. The circadian changes in the dimension density enable us to distinguish almost completely between real data and computer-generated data from the CiC 2002 challenge using only one parameter. In the second part we analyzed the data of 15 patients with atrial fibrillation (AF), 15 patients with congestive heart failure (CHF), 15 elderly healthy subjects (EH), as well as 18 young and healthy persons (YH). With our method we are able to separate completely the AF (rho(mu)(ls)=0.97 +/- 0.02) group from the others and, especially during daytime, the CHF patients show significant differences from the young and elderly healthy volunteers (CHF, 0.65 +/- 0.13; EH, 0.54 +/- 0.05; YH, 0.57 +/- 0.05; p < 0.05 for both comparisons). Moreover, for the CHF patients we find no circadian changes in rho(mu)(ls) (day, 0.65 +/- 0.13; night, 0.66 +/- 0.12; n.s.) in contrast to healthy controls (day, 0.54 +/- 0.05; night, 0.61 +/- 0.05; p=0.002). Correlation analysis showed no statistical significant relation between standard HRV and circadian LASDID, demonstrating a possibly independent application of our method for clinical risk stratification
Experimental evidence of anomalous phase synchronization in two diffusively coupled Chua oscillators
(2006)
We study the transition to phase synchronization in two diffusively coupled, nonidentical Chua oscillators. In the experiments, depending on the used parameterization, we observe several distinct routes to phase synchronization, including states of either in-phase, out-of-phase, or antiphase synchronization, which may be intersected by an intermediate desynchronization regime with large fluctuations of the frequency difference. Furthermore, we report the first experimental evidence of an anomalous transition to phase synchronization, which is characterized by an initial enlargement of the natural frequency difference with coupling strength. This results in a maximal frequency disorder at intermediate coupling levels, whereas usual phase synchronization via monotonic decrease in frequency difference sets in only for larger coupling values. All experimental results are supported by numerical simulations of two coupled Chua models
Spatial recurrence plots
(2006)
We propose an extension of the recurrence plot concept to perform quantitative analyzes of roughness and disorder of spatial patterns at a fixed time. We introduce spatial recurrence plots (SRPs) as a graphical representation of the pointwise correlation matrix, in terms of a two-dimensional spatial return plot. This technique is applied to the study of complex patterns generated by coupled map lattices, which are characterized by measures of complexity based on SRPs. We show that the complexity measures we propose for SRPs provide a systematic way of investigating the distribution of spatially coherent structures, such as synchronization domains, in lattice profiles. This approach has potential for many more applications, e.g., in surface roughness analyzes
We present two different approaches to detect and quantify phase synchronization in the case of coupled non- phase coherent oscillators. The first one is based on the general idea of curvature of an arbitrary curve. The second one is based on recurrences of the trajectory in phase space. We illustrate both methods in the paradigmatic example of the Rossler system in the funnel regime. We show that the second method is applicable even in the case of noisy data. Furthermore, we extend the second approach to the application of chains of coupled systems, which allows us to detect easily clusters of synchronized oscillators. In order to illustrate the applicability of this approach, we show the results of the algorithm applied to experimental data from a population of 64 electrochemical oscillators
In this Letter, we show that coherence and phase synchronization analysis are sensitive but not specific in detecting the correct class of underlying dynamics. We propose procedures to increase specificity and demonstrate the power of the approach by application to paradigmatic dynamic model systems. (c) 2006 Elsevier B.V. All rights reserved
Human comment is studied using data from 'tianya' which is one of the most popular on-line social systems in China. We found that the time interval between two consecutive comments on the same topic, called inter-event time, follows a power-law distribution. This result shows that there is no characteristic decay time on a topic. It allows for very long periods without comments that separate bursts of intensive comments. Furthermore, the frequency of a different ID commenting on a topic also follows a power-law distribution. It indicates that there are some "hubs" in the topic who lead the direction of the public opinion. Based on the personal comments habit, a model is introduced to explain these phenomena. The numerical simulations of the model fit well with the empirical results. Our findings are helpful for discovering regular patterns of human behavior in on-line society and the evolution of the public opinion on the virtual as well as real society.