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PubMed2001

Effects of spatial inhomogeneities on the dynamics of cavity solitons in quadratically nonlinear media.

We study the dynamics of cavity solitons under the influence of spatial inhomogeneities and derive generalized equations of motions. For perturbations large compared to the soliton size we find the modulus of the soliton velocity to be proportional to the gradient of the respective perturbation and that the proportionality coefficient changes sign when the soliton peak power drives the cavity beyond the resonance. For short scale perturbations solitons may be trapped at the extrema of the inhomogeneities. Shape and stability of these trapped solitons can be quasianalytically described by means of a perturbation theory. If both types of perturbations act solitons are either trapped or move depending on the strength of the respective perturbation. In the framework of a quasiparticle approach this dynamics is governed by a differential equation that holds for particle motion in a strongly viscous fluid under the action of a constant and harmonically varying force. We also show that in addition to acquiring a velocity the very existence conditions of the solitons (hysteresis curve) are affected by both kinds of perturbations. We find good quantitative agreement between our analytical results and numerical findings, which were obtained for a two wave interaction in a cavity filled with a quadratically nonlinear material.

Journal Article↗

PubMed2005

Identifying ill-behaved nonlinear processes without metrics: use of symbolic dynamics.

Given ill-behaved psychological data that are unlikely to satisfy metric axioms, the use of encoding in symbolic dynamics, and hence leading into Markov analyses, is explored. Various measures of entropy are calculated. The tractability of entropic measures for categorizing the trajectories of nonlinear dynamics that may be present and chaotic is considered, with a focus on the case where there are two attractors and at least one heteroclinic orbit between them. Fast/slow dynamics are treated as a special case. The problem of identification is in other contexts the problem of diagnosis in time-varying pathologies. Some real data, selected for their psychological relevance in clinical, forensic and psychophysical processes, that are apparently edge-of chaos and nonstationary, are for comparison analysed both as metric and discrete and in symbolic encoding.

Humans↗

PubMed2003

Nonlinear synchronization in EEG and whole-head MEG recordings of healthy subjects.

According to Friston, brain dynamics can be modelled as a large ensemble of coupled nonlinear dynamical subsystems with unstable and transient dynamics. In the present study, two predictions from this model (the existence of nonlinear synchronization between macroscopic field potentials and itinerant nonlinear dynamics) were investigated. The dependence of nonlinearity on the method of measuring brain activity (EEG vs. MEG) was also investigated. Dataset I consisted of 10 MEG recordings in 10 healthy subjects. Dataset II consisted of simultaneously recorded MEG (126 channels) and EEG (19 channels) in 5 healthy subjects. Nonlinear coupling was assessed with the synchronization likelihood S and dynamic itinerancy with the synchronization entropy Hs. Significance was assessed with a bootstrap procedure ("surrogate data testing"), comparing S and Hs with their distribution under the null hypothesis of stationary, linear dynamics. Significant nonlinear synchronization was detected in 14 of 15 subjects. The nonlinear dynamics were associated with a high index of itinerant behaviour. Nonlinear interdependence was significantly more apparent in MEG data than EEG. Synchronous oscillations in MEG and EEG recordings contain a significant nonlinear component that exhibits characteristics of unstable and itinerant behaviour. These findings are in line with Friston's proposal that the brain can be conceived as a large ensemble of coupled nonlinear dynamical subsystems with labile and unstable dynamics. The spatial scale and physical properties of MEG acquisition may increase the sensitivity of the data to underlying nonlinear structure.

Adult↗

PubMed2002

Comparison of nonequilibrium molecular dynamics with experimental measurements in the nonlinear shear-thinning regime.

Nonequilibrium molecular dynamics of a low-molecular-weight fluid (squalane) are compared with experimental measurements in both the linear (Newtonian) and nonlinear (non-Newtonian) regimes. The experimental and simulation data are shown to follow the same time-temperature superposition master curve. This represents the first comparison of the nonlinear rheology predicted by nonequilibrium molecular dynamics with experiment, and is thus the first experimental test of nonequilibrium molecular dynamics simulations in the nonlinear regime.

Journal Article↗

PubMed2001

A biologically inspired neural network for dynamic programming.

An artificial neural network with a two-layer feedback topology and generalized recurrent neurons, for solving nonlinear discrete dynamic optimization problems, is developed. A direct method to assign the weights of neural networks is presented. The method is based on Bellmann's Optimality Principle and on the interchange of information which occurs during the synaptic chemical processing among neurons. The neural network based algorithm is an advantageous approach for dynamic programming due to the inherent parallelism of the neural networks; further it reduces the severity of computational problems that can occur in methods like conventional methods. Some illustrative application examples are presented to show how this approach works out including the shortest path and fuzzy decision making problems.

Algorithms↗

PubMed1997

Dynamic consequences of reproductive delay in Leslie matrix models with nonlinear survival probabilities.

The dynamic consequences of reproductive delay in Leslie matrix models with nonlinear survival probabilities p are analyzed. In consideration of two-age classes, proof is presented for a wide range of p functions that, outside the strongly resonant cases, the transfer from stability to instability goes through a supercritical Hopf bifurcation and, moreover, that the nonlinear development has a strong resemblance of three or four cycles, either exact or approximate. In three-age class models, the tendency toward four-periodical dynamics is shown to be even more pronounced, a qualitative finding that gradually disappears as we turn to the higher-dimensional cases. We also prove that for models of any dimension n > 1 theme are regions in parameter space where the equilibrium is unstable at its creation and we demonstrate that the dynamics in this age-class extinguishing case is 2k.n cyclic.

Age Factors↗

PubMed1997

Melnikov analysis of chaos in a simple epidemiological model.

Melnikov's method is applied to an SIR model of epidemic dynamics with a periodically modulated nonlinear incidence rate. This analysis establishes mathematically, for the first time, the existence of chaotic motion in these models. A related technique also makes it possible to prove that homoclinic bifurcations occurs in the model.

Disease Outbreaks↗

PubMed1999

Damping and modification of the multiquanta Davydov-like solitons in molecular chains.

Relaxation of the multivibron soliton in molecular chain on lattice vibrations is investigated within the simple microscopic model. It was shown that its dynamics is governed by the nonlinear Schrödinger equation containing damping term. Its appearance is the consequence of the emission and absorption of real phonons arising when soliton velocity approaches the phase speed of sound. Explicit time dependence of the soliton parameters results as a consequence of these perturbations. In particular, soliton amplitude decreases while its width increases.

Nonlinear Dynamics↗

PubMedDate not supplied

A novel method for the topographic analysis of neural activity reveals formation and dissolution of 'Dynamic Cell Assemblies'.

The study of synchronous oscillations in neural systems is a very active area of research. However, cognitive function may depend more crucially upon a dynamic alternation between synchronous and desynchronous activity rather than synchronous behaviour per se. The principle aim of this study is to develop and validate a novel method of quantifying this complex process. The method permits a direct mapping of phase synchronous dynamics and desynchronizing bursts in the spatial and temporal domains. Two data sets are analyzed: Numeric data from a model of a sparsely coupled neural cell assembly and experimental data consisting of scalp-recorded EEG from 40 human subjects. In the numeric data, the approach enables the demonstration of complex relationships between cluster size and temporal duration that cannot be detected with other methods. Dynamic patterns of phase-clustering and desynchronization are also demonstrated in the experimental data. It is further shown that in a significant proportion of the recordings, the pattern of dynamics exhibits nonlinear structure. We argue that this procedure provides a 'natural partitioning' of ongoing brain dynamics into topographically distinct synchronous epochs which may be integral to the brain's adaptive function. In particular, the character of transitions between consecutive synchronous epochs may reflect important aspects of information processing and cognitive flexibility.

Adaptation, Ocular↗

PubMed2002

A theoretical framework of ecological phase transitions for characterizing tree-grass dynamics.

This paper describes a theoretical framework of ecological phase transitions for modeling tree-grass dynamics and analyzing the shifts or phase transitions from one vegetation structure to another in the southern Texas landscape. This framework implements the integration of percolation theory, fractal geometry and phase transition theory as a method for modeling the spatial patterns of tree-grass dynamics, and nonlinear Markov non-equilibrium thermodynamic stability theory as a method for characterizing temporal tree-grass dynamics and phase transition. An historical sequence of aerial photographs at a Prosopis-thornscrub savanna parkland site in southern Texas was used to determine the parameters of the models. The preliminary analytical result accords well with current understanding and field survey of vegetation dynamics in the southern Texas landscape. The potential of such approaches and other relevant theories such as self-organized criticality and synergetics to vegetation dynamics is also discussed.

Ecosystem↗

PubMed1994

Lack of evidence for low-dimensional chaos in heart rate variability.

INTRODUCTION: The term chaos is used to describe erratic or apparently random time-dependent behavior in deterministic systems. It has been suggested that the variability observed in the normal heart rate may be due to chaos, but this question has not been settled. METHODS AND RESULTS: Heart rate variability was assessed by recordings of consecutive RR intervals in ten healthy subjects using ambulatory ECG. All recordings were performed with the subjects at rest in the supine position. To test for the presence of nonlinearities and/or chaotic dynamics, ten surrogate time series were constructed from each experimental dataset. The surrogate data were tailored to have the same linear dynamics and the same amplitude distribution as the original data. Experimental and surrogate data were then compared using various nonlinear measures. Power spectral analysis of the RR intervals showed a 1/f pattern. The correlation dimension differed only slightly between the experimental and the surrogate data, indicating that linear correlations, and not a "strange" attractor, are the major determinants of the calculated correlation dimension. A test for nonlinear predictability showed coherent nonlinear dynamic structure in the experimental data, but the prediction error as a function of the prediction length increased at a slower rate than characteristic of a low-dimensional chaotic system. CONCLUSION: There is no evidence for low-dimensional chaos in the time series of RR intervals from healthy human subjects. However, nonlinear determinism is present in the data, and various mechanisms that could generate such determinism are discussed.

Adolescent↗

PubMed1996

Nonlinear prediction as a tool for tracking the dynamics of single-trial readiness potentials.

The performance of a voluntary act is preceded by an intrinsic process of intention and preparation accompanied by specific patterns of scalp-recorded EEG. The baseline shift to negativity and the decrease of alpha and beta oscillations prior to movement performance are considered to reflect the motor preparation and are observable even in single-trial EEG records during repetitive voluntary movements. The single-trial features of these patterns are of importance since they would reflect the dynamics of EEG activity during movement performance throughout the experimental session. Since this process is very complex and its dynamics is unclear, we applied a more general approach, i.e. the method of nonlinear prediction (NP) recommended for detection of chaos in the behavior of nonlinear dynamical systems. The NP is based on a library of past patterns of the time series used to compute the prediction of future pattern. If the time series is chaotic, the correlation coefficient between predicted and actual values would decrease as a function of the prediction step. The NP was applied to EEG records containing readiness potentials (RPs) at Fz, Cz, Pz, C3 and C4 during repetitive voluntary finger flexion. Because of nonstationarity, the prediction and the library spanned the same time period. As a result (1) chaotic segments in EEG records were detected; and (2) the sharp jumps of the absolute error between predicted and actual values indicated instances of poor prediction, which were interpreted as indicating new stages of EEG activity related to the preparatory process of motor activity.

Adult↗

PubMed1993

Chaos, population biology, and epidemiology: some research implications.

In this article I aim to provide some feeling of the new paradigm of disease causation (chaos) as it applies to the field of population biology and epidemiology. A secondary objective is to show, with the aid of qualitative methods, how one can approach chaos in time-series data. The multifactorial stochastic paradigm of causation is contrasted with the new deterministic approach. This approach is embedded in the theory of nonlinear system dynamics. Chaos implies that randomness is intrinsic to a nonlinear deterministic system; this is true despite the extent of knowledge of the intervening causes and, ultimately, despite determinism. Three research avenues are discussed in depth from the standpoint of chaos theory. First, the topic of sporadic epidemics is dealt with. I argue that the space-time clustering of cases from a starting epidemic is due to a sudden and high increase of the contact rate beyond a threshold. Interaction rather than main effects and nonlinear rather than linear dynamics are involved. Second, the incubation period of disease is studied. I advocate that an individual-level deterministic process underlies Sartwell's model of the incubation period. This accounts for the robustness of the model vis-à-vis confounding variables. Third, monozygotic twinning is analyzed. Assumed by some to be a random process, monozygotic twinning proves to be dynamically different from dizygotic or single-maternity processes; its dynamics can actually be chaotic. Throughout the provided examples, the point is made that chancelike phenomena are primarily concerned with chaos theory. For biological problems showing recurrent inconsistencies by stochastic modeling, dynamic modeling should be envisaged. Inconsistencies can suggest that the relevant factors are out of the model and that they are related deterministically. Finally, spectral analysis and attractors in the phase space are presented; these tools can aid the population biologist in tracing out chaos from time-series data sets. Several time-series data sets are simulated according to a simple nonlinear difference equation that bears some relationship to the basics of the dynamics of infections in the population. I show how the series can be analyzed and interpreted. Much research remains to be carried out until the nonlinear effects of risk factors can be validated. The undertaking is worth the effort, as a new paradigm of causation is at stake.

Causality↗

PubMed1999

Linear and nonlinear analysis of human dynamic cerebral autoregulation.

The linear dynamic relationship between systemic arterial blood pressure (ABP) and cerebral blood flow velocity (CBFV) was studied by time- and frequency-domain analysis methods. A nonlinear moving-average approach was also implemented using Volterra-Wiener kernels. In 47 normal subjects, ABP was measured with Finapres and CBFV was recorded with Doppler ultrasound in both middle cerebral arteries at rest in the supine position and also during ABP drops induced by the sudden deflation of thigh cuffs. Impulse response functions estimated by Fourier transfer function analysis, a second-order mathematical model proposed by Tiecks, and the linear kernel of the Volterra-Wiener moving-average representation provided reconstructed velocity model responses, for the same segment of data, with significant correlations to CBFV recordings corresponding to r = 0.52 +/- 0.19, 0.53 +/- 0.16, and 0.67 +/- 0.12 (mean +/- SD), respectively. The correlation coefficient for the linear plus quadratic kernels was 0.82 +/- 0.08, significantly superior to that for the linear models (P < 10(-6)). The supine linear impulse responses were also used to predict the velocity transient of a different baseline segment of data and of the thigh cuff velocity response with significant correlations. In both cases, the three linear methods provided equivalent model performances, but the correlation coefficient for the nonlinear model dropped to 0.26 +/- 0.25 for the baseline test set of data and to 0.21 +/- 0.42 for the thigh cuff data. Whereas it is possible to model dynamic cerebral autoregulation in humans with different linear methods, in the supine position a second-order nonlinear component contributes significantly to improve model accuracy for the same segment of data used to estimate model parameters, but it cannot be automatically extended to represent the nonlinear component of velocity responses of different segments of data or transient changes induced by the thigh cuff test.

Adult↗

PubMed2004

Relationship between noradrenaline and nonlinear indexes of blood pressure dynamics in normotensive and spontaneously hypertensive rats.

Recent studies in normotensive Wistar-Kyoto (WKY) and spontaneously hypertensive (SHR) rats show that some nonlinear indexes derived from the recurrence plot method may be better markers of sympathetic activity than the spectral powers of blood pressure (BP). We herein investigated the relationships between nonlinear indexes and plasma noradrenaline concentration in conscious WKY rats and SHRs. Blood pressure was recorded for 30 min after intravenous injection of saline (0.9% NaCl, 100 microL/kg), hexamethonium (20 mg/kg), atropine (0.5 mg/kg), atenolol (1 mg/kg) or prazosin (1 mg/kg). Spectral power in the low-frequency (LF) band and the nonlinear index (L(max)), calculated on diastolic (DBP) and systolic blood pressures (SBP), were used to analyse the BP signal. Noradrenaline concentration was determined by radioenzymatic technique. A robust stepwise regression analysis - using noradrenaline concentration as dependent variable, and LF, L(max) and treatment, as independent variables -shows that treatment is the main variable explaining the variance of noradrenaline level in WKY rats, excluding the use of the pooled data to explore the relationship between noradrenaline concentration and LF or L(max). In contrast, in SHRs, treatment has no effect on the variance of noradrenaline concentration and the pooled data were then used. In this group, no correlation was observed between noradrenaline concentration and LF. In contrast, very high positive correlation was observed between noradrenaline level and L(max)-DBP (r = 0.59; P = 0.0005) or L(max)-SBP (r = 0.53; P < 0.002). The results strengthen our previous suggestion that nonlinear indexes may be better tools than spectral powers to investigate the sympathetic nervous system.

Animals↗

PubMed2004

Nonlinear space-time dynamics of ultrashort wave packets in water.

We have monitored the space-time transformation of a 150-fs pulse undergoing self-focusing and filamentation in water, by means of the nonlinear gating technique. We have observed that pulse splitting and subsequent recombination apply to axial temporal intensity only, whereas the space-integrated pulse profile preserves its original shape.

Journal Article↗

PubMed1995

Neural network control of functional neuromuscular stimulation systems: computer simulation studies.

A neural network control system has been designed for the control of cyclic movements in Functional Neuromuscular Stimulation (FNS) systems. The design directly addresses three major problems in FNS control systems: customization of control system parameters for a particular individual, adaptation during operation to account for changes in the musculoskeletal system, and attaining resistance to mechanical disturbances. The control system was implemented by a two-stage neural network that utilizes a combination of adaptive feedforward and feedback control techniques. A new learning algorithm was developed to provide rapid customization and adaptation. The control system was evaluated in a series of studies on a computer simulated musculoskeletal model. The model of electrically stimulated muscle used in the study included nonlinear recruitment, linear dynamics, and multiplicative nonlinear torque-angle and torque-velocity scaling factors. The skeletal model consisted of a one-segment planar system with passive constraints on joint movement. Results of the evaluation have demonstrated that the control system can provide automated customization of the feedforward controller parameters for a given musculoskeletal system. It can account for changes in the musculoskeletal system by adapting the feedforward controller parameters on-line and it can resist the effects of mechanical disturbances. These results suggest that this design may be suitable for the control of FNS systems and other dynamic systems.

Adaptation, Physiological↗

PubMed1999

A nonlinear axisymmetric model with fluid-wall interactions for steady viscous flow in stenotic elastic tubes.

Arteries with high-grade stenoses may compress under physiologic conditions due to negative transmural pressure caused by high-velocity flow passing through the stenoses. To quantify the compressive conditions near the stenosis, a nonlinear axisymmetric model with fluid-wall interactions is introduced to simulate the viscous flow in a compliant stenotic tube. The nonlinear elastic properties of the tube (tube law) are measured experimentally and used in the model. The model is solved using ADINA (Automatic Dynamic Incremental Nonlinear Analysis), which is a finite element package capable of solving problems with fluid-structure interactions. Our results indicate that severe stenoses cause critical flow conditions such as negative pressure and high and low shear stresses, which may be related to artery compression, plaque cap rupture, platelet activation, and thrombus formation. The pressure filed near a stenosis has a complex pattern not seen in one-dimensional models. Negative transmural pressure as low as -24 mmHg for a 78 percent stenosis by diameter is observed at the throat of the stenosis for a downstream pressure of 30 mmHg. Maximum shear stress as a high as 1860 dyn/cm2 occurs at the throat of the stenoses, while low shear stress with reversed direction is observed right distal to the stenosis. Compressive stresses are observed inside the tube wall. The maximal principal stress and hoop stress in the 78 percent stenosis are 80 percent higher than that from the 50 percent stenosis used in our simulation. Flow rates under different pressure drop conditions are calculated and compared with experimental measurements and reasonable agreement is found for the prebuckling stage.

Animals↗