Index of content:
Volume 5, Issue 1, March 1995
5(1995); http://dx.doi.org/10.1063/1.166069View Description Hide Description
Dynamical diseases are characterized by sudden changes in the qualitative dynamics of physiological processes, leading to abnormal dynamics and disease. Thus, there is a natural matching between the mathematical field of nonlinear dynamics and medicine. This paper summarizes advances in the study of dynamical disease with emphasis on a NATO Advanced Research Worshop held in Mont Tremblant, Québec, Canada in February 1994. We describe the international effort currently underway to identify dynamical diseases and to study these diseases from a perspective of nonlinear dynamics. Linear and nonlinear time series analysis combined with analysis of bifurcations in dynamics are being used to help understand mechanisms of pathological rhythms and offer the promise for better diagnostic and therapeutic techniques.
5(1995); http://dx.doi.org/10.1063/1.166103View Description Hide Description
Thirty‐two (32) periodic diseases of the nervous system are identified in which symptoms and/or signs recur. In 10/32, the recurrence of a symptom complex is one of the defining features of the illness, whereas in 22/32 oscillatory signs occur in the setting of an ongoing nervous system disorder. We discuss the possibility that these disorders may be dynamic diseases.
5(1995); http://dx.doi.org/10.1063/1.166067View Description Hide Description
Breathing is regulated by a central neural oscillator that produces rhythmic output to the respiratory muscles. Pathological disturbances in rhythm (dysrhythmias) are observed in the breathing pattern of children and adults with neurological and cardiopulmonary diseases. The mechanisms responsible for genesis of respiratory dysrhythmias are poorly understood. The present studies take a novel approach to this problem. The basic postulate is that the rhythm of the respiratory oscillator can be altered by a variety of stimuli. When the oscillator recovers its rhythm after such perturbations, its phase may be reset relative to the original rhythm. The amount of phase resetting is dependent upon stimulus parameters and the level of respiratory drive. The long‐range hypothesis is that respiratory dysrhythmias can be induced by stimuli that impinge upon or arise within the respiratory oscillator with certain combinations of strength and timing relative to the respiratory cycle. Animal studies were performed in anesthetized or decerebrate preparations. Neural respiratory rhythmicity is represented by phrenic nerve activity, allowing use of open‐loop experimental conditions which avoid negative chemical feedback associated with changes in ventilation.
In animal experiments, respiratory dysrhythmias can be induced by stimuli having specific combinations of strength and timing. Newborn animals readily exhibit spontaneous dysrhythmias which become more prominent at lower respiratory drives. In human subjects, swallowing was studied as a physiological perturbation of respiratory rhythm, causing a pattern of phase resetting that is characterized topologically as type 0. Computational studies of the Bonhoeffer–van der Pol (BvP) equations, whose qualitative behavior is representative of many excitable systems, supports a unified interpretation of these experimental findings. Rhythmicity is observed when the BvP model exhibits recurrent periods of excitation alternating with refractory periods. The same system can be perturbed to a state in which amplitude of oscillation is attenuated or abolished. We have characterized critical perturbations which induce transitions between these two states, giving rise to patterns of dysrhythmic activity that are similar to those seen in the experiments. We illustrate the importance of noise in initiation and termination of rhythm, comparable to normal respiratory rhythm intermixed with spontaneous dysrhythmias. In the BvP system the incidence and duration of dysrhythmia is shown to be strongly influenced by the level of noise. These studies should lead to greater understanding of rhythmicity and integrative responses of the respiratory control system, and provide insight into disturbances in control mechanisms that cause apnea and aspiration in clinical disease states.
5(1995); http://dx.doi.org/10.1063/1.166078View Description Hide Description
Irregularities in voiced speech are often observed as a consequence of vocal fold lesions, paralyses, and other pathological conditions. Many of these instabilities are related to the intrinsic nonlinearities in the vibrations of the vocal folds. In this paper, bifurcations in voice signals are analyzed using narrow‐band spectrograms. We study sustained phonation of patients with laryngeal paralysis and data from an excised larynx experiment. These spectrograms are compared with computer simulations of an asymmetric 2‐mass model of the vocal folds.
5(1995); http://dx.doi.org/10.1063/1.166082View Description Hide Description
Experimental evidence has shown a plethora of short‐term fluctuations in patients with Parkinson’s disease. We investigate these transitory events using the concept of dynamical disease. Several examples of short‐term fluctuations in tremor are analyzed, and in two cases, other systemic variables (i.e., respiration and blood pressure) are examined as well. A model for tremor, based on negative feedback with delays is proposed, and the transient events are simulated. The theoretical implications of the model suggest that interactions between the central and peripheral loops, as well as interactions between the control loops and other systemic signals, can give rise to transitory events in tremor, both in the pathological and in the normal case.
5(1995); http://dx.doi.org/10.1063/1.166083View Description Hide Description
The movement disorder syndrome of tardive dyskinesia arises as a consequence of prolonged regimens of neuroleptic medication, and is characterized, although not exclusively, by jerky and sometimes rhythmical stereotypical motions in a wide range of muscle systems. It is well established that the degree and variability of tremor in tardive dyskinesia is greater than that in normal age‐matched subjects. The findings from the current experiment show that the dimension of the tardive dyskinetic finger tremor time series is systematically lower than that evident in normal finger tremor. Furthermore, the variability of finger motion in both groups is inversely related to the dimension of the respective attractor dynamic. The neuroleptic medication appears to constrain the degrees of freedom regulated in organization of the motor system.
5(1995); http://dx.doi.org/10.1063/1.166084View Description Hide Description
The separation between physiologic tremor (PT) in normal subjects and the pathological tremors of essential tremor (ET) or Parkinson’s disease (PD) was investigated on the basis of monoaxial accelerometric recordings of 35 s hand tremor epochs. Frequency and amplitude were insufficient to separate between these conditions, except for the trivial distinction between normal and pathologic tremors that is already defined on the basis of amplitude. We found that waveform analysis revealed highly significant differences between normal and pathologic tremors, and, more importantly, among different forms of pathologic tremors. We found in our group of 25 patients with PT and 15 with ET a reasonable distinction with the third momentum and the time reversal invariance. A nearly complete distinction between these two conditions on the basis of the asymmetric decay of the autocorrelation function. We conclude that time series analysis can probably be developed into a powerful tool for the objective analysis of tremors.
5(1995); http://dx.doi.org/10.1063/1.166085View Description Hide Description
This paper describes our analysis procedure for long‐term tremor EMG recordings, as well as three examples of applications. The description of the method focuses on how characteristics of the tremor (e.g. frequency, intensity, agonist–antagonist interaction) can be defined and calculated based on surface EMG data. The resulting quantitative characteristics are called ‘‘tremor parameters.’’ We discuss sinusoidally modulated, band‐limited white noise as a model for pathological tremor‐EMG, and show how the basic parameters can be extracted from this class of signals. The method is then applied to (1) estimate tremor severity in clinical studies, (2) quantify agonist–antagonist interaction, and (3) investigate the variations of the tremor parameters using simple methods from time‐series analysis.
Upright, correlated random walks: A statistical‐biomechanics approach to the human postural control system5(1995); http://dx.doi.org/10.1063/1.166086View Description Hide Description
The task of maintaining erect stance involves a complex sensorimotor control system, the output of which can be highly irregular. Even when a healthy individual attempts to stand still, the center of gravity of his or her body and the center of pressure (COP) under his or her feet continually move about in an erratic fashion. In this study, we approach the problem of characterizing postural sway from the perspective of random‐walk theory. Specifically, we analyze COP trajectories as one‐dimensional and two‐dimensional random walks. These analyses reveal that over short‐term intervals of time during undisturbed stance the COP behaves as a positively correlated random walk, whereas over long‐term intervals of time it resembles a negatively correlated random walk. We interpret this novel finding as an indication that during quiet standing the postural control system utilizes open‐loop and closed‐loop control schemes over short‐term and long‐term intervals, respectively. From this perspective, our approach, known as stabilogram‐diffusion analysis, has the advantage that it leads to the extraction of COP parameters which can be directly related to the steady‐state behavior and functional interaction of the neuromuscular mechanisms underlying the maintenance of erect stance.
5(1995); http://dx.doi.org/10.1063/1.166087View Description Hide Description
Using a sensorimotor coordination task in conjunction with an array of SQUIDs (Superconducting Quantum Interference Devices) we demonstrate critical instabilities in human brain activity patterns. Analysis of the dominant spatial pattern of the brain and its time‐varying amplitude displays a task‐dependent geometry characteristic of Šil’nikov‐like chaos, which changes qualitatively at the transition.
5(1995); http://dx.doi.org/10.1063/1.166088View Description Hide Description
After listening to a sound that is presented repeatedly, subjects report hearing different transforms of the original sound. The frequency of reported transforms is a sensitive index of some speech disorders as well as cognitive flexibility in aging. In this paper, we propose and investigate quantitative measures that characterize the dynamics of this phenomenon, known as the verbal transformation effect. In particular, we show that the distribution of the dwell time, the time spent perceiving a string of a given phonemic form before switching to another form, obeys a power law for normal subjects with an exponent valued between 1 and 2. This result suggests that within this paradigm there is no characteristic time scale for the perceptual process. Additionally, we analyze the correlation properties of the transforms. We suggest that the complexity measures and techniques introduced here might be useful diagnostic tools for a number of speech and cognitive disorders.
5(1995); http://dx.doi.org/10.1063/1.166089View Description Hide Description
In many biological systems,information is transferred by hormonal ligands, and it is assumed that these hormonal signals encode developmental and regulatory programs in mammalian organisms. In contrast to the dogma of endocrine homeostasis, it could be shown that the biological information in hormonal networks is not only present as a constant hormone concentration in the circulation pool. Recently, it has become apparent that hormone pulses contribute to this hormonal pool, which modulates the responsiveness of receptors within the cell membrane by regulation of the receptor synthesis, movement within the membrane layer, coupling to signal transductionproteins and internalization. Phase space analysis of dynamic parathyroid hormone (PTH) secretion allowed the definition of a (in comparison to normal subjects) relatively quiet ‘‘low dynamic’’ secretory pattern in osteoporosis, and a ‘‘high dynamic’’ state in hyperparathyroidism. We now investigate whether this pulsatile secretion of PTH in healthy men exhibits characteristics of nonlinear determinism. Our findings suggest that this is conceivable, although on the basis of presently available data and techniques, no proof can be established. Nevertheless, pulsatile secretion of PTH might be a first example of nonlinear deterministic dynamics in an apparently irregular hormonal rhythm in human physiology.
5(1995); http://dx.doi.org/10.1063/1.166141View Description Hide Description
The healthy heartbeat is traditionally thought to be regulated according to the classical principle of homeostasis whereby physiologic systems operate to reduce variability and achieve an equilibrium‐like state [Physiol. Rev. 9, 399–431 (1929)]. However, recent studies [Phys. Rev. Lett. 70, 1343–1346 (1993); Fractals in Biology and Medicine (Birkhauser‐Verlag, Basel, 1994), pp. 55–65] reveal that under normal conditions, beat‐to‐beat fluctuations in heart rate display the kind of long‐range correlations typically exhibited by dynamical systems far from equilibrium [Phys. Rev. Lett. 59, 381–384 (1987)]. In contrast, heart rate time series from patients with severe congestive heart failure show a breakdown of this long‐range correlation behavior. We describe a new method—detrended fluctuation analysis (DFA)—for quantifying this correlation property in non‐stationary physiological time series. Application of this technique shows evidence for a crossover phenomenon associated with a change in short and long‐range scaling exponents. This method may be of use in distinguishing healthy from pathologic data sets based on differences in these scaling properties.
5(1995); http://dx.doi.org/10.1063/1.166090View Description Hide Description
In the modern industrialized countries every year several hundred thousands of people die due to sudden cardiac death. The individual risk for this sudden cardiac death cannot be defined precisely by common available, noninvasive diagnostic tools like Holter monitoring, highly amplified ECG and traditional linear analysis of heart rate variability (HRV). Therefore, we apply some rather unconventional methods of nonlinear dynamics to analyze the HRV. Especially, some complexity measures that are based on symbolic dynamics as well as a new measure, the renormalized entropy, detect some abnormalities in the HRV of several patients who have been classified in the low risk group by traditional methods. A combination of these complexity measures with the parameters in the frequency domain seems to be a promising way to get a more precise definition of the individual risk. These findings have to be validated by a representative number of patients.
5(1995); http://dx.doi.org/10.1063/1.166104View Description Hide Description
In recent years there has been an increasing number of papers in the literature, applying the methods and techniques of Nonlinear Dynamics to the time series of electrical activity in normal electrocardiograms (ECGs) of various human subjects. Most of these studies are based primarily on correlation dimension estimates, and conclude that the dynamics of the ECG signal is deterministic and occurs on a chaotic attractor, whose dimension can distinguish between healthy and severely malfunctioning cases. In this paper, we first demonstrate that correlation dimension calculations must be used with care, as they do not always yield reliable estimates of the attractor’s ‘‘dimension.’’ We then carry out a number of additional tests (time differencing, smoothing, principal component analysis, surrogate data analysis, etc.) on the ECGs of three ‘‘normal’’ subjects and three ‘‘heavy smokers’’ at rest and after mild exercising, whose cardiac rhythms look very similar. Our main conclusion is that no major dynamical differences are evident in these signals. A preliminary estimate of three to four basic variables governing the dynamics (based on correlation dimension calculations) is updated to five to six, when temporal correlations between points are removed. Finally, in almost all cases, the transition between resting and mild exercising seems to imply a small increase in the complexity of cardiac dynamics.
Age‐related changes in the ‘‘complexity’’ of cardiovascular dynamics: A potential marker of vulnerability to disease5(1995); http://dx.doi.org/10.1063/1.166091View Description Hide Description
Healthy physiologic control of cardiovascular function is a result of complex interactions between multiple regulatory processes that operate over different time scales. These include the sympathetic and parasympathetic nervous systems which regulate beat‐to‐beat heart rate (HR) and blood pressure (BP), as well as extravascular volume, body temperature, and sleep which influence HR and BP over the longer term. Interactions between these control systems generate highly variable fluctuations in continuous HR and BP signals. Techniques derived from nonlinear dynamics and chaos theory are now being adapted to quantify the dynamic behavior of physiologic time series and study their changes with age or disease. We have shown significant age‐related changes in the 1/f x relationship between the log amplitude and log frequency of the heart rate power spectrum, as well as declines in approximate dimension and approximate entropy of both heart rate and blood pressure time series. These changes in the ‘‘complexity’’ of cardiovascular dynamics reflect the breakdown and decoupling of integrated physiologic regulatory systems with aging, and may signal an impairment in cardiovascular ability to adapt to external and internal perturbations. Studies are currently underway to determine whether the complexity of HR or BP time series can distinguish patients with fainting spells due to benign vasovagal reactions from those due to life‐threatening cardiac arrhythmias. Thus, measures of the complexity of physiologic variability may provide novel methods to monitor cardiovascular aging and test the efficacy of specific interventions to improve adaptive capacity in old age.
5(1995); http://dx.doi.org/10.1063/1.166092View Description Hide Description
Approximate entropy (ApEn) is a recently developed statistic quantifying regularity and complexity, which appears to have potential application to a wide variety of relatively short (greater than 100 points) and noisy time‐series data. The development of ApEn was motivated by data length constraints commonly encountered, e.g., in heart rate, EEG, and endocrine hormone secretion data sets. We describe ApEn implementation and interpretation, indicating its utility to distinguish correlated stochastic processes, and composite deterministic/ stochastic models. We discuss the key technical idea that motivates ApEn, that one need not fully reconstruct an attractor to discriminate in a statistically valid manner—marginal probability distributions often suffice for this purpose. Finally, we discuss why algorithms to compute, e.g., correlation dimension and the Kolmogorov–Sinai (KS) entropy, often work well for true dynamical systems, yet sometimes operationally confound for general models, with the aid of visual representations of reconstructed dynamics for two contrasting processes.
5(1995); http://dx.doi.org/10.1063/1.166093View Description Hide Description
To compare direct tests for detecting determinism in chaotic time series, data from Hénon, Lorenz, and Mackey–Glass equations were contaminated with various levels of additive colored noise. These data were analyzed with a variety of recently developed tests for determinism, and the results compared.
5(1995); http://dx.doi.org/10.1063/1.166094View Description Hide Description
According to a theorem of Takens [Lecture Notes in Mathematics (Springer‐Verlag, Berlin, 1981), Vol. 898], dynamical state information can be reproduced from a time series of amplitude measurements. In this paper we investigate whether the same information can be reproduced from interspike interval (ISI) measurements. Assuming an integrate‐and‐fire model coupling the dynamical system to the spike train, there is a one‐to‐one correspondence between the system states and interspike interval vectors of sufficiently large dimension. The correspondence implies in particular that a data series of interspike intervals, formed in this manner, can be forecast from past history. This capability is demonstrated using a nonlinear prediction algorithm, and is found to be robust to noise. A set of interspike intervals measured from a simple neuronal circuit is studied for deterministic structure using a prediction error statistic.