Dynamic equilibrium in granular flow obtained by a nonlinear dynamic equation.
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Computational design and nonlinear dynamics of a recurrent network model of the primary visual cortex.
Recurrent interactions in the primary visual cortex make its output a complex nonlinear transform of its input. This transform serves preattentive visual segmentation, that is, autonomously processing visual inputs to give outputs that selectively emphasize certain features for segmentation. An analytical understanding of the nonlinear dynamics of the recurrent neural circuit is essential to harness its computational power. We derive requirements on the neural architecture, components, and connection weights of a biologically plausible model of the cortex such that region segmentation, figure-ground segregation, and contour enhancement can be achieved simultaneously. In addition, we analyze the conditions governing neural oscillations, illusory contours, and the absence of visual hallucinations. Many of our analytical techniques can be applied to other recurrent networks with translation-invariant neural and connection structures.
Animals↗
Measurement of heart rate variability by methods based on nonlinear dynamics.
Heart rate (HR) variability has been conventionally analyzed with time and frequency domain methods, which measure the overall magnitude of R-R interval fluctuations around its mean value or the magnitude of fluctuations in some predetermined frequencies. Analysis of HR dynamics by methods based on chaos theory and nonlinear system theory has gained recent interest. This interest is based on observations suggesting that the mechanisms involved in cardiovascular regulation likely interact with each other in a nonlinear way. Furthermore, recent observational studies suggest that some indexes describing nonlinear HR dynamics, such as fractal scaling exponents, may provide more powerful prognostic information than the traditional HR variability indexes. In particular, short-term fractal scaling exponent measured by detrended fluctuation analysis method has been shown to predict fatal cardiovascular events in various populations. Approximate entropy, a nonlinear index of HR dynamics, which describes the complexity of R-R interval behavior, has provided information on the vulnerability to atrial fibrillation. There are many other nonlinear indexes, eg, Lyapunov exponent and correlation dimensions, which also give information on the characteristics of HR dynamics, but their clinical utility is not well established. Although concepts of chaos theory, fractal mathematics, and complexity measures of HR behavior in relation to cardiovascular physiology or various cardiovascular events are still far away from clinical medicine, they are a fruitful area for future research to expand our knowledge concerning the behavior of cardiovascular oscillations in normal healthy conditions as well as in disease states.
Atrial Fibrillation↗
Renal blood flow regulation and arterial pressure fluctuations: a case study in nonlinear dynamics.
The arterial blood pressure, a physiological variable on which all renal excretory processes depend, fluctuates over a wide range of amplitudes and frequencies. Much of this variation originates in nonrenal vascular beds to support nonrenal tasks, and the fluctuations provide a noisy environment in which the kidney is obliged to operate. Were it not for renal blood flow autoregulation, it would be difficult to regulate renal excretory processes so as to maintain whole body variables within narrow bounds. Autoregulation is the noise filter on which other renal processes depend for maintaining a relatively noise-free environment in which to work. Because of the time-varying nature of the blood pressure, we have concentrated in this review on the now substantial body of work on the dynamics of renal blood flow regulation and the underlying mechanisms. Renal vascular control mechanisms are not simply reactive but have their own spontaneous dynamics. Both TGF and the myogenic mechanism oscillate autonomously. The TGF oscillation is the better understood of the two. There is an oscillation of tubular pressure, proximal tubular flow, early distal Cl- concentration, and efferent arteriolar blood flow at approximately 35 mHz; all these variables are synchronized when the measurements are made in a single tubule. The autonomous nature of the oscillation is supported by simulations of the nephron and its vasculature, which show that for a reasonable representation of the dynamics of these structures and of the parameters that govern their behavior, the solutions of the equation set are periodic at the frequency of the observed oscillation, and with the same phase relationships among its variables. The simulations also show that the critical variables for the development of the oscillation are the open-loop gain of the feedback system, and the various delays in the system of which convective transport in the axis of the thick ascending limb and signal transmission between the macula densa and the afferent arteriole are the most important. The oscillation in TGF is an example of nonlinear dynamical behavior and is yet another in a long list of oscillations and related dynamics arising in the inherently nonlinear properties of living systems. Some nonlinear systems can bifurcate to states known collectively as deterministic chaos, and TGF is a clear example of such a system. Rats with two different and unrelated forms of experimental hypertension provide tubular pressure records that pass statistical tests for ordered structure and sensitive dependence on initial conditions in the reconstructed state space, two of the hallmarks of deterministic chaos. These records also pass recent more stringent tests for chaos. The significance of deterministic chaos in the context of renal blood flow regulation is that the system regulating blood flow undergoes a physical change to a different dynamical state, and because the change is deterministic, there is every expectation that the critical change will yield itself to experimental discovery.(ABSTRACT TRUNCATED AT 400 WORDS)
Animals↗
An Approach of a Nonlinear Dynamic Model for Micellar Effects on Bimolecular Reactions.
The effects of micellar solutions on bimolecular reactions were studied in detail using a newly established nonlinear dynamic model with a far-from-equilibrium reaction scheme, by which the incorporation of organic substrates with the micellar phase was imitated. Through the linear stability analysis of the steady state solutions given by the rate equation of the change of substrate concentration within a micellar phase with time, it leads to some explicit relationships between the incorporated substrate concentration and the relevant parameters. Based on this model the micellar effects on the kinetics of the alkaline hydrolysis of esters have been interpreted. Copyright 1999 Academic Press.
Journal Article↗
Nonlinear dynamic behavior of the human knee joint--Part II: Time-domain analyses: effects of structural damage in postmortem experiments.
A description is given of the results obtained for step excitation for two human knee joint specimens using a time-domain analysis technique. As was expected from the results of a previous study, the magnitude of the dynamic load applied has a marked influence upon the stiffness and damping values for the two observed vibration modes. Deliberate damaging of selected joint elements also yields a well observable change in the dynamic behavior of the joint although these changes are difficult to interpret. Here the use of a nonlinear dynamic numerical model of the knee joint seems indispensable. An important observation is, however, that the experimental method discussed here enables to quantify the behavior of the joint and therefore may provide a valuable tool for validation of such a model.
Anterior Cruciate Ligament↗
Nonlinear dynamics of a semiconductor laser with filtered optical feedback and the influence of noise.
We investigate the influence of quantum noise on the highly complex nonlinear dynamics that arise in a single-mode semiconductor laser subject to filtered optical feedback. Our numerical study, which utilizes rate equations that are augmented by Langevin noise terms to account for the spontaneous-recombination noise, shows that for relatively broad filters the noise may lead to qualitatively different dynamics than predicted by a deterministic analysis. In particular, we find that certain attractors that are predicted in the absence of noise may no longer be available when the effects of noise are correctly incorporated. For narrow bandwidth filters we demonstrate optical-injection-like behavior and identify locking of the semiconductor laser to the relaxation oscillation side peaks. In general, the results indicate that shot noise in the laser can influence the dynamics quite substantially.
Journal Article↗
General measures for signal-noise separation in nonlinear dynamical systems.
We propose the straight phi divergences from statistics and information theory (IT) as a set of separation indices between signal and noise in stochastic nonlinear dynamical systems (SNDS). The straight phi divergences provide a more informative alternative to the signal-to-noise ratio (SNR) and have the advantage of being applicable to virtually any kind of stochastic system. Moreover, straight phi divergences are intimately connected to various fundamental limits in IT. Using the properties of straight phi divergences, we show that the classical stochastic resonance (SR) curve can be interpreted as the performance of a nonoptimal, or mismatched, detector applied to the output of a SNDS. Indeed, for a prototype double-well system with forcing in the form of white Gaussian noise plus a possible embedded signal, the whole information loss can be attributed to this mismatch; an optimal detection procedure (for the signal) gives the same performance when based on the output as when based on the input of the system. More generally, it follows that, when characterizing signal-noise separation (or system performance) of SNDS in terms of criteria that do not correspond to IT limits, the choice of criterion can be crucial. The indicated figure of merit will then not be universal and will be relevant only to some family of applications, such as the classical (narrow-band SNR) SR criterion, which is relevant for narrow-band post processing. We illustrate the theory using simple SNDS excited by both wide- and narrow-band signals; however, we stress that the results are applicable to a much larger class of signals and systems.
Journal Article↗
A stochastic nonlinear autoregressive algorithm reflects nonlinear dynamics of heart-rate fluctuations.
Current methods for detecting nonlinear determinism in a time series require long and stationary data records, as most of them assume that the observed dynamics arise only from the internal, deterministic workings of the system, and the stochastic portion of the signal (the noise component) is assumed to be negligible. To explicitly account for the stochastic portion of the data we recently developed a method based on a stochastic nonlinear autoregressive (SNAR) algorithm. The method iteratively estimates nonlinear autoregressive models for both the deterministic and stochastic portions of the signal. Subsequently, the Lyapunov exponents (LE) are calculated for the estimated models in order to examine if nonlinear determinism is present in the deterministic portion of the fitted model. To determine if nonlinear dynamic analysis of heart-rate fluctuations can be used to assess arrhythmia susceptibility by predicting the outcome of invasive cardiac electrophysiologic study (EPS), we applied the SNAR algorithm to noninvasively measured resting sinus-rhythm heart-rate signals obtained from 16 patients. Our analysis revealed that a positive LE was highly correlated to a patient with a positive outcome of EPS. We found that the statistical accuracy of the SNAR algorithm in predicting the outcome of EPS was 88% (sensitivity=100%, specificity=75%, positive predictive value=80%, negative predictive value=100%, p=0.0019). Our results suggest that the SNAR algorithm may serve as a noninvasive probe for screening high-risk populations for malignant cardiac arrhythmias.
Adult↗
Nonlinear dynamics of periodically focused intense particle beams.
We extend a previous study [R. Pakter and F. B. Rizzato, Phys. Rev. Lett. 87, 044801 (2001)] and investigate the nonlinear dynamics of periodically focused intense particle beams. We show that (i) the scenario as the focusing field increases is not the existence of a single threshold above which stable matched (equilibrium) solutions are absent, as believed so far, but the existence of successive regions of stability interrupted by gaps where periodic solutions are either unstable or simply do not exist; (ii) the beam can be focused to tighter radii using stable matched solutions found for focusing field strengths greater than the previous threshold. A comprehensive analysis is carried out as a function of the relevant parameters of the system. Self-consistent simulations validate the findings. The gaps are of crucial importance because they must be avoided if the goal is beam confinement with matched solutions; we develop an analytical model to determine the gap structure, which agrees well with computer simulations.
Journal Article↗
Nonlinear dynamics of paced cardiac cells.
When a cardiac cell is rapidly paced it can exhibit a beat-to-beat alternation in the action potential duration (APD) and the intracellular calcium transient. This dynamical instability at the cellular level has been shown to correlate with the genesis of cardiac arrhythmias and has motivated the application of nonlinear dynamics in cardiology. In this article, we review mathematical approaches to describe the underlying mechanisms for alternans using beat-to-beat iterated maps. We explain the development and properties of these maps, and show that they provide a fruitful framework to understand dynamical instabilities of voltage and calcium in paced cardiac cells.
Action Potentials↗
Comment on "Social contagion, adolescent sexual behavior, and pregnancy: a nonlinear dynamic EMOSA model".
The epidemic model of the onset of social activities (EMOSA model) for adolescent sexuality by J. L. Rodgers, D. C. Rowe, and M. Buster (1998) is examined. Strengths of the model include the nonlinear dynamic specification that seems reasonably realistic and the potential when used as a theoretical tool to generate new hypotheses for further testing at the individual level. Limitations include the lack of a well-developed statistical framework, the inability to include a variety of predictors to capture both social contagion and individual differences in the same model, and the focus on aggregate-level data. The EMOSA model for sexuality can be reparameterized to a hazard rate or an event history model that eliminates these shortcomings. Growth curve analysis represents another analytic alternative for the EMOSA model that also does not have these limitations.
Adolescent↗
Nonlinear dynamics of traveling waves in rotating Rayleigh-Bénard convection: effects of the boundary conditions and of the topology.
Motivated by the experimental results of Liu and Ecke (1997, 1999), different models are developed to analyze the weakly nonlinear dynamics of the traveling-wave sidewall modes appearing in rotating Rayleigh-Bénard convection. These models assume fully rigid boundary conditions for the velocity field. At the linear level, this influences most strongly the critical frequencies: they appear to be proportional to the logarithm of the Coriolis number, which is twice the inverse of the Ekman number. An annular flow domain is considered. This multiply connected geometry is shown to lead generally to the existence of a global mean-flow mode proportional to the average, over the azimuthal coordinate, of the square of the modulus of the envelope of the waves. Because this mode feeds back on the active wave modes at cubic order, the resulting Ginzburg-Landau envelope equation contains a nonlocal term. This new term, however, vanishes in the large-gap limit relevant to the experiments of Liu and Ecke. As compared with previous theoretical work, the present models lead to reduced discrepancies with the results of these experiments concerning the coefficients of the envelope equation. It is also shown that the new nonlocal effects may be realized experimentally in a small-gap annular geometry if a small-Prandtl-number fluid is used, despite the fact that no regime of Benjamin-Feir instability is predicted to occur.
Journal Article↗
Dynamic nonlinear X waves for femtosecond pulse propagation in water.
Recent experiments involving femtosecond pulses in water displayed long-distance propagation analogous to that reported in air. We verify this phenomenon numerically and show that the propagation is dynamic as opposed to self-guided. Furthermore, we demonstrate that the propagation can be interpreted as being due to dynamic nonlinear X waves whose robustness and role in long-distance propagation follows from the interplay between nonlinearity and chromatic dispersion.
Journal Article↗
Nonlinear dynamics of eucaryotic pyruvate dehydrogenase multienzyme complex: decarboxylation rate, oscillations, and multiplicity.
Pyruvate conversion to acetyl-CoA by the pyruvate dehydrogenase (PDH) multienzyme complex is known as a key node in affecting the metabolic fluxes of animal cell culture. However, its possible role in causing possible nonlinear dynamic behavior such as oscillations and multiplicity of animal cells has received little attention. In this work, the kinetic and dynamic behavior of PDH of eucaryotic cells has been analyzed by using both in vitro and simplified in vivo models. With the in vitro model the overall reaction rate (nu(1)) of PDH is shown to be a nonlinear function of pyruvate concentration, leading to oscillations under certain conditions. All enzyme components affect nu(1) and the nonlinearity of PDH significantly, the protein X and the core enzyme dihydrolipoamide acyltransferase (E2) being mostly predominant. By considering the synthesis rates of pyruvate and PDH components the in vitro model is expanded to emulate in vivo conditions. Analysis using the in vivo model reveals another interesting kinetic feature of the PDH system, namely, multiple steady states. Depending on the pyruvate and enzyme levels or the operation mode, either a steady state with high pyruvate decarboxylation rate or a steady state with significantly lower decarboxylation rate can be achieved under otherwise identical conditions. In general, the more efficient steady state is associated with a lower pyruvate concentration. A possible time delay in the substrate supply and enzyme synthesis can also affect the steady state to be achieved and leads to oscillations under certain conditions. Overall, the predictions of multiplicity for the PDH system agree qualitatively well with recent experimental observations in animal cell cultures. The model analysis gives some hints for improving pyruvate metabolism in animal cell culture.
Acetyl Coenzyme A↗
Nonlinear dynamical factor analysis for state change detection.
Changes in a dynamical process are often detected by monitoring selected indicators directly obtained from the process observations, such as the mean values or variances. Standard change detection algorithms such as the Shewhart control charts or the cumulative sum (CUSUM) algorithm are often based on such first- and second-order statistics. Much better results can be obtained if the dynamical process is properly modeled, for example by a nonlinear state-space model, and then the accuracy of the model is monitored over time. The success of the latter approach depends largely on the quality of the model. In practical applications like industrial processes, the state variables, dynamics, and observation mapping are rarely known accurately. Learning from data must be used; however, methods for the simultaneous estimation of the state and the unknown nonlinear mappings are very limited. We use a novel method of learning a nonlinear state-space model, the nonlinear dynamical factor analysis (NDFA) algorithm. It takes a set of multivariate observations over time and fits blindly a generative dynamical latent variable model, resembling nonlinear independent component analysis. We compare the performance of the model in process change detection to various traditional methods. It is shown that NDFA outperforms the classical methods by a wide margin in a variety of cases where the underlying process dynamics changes.
Factor Analysis, Statistical↗
Nonlinear dynamics of physiological function and control.
Complex rhythms are observed in the physiological systems that control and carry out vital bodily functions. Theoretical approaches to analyze the physiological systems include control theory and computation theory. Complementary to these approaches is nonlinear dynamics, which offers ways to classify both normal and abnormal dynamics, and to analyze bifurcations occurring in physiological dynamics.
Journal Article↗
Applications of nonlinear dynamics to clinical cardiology.
(1) Nonlinear mechanisms may apply both to the understanding of SA-AV node interactions and to bifurcations leading to certain types of AV block. (2) The fractal His-Purkinje system serves as the structural substrate for the generation of the broadband, inverse power-law spectrum of the stable ventricular depolarization (QRS) waveform. (3) Fractal anatomy is also seen in multiple other systems: pulmonary, hepatobiliary, renal, etc. Fractal morphogenesis may reflect a type of critical phenomenon that results in the generation of these irregular, but self-similar structures. (4) Self-similar (fractal) scaling may underlie the 1/f-like spectra seen in multiple systems (e.g., interbeat interval variability, daily neutrophil fluctuations). This fractal scaling may provide a mechanism for the "constrained randomness" that appears to underlie physiological variability and adaptability. (5) Behavior consistent with subharmonic bifurcations is seen in cardiac electrophysiology (e.g., sick sinus syndrome) and hemodynamic perturbations (e.g., swinging heart phenomenon in pericardial tamponade). (6) Ventricular tachyarrhythmias associated with sudden cardiac death (e.g., torsades de pointes, ventricular fibrillation) appear to reflect relatively periodic, not chaotic (turbulent) processes resulting from disruption of the physiologic fractal depolarization sequence. (7) Spectral analysis of Holter monitor data may help in the detection of patients at high risk for sudden death.
Cardiovascular Diseases↗