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PubMed1999

Nonlinear analysis of the performance and reliability of wavelet singularity detection based denoising for Doppler ultrasound fetal heart rate signals.

Many studies on the physiology of the cardiovascular system revealed that nonlinear chaotic dynamics govern the generation of the heart rate signal. This is also valid for the fetal heart rate (FHR) variability, where however the variability is affected by many more factors and is significantly more complicated than for the adult case. Recently an adaptive wavelet denoising method for the Doppler ultrasound FHR recordings has been introduced. In this paper the performance and reliability of that method is confirmed by the observation that for the wavelet denoised FHR signal, a deterministic nonlinear structure, which was concealed by the noise, becomes apparent. It provides strong evidence that the denoising process removes actual noise components and can therefore be utilized for the improvement of the signal quality. Hence by observing after denoising a significant improvement of the 'chaoticity' of the FHR signal we obtain strong evidence for the reliability and efficiency of the wavelet based denoising method. The estimation of the chaoticity of the FHR signal before and after the denoising is approached with three nonlinear analysis methods. First, the rescaled scale analysis (RSA) technique reveals that the denoising process increases the Hurst exponent parameter as happens when additive noise is removed from a chaotic signal. Second, the nonlinear prediction error evaluated with radial basis function (RBF) prediction networks is significantly lower at the denoised signal. The significant gain in predictability can be attributed to the drastic reduction of the additive noise from the signal by the denoising algorithm. Moreover, the evaluation of the correlation coefficient between actual and neural network predicted values as a function of the prediction time displays characteristics of chaos only for the denoised signal. Third, a chaotic attractor, reconstructed with the embedding dimension technique, becomes evident for the denoised signal, while it is completely obscured for the original signals. The correlation dimension of the reconstructed attractor for the denoised signal tends to reach a value independent of the embedding dimension, a sign of deterministic chaotic signal. In contrast for the original signal the correlation dimension increases steadily with the embedding dimension, a fact that indicates strong contribution of noise.

Adult↗

PubMed2003

Self-organization and complexity: a new age for theory, computation and experiment.

I first describe the notion of self-organization as a property of far-from-equilibrium nonlinear dissipative dynamical systems. Rather than describing such complex systems at a purely phenomenological level, however, I focus attention on the emergent nature of this complexity, by analysing a few examples of physical and physicochemical systems with simple underlying microscopic dynamics yet complex, self-organizing macroscopic properties. These include several mesoscopic models of fluid dynamics as well as a modern approach to nucleation and growth phenomena. Finally, I discuss how the advent of computational grids is set to provide a major boost to the study of such complex, self-organizing systems.

Adaptation, Physiological↗

PubMed1989

A physiological view of in vivo calcium dynamics: the regulation of a nonlinear self-organized system.

Our aim is neither to re-evaluate the term homeostasis, nor to summarize the conventional concepts in the field of calcium metabolism and its regulation, nor even to comment on their advantages and their limitations (excellent recent reviews have been published). This paper is rather a position article and references to the current literature will be made only if they contribute to a better understanding of our proposals; in contrast, emphasis will be placed on a literature which has, until now, remained unfamiliar to the field of calcium metabolism. The text is organized around three related features which are largely dictated by the characteristics of our recently published compartmental self-oscillatory model for rat calcium metabolism: (a) The circadian behavior associated with calcium dynamics in vivo may be viewed as a "key" temporal behavior for investigating the spatiotemporal organization of calcium metabolism in the normal rat. Within the bone, a large part of this circadian behavior should stem from the physico-chemical properties of the transformations of calcium-phosphate associations at the extracellular fluid (ECF)/mature bone interface; (b) an important part of the maintenance of a nearly constant plasma calcium concentration (homeostasis) results from interaction between nonlinear oscillators belonging to both calcium metabolism and calcium-regulating hormones. This implies that: firstly, calcium metabolism, like any biological system, is a complex dynamic system which has evolved over a long period and whose metabolic components--gut, kidney, bone--are organized as dynamic entities, adapted to periodic relationships with the external environment. The intrinsic nature of the circadian behavior of bone calcium efflux proposed here is a sufficient demonstration of this. Secondly, the existence of rhythmic variations in the main calcium regulating hormones, parathyroid hormone (PTH), calcitonin (CT) and vitamin D (VitD), are in agreement with this argument. As developed below, fascinating properties emerge from interaction between oscillators (hormones and target organs) which provide a new perspective on calcium regulation; and (c) one of the striking properties of the kind of nonlinear dynamic system required for this representation of calcium metabolism is that periodicity is only one of many temporal expressions. Thus, qualitative diversity in the temporal expression of calcium metabolism can be expected with varying experimental situations and different modes of temporal regulation of calcium metabolism might be physiologically effective, depending on the species studied.(ABSTRACT TRUNCATED AT 400 WORDS)

Animals↗

PubMed2003

Dynamic finite element implementation of nonlinear, anisotropic hyperelastic biological membranes.

We present a novel method for the implementation of hyperelastic finite strain, non-linear strain-energy functions for biological membranes in an explicit finite element environment. The technique is implemented in LS-DYNA but may also be implemented in any suitable non-linear explicit code. The constitutive equations are implemented on the foundation of a co-rotational uniformly reduced Hughes-Liu shell. This shell is based on an updated-Lagrangian formulation suitable for relating Cauchy stress to the rate-of-deformation, i.e. hypo-elasticity. To accommodate finite deformation hyper-elastic formulations, a co-rotational deformation gradient is assembled over time, resulting in a formulation suitable for pseudo-hyperelastic constitutive equations that are standard assumptions in biomechanics. Our method was validated by comparison with (1) an analytic solution to a spherically-symmetric dynamic membrane inflation problem, incorporating a Mooney-Rivlin hyperelastic equation and (2) with previously published finite element solutions to a non-linear transversely isotropic inflation problem. Finally, we implemented a transversely isotropic strain-energy function for mitral valve tissue. The method is simple and accurate and is believed to be generally useful for anyone who wishes to model biologic membranes with an experimentally driven strain-energy function.

Animals↗

PubMed1998

A new method to analyze lung compliance when pressure-volume relationship is nonlinear.

Changes in dynamic lung compliance during inspiration and expiration cannot be modeled accurately with conventional algorithms. We developed a simple method to analyze pressure-volume (P/V) relationships under condition of nonlinearity (APVNL) and tested it in a lung model with known resistance and nonlinear P/V relationship. In addition, pulmonary mechanics in 22 infants, 11 of them with nonlinear P/V relationships, were analyzed with the new method. The findings were compared with those obtained by a recently introduced algorithm, multiple linear regression analysis (MLR) of the equation of motion. The APVNL method described the changing compliance (C) of the lung model accurately, whereas the MLR method underestimated C especially in the first half of the breath. In infants the MLR method gave highly variable, often nonphysiological C values in the beginning of a breath. In contrast, the coefficient of variability of measurements obtained by the APVNL method was significantly smaller (p < 0.02), and the indices of model-fit showed better agreement between calculated and observed pressure than for the MLR method (p < 0.02). We conclude that the APVNL method accurately describes nonlinear P/V relationships present during spontaneous breathing or mechanical ventilation. The method may be helpful in identifying and preventing pulmonary overdistention.

Airway Resistance↗

PubMed2001

Complex adaptive systems, aging and longevity.

Mortality and reproduction are intimately entwined in the study of aging and longevity. I apply the modern theory of complex adaptive systems (nonlinear, stochastic, dynamic methods) to questions of aging and longevity. I begin by highlighting major questions that must be answered in order to obtain a deeper understanding of aging. These are: (i) What should (in an evolutionary sense) mortality trajectories look like? (ii) Why does caloric restriction slow aging? (iii) Why does reproduction cause delayed mortality? (iv) Why does compensatory growth cause delayed mortality? I show how dynamic state variable models based on stochastic dynamic programming (Clark & Mangel, 2000) can be used to embed genetic theories of senescence (either mutation accumulation or antagonistic pleiotropy) in the somatic environment, as George Williams called for in 1957, and how they make the disposable soma theory of aging operational. Such models will allow unification of genetic and phenotypic theories of aging.

Adaptation, Physiological↗

PubMedDate not supplied

The chaotic component of human heart rate variability shows a circadian periodicity as documented by the correlation dimension of the time-qualified sinusal R-R intervals.

OBJECTIVE: This study investigates the hypothesis that the nonlinear component of human heart rate (HR) variability might show a periodic structure over the 24-h span. Such a postulate could explain how the chaotic component might coexist with the deterministic periodic variability of instantaneous HR in beat per minute. MATERIALS AND METHODS: The sinusal R-R intervals (sRRi) of the Holter EKG of 10 clinically healthy subjects (5 M, 5 F, 23-30 years) were analyzed per each hour of the day-night span according to two methods for the nonlinear chaotic variability, i.e., the correlation dimension method, and the linear periodic variability, i.e., periodic regression analysis. RESULTS: The hourly-qualified correlation integrals were found to show a significant circadian rhythm, with an acrophase located during the night in coincidence with the longest duration of the sRRi and the lowest rate of cardiac pulse. CONCLUSIONS: The rhythmic structure of the chaotic component of the human HR variability let us to think that a deterministic periodic chaos of fractal type regulates the nonlinear cardiac dynamics. Such a periodic structure allows the chaos to be compatible with the deterministic linear periodicity of circadian type which characterizes the within-day variability of human HR.

Adult↗

PubMed2006

Closed-loop nonlinear system identification via the vector optimal parameter search algorithm: application to heart rate baroreflex control.

The vector optimal parameter search (VOPS) and the constrained optimal parameter search (COPS) are recently developed algorithms for closed-loop linear system identification. We extend both algorithms to be applicable to a closed-loop nonlinear system, which is characterized by a vector nonlinear autoregressive model. Monte Carlo simulations of nonlinear closed-loop systems were performed to compare the performance of the VOPS to the widely utilized vector least squares (VLS), the COPS and the total least squares (TLS) approaches. The relative error and linear transfer functions are computed to determine the accuracy of each method. The comparative results show that both the VOPS and COPS algorithms provide far superior parameter estimates than does the VLS for all simulation examples considered. The TLS provides better estimates than the VOPS, COPS and VLS when there is only observation noise present in the data. However, the performance of the TLS degrades considerably when the data are corrupted by dynamic noise. The clinical applicability of the two extended methods is examined by applying them to a classical physiological closed-loop system, the heart rate baroreflex. It was found that while both control and blockade of parasympathetic system conditions are dominated by linear dynamics, more nonlinearity was observed in the latter. This observation is statistically supported by the calculation of the mutual information of the data and their surrogates.

Algorithms↗

PubMed2003

Two-state theory of nonlinear stochastic resonance.

An amenable, analytical two-state description of the nonlinear population dynamics of a noisy bistable system driven by a rectangular subthreshold signal is put forward. Explicit expressions for the driven population dynamics, the correlation function (its coherent and incoherent parts), the signal-to-noise ratio (SNR), and the stochastic resonance (SR) gain are obtained. Within a suitably chosen range of parameter values this reduced description yields anomalous SR gains exceeding unity and, simultaneously, gives rise to a nonmonotonic behavior of the SNR vs the noise strength. The analytical results agree well with those obtained from numerical solutions of the Langevin equation.

Nonlinear Dynamics↗

PubMed2006

Nonlinear effects on solvation dynamics in simple mixtures.

The authors applied the time dependent density functional method (TDDFM) and a linear model to solvation dynamics in simple binary solvents. Changing the solute-solvent interactions at t=0, the authors calculated the time evolution of density fields for solvent particles after the change (t>0) by the TDDFM and linear model. First, the authors changed the interaction of only one component of solvents. In this case, the TDDFM showed that the solvation time decreased monotonically with a mole fraction of the solvent strongly interacting with the solute. The monotonical decreases agreed with experimental results, while the linear model did not reproduce these results. The authors also calculated the solvation time by changing the interaction of both components. The calculation showed that the mole fraction dependence had the peak. The TDDFM presented a much higher peak than the linear model. The difference between the TDDFM and the linear model was caused by a nonlinear effect on an exchange process of solvent particles.

Journal Article↗

PubMedDate not supplied

Issues in the design of a multirate model-based controller for a nonlinear drug infusion system.

Multivariable controller design for the regulation of mean arterial pressure (MAP) and cardiac output (CO) in congestive heart failure patients is restricted by the limited frequency of CO sampling. Performance criteria for the controller specify maximum allowable transient settling times for both variables, and the design should account for the inherent multirate nature of the process in order to satisfy these criteria. We present a multirate model predictive control (MPC) design for MAP and CO regulation by combined infusion of sodium nitroprusside and dopamine, based on a comprehensive nonlinear model of the system. The multirate MPC algorithm is based on nonlinear quadratic dynamic matrix control. To reduce computation time, we introduce a selective linearization technique that linearizes the model on the basis of trends in the plant-model mismatch. The problem is complicated by restrictions on initial dopamine infusion, prescribed to avoid extremely slow responses. We present a novel rule-based override (RBO) to the MPC controller that uses a set of heuristics to initialize dopamine. The performance of the MPC/RBO controller is illustrated using simulation results.

Algorithms↗

PubMed2006

Stability of dynamic trunk movement.

STUDY DESIGN: Nonlinear systems analyses of trunk kinematics were performed to estimate control of dynamic stability during repetitive flexion and extension movements. OBJECTIVE: Determine whether movement pace and movement direction of dynamic trunk flexion and extension influence control of local dynamic stability. SUMMARY OF BACKGROUND DATA: Spinal stability has been previously characterized in static, but not in dynamic movements. Biomechanical models make inferences about static spinal stability, but existing analyses provide limited insight into stability of dynamic movement. Stability during dynamic movements can be estimated from Lyapunov analyses of empirical data. METHODS: There were 20 healthy subjects who performed repetitive trunk flexion and extension movements at 20 and 40 cycles per minute. Maximum Lyapunov exponents describing the expansion of the kinematic state-space were calculated from the measured trunk kinematics to estimate stability of the dynamic system. RESULTS: The complexity of torso movement dynamics required at least 5 embedded dimensions, which suggests that stability components of lumbar lordosis may be empirically measurable in addition to global stability of trunk dynamics. Repeated trajectories from fast paced movements diverged more quickly than slower movement, indicating that local dynamic stability is limited in fast movements. Movements in the midsagittal plane showed higher multidimensional kinematic divergence than asymmetric movements. CONCLUSION: Nonlinear dynamic systems analyses were successfully applied to empirically measured data, which were used to characterize the neuromuscular control of stability during repetitive dynamic trunk movements. Movement pace and movement direction influenced the control of spinal stability. These stability assessment techniques are recommended for improved workplace design and the clinical assessment of spinal stability in patients with low back pain.

Adult↗

PubMed2006

The most likely voltage path and large deviations approximations for integrate-and-fire neurons.

We develop theory and numerical methods for computing the most likely subthreshold voltage path of a noisy integrate-and-fire (IF) neuron, given observations of the neuron's superthreshold spiking activity. This optimal voltage path satisfies a second-order ordinary differential (Euler-Lagrange) equation which may be solved analytically in a number of special cases, and which may be solved numerically in general via a simple "shooting" algorithm. Our results are applicable for both linear and nonlinear subthreshold dynamics, and in certain cases may be extended to correlated subthreshold noise sources. We also show how this optimal voltage may be used to obtain approximations to (1) the likelihood that an IF cell with a given set of parameters was responsible for the observed spike train; and (2) the instantaneous firing rate and interspike interval distribution of a given noisy IF cell. The latter probability approximations are based on the classical Freidlin-Wentzell theory of large deviations principles for stochastic differential equations. We close by comparing this most likely voltage path to the true observed subthreshold voltage trace in a case when intracellular voltage recordings are available in vitro.

Action Potentials↗

PubMed2005

The role of seasonality in the dynamics of deer tick populations.

In this paper, we formulate a nonlinear system of difference equations that models the three-stage life cycle of the deer tick over four seasons. We study the effect of seasonality on the stability and oscillatory behavior of the tick population by comparing analytically the seasonal model with a non-seasonal one. The analysis of the models reveals the existence of two equilibrium points. We discuss the necessary and sufficient conditions for local asymptotic stability of the equilibria and analyze the boundedness and oscillatory behavior of the solutions. A main result of the mathematical analysis is that seasonality in the life cycle of the deer tick can have a positive effect, in the sense that it increases the stability of the system. It is also shown that for some combination of parameters within the stability region, perturbations will result in a return to the equilibrium through transient oscillations. The models are used to explore the biological consequences of parameter variations reflecting expected environmental changes.

Algorithms↗

PubMed1996

Modeling of cardiovascular variability using a differential delay equation.

The influence of time delay in the baroreflex control of the heart activity is analyzed by using a simple mathematical model of the short-term pressure regulation. The mean arterial pressure in a Windkessel model is controlled by a nonlinear feedback driving a nonpulsatile model of the cardiac pump in accordance with the steady-state characteristics of the arterial baroreceptor reflex. A pure time delay is placed in the feedback branch to simulate the latent period of the baroreceptor regulation. Because of system nonlinearity model dynamics is found to be highly sensitive to time delay and changes of this parameter within a physiological range cause the model to exhibit different patterns of behavior. For low values of time delay (shorter than 0.5 s) the model remains in a steady state. When time delay is longer than 0.5 s, a Hopf bifurcation is crossed and spontaneous oscillations occur with frequencies in the high-frequency (HF) band. Further increases of time delay above 1.2s cause the oscillations to become more complex, and following the typical Feigenbaum cascade, the system becomes chaotic. In this condition heart rate, and flow show evident variability. The heart rate power spectrum exhibits a peak whose frequency moves from the HF to LF band depending on whether simulated time delay is as short as the vagal-mediated control or long as the sympathetic one.

Baroreflex↗

PubMed2006

Analytical solutions for nonlinear cable equations with calcium dynamics. I: Derivations.

The interaction between membrane potential and internal calcium concentration plays many important roles in regulating synaptic integration and neuronal firing. In order to gain a better theoretical understanding between the voltage-calcium interaction, a nonlinear cable equation with calcium dynamics is solved analytically. This general reaction-diffusion system represents a model of a cylindrical dendritic segment in which calcium diffuses internally in the presence of buffers, pumps and exchangers, and where ion channels are sparsely distributed over the membrane,in the form of hotspots, acting as point current sources along the dendritic membrane. In order to proceed, the reaction-diffusion system is recast into a system of coupled nonlinear integral equations, with which a perturbative expansion in dimensionless voltage and calcium concentration are used to find analytical solutions to this general system. The resulting solutions can be used to investigate, the interaction between the membrane potential and underlying calcium dynamics in a natural (non-discretized) setting.

Animals↗

PubMed2005

Neural network-based adaptive dynamic surface control for a class of uncertain nonlinear systems in strict-feedback form.

The dynamic surface control (DSC) technique was developed recently by Swaroop et al. This technique simplified the backstepping design for the control of nonlinear systems in strict-feedback form by overcoming the problem of "explosion of complexity." It was later extended to adaptive backstepping design for nonlinear systems with linearly parameterized uncertainty. In this paper, by incorporating this design technique into a neural network based adaptive control design framework, we have developed a backstepping based control design for a class of nonlinear systems in strict-feedback form with arbitrary uncertainty. Our development is able to eliminate the problem of "explosion of complexity" inherent in the existing method. In addition, a stability analysis is given which shows that our control law can guarantee the uniformly ultimate boundedness of the solution of the closed-loop system, and make the tracking error arbitrarily small.

Algorithms↗

PubMed1999

Consequences of deterministic and random dynamics for the course of affective disorders.

BACKGROUND: Uni- and bipolar affective disorders tend to be recurrent and progressive. Illness patterns can evolve from isolated episodes to more rapid, rhythmic, and "chaotic" mood patterns. Nonlinear deterministic dynamics are currently proposed to explain this progression. However, most natural systems are nonlinear and noisy, and cooperative behavior of possible clinical relevance can result. METHODS: The latter issue has been studied with a mathematical model for progression of disease patterns in affective disorders. RESULTS: Deterministic dynamics can reproduce a progression from stable, to periodic, to chaotic patterns. Noise increases the spectrum of dynamic behaviors, enhances the responsiveness to weak activations, and facilitates the occurrence of aperiodic patterns. CONCLUSIONS: Noise might amplify subclinical vulnerabilities into disease onset and could induce transitions to rapid-changing dysrhythmic mood patterns. We suggest that noise-mediated cooperative behavior, including stochastic resonance, should be considered in appropriate models for affective illness.

Disease Progression↗