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PubMed1996

Nonlinear dynamics in ventricular fibrillation.

Electrogram recordings of ventricular fibrillation appear complex and possibly chaotic. However, sequences of beat-to-beat intervals obtained from these recordings are generally short, making it difficult to explicitly demonstrate nonlinear dynamics. Motivated by the work of Sugihara on atmospheric dynamics and the Durbin-Watson test for nonlinearity, we introduce a new statistical test that recovers significant dynamical patterns from smoothed lag plots. This test is used to show highly significant nonlinear dynamics in a stable canine model of ventricular fibrillation.

Algorithms↗
PubMed2005

Nonlinear dynamical analysis of EEG and MEG: review of an emerging field.

Many complex and interesting phenomena in nature are due to nonlinear phenomena. The theory of nonlinear dynamical systems, also called 'chaos theory', has now progressed to a stage, where it becomes possible to study self-organization and pattern formation in the complex neuronal networks of the brain. One approach to nonlinear time series analysis consists of reconstructing, from time series of EEG or MEG, an attractor of the underlying dynamical system, and characterizing it in terms of its dimension (an estimate of the degrees of freedom of the system), or its Lyapunov exponents and entropy (reflecting unpredictability of the dynamics due to the sensitive dependence on initial conditions). More recently developed nonlinear measures characterize other features of local brain dynamics (forecasting, time asymmetry, determinism) or the nonlinear synchronization between recordings from different brain regions. Nonlinear time series has been applied to EEG and MEG of healthy subjects during no-task resting states, perceptual processing, performance of cognitive tasks and different sleep stages. Many pathologic states have been examined as well, ranging from toxic states, seizures, and psychiatric disorders to Alzheimer's, Parkinson's and Cre1utzfeldt-Jakob's disease. Interpretation of these results in terms of 'functional sources' and 'functional networks' allows the identification of three basic patterns of brain dynamics: (i) normal, ongoing dynamics during a no-task, resting state in healthy subjects; this state is characterized by a high dimensional complexity and a relatively low and fluctuating level of synchronization of the neuronal networks; (ii) hypersynchronous, highly nonlinear dynamics of epileptic seizures; (iii) dynamics of degenerative encephalopathies with an abnormally low level of between area synchronization. Only intermediate levels of rapidly fluctuating synchronization, possibly due to critical dynamics near a phase transition, are associated with normal information processing, whereas both hyper-as well as hyposynchronous states result in impaired information processing and disturbed consciousness.

Animals↗
PubMed2006

The nonlinear dynamical hypothesis in science education problem solving: a catastrophe theory approach.

The current study tests the nonlinear dynamical hypothesis in science education problem solving by applying catastrophe theory. Within the neo-Piagetian framework a cusp catastrophe model is proposed, which accounts for discontinuities in students' performance as a function of two controls: the functional M-capacity as asymmetry and the degree of field dependence/independence as bifurcation. The two controls have functional relation with two opponent processes, the processing of relevant information and the inhibitory process of dis-embedding irrelevant information respectively. Data from achievement scores of freshmen at a technological college were measured at two points in time, and were analyzed using dynamic difference equations and statistical regression techniques. The cusp catastrophe model proved superior (R(2)=0.77) comparing to the pre-post linear counterpart (R(2)=0.46). Besides the empirical evidence, theoretical analyses are provided, which attempt to build bridges between NDS-theory concepts and science education problem solving and to neo-Piagetian theories as well. This study sets a framework for the application of catastrophe theory in education.

Humans↗
PubMed1996

Detection of seizures from small samples using nonlinear dynamic system theory.

The electroencephalogram (EEG), like many other biological phenomena, is quite likely governed by nonlinear dynamics. Certain characteristics of the underlying dynamics have recently been quantified by computing the correlation dimensions (D2) of EEG time series data. In this paper, D2 of the unbiased autocovariance function of the scalp EEG data was used to detect electrographic seizure activity. Digital EEG data were acquired at a sampling rate of 200 Hz per channel and organized in continuous frames (duration 2.56 s, 512 data points). To increase the reliability of D2 computations with short duration data, raw EEG data were initially simplified using unbiased autocovariance analysis to highlight the periodic activity that is present during seizures. The D2 computation was then performed from the unbiased autocovariance function of each channel using the Grassberger-Procaccia method with Theiler's box-assisted correlation algorithm. Even with short duration data, this preprocessing proved to be computationally robust and displayed no significant sensitivity to implementation details such as the choices of embedding dimension and box size. The system successfully identified various types of seizures in clinical studies.

Algorithms↗
PubMed2002

Noise activated nonlinear dynamic sensors.

We introduce a novel dynamical description for a wide class of nonlinear physical sensors operating in a noisy environment. The presence of unknown physical signals is assessed via the monitoring of the residence times in the metastable attractors of the system. We show that the presence of ambient noise, far from degrading the sensor operation, can actually improve its sensitivity and provide a greatly simplified readout scheme, as well as significantly reduce processing procedures for this new class of devices that we propose to call noise activated nonlinear dynamic sensors. Such devices can also show interesting dynamical features such as the resonant trapping effect.

Journal Article↗
PubMed1994

Nonlinear dynamical aspects of the human sleep EEG.

This article deals with the application of methods from the theory of nonlinear dynamical systems to EEG signals. Theoretical background, mathematical concepts and algorithms for the calculation of "non-linear parameters" are reviewed and influences of the structure of reconstructed data sets on the calculations are pointed out. We present results for the estimation of the correlation dimension D2 and the principal Lyapunov-exponent lambda 1 for sleep EEG data respectively from 10 and 15 healthy subjects corresponding to different sleep stages. Essentially, we found a statistically significant decrease of both D2 and lambda 1 as sleep moves towards slow wave stages. The values for REM sleep lie between the values of stage I and II. Moreover, for one subject as an example we present calculations of the principal Lyapunov-exponent lambda 1 and the sum of the two largest Lyapunov-exponents lambda 1 + lambda 2 for EEG segments following subsequently during the night. Finally, we compare our results with investigations of other groups and discuss difficulties and opportunities of the nonlinear approach to human EEG signals.

Algorithms↗
PubMed1996

Using fractals and nonlinear dynamics to determine the physical properties of ion channel proteins.

Three examples are given of how concepts from fractals and nonlinear dynamics have been used to analyze the voltages and currents recorded through ion channels in an attempt to determine the physical properties of ion channel proteins. (1) Early models had assumed that the switching of the ion channel protein from one conformational state to another can be represented by a Markov process that has no long-term correlations. However, one support for the existence of long-term correlations in channel function is that the currents recorded through individual ion channels have self-similar properties. These fractal properties can be characterized by a scaling function determined from the distribution of open and closed time intervals, which provides information on the distribution of activation energy barriers between the open and closed conformational substates of the ion channel protein and/or on how those energy barriers change in time. (2) Another support for such long-term correlations is that the whole-cell membrane voltage recorded across many channels at once may also have a fractal form. The Hurst rescaled range analysis of these fluctuations provides information on the type and degree of correlation in time of the functioning of ion channels. (3) The early models had also assumed that the switching from one state to another is an inherently random process driven by the energy from thermal fluctuations. More recently developed models have shown that deterministic dynamics may also produce the same distributions of open and closed times as those previously attributed to random events. This raises the possibility that the deterministic atomic and electrostatic forces play a role in switching the channel protein from one conformational shape to another. Debate exists about whether random, fractal, or deterministic models best represent the functioning of ion channels. However, fractal and deterministic dynamics provide a new approach to the study of ion channels that should be seriously considered by neuroscientists.

Animals↗
PubMed1993

Nonlinear dynamics in a model neuron provide a novel mechanism for transient synaptic inputs to produce long-term alterations of postsynaptic activity.

1. A mathematical model of a bursting molluscan neuron has been found to possess multiple modes of electrical activity, such as periodic beating (tonic firing), periodic bursting (bursts of action potentials separated by quiescent periods), and potentially chaotic bursting, all at a single set of parameters. The multiple modes correspond to multiple stable attractors, whose existence is an emergent property of the nonlinear dynamics of the system. 2. Transient synaptic inputs can switch the activity of the neuron between different modes. These mode transitions, which do not require any changes in the biochemical or biophysical parameters of the neuron, provide an enduring response to a transient input, as well as a mechanism for phasic sensitivity (i.e., temporal specificity). 3. These results provide new insights into the role of nonlinear dynamics in information processing and storage at the level of the single neuron.

Animals↗
PubMed2006

Nonlinear dynamics of an interface between shear bands.

We study numerically the nonlinear dynamics of a shear banding interface in two-dimensional planar shear flow, within the nonlocal Johnson-Segalman model. Consistent with a recent linear stability analysis, we find that an initially flat interface is unstable with respect to small undulations for a sufficiently small ratio of the interfacial width l to cell length L(x). The instability saturates in finite amplitude interfacial fluctuations. For decreasing l/L(x) these undergo a nonequilibrium transition from simple traveling interfacial waves with constant average wall stress, to periodically rippling waves with a periodic stress response. When multiple shear bands are present we find erratic interfacial dynamics and a stress response suggesting low dimensional chaos.

Journal Article↗
PubMed2004

Nonlinear dynamics of gravity and matter creation in a cosmology with an unbounded Hamiltonian.

Nonlinear dynamics and stability properties of a cosmological model of spatially homogeneous coupled gravity and matter fields is analyzed using methods of classical mechanics. The system exhibits regions of chaos and dramatic changes in structural stability as the strength of the coupling between the fields is varied. Numerical simulations suggest that Hamiltonian systems with structure appropriate for describing matter-gravity interaction constitute a new class of nonlinear systems with very unusual and rich dynamics.

Journal Article↗
PubMed1998

Nonlinear dynamical analysis of periodic lateralized epileptiform discharges.

Nonlinear time series analysis can be used to investigate the dynamics underlying the generation of EEG signal. In the present study we used this approach to study the pathophysiology of PLEDs. We calculated the correlation dimension D2 of an EEG with typical PLEDs, and compared the results with those obtained for surrogate data. These surrogate data have the same power spectrum and amplitude distribution as the original EEG data, but are otherwise random. By construction, such surrogate data can be described by a linear model. Our results showed that D2 estimations for PLEDs were low, on the order of one, and that the results for EEG and the surrogate data were clearly different, indicating that the EEG with PLEDs reflects nonlinear dynamics of the underlying neural networks.

Electroencephalography↗
PubMed1995

Locally activated neural networks and stable neural controller design for nonlinear dynamic systems.

A stable neural control scheme using a locally activated neural network has been proposed for a class of nonlinear dynamic systems. The locally activated neural network, for a given input, essentially selects a small subset of the network hidden nodes for output computation using the CMAC-like content addressing mechanism. This network aims to maintain local representations of the system dynamics. Thus, the global control performance in the concerned state space is achieved by the cooperation of many local control efforts and furthermore, real-time control can be facilitated because only a small sized network is involved to control and learn at any given time. The proposed control scheme is composed of two stages: (1) prediction error based learning in which the network attempts to learn the nonlinear basis functions of the plant inverse dynamics by a modified backpropagation learning rule; and (2) tracking error based learning in which the network weights are further fine-tuned using the basis set obtained in (1). This basis set spans the locally partitioned vector space of the system inverse dynamics when the prediction error based learning is achieved within a prescribed error tolerance. For uniform stability, the sliding mode control is introduced as a safety mechanism when the network has not sufficiently learned the plant dynamics yet. With suitable assumptions on the controlled plant, global stability and tracking error convergence proof has been given. Finally, the proposed control scheme is verified with computer simulation.

Adaptation, Physiological↗
PubMed2005

Origins of group coordination: nonlinear dynamics and the role of verbalization.

Coordination occurs when two or more people do the same or complimentary tasks simultaneously; its explanation game theory, nonlinear dynamics, and implicit learning theory. In the experiment, 12 four-person groups were allowed to discuss the coordination (card game) task while performing it; 12 other groups worked nonverbally. One to three group members were replaced during the game. Split-plot ANOVA showed that verbalizing groups performed better than nonverbalizing groups overall and showed more acute coordination learning curves, but verbalization did not compensate for the replacement of personnel. Groups that changed one or two players showed positive coordination transfer, but groups that changed three players did not. Nonlinear regression for temporal dynamics within verbalizing and nonverbalizing groups showed asymptotic stability for initial coordination learning and transfer to a difficult rule, a chaotic function when replacements were introduced, and asymptotic stability again when the team with replacements switched to the difficult rule.

Cooperative Behavior↗
PubMed1993

A nonlinear dynamical model for atmospheric boundary layer turbulence.

Using the qualitative theory of nonlinear dynamical systems and the ergodic theory of chaos and strange attractors, we study a truncated-spectrum model of dynamical equations of the atmosphere. In the parameter plane (Re, Ri), the atmospheric motion states can be divided into four regions: O (basic), P (periodic), T (turbulent or chaotic), and T-P (transition of T and P). We analyze the routes to turbulence during the day and at night. Finally, we discuss the physical aspects of the occurrence of turbulence.

Journal Article↗
PubMed2005

Reconstruction of stochastic nonlinear dynamical models from trajectory measurements.

An algorithm is presented for reconstructing stochastic nonlinear dynamical models from noisy time-series data. The approach is analytical; consequently, the resulting algorithm does not require an extensive global search for the model parameters, provides optimal compensation for the effects of dynamical noise, and is robust for a broad range of dynamical models. The strengths of the algorithm are illustrated by inferring the parameters of the stochastic Lorenz system and comparing the results with those of earlier research. The efficiency and accuracy of the algorithm are further demonstrated by inferring a model for a system of five globally and locally coupled noisy oscillators.

Journal Article↗
PubMed2006

Nonlinear dynamical behavior of surface electromyographical signals of biceps muscle under two simulated static work postures.

The primary objective of this study was to explore nonlinear characteristics, if any, of the surface myoelectrical (EMG) activity of the biceps muscle during simulated maximum voluntary contraction at a reference arm posture (MVCRP), and at an elevated static arm work posture (EAWP) observed on the manufacturing assembly line. The results showed that positive Lyapunov exponents existed for each of the trials in both work postures. The statistical test of significance showed that surface EMG of the biceps brachii was more chaotic under the MVCRP condition as compared to the EWAP condition. Based on the Kaplan-Yorke dimension, it was observed that the recorded two types of time series EMG data were similar in complexity. The study also found a significant difference in fractal dimensions between the two postures. It was concluded that the nonlinear dynamical EMG properties expressed in terms of the Lyapunov exponents, Kaplan-Yorke dimension, and fractal dimensions could be used as sensitive indices for evaluation of the level of muscular efforts under static work conditions.

Adult↗
PubMed1995

Spatial heterogeneity, nonlinear dynamics and chaos in infectious diseases.

There is currently considerable interest in the role of nonlinear phenomena in the population dynamics of infectious diseases. Childhood diseases such as measles are particularly well documented dynamically, and have recently been the subject of analyses (of both models and notification data) to establish whether the pattern of epidemics is chaotic. Though the spatial dynamics of measles have also been extensively studied, spatial and nonlinear dynamics have only recently been brought together. The present review concentrates mainly on describing this synthesis. We begin with a general review of the nonlinear dynamics of measles models, in a spatially homogeneous environment. Simple compartmental models (specifically the SEIR model) can behave chaotically, under the influence of strong seasonal 'forcing' of infection rate associated with patterns of schooling. However, adding observed heterogeneities such as age structure can simplify the deterministic dynamics back to limit cycles. By contrast all current strongly seasonally forced stochastic models show large amplitude irregular fluctuations, with many more 'fadeouts' of infection that is observed in real communities of similar size. This indicates that (social and/or geographical) spatial heterogeneity is needed in the models. We review the exploration of this problem with nonlinear spatiotemporal models. The few studies to date indicate that spatial heterogeneity can help to increase the realism of models. However, a review of nonlinear analyses of spatially subdivided measles data show that more refinements of the models (particularly in representing the impact of human demographic changes on infection dynamics) are required. We conclude with a discussion of the implication of these results for the dynamics of infectious diseases in general and, in particular, the possibilities of cross fertilization between human disease epidemiology and the study of plant and animal diseases.

Communicable Diseases↗
PubMed2005

Relation between observability and differential embeddings for nonlinear dynamics.

In the analysis of a scalar time series, which lies on an m-dimensional object, a great number of techniques will start by embedding such a time series in a d-dimensional space, with d>m. Therefore there is a coordinate transformation phi(s) from the original phase space to the embedded one. The embedding space depends on the observable s(t). In theory, the main results reached are valid regardless of s(t). In a number of practical situations, however, the choice of the observable does influence our ability to extract dynamical information from the embedded attractor. This may arise in problems in nonlinear dynamics such as model building, control and synchronization. To some degree, ease of success will depend on the choice of the observable simply because it is related to the observability of the dynamics. In this paper the observability matrix for nonlinear systems, which uses Lie derivatives, is revisited. It is shown that such a matrix can be interpreted as the Jacobian matrix of phi(s)--the map between the original phase space and the differential embedding induced by the observable--thus establishing a link between observability and embedding theory.

Journal Article↗