Nonlinear dynamics of heart rate variability during experimental hemorrhage in ketamine-anesthetized rats.

Indexes of heart rate variability (HRV) based on linear stochastic models are independent risk factors for arrhythmic death (AD). An index based on a nonlinear deterministic model, a reduction in the point correlation dimension (PD2i), has been shown in both animal and human studies to have a higher sensitivity and specificity for predicting AD. Dimensional reduction subsequent to transient ischemia was examined previously in a simple model system, the intrinsic nervous system of the isolated rabbit heart. The present study presents a new model system in which the higher cerebral centers are blocked chemically (ketamine inhibition of N-methyl-D-aspartate receptors) and the system is perturbed over a longer 15-min interval by continuous hemorrhage. The hypothesis tested was that dimensional reduction would again be evoked, but in association with a more complex relationship between the system variables. The hypothesis was supported, and we interpret the greater response complexity to result from the larger autonomic superstructure attached to the heart. The complexities observed in the nonlinear heartbeat dynamics constitute a new genre of autonomic response, one clearly distinct from a hardwired reflex or a cerebrally determined defensive reaction.

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.

Event-related dimensional reductions in the primary auditory cortex of the conscious cat are revealed by new techniques for enhancing the non-linear dimensional algorithms.

The analytic algorithms derived from non-linear deterministic models may be more sensitive to differences in physiological data than those based on linear stochastic models. Among the non-linear algorithms the time-dependent dimensional ones appear to be the most sensitive discriminators. In the present study dimensional responses were examined in both electronically and mathematically generated data and in high-resolution physiological data. The latter were event-related potentials (ERPs) recorded from the primary auditory cortex of cats during classical conditioning. Two techniques were found to lengthen and stabilize the linear scaling region in the correlation integral of the dimensional algorithms: (1) linking trials to increase data length; and (2) gain reduction to lower integer-values of noise, combined with algorithmic setting of slopes < 0.5 to zero. Of the three dimensional algorithms examined, only the time-dependent Point Correlation Dimension (PD2i) showed low error rates when tracking the dimensional shifts in non-stationary generated data. This algorithm also uniquely distinguished between the conditioned and unconditioned physiological responses. The ERPs had corresponding PD2i's that were significantly different from each other as well as from their own randomized-phase surrogates. The brief dimensional reduction that follows a conditioned stimulus is interpreted to be related to 'cooperativity' among the underlying cortical neurons that contribute to its electrogenesis.

Is the heart preadapted to hypoxia? Evidence from fractal dynamics of heartbeat interval fluctuations at high altitude (5,050 m).

The dynamics of heartbeat interval time series over large time scales were studied by a modified random walk analysis introduced recently as Detrended Fluctuation Analysis. In this analysis, the intrinsic fractal long-range power-law correlation properties of beat-to-beat fluctuations generated by the dynamical system (i.e., cardiac rhythm generator), after decomposition from extrinsic uncorrelated sources, can be quantified by the scaling exponent (alpha) which, in healthy subjects, for time scales of approximately 10(4) beats is approximately 1.0. The effects of chronic hypoxia were determined from serial heartbeat interval time series of digitized twenty-four-hour ambulatory ECGs recorded in nine healthy subjects (mean age thirty-four years old) at sea level and during a sojourn at 5,050 m for thirty-four days (EvK2-CNR Pyramid Laboratory, Sagarmatha National Park, Nepal). The group averaged alpha exponent (+/- SD) was 0.99 +/- 0.04 (range 0.93-1.04). Longitudinal assessment of alpha in individual subjects did not reveal any effect of exposure to chronic high altitude hypoxia. The finding of alpha approximately 1 indicating scale-invariant long-range power-law correlations (1/f noise) of heartbeat fluctuations would reflect a genuinely self-similar fractal process that typically generates fluctuations on a wide range of time scales. Lack of a characteristic time scale along with the absence of any effect from exposure to chronic hypoxia on scaling properties suggests that the neuroautonomic cardiac control system is preadapted to hypoxia which helps prevent excessive mode-locking (error tolerance) that would restrict its functional responsiveness (plasticity) to hypoxic or other physiological stimuli.

Stability of heartbeat interval distributions in chronic high altitude hypoxia.

Recent studies of nonlinear dynamics of the long-term variability of heart rate have identified nontrivial long-range correlations and scale-invariant power-law characteristics (l/f noise) that were remarkably consistent between individuals and were unrelated to external or environmental stimuli (Meyer et al., 1998a). The present analysis of complex nonstationary heartbeat patterns is based on the sequential application of the wavelet transform for elimination of local polynomial nonstationary behavior and an analytic signal approach by use of the Hilbert transform (Cumulative Variation Amplitude Analysis). The effects of chronic high altitude hypoxia on the distributions and scaling functions of cardiac intervals over 24 hr epochs and 4 hr day/nighttime subepochs were determined from serial heartbeat interval time series of digitized 24 hr ambulatory ECGs recorded in 9 healthy subjects (mean age 34 yrs) at sea level and during a sojourn at high altitude (5,050 m) for 34 days (Ev-K2-CNR Pyramid Laboratory, Sagarmatha National Park, Nepal). The results suggest that there exists a hidden, potentially universal, common structure in the heterogeneous time series. A common scaling function with a stable Gamma distribution defines the probability density of the amplitudes of the fluctuations in the heartbeat interval time series of individual subjects. The appropriately rescaled distributions of normal subjects at sea level demonstrated stable Gamma scaling consistent with a single scaled plot (data collapse). Longitudinal assessment of the rescaled distributions of the 24 hr recordings of individual subjects showed that the stability of the distributions was unaffected by the subject's exposure to a hypobaric (hypoxic) environment. The rescaled distributions of 4 hr subepochs showed similar scaling behavior with a stable Gamma distribution indicating that the common structure was unequivocally applicable to both day and night phases and, furthermore, did not undergo systematic changes in response to high altitude. In contrast, a single function stable over a wide range of time scales was not observed in patients with congestive heart failure or patients after cardiac transplantation. The functional form of the scaling in normal subjects would seem to be attributable to the underlying nonlinear dynamics of cardiac control. The results suggest that the observed Gamma scaling of the distributions in healthy subjects constitutes an intrinsic dynamical property of normal heart function that would not undergo early readjustment or late acclimatization to extrinsic environmental physiological stress, e.g., chronic hypoxia.

Dimensional complexity of the EEG in subcortical stroke--a case study.

The conventional electrophysiological methods used for the analysis of the functional characteristics of the nervous system are not able to grasp its non-linear and random features. Of the methods based on the application of chaos-theory the correlation dimension analysis can be used to quantify the complexity of the analyzed signal, such as the electroencephalogram (EEG). The new version (point-correlation dimension, PD2) was used in this study, which is more accurate than the other, currently used algorithms. The purpose of the present investigation was to compare the sensitivity of the methods based on chaos-theory with the traditional electrophysiological ones in a case when no apparent abnormality was present as judged on the basis of this latter methodology. The PD2 was calculated from the EEG recorded in 13 healthy control subjects and in a patient who suffered a small subcortical stroke 2 years prior to the investigation and who was free of neurological symptoms at the time of recording. Compared to that seen in the control group, in the Z-score maps of the scalp distribution of the PD2, a marked asymmetry was seen and the absolute PD2 values showed a low-dimensional area in the parietal region, ipsilateral to the stroke. A relative decrease of the gamma band was found in the frequency power spectra in the same area. It is suggested that the additional information extracted from the EEG by non-linear analysis may increase the sensitivity of electrophysiological methods for detecting brain pathology.

Heart rate variability in the human transplanted heart: nonlinear dynamics and QT vs RR-QT alterations during exercise suggest a return of neurocardiac regulation in long-term recovery.

RATIONALE: Functional reinnervation of the transplanted human heart by the autonomic nervous system has not been demonstrated. A lack of autonomic control of the transplanted allograft is reflected by an increased resting heart rate, a sluggish heart rate response to dynamical exercise and a reduced heart rate variability. Recent evidence suggests that a measure of deterministic chaos in the heartbeat interval series (point correlation dimension, PD2i) is superior to the conventional power spectrum or other stochastic measures in detecting changes in the mechanism underlying heartbeat generation. METHODS: The PD2i is based on the presumption that the variability is determined and patterned, whereas the stochastic measures all assume that the variability is around a stationary mean and is noise. The PD2i reconstructs the degrees of freedom (number of independent variables) in the system that generates the time series examined, and does this irrespective of whether the system is stochastic or deterministic and is stationary in time. RESULTS: PD2i was determined for heartbeat intervals (RR, ECG digitized at 1200 Hz; supine posture) of 23 heart transplant recipients (HTR: 9 adults, 14 children; 0.04-7.7 years after transplantation) and 21 healthy control subjects (CTL; 13 adults, 8 children). The PD2i (+/-SD) averaged 5.4 +/- 0.7 for the CTL adults and 5.4 +/- 0.6 for the CTL children. Mean PD2i was reduced after transplantation to 1.1 +/- 0.1 in 6 HTRs recorded within 1 year after surgery; in one HTR recorded 2 weeks after surgery the mean PD2i was 3.7. Between 1 to 2 years PD2i was found increased in 2 of 3 subjects and between 2 to 8 years it was increased in 13 of 13, but not to control levels. In normal hearts the QT subinterval of each heartbeat cycle is associated with inotropy and the RR-QT remainder with chronotropy (i.e., the dyastolic interval during which RR is primarily regulated). To examine more closely the residual and returning heartbeat dynamics of the HTR subjects, these subinterval series were examined during mild exercise (40 to 90 Watts) and its recovery. In recent HTRs, resting QT and RR-QT were moderately reduced and modulated by exercise and recovery, but with an approximate 100 beat latency. In long-term (7-8 years) HTR subjects there was a rapid and larger response to exercise/recovery, but compared to normal the range was smaller and the complexity of the subinterval trajectories in time was simpler. CONCLUSIONS: Recurrence of low-dimensional deterministic dynamics after transplantation suggests recovery of neurocardiac control attributable to 1) reorganization of the viable intrinsic cardiac nervous system, 2) reinnervation by the extrinsic autonomic nervous system, or 3) both.

Application of chaos theory to a model biological system: evidence of self-organization in the intrinsic cardiac nervous system.

The neutral organization that determines the specific beat-to-beat pattern of cardiac behavior is expected to be demonstrated in the independent regulation of the RR intervals (chronotropy) and the corresponding QT subintervals (inotropy), as the former defines the rate of contraction and the latter has a linear negative correlation with the peak pressure inside the contracting ventricular muscles. The neurons of the isolated cardiac nervous system, many of which are located in the fat-pads of the heart, exhibit the same types of mechanical and chemical receptors and the same types of cholinergic and noradrenergic effectors as those found in the neural superstructure. In the surgically isolated and perfused rabbit heart we studied the responses of the QT and RR intervals evoked by block of coronary blood flow. We found that if we separated each RR cycle into QT and RR-QT components, then the dynamics of variation for each subinterval series often had the same fractional number of degrees of freedom (i.e., chaotic dimensions), a finding which suggests they are both regulated by the same underlying system. The ischemia/anoxia evoked transient dimensional increases and separations between the two subinterval series that, after the temporary divergence, reconverged to having the same lower value. The dimensional fluctuations occurred repeatedly and preceded or coincided with alterations in the magnitude and sign of the slope of QT vs RR-QT. We interpret the dimensional fluctuations of the two subinterval series as correlates of adaptation-dependent self-organization and reorganization in the underlying intrinsic cardiac nervous system during accumulating ischemia/anoxia. Such attempts at functional reorganization in this simple neurocardiac system may explain the transient dimensional changes in the RR intervals that precedes by 24 hrs the occurrences of fatal ventricular fibrillation in high-risk cardiac patients.

Correlation dimension changes accompanying the occurrence of the mismatch negativity and the P3 event-related potential component.

The aim of this study was to apply recently developed mathematical tools of chaos theory to the analysis of event-related potentials (ERPs) recorded in paradigms in which the mismatch negativity (MMN) and the P3 component appeared. A new method, the point correlation dimension (PD2i), was used for data analysis, which is more accurate than other algorithms for the calculation of the correlation dimension (D2), which latter is a measure of the complexity of the generator(s) responsible for producing the analyzed time series, i.e., the EEG. ERPs were recorded from Fz, Cz and Pz in 6 subjects. With respect to baseline, the PD2i decreased significantly both during the event-related potentials in which the MMN and also in which the P3 was present, but the pattern and magnitude of this decrease was different between these two situations. The pattern of PD2i changes during the occurrence of deviant stimuli eliciting the MMN suggests the presence of a frontal MMN generator. The conspicuous PD2i decrease during the occurrence of the P3 wave may support the "context closure" hypothesis concerning its functional significance.

The role of the central nervous system in sudden cardiac death: heartbeat dynamics in conscious pigs during coronary occlusion, psychologic stress and intracerebral propranolol.

Electrocardiograms were recorded in conscious pigs during psychological stress (touching), left anterior descending coronary artery occlusion, intracerebral levopropranolol (0.05 mg/kg), and their respective controls. The R-R intervals were evaluated with both a deterministic measure (point correlation dimension) and a stochastic one (mean). Only the deterministic measure was sensitive to between-subject responses (P < .01), whereas both measures were sensitive to within-subject changes. The results are explained by a deterministic model of heartbeat generation.

The point correlation dimension: performance with nonstationary surrogate data and noise.

The dynamics of many biological systems have recently been attributed to low-dimensional chaos instead of high-dimensional noise, as previously though. Because biological data are invariably nonstationary, especially when recorded over a long interval, the conventional measures of low-dimensional chaos (e.g., the correlation dimension algorithms) cannot be applied. A new algorithm, the point correction dimension (PD2i) was developed to deal with this fundamental problem. In this article we describe the details of the algorithm and show that the local mean PD2i will accurately track dimension in nonstationary surrogate data.

Low-dimensional chaos in biological systems.

During the past five years general rules have been developed for the application of chaos theory to biology and medicine, which enable investigators to avoid the pitfalls that invalidated and trivialized many earlier results. The importance of biological chaos is that the variables governing the spatial and temporal geometries of the system may be few in number, fractional in dimension, and thus enable low-energy control with complex deterministic consequences. The complexity of control inherent in chaotic systems may be important in the dynamics of gene expression and translation. Extending these ideas may lead to completely novel ways to modulate protein production by introducing simple pulses at critical times or places.

Chaos and physiology: deterministic chaos in excitable cell assemblies.

In this review we examined the emerging science of deterministic chaos (nonlinear systems theory) and its application to selected physiological systems. Although many of the popular images of fractals represent fascination and beauty that by analogy corresponds to nature as we see it, the question remains as to its ultimate meaning for physiological processes. It was our intent to help clarify this somewhat popular, somewhat obscure area of nonlinear dynamics in the context of an ever-changing procedural base. We examined not only the basic concepts of chaos, but also its applications ranging from observations in single cells to the complexity of the EEG. We have not suggested that nonlinear dynamics will answer all of our questions; however, we did attempt to illustrate ways in which this approach may help us to answer new questions and to rearticulate old ones. Chaos is revolutionary in that the overall approach requires us to adopt a different frame of reference which, at times, may move us away from previous concerns and methods of data analysis. In sections I-IV, we summarized the nonlinear dynamics approach and described its application to physiology and neural systems. First, we presented a general overview of the application of nonlinear dynamical techniques to neural systems. We discussed the manner in which even apparently simple deterministic systems can behave in an unpredictable manner. Second, we described the principles of nonlinear dynamical systems including the derived analytical techniques. We now see a variety of procedures for delineating whether frenetic chaotic behavior results from a nonlinear dynamical system with a few degrees of freedom, or whether it is caused by an infinite number of variables, i.e., noise. Third, we approached the applications of nonlinear procedures to the cardiovascular systems and to the neurosciences. In terms of time series, we described initial studies which applied the now "traditional" measures of dimensionality (e.g., based on the algorithm by Grassberger and Procaccia) and information change (e.g., Lyapunov exponents). Examples include our own work and that of Pritchard et al., demonstrating that the dynamics of neural mass activity reflect psychopathological states. Today, however, the trend has expanded to include the use of surrogate data and statistical null hypotheses testing to examine whether a given time series can be considered different from that of white or colored noise (cf. Ref. 262). One of the most important potential applications is that of quantifying changes in nonlinear dynamics to predict future states of the system.(ABSTRACT TRUNCATED AT 400 WORDS)

Conventional heart rate variability analysis of ambulatory electrocardiographic recordings fails to predict imminent ventricular fibrillation.

OBJECTIVES: The purpose of this report was to study heart rate variability in Holter recordings of patients who experienced ventricular fibrillation during the recording. BACKGROUND: Decreased heart rate variability is recognized as a long-term predictor of overall and arrhythmic death after myocardial infarction. It was therefore postulated that heart rate variability would be lowest when measured immediately before ventricular fibrillation. METHODS: Conventional indexes of heart rate variability were calculated from Holter recordings of 24 patients with structural heart disease who had ventricular fibrillation during monitoring. The control group consisted of 19 patients with coronary artery disease, of comparable age and left ventricular ejection fraction, who had nonsustained ventricular tachycardia but no ventricular fibrillation. RESULTS: Heart rate variability did not differ between the two groups, and no consistent trends in heart rate variability were observed before ventricular fibrillation occurred. CONCLUSIONS: Although conventional heart rate variability is an independent long-term predictor of adverse outcome after myocardial infarction, its clinical utility as a short-term predictor of life-threatening arrhythmias remains to be elucidated.

Neurocardiology. Brain mechanisms underlying fatal cardiac arrhythmias.

Chaos theory may have a widespread application in medicine, from the analysis of protein structure at one end of the spectrum to fetal monitoring and the measurement of aging on the other. This application is especially on firm ground in cardiology, where it is simple and precise for the experimenter, and a primer exists for the clinician. As just presented, the point-correlation-dimension analysis of heartbeat variability is able to characterize the patterns of low-dimensional chaos produced by the heartbeat generator, a mechanism that is a composite of the voltage-dependent, neurotransmitter-dependent, and circulation-dependent ionic conductances that are all located in the myocardium where the heartbeat is formed. Most importantly, this deterministic measure of heartbeat variability is able to predict imminent lethal arrhythmogenesis with an accuracy that the more familiar stochastic measures do not have. This may arise because the PD2 is sensitive to the net degrees of freedom of the heartbeat generator, which, under the influence of the nerves and the coronary circulation, can be shifted so the resulting dynamics cause the initiation of lethal arrhythmogenesis. Application of chaos theory in neurology may be equally fruitful because the deterministic measures can discriminate among neuronal firing patterns, reveal subtle changes in brain waves, and be related to higher cognitive processes. Psychiatry also seems quite likely to benefit more and more from the application because the algorithms can discriminate schizophrenic brain functions from normal ones. Thus it can be expected that future applications will enable observation of biologic processes all along the brain-heart axis by which lethal ventricular fibrillation is regulated and perhaps even caused by its determination of the heartbeat dynamics. The subfield of neurocardiology has come a long way since it was first positively identified as an important research area. It took a lot of experiments to show how lethal cardiac arrhythmogenesis is involved with cerebral activities delivered over autonomic pathways. Although we do not yet fully understand the causal mechanism of lethal arrhythmogenesis, we may be getting close. It will become increasingly imperative for the clinical neurologist, along with the cardiologist, to understand the importance of neurocardiology in medical therapy and for the neurologist, along with the cardiologist and psychiatrist, to understand its importance in preventive treatments.

A reduction in the correlation dimension of heartbeat intervals precedes imminent ventricular fibrillation in human subjects.

Reduced reflexive control of heartbeat intervals occurs with advanced heart disease and is an independent risk factor for mortality. Based on a previous study of experimental myocardial infarction in pigs, we hypothesized that a deterministic measure of heartbeat dynamics, the correlation dimension of R-R intervals (D2), may be a better predictor of risk than a stochastic measure, such as the standard deviation (SD). We determined the point estimates of the heartbeat D2 (i.e., PD2s) in Holter electrocardiographic recordings from 11 high-risk patients who manifested ventricular fibrillation (VF) during the recording and in high-risk controls having only nonsustained ventricular tachycardia (14 patients) or premature ventricular complexes (13 patients). We found that PD2 reduction (i.e., PD2s < 1.2) precedes lethal arrhythmias by hours, but is not reduced in high-risk controls (p < 0.001; sensitivity, 91%; specificity, 85%). Heartbeat SD did not discriminate among the patients. Thus PD2 of heartbeat intervals may provide an important diagnostic test and early warning sign of VF.

Low-dimensional chaos maps learning in a model neuropil (olfactory bulb).

Quantification of a chaotic system can be made by calculating the correlation dimension (D2) of the data that the system generates (Packard et al., 1980). The D2 algorithm, however, requires stationarity of the generator, a feature that biological data rarely reflect (Mayer-Kress et al., 1988). So we developed the "point correlation dimension" (PD2), an algorithm that accurately tracks D2 in linked data of different dimensions (Carpeggiani et al., 1991). We now present a mathematical argument that, for stationary data, individual PD2s converge to D2 and we demonstrate that the algorithm rejects contributions made by bursts of noise. Data were obtained from the surface of the olfactory bulb of the conscious rabbit (64 electrodes, 640 Hz each, 1.3 sec epochs) before and after presentation of a novel or habituated odor. D2 could be calculated in only 1 of 10 novel-odor trials, whereas PD2 could be calculated in all. Both algorithms indicated that a novel odor evokes a spatially uniform dimensional increase. The PD2 uniquely exhibited the dimensional decreases that occur during inspiration and the gradients of mean dimension present during the nonstimulated control state. These control gradients remained unchanged without odor experience, but showed spatially specific PD2 increases following odor habituation. It is interpreted that, 1) the PD2 is sensitive, accurate, and appropriate for dimensional assessment of biological data, 2) that during analysis of unfamiliar information a single global process is transiently evoked in the neuropil, and 3) after experience multiple spatially specific processes tonically map the sites of learning.