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PubMed1990

Fascinating rhythm: a primer on chaos theory and its application to cardiology.

Nonlinear dynamics is an exciting new way of looking at peculiarities that in the past have been ignored or explained away. We have attempted to give a general introduction to the basics of the mathematics, applications to cardiology, and a brief review of the new tools needed to use the concepts of nonlinear mathematics. The careful mathematical approach to problems in cardiac electrical dynamics and blood flow is opening a window on behaviors and mechanisms previously inaccessible.

Animals↗

PubMed1997

Nonlinear analysis of the pharmacological conversion of sustained atrial fibrillation in conscious goats by the class Ic drug cibenzoline.

Methods from nonlinear dynamics were applied to test the hypothesis that the dynamics of sustained atrial fibrillation (AF) is modified by the class Ic drug cibenzoline during pharmacological conversion. The experiments were performed in conscious goats in which sustained AF was induced by continuous maintenance of AF via programmed electrical stimulation. Data were collected from electrophysiological experiments in five goats to terminate sustained AF by continuous infusion of cibenzoline. Sets of five unipolar epicardial electrograms of one minute duration were recorded from the left and right atrial free wall during sustained AF (control), and at three episodes during infusion of cibenzoline, when the mean AF interval had been prolonged to 25%, 50% and 85% with respect to control. Ventricular far-field potentials were removed from atrial electrograms by a coherent averaging procedure. Using the Grassberger-Procaccia method, the dynamics of the local atrial electrograms was investigated by estimating the (coarse-grained) correlation dimension and correlation entropy from the correlation integral. The results were related to a recently proposed classification (types I-III) of AF based on the degree of complexity of atrial activation patterns. The coarse-grained correlation dimension D(cg) and entropy K(cg) indicated that sustained AF corresponded to type II. During drug administration the coarse-grained parameters were not significantly different from control. Scaling regions in the correlation integral were observed after infusion of cibenzoline (3 out of 5 goats) suggesting that the drug introduced low-dimensional features (type I) in the dynamics of AF (correlation dimension D ranging from 2.8 to 4.4 and correlation entropy K from 1.6 to 6.2 nats/s). Sinus rhythm recorded shortly after cardioversion was very regular (D<2 and K<3 nats/s). The hypothesis that the electrograms during AF and sinus rhythm were generated by a static transformation of a linear Gaussian random process was rejected using a test for time reversibility. The nonlinear analysis revealed that cibenzoline does not significantly alter the dynamics of sustained AF during pharmacological conversion other than a slowing down of the atrial activation and a somewhat increasing global organization of the atrial activation pattern. The sudden change in the dynamical behavior at cardioversion suggests a mechanism that is reminiscent of a bifurcation. (c) 1997 American Institute of Physics.

Journal Article↗

PubMed2000

Modeling observed chaotic oscillations in bursting neurons: the role of calcium dynamics and IP3.

Chaotic bursting has been recorded in synaptically isolated neurons of the pyloric central pattern generating (CPG) circuit in the lobster stomatogastric ganglion. Conductance-based models of pyloric neurons typically fail to reproduce the observed irregular behavior in either voltage time series or state-space trajectories. Recent suggestions of Chay [Biol Cybern 75: 419-431] indicate that chaotic bursting patterns can be generated by model neurons that couple membrane currents to the nonlinear dynamics of intracellular calcium storage and release. Accordingly, we have built a two-compartment model of a pyloric CPG neuron incorporating previously described membrane conductances together with intracellular Ca2+ dynamics involving the endoplasmic reticulum and the inositol 1,4,5-trisphosphate receptor IP3R. As judged by qualitative inspection and quantitative, nonlinear analysis, the irregular voltage oscillations of the model neuron resemble those seen in the biological neurons. Chaotic bursting arises from the interaction of fast membrane voltage dynamics with slower intracellular Ca2+ dynamics and, hence, depends on the concentration of IP3. Despite the presence of 12 independent dynamical variables, the model neuron bursts chaotically in a subspace characterized by 3-4 active degrees of freedom. The critical aspect of this model is that chaotic oscillations arise when membrane voltage processes are coupled to another slow dynamic. Here we suggest this slow dynamic to be intracellular Ca2+ handling.

Action Potentials↗

PubMed2001

Dynamic analysis of renal nerve activity responses to baroreceptor denervation in hypertensive rats.

Sinoaortic and cardiac baroreflexes exert important control over renal sympathetic nerve activity. Alterations in these reflex mechanisms contribute to renal sympathoexcitation in hypertension. Nonlinear dynamic analysis was used to examine the chaotic behavior of renal sympathetic nerve activity in normotensive Sprague-Dawley and Wistar-Kyoto rats and spontaneously hypertensive rats before and after complete baroreceptor denervation (sinoaortic and cardiac baroreceptor denervation). The peak interval sequence of synchronized renal sympathetic nerve discharge was extracted and used for analysis. In all rat strains, this yielded systems whose correlation dimensions converged to similar low values over the embedding dimension range of 10 to 15 and whose greatest Lyapunov exponents were positive. In Sprague-Dawley and Wistar-Kyoto rats, compete baroreceptor denervation was associated with decreases in the correlation dimensions (Sprague-DAWLEY: 2.42+/-0.04 to 2.16+/-0.04; Wistar-KYOTO: 2.44+/-0.04 to 2.34+/-0.04) and in the greatest Lyapunov exponents (Sprague-DAWLEY: 0.199+/-0.004 to 0.130+/-0.015; Wistar-KYOTO: 0.196+/-0.002 to 0.136+/-0.010). Spontaneously hypertensive rats had a similar correlation dimension, which was unaffected by complete baroreceptor denervation (2.42+/-0.02 versus 2.42+/-0.03), and a lower value for the greatest Lyapunov exponent, which decreased to a lesser extent after complete baroreceptor denervation (0.183+/-0.006 versus 0.158+/-0.006). These results indicate that removal of sinoaortic and cardiac baroreceptor regulation of renal sympathetic nerve activity is associated with a greater decrease in the chaotic behavior of renal sympathetic nerve activity in normotensive compared with hypertensive rats. This suggests that the central neural mechanisms that regulate renal sympathetic nerve activity in response to alterations in cardiovascular reflex inputs are different in spontaneously hypertensive rats from those in Sprague-Dawley and Wistar-Kyoto rats.

Anesthesia↗

PubMed1986

System analysis of Phycomyces light-growth response with sum-of-sinusoids test stimuli.

The light-growth response of Phycomyces has been studied with the sum-of-sinusoids method of nonlinear system identification (Victor, J.D., and R.M. Shapley, 1980, Biophys. J., 29:459). This transient response of the sporangiophore has been treated as a black-box system with one input (logarithm of the light intensity, I) and one output (elongation rate). The light intensity was modulated so that log I, as a function of time, was a sum of sinusoids. The log-mean intensity was 10(-4) W m-2 and the wavelength was 477 nm. The first- and second-order frequency kernels, which represent the linear and nonlinear behavior of the system, were obtained from the Fourier transform of the response at the appropriate component and combination frequencies. Although the first-order kernel accounts for most of the response, there remains a significant nonlinearity beyond the logarithmic transducer presumed to occur at the input of the sensory transduction chain. From the analysis of the frequency kernels, we have derived a dynamic nonlinear model of the light-growth response system. The model consists of a nonlinear subsystem followed by a linear subsystem. The model parameters were estimated from a combined nonlinear least-squares fit to the first- and second-order frequency kernels.

Kinetics↗

PubMedDate not supplied

Period multupling-evidence for nonlinear behaviour of the canine heart.

Although there has recently been considerable interest in applying the theory of nonlinear dynamics to the analysis of complex systems, as yet applications of the theory to biological systems in vivo have been very limited. We report here evidence of nonlinear behaviour in the electrocardiogram and arterial blood pressure traces of the noradrenaline-treated dog. Noradrenaline produces variations in these traces that repeat themselves with regular periods of integral numbers of heart-beats (period multupling), an effect that resembles the 'period-doubling' and other 'bifurcative' behaviour observed when the driving frequency of a nonlinear oscillator is increased above a critical value. The simplest type of periodic variation that we report is the so-called 'electrical alternans', which has long been known as one response of cardiac electrical activity to certain stresses and disease states.

Animals↗

PubMed2003

Noise-induced unstable dimension variability and transition to chaos in random dynamical systems.

Results are reported concerning the transition to chaos in random dynamical systems. In particular, situations are considered where a periodic attractor coexists with a nonattracting chaotic saddle, which can be expected in any periodic window of a nonlinear dynamical system. Under noise, the asymptotic attractor of the system can become chaotic, as characterized by the appearance of a positive Lyapunov exponent. Generic features of the transition include the following: (1) the noisy chaotic attractor is necessarily nonhyperbolic as there are periodic orbits embedded in it with distinct numbers of unstable directions (unstable dimension variability), and this nonhyperbolicity develops as soon as the attractor becomes chaotic; (2) for systems described by differential equations, the unstable dimension variability destroys the neutral direction of the flow in the sense that there is no longer a zero Lyapunov exponent after the noisy attractor becomes chaotic; and (3) the largest Lyapunov exponent becomes positive from zero in a continuous manner, and its scaling with the variation of the noise amplitude is algebraic. Formulas for the scaling exponent are derived in all dimensions. Numerical support using both low- and high-dimensional systems is provided.

Journal Article↗

PubMed2005

Spontaneous optical fractal pattern formation.

We report, for the first time, spontaneous nonlinear optical spatial fractals. The proposed generic mechanism employs intrinsic nonlinear dynamics both to generate an initial pattern seed and to fill out structure across decades of spatial scale. We demonstrate this in one of the simplest of nonlinear optical systems, composed of a Kerr slice and a single-feedback mirror. In this case, the smallest pattern scales are limited by either the optical wavelength or the diffusion length of the medium photoexcitation. The dimension characteristics of these particular fractals are also derived.

Journal Article↗

PubMed1998

Dynamics of membrane excitability determine interspike interval variability: a link between spike generation mechanisms and cortical spike train statistics.

We propose a biophysical mechanism for the high interspike interval variability observed in cortical spike trains. The key lies in the nonlinear dynamics of cortical spike generation, which are consistent with type I membranes where saddle-node dynamics underlie excitability (Rinzel & Ermentrout, 1989). We present a canonical model for type I membranes, the theta-neuron. The theta-neuron is a phase model whose dynamics reflect salient features of type I membranes. This model generates spike trains with coefficient of variation (CV) above 0.6 when brought to firing by noisy inputs. This happens because the timing of spikes for a type I excitable cell is exquisitely sensitive to the amplitude of the suprathreshold stimulus pulses. A noisy input current, giving random amplitude "kicks" to the cell, evokes highly irregular firing across a wide range of firing rates; an intrinsically oscillating cell gives regular spike trains. We corroborate the results with simulations of the Morris-Lecar (M-L) neural model with random synaptic inputs: type I M-L yields high CVs. When this model is modified to have type II dynamics (periodicity arises via a Hopf bifurcation), however, it gives regular spike trains (CV below 0.3). Our results suggest that the high CV values such as those observed in cortical spike trains are an intrinsic characteristic of type I membranes driven to firing by "random" inputs. In contrast, neural oscillators or neurons exhibiting type II excitability should produce regular spike trains.

Algorithms↗

PubMed1991

Intracellular nonlinear frequency response measurements in the cockroach tactile spine neuron.

The threshold of the cockroach tactile neuron increases strongly with depolarization by a process involving at least two time constants. This effect is probably responsible for the rapid and complete adaptation of the neuron's response to step inputs. A technique for intracellular recording and stimulation of the neuron has recently been established and this allows direct observation of the dynamic response of the neuronal encoder. A white noise stimulus was used to modulate the membrane potential of the neuron. The first-order frequency response function between membrane potential and action potential discharge could be explained by a variable threshold model with two time constants. Second-order frequency response functions could be accounted for by a Wiener cascade model. The dynamic nonlinear behavior of the encoder can therefore be explained by a unidirectional threshold which increases linearly and dynamically with membrane potential.

Acoustic Stimulation↗

PubMed2004

Testing time symmetry in time series using data compression dictionaries.

Time symmetry, often called statistical time reversibility, in a dynamical process means that any segment of time-series output has the same probability of occurrence in the process as its time reversal. A technique, based on symbolic dynamics, is proposed to distinguish such symmetrical processes from asymmetrical ones, given a time-series observation of the otherwise unknown process. Because linear stochastic Gaussian processes, and static nonlinear transformations of them, are statistically reversible, but nonlinear dynamics such as dissipative chaos are usually statistically irreversible, a test will separate large classes of hypotheses for the data. A general-purpose and robust statistical test procedure requires adapting to arbitrary dynamics which may have significant time correlation of undetermined form. Given a symbolization of the observed time series, the technology behind adaptive dictionary data compression algorithms offers a suitable estimate of reversibility, as well as a statistical likelihood test. The data compression methods create approximately independent segments permitting a simple and direct null test without resampling or surrogate data. We demonstrate the results on various time-series-reversible and irreversible systems.

Journal Article↗

PubMed2005

The relationship of normal and abnormal microstructural proliferation to the mitral valve closure sound.

BACKGROUND: Many diseases that affect the mitral valve are accompanied by the proliferation or degradation of tissue microstructure. The early acoustic detection of these changes may lead to the better management of mitral valve disease. In this study, we examine the nonstationary acoustic effects of perturbing material parameters that characterize mitral valve tissue in terms of its microstructural components. Specifically, we examine the influence of the volume fraction, stiffness and splay of collagen fibers as well as the stiffness of the nonlinear matrix in which they are embedded. METHODS AND RESULTS: To model the transient vibrations of the mitral valve apparatus bathed in a blood medium, we have constructed a dynamic nonlinear fluid-coupled finite element model of the valve leaflets and chordae tendinae. The material behavior for the leaflets is based on an experimentally derived structural constitutive equation. The gross movement and small-scale acoustic vibrations of the valvular structures result from the application of physiologic pressure loads. Material changes that preserved the anisotropy of the valve leaflets were found to preserve valvular function. By contrast, material changes that altered the anisotropy of the valve were found to profoundly alter valvular function. These changes were manifest in the acoustic signatures of the valve closure sounds. Abnormally, stiffened valves closed more slowly and were accompanied by lower peak frequencies. CONCLUSION: The relationship between stiffness and frequency, though never documented in a native mitral valve, has been an axiom of heart sounds research. We find that the relationship is more subtle and that increases in stiffness may lead to either increases or decreases in peak frequency depending on their relationship to valvular function.

Animals↗

PubMed1991

Nonlinear analysis of EEG sleep states.

Recent advances in the field of nonlinear dynamics have provided new conceptual models as well as novel analytical techniques applicable to neuropsychopharmacologic studies. One measurement technique that has been recently developed in an attempt to characterize nonlinear systems in physics and biology is the estimation of dimension. Dimension may be seen as a measure of the information required to describe the current behavior of a system. We have applied these techniques to the analysis of the sleep EEG, and have found that the dimension of rapid eye movement (REM) sleep is significantly higher than non-rapid-eye-movement (NREM) sleep. These data support a preliminary hypothesis that EEG dimension may represent the number of nonlinear modes activated in the brain. Thus, sleep states of low arousal or low input would be envisioned as having low dimension (e.g., slow-wave sleep) whereas increased arousal (REM) would activate more nonlinear modes. Although more investigations will be needed to explore this hypothesis, these studies suggest that further development of nonlinear approaches to the analysis of brain systems are likely to generate new clinical measures as well as new ways of viewing brain electrical function.

Adult↗

PubMed1995

EEG dimension in sons of alcoholics.

Recent advances in the field of nonlinear dynamics have provided new conceptual models, as well as novel analytical techniques applicable to studies in alcohol research. One measurement technique, which has been developed in an attempt to characterize nonlinear systems in physics and biology, is the estimation of attractor dimension. This dimension may be seen as a measure of the information required to describe the current behavior of a system. We have applied these techniques to the analysis of EEG collected from 17 men with alcoholic fathers and 19 men with no alcoholic relatives. The men with alcoholic fathers were found to have a lower EEG attractor dimension than their matched controls. This suggests that the EEG of young men with alcoholic fathers may be "more organized" or "less complex" than men with no alcoholic relatives. Although more studies will be needed to explore this hypothesis, these studies suggest that further development of nonlinear approaches to the analysis of brain systems are likely to generate new clinical measures, as well as new ways of viewing brain electrical function and alcoholism.

Action Potentials↗

PubMed1988

Dynamic pattern generation in behavioral and neural systems.

In the search for principles of pattern generation in complex biological systems, an operational approach is presented that embraces both theory and experiment. The central mathematical concepts of self-organization in nonequilibrium systems (including order parameter dynamics, stability, fluctuations, and time scales) are used to show how a large number of empirically observed features of temporal patterns can be mapped onto simple low-dimensional (stochastic, nonlinear) dynamical laws that are derivable from lower levels of description. The theoretical framework provides a language and a strategy, accompanied by new observables, that may afford an understanding of dynamic patterns at several scales of analysis (including behavioral patterns, neural networks, and individual neurons) and the linkage among them.

Animals↗

PubMed2006

Dynamic effects on nonlinear alternating current responses in electrorheological fluids.

By using a perturbation approach, we investigate dynamic effects on nonlinear alternating current (ac) responses in electrorheological (ER) fluids under an ac or direct current electric field. We show that the dynamic effect due to a shear flow, which exerts a torque on ER particles and thus leads to the rotation of the particles about their centers, plays a significant role in the responses. Our results can be well interpreted in the dielectric dispersion spectral representation, and they offer a convenient method to determine the relaxation time and rotation velocity of ER particles by measuring the nonlinear ac responses.

Journal Article↗

PubMed2005

Dissipative quantum chaos: transition from wave packet collapse to explosion.

Using the quantum trajectories approach, we study the quantum dynamics of a dissipative chaotic system described by the Zaslavsky map. For strong dissipation the quantum wave function in the phase space collapses onto a compact packet which follows classical chaotic dynamics and whose area is proportional to the Planck constant. At weak dissipation the exponential instability of quantum dynamics on the Ehrenfest time scale dominates and leads to wave packet explosion. The transition from collapse to explosion takes place when the dissipation time scale exceeds the Ehrenfest time. For integrable nonlinear dynamics the explosion practically disappears leaving place to collapse.

Journal Article↗

PubMedDate not supplied

A priori identifiability of distributed models of blood-tissue exchange.

A priori identifiability deals with the uniqueness of the solution for the unknown parameters of a dynamic model from a given input-output experiment, and is a prerequisite for well posedness of parameter estimation from the data. Identifiability has been extensively investigated for lumped parameter, linear, and nonlinear dynamic models, in particular, compartmental models of biological systems. Much less attention has been devoted to distributed parameter model, in particular, those describing blood-tissue exchange, which are normally used to interpret regional multiple tracer dilution experiments. In this paper, we study a priori identifiability of distributed parameter models of transcapillary exchange, focusing first on a single capillary (one-region) model, then moving on to a single capillary-interstitial fluid (two-region) model, and finally to an organ model also describing flow heterogeneity.

Capillaries↗