Dynamics of Dispersive and Nonlinear Media.
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Separable least squares identification of nonlinear Hammerstein models: application to stretch reflex dynamics.
The Hammerstein cascade, consisting of a zero-memory nonlinearity followed by a linear filter, is often used to model nonlinear biological systems. This structure can represent some high-order nonlinear systems accurately with relatively few parameters. However, it is not possible, in general, to estimate the parameters of a Hammerstein cascade in closed form. The most effective method available to date uses an iterative approach, which alternates between estimating the linear element from a crosscorrelation, and then fitting a polynomial to the nonlinearity via linear regression. This paper proposes the use of separable least squares optimization methods to estimate the linear and nonlinear elements simultaneously in a least squares framework. A separable least squares algorithm for the identification of Hammerstein cascades is developed and used to analyze stretch reflex electromyogram data from two experimental subjects. The results show that in each case the proposed algorithm produced a better model, in that it predicted the system's response to novel inputs more accurately, than did models estimated using the traditional iterative algorithm. Monte-Carlo simulations demonstrated that when the input is a non-Gaussian, nonwhite signal, as is often the case experimentally, the traditional iterative identification approach produces biased models, whereas the separable least squares approach proposed in this paper does not.
Algorithms↗
Ratchet behavior in nonlinear Klein-Gordon systems with pointlike inhomogeneities.
We investigate the ratchet dynamics of nonlinear Klein-Gordon kinks in a periodic, asymmetric lattice of pointlike inhomogeneities. We explain the underlying rectification mechanism within a collective coordinate framework, which shows that such a system behaves as a rocking ratchet for point particles. Careful attention is given to the kink width dynamics and its role in the transport. We also analyze the robustness of our kink rocking ratchet in the presence of noise. We show that the noise activates unidirectional motion in a parameter range where such motion is not observed in the noiseless case. This is subsequently corroborated by the collective variable theory. An explanation for this phenomenon is given.
Journal Article↗
Nonlinear analyses of heart rate variability in normal and growth-restricted fetuses.
BACKGROUND: Many studies on the physiology of the cardiovascular system reported that nonlinear chaotic dynamics may govern the generation of the heart rate signal. OBJECTIVE: To examine whether the heart rate dynamics of an intrauterine growth restricted (IUGR) fetus is different from a healthy normal fetus by nonlinear methods of time series analysis. DESIGN OF THE STUDY: One hundred nineteen fetal heart rate (FHR) recordings from healthy normal fetuses, and 69 recordings from IUGR fetuses were analyzed. Nonlinear analyses included attractor reconstruction, calculation of the largest Lyapunov exponents using the Wolf algorithm, and estimation of correlation dimension. The largest Lyapunov exponents from normal fetuses were checked by means of surrogate-data test. RESULTS: Abnormal FHR patterns of IUGR fetuses such as decreased variability and repetitive late decelerations presented a remarkably different structure in the reconstructed attractor. Surrogate data suggest that the FHR of healthy normal fetuses has unique nonlinear characteristics. The largest Lyapunov exponents were positive for all of 119 healthy normal fetuses, indicating that the FHR control system is sensitive to initial conditions. The values of IUGR fetuses were significantly lower than those of normal subjects. In normal fetuses, significant changes of correlation dimension according to gestational age were observed. In IUGR fetuses, however, such changes were not found. CONCLUSIONS: The heart rate dynamics of IUGR fetuses is less chaotic than that of normal fetuses. Decreased system complexity suggested by correlation dimension may limit the IUGR fetuses' ability to maintain cardiovascular integrity, and therefore, to adapt to the variety of internal and external cardiovascular stresses.
Female↗
Generalizations of the Moran effect explaining spatial synchrony in population fluctuations.
The Moran effect for populations separated in space states that the autocorrelations in the population fluctuations equal the autocorrelation in environmental noise, assuming the same linear density regulation in all populations. Here we generalize the Moran effect to include also nonlinear density regulation with spatial heterogeneity in local population dynamics as well as in the effects of environmental covariates by deriving a simple expression for the correlation between the sizes of two populations, using diffusion approximation to the theta-logistic model. In general, spatial variation in parameters describing the dynamics reduces population synchrony. We also show that the contribution of a covariate to spatial synchrony depends strongly on spatial heterogeneity in the covariate or in its effect on local dynamics. These analyses show exactly how spatial environmental covariation can synchronize fluctuations of spatially segregated populations with no interchange of individuals even if the dynamics are nonlinear.
Demography↗
Nonlinear mechanisms of cortical oscillations.
Not only theoretical consideration but also analyses of MEG or EEG recordings prove the nonlinear character of cortical dynamics. For instance, an averaged local Lyapunov Exponents (ILE) have positive value that is characteristic for chaotic dynamics. Also a test for nonlinearity (or determinism)--so called surrogate data test distinguishes between original- and randomized-phase time-series proving that recorded signals are nonlinear. These facts are a very strong experimental evidence to support the hypothesis that brain oscillators are governed by the deterministic, nonlinear, low-dimensional dynamics. The experimental manifestations of nonlinear cortical oscillations in the healthy and pathologically altered human brain and their deterministic character seems to be an important step in the understanding brain dynamics in the language of nonlinear systems theory. Clinical application may use nonlinear measures (especially ILE, and PD2i) for classification of pathologies and rough localization of the functional disturbance in the brain.
Cerebral Cortex↗
Boundary force effects exerted on solitons in highly nonlinear media.
We study the effects caused by remote boundaries on soliton dynamics in nonlinear media with a large range of nonlocality, and demonstrate theoretically and experimentally how asymmetric boundary forces can lead to soliton steering and oscillation in predetermined trajectories.
Journal Article↗
Bispectral analysis as a tool to investigate dynamics of cardiorespiratory physiology.
INTRODUCTION: Recent studies show that cardiorespiratory variables are nonlinearly coupled. In the present study, we tested the feasibility of using bispectral analysis to estimate changes in nonlinear phase coupling. METHODS: We stimulated chemo- and baro-reflexes by switching inspired gas between air and air with 5% CO2 added during supine and 70 degrees head-up tilt. Bispectra were estimated using Fourier transform of the triple cumulant and were integrated between 0.04 and 0.3 Hz. Variables analyzed included cardiac RR intervals, systolic BP (SBP), tidal volume (VT), end tidal pressure of CO2 (PETCO2) and mean cerebral blood flow velocity (CBFM). Data were collected in 30 healthy adults who were then grouped according to whether they become presyncopal (PS) during tilt or remained normal with no signs or symptoms of presyncope (NPS). RESULTS: Six subjects (20%) developed presyncope. During tilt with room air breathing, phase coupling in RR, SBP, and between RR-SBP and RR-VT was lower in PS than in NPS, while phase coupling between CBFM-SBP was higher. In contrast, during tilt with 5% inhaled CO2, phase coupling between CBFM-PETCO2 was lower for PS than for NPS. DISCUSSION: Perturbations to baro- and chemo-reflexes led to changes in nonlinear phase coupling that were altered in presyncope. Although physiological interpretation of changes in phase coupling are less than clear at this time, our results show that such nonlinear indexes may provide a helpful perspective in understanding the complex phenomenon of orthostatically mediated syncope.
Adult↗
Quantification of emotion by nonlinear analysis of the chaotic dynamics of electroencephalograms during perception of 1/f music.
The goal of this study is to quantify and determine the way in which the emotional response to music is reflected in the electrical activities of the brain. When the power spectrum of sequences of musical notes is inversely proportional to the frequency on a log-log plot, we call it 1/f music. According to previous research, most listeners agree that 1/f music is much more pleasing than white (1/f0) or brown (1/f2) music. Based on these studies, we used nonlinear methods to investigate the chaotic dynamics of electroencephalograms (EEGs) elicited by computer-generated 1/f music, white music, and brown music. In this analysis, we used the correlation dimension and the largest Lyapunov exponent as measures of complexity and chaos. We developed a new method that is strikingly faster and more accurate than other algorithms for calculating the nonlinear invariant measures from limited noisy data. At the right temporal lobe, 1/f music elicited lower values of both the correlation dimension and the largest Lyapunov exponent than white or brown music. We observed that brains which feel more pleased show decreased chaotic electrophysiological behavior. By observing that the nonlinear invariant measures for the 1/f distribution of the rhythm with the melody kept constant are lower than those for the 1/f distribution of melody with the rhythm kept constant, we could conclude that the rhythm variations contribute much more to a pleasing response to music than the melody variations do. These results support the assumption that chaos plays an important role in brain function, especially emotion.
Auditory Perception↗
Heart rate dynamics in low risk human fetuses.
Evaluation of nonlinear heart rate (HR) dynamics has received considerable attention in the pediatric literature because such analyses not only provide insight into underlying control mechanisms, but may also help to differentiate between normal and abnormal infants. The purpose of this study was to determine, in eight low risk human fetuses, if nonlinear HR dynamics could be identified by analyzing the dispersion of interbeat intervals at slow (Ds) and fast (Df) HRs. The fetal cardiac electrical signal was captured transabdominally at a resolution of +/- 1 ms. To test the null hypothesis, that the time series is the result of a linear stochastic process, Ds and Df for the original time series were compared with the values calculated for three linear models. The linear models were constructed to preserve the major statistical properties of the original time series, including the mean, SD, and the Fourier power spectrum. For each fetus, there was no evidence of nonlinear cardiac dynamics based on analyses of Ds and Df. In contrast, the distribution of adjacent R-R intervals and the pattern of change across three successive interbeat intervals both revealed significant nonlinearities in HR control in each fetus. If the difference between normal and abnormal infants is the result of aberrant control of nonlinear processes, then our findings indicate that parameters which describe the nonlinearity may be more useful then Ds and Df in assigning a risk status.
Apgar Score↗
Long-time dynamics of modulated waves in a nonlinear discrete LC transmission line.
The long-time dynamics of modulated waves in a nonlinear LC transmission line is investigated. Considering the higher-order nonlinear Schrödinger equation, we define analytically the conditions leading to the instability of modulated waves. We show that two kinds of instabilities may develop in the network depending on the frequency range of the chosen carrier wave and on the magnitude of its initial amplitude, which is confirmed by our numerical simulations. The nonreproducibility of numerical experiments on modulated waves is also considered.
Journal Article↗
Strain-induced nonlinearity of filled rubbers.
Dynamic strain-induced nonlinearity in the modulus of filled rubbers shows a striking similarity to what is known about the glass transition of solid materials and the jamming transition of granular materials. This analogy stems from the reality that shear strain in dynamic mechanical measurements introduces fluctuations in a filler network by forcing the system to explore different configurations. Such fluctuations can be described by an "effective temperature" that has many attributes of a true temperature, and particularly is proportional to the strain amplitude. Thus, filled rubbers with respect to strain will display many unusual phenomena that are usually observed in glass-forming materials, but now demonstrated in filled rubbers, including asymmetric kinetics, crossover effects, and glasslike kinetic transitions. The nonlinearity in the modulus of filled rubbers simply reflects a dejamming transition of fillers in rubber matrices. The agglomeration of filler in an elastomeric matrix shares a common ground of physics with the jamming process and glass formation.
Journal Article↗
Bivariate nonlinear prediction to quantify the strength of complex dynamical interactions in short-term cardiovascular variability.
A nonlinear prediction method for investigating the dynamic interdependence between short length time series is presented. The method is a generalization to bivariate prediction of the univariate approach based on nearest neighbor local linear approximation. Given the input and output series x and y, the relationship between a pattern of samples of x and a synchronous sample of y was approximated with a linear polynomial whose coefficients were estimated from an equation system including the nearest neighbor patterns in x and the corresponding samples in y. To avoid overfitting and waste of data, the training and testing stages of the prediction were designed through a specific out-of-sample cross validation. The robustness of the method was assessed on short realizations of simulated processes interacting either linearly or nonlinearly. The predictor was then used to characterize the dynamical interaction between the short-term spontaneous fluctuations of heart period (RR interval) and systolic arterial pressure (SAP) in healthy young subjects. In the supine position, the predictability of RR given SAP was low and influenced by nonlinear dynamics. After head-up tilt the predictability increased significantly and was mostly due to linear dynamics. These findings were related to the larger involvement of the baroreflex regulation from SAP to RR in upright than in supine humans, and to the simplification of the RR-SAP coupling occurring with the tilt-induced alteration of the neural regulation of the cardiovascular rhythms.
Blood Pressure↗
Non-random fluctuations and multi-scale dynamics regulation of human activity.
We investigate if known extrinsic and intrinsic factors fully account for the complex features observed in recordings of human activity as measured from forearm motion in subjects undergoing their regular daily routine. We demonstrate that the apparently random forearm motion possesses dynamic patterns characterized by robust scale-invariant and nonlinear features. These patterns remain stable from one subject to another and are unaffected by changes in the average activity level that occur within individual subjects throughout the day and on different days of the week, since they persist during daily routine and when the same subjects undergo time-isolation laboratory experiments designed to account for the circadian phase and to control the known extrinsic factors. Further, by modeling the scheduled events imposed throughout the laboratory protocols, we demonstrate that they cannot account for the observed scaling patterns in activity fluctuations. We attribute these patterns to a previously unrecognized intrinsic nonlinear multi-scale control mechanism of human activity that is independent of known extrinsic factors such as random and scheduled events, as well as the known intrinsic factors which possess a single characteristic time scale such as circadian and ultradian rhythms.
Adult↗
The relationship between electrical stimulus and joint torque: a dynamic model.
The knowledge of the behavior of electrically activated muscles is an important requisite for the development of functional electrical stimulation (FES) systems to restore mobility to persons with paralysis. The aim of this work was to develop a model capable of relating electrical parameters to dynamic joint torque for FES applications. The knee extensor muscles, stimulated using surface electrodes, were used for the experimental preparation. Both healthy subjects and people with paraplegia were tested. The dynamics of the lower limb were represented by a nonlinear second order model, which took account of the gravitational and inertial characteristics of the anatomical segments as well as the damping and stiffness properties of the knee joint. The viscous-elastic parameters of the system were identified experimentally through free pendular movements of the leg. Leg movements induced by quadriceps stimulation were acquired too, using a motion analysis system. Results showed that, for the considered experimental conditions, a simple one-pole transfer function is able to model the relationship between stimulus pulsewidth (PW) and active muscle torque. The time constant of the pole was found to depend on the stimulus pattern (ramp or step) while gain was directly dependent on stimulation frequency.
Bias↗
Convergence of the Integration Dynamics of the Construction-Integration Model.
The connectionist network corresponding to the nonlinear integration dynamical system associated with Kintsch's construction-integration (CI) model is analysed with linear algebra tools. This addresses some theoretical questions raised and left unanswered by Rodenhausen (1992, Psychological Review 99, 547-549). A mathematical characterization for equilibrium points, which allows an a priori enumeration of all possible asymptotic states for the integration dynamical system, given a connectivity matrix, is given. The dynamics of convergence of the integration dynamical system is characterized in some detail as well. This provides a tool for understanding CI simulations and helps in particular to let us know to what extent the outcome will depend on the initial conditions. The criteria also provide a new mathematical analysis which allows for the explicit calculation of asymptotic states of the integration process without requiring computer simulation experiments. The new mathematical analysis should facilitate comparisons of the model's predictions with behavioural data. Copyright 2001 Academic Press.
Journal Article↗
The role of the thalamic reticular neurons in alpha- and gamma-oscillations in neocortex: a mechanism for selective perception and stimulus binding.
The long-term objective is to understand how large masses of neurons in the brain process information during various learning and memory paradigms. Both time- and space-dependent processes have been identified in animals through computer-based analytic quantifications of event-related extracellular potentials. New nonlinear analyses have been introduced that presume that the fine-grain variation in the signal is determined and patterned in phase-space. Some neurons in the primary visual cortex manifest gamma-band oscillations. These cells show both a nonspecific phase-alignment (response synchrony) and a specific tuning (orientation tuning) when stimuli are presented to their receptive fields. This dual regulation of the sensory cells is proposed to underlie stimulus binding, a theoretical mechanism for "object" perception. Nonlinear analytic results from gamma-activities in a simple model neuropil (olfactory bulb) suggest that neuroplasticity may arise through self-organization, a process in which a nonlinear change in the dynamics of the oscillatory field potentials is the hallmark. This self-organization may follow simple dynamical laws in which global cooperativity among the neurons is transiently brought about that, over trials, results in enduring changes in the nonlinear dynamics of some neurons. In conclusion, the sculpturing of the synaptic throughput in the sensory cortex (stimulus binding) may be associated with the irregular phases of the gamma-activities and may result from both specific and nonspecific systems operating together in a nonlinear self-organizing manner.
Alpha Rhythm↗
Dynamics of solitons and quasisolitons of the cubic third-order nonlinear Schrödinger equation.
The dynamics of soliton and quasisoliton solutions of the cubic third-order nonlinear Schrödinger equation is studied. Regular solitons exist due to a balance between the nonlinear terms and (linear) third-order dispersion; they are not important at small alpha(3) (alpha(3) is the coefficient in the third derivative term) and vanish at alpha(3)-->0. The most essential, at small alpha(3), is a quasisoliton emitting resonant radiation (resonantly radiating soliton). Its relationship with the other (steady) quasisoliton, called embedded soliton, is studied analytically and also in numerical experiments. It is demonstrated that the resonantly radiating solitons emerge in the course of nonlinear evolution, which shows their physical significance.
Journal Article↗