Entropy, as it relates to dynamical systems, is the rate of information production. Methods for estimation of the entropy of a system represented by a time series are not, however, well suited to analysis of the short and noisy data sets encountered in cardiovascular and other biological studies. Pincus introduced approximate entropy (ApEn), a set of measures of system complexity closely related to entropy, which is easily applied to clinical cardiovascular and other time series. ApEn statistics, however, lead to inconsistent results. We have developed a new and related complexity measure, sample entropy (SampEn), and have compared ApEn and SampEn by using them to analyze sets of random numbers with known probabilistic character. We have also evaluated cross-ApEn and cross-SampEn, which use cardiovascular data sets to measure the similarity of two distinct time series. SampEn agreed with theory much more closely than ApEn over a broad range of conditions. The improved accuracy of SampEn statistics should make them useful in the study of experimental clinical cardiovascular and other biological time series.

Physiological time-series analysis using approximate entropy and sample entropy

Heart rate variability (HRV) is a reliable reflection of the many physiological factors modulating the normal rhythm of the heart. In fact, they provide a powerful means of observing the interplay between the sympathetic and parasympathetic nervous systems. It shows that the structure generating the signal is not only simply linear, but also involves nonlinear contributions. Heart rate (HR) is a nonstationary signal; its variation may contain indicators of current disease, or warnings about impending cardiac diseases. The indicators may be present at all times or may occur at random-during certain intervals of the day. It is strenuous and time consuming to study and pinpoint abnormalities in voluminous data collected over several hours. Hence, HR variation analysis (instantaneous HR against time axis) has become a popular noninvasive tool for assessing the activities of the autonomic nervous system. Computer based analytical tools for in-depth study of data over daylong intervals can be very useful in diagnostics. Therefore, the HRV signal parameters, extracted and analyzed using computers, are highly useful in diagnostics. In this paper, we have discussed the various applications of HRV and different linear, frequency domain, wavelet domain, nonlinear techniques used for the analysis of the HRV.

Heart rate variability: a review

Traditional approaches to measuring the complexity of biological signals fail to account for the multiple time scales inherent in such time series. These algorithms have yielded contradictory findings when applied to real-world datasets obtained in health and disease states. We describe in detail the basis and implementation of the multiscale entropy (MSE) method. We extend and elaborate previous findings showing its applicability to the fluctuations of the human heartbeat under physiologic and pathologic conditions. The method consistently indicates a loss of complexity with aging, with an erratic cardiac arrhythmia (atrial fibrillation), and with a life-threatening syndrome (congestive heart failure). Further, these different conditions have distinct MSE curve profiles, suggesting diagnostic uses. The results support a general "complexity-loss" theory of aging and disease. We also apply the method to the analysis of coding and noncoding DNA sequences and find that the latter have higher multiscale entropy, consistent with the emerging view that so-called "junk DNA" sequences contain important biological information.

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Multiscale entropy analysis of biological signals

Approximate entropy (ApEn) is a recently developed statistic quantifying regularity and complexity that appears to have potential application to a wide variety of physiological and clinical time-series data. The focus here is to provide a better understanding of ApEn to facilitate its proper utilization, application, and interpretation. After giving the formal mathematical description of ApEn, we provide a multistep description of the algorithm as applied to two contrasting clinical heart rate data sets. We discuss algorithm implementation and interpretation and introduce a general mathematical hypothesis of the dynamics of a wide class of diseases, indicating the utility of ApEn to test this hypothesis. We indicate the relationship of ApEn to variability measures, the Fourier spectrum, and algorithms motivated by study of chaotic dynamics. We discuss further mathematical properties of ApEn, including the choice of input parameters, statistical issues, and modeling considerations, and we conclude with a section on caveats to ensure correct ApEn utilization.

https://www.physiology.org/doi/pdf/10.1152/ajpheart.1994.266.4.H1643

Physiological time-series analysis: What does regularity quantify?

During the last decade, the GH axis has become the compelling focus of remarkably active and broad-ranging basic and clinical research. Molecular and genetic models, the discovery of human GHRH and its receptor, the cloning of the GHRP receptor, and the clinical availability of recombinant GH and IGF-I have allowed surprisingly rapid advances in our knowledge of the neuroregulation of the GH-IGF-I axis in many pathophysiological contexts. The complexity of the GHRH/somatostatin-GH-IGF-I axis thus commends itself to more formalized modeling (154, 155), since the multivalent feedback-control activities are difficult to assimilate fully on an intuitive scale. Understanding the dynamic neuroendocrine mechanisms that direct the pulsatile secretion of this fundamental growth-promoting and metabolic hormone remains a critical goal, the realization of which is challenged by the exponentially accumulating matrix of experimental and clinical data in this arena. To the above end, we review here the pathophysiology of the GHRH somatostatin-GH-IGF-I feedback axis consisting of corresponding key neurotransmitters, neuromodulators, and metabolic effectors, and their cloned receptors and signaling pathways. We propose that this system is best viewed as a multivalent feedback network that is exquisitely sensitive to an array of neuroregulators and environmental stressors and genetic restraints. Feedback and feedforward mechanisms acting within the intact somatotropic axis mediate homeostatic control throughout the human lifetime and are disrupted in disease. Novel effectors of the GH axis, such as GHRPs, also offer promise as investigative probes and possible therapeutic agents. Further understanding of the mechanisms of GH neuroregulation will likely allow development of progressively more specific molecular and clinical tools for the diagnosis and treatment of various conditions in which GH secretion is regulated abnormally. Thus, we predict that unexpected and enriching insights in the domain of the neuroendocrine pathophysiology of the GH axis are likely be achieved in the succeeding decades of basic and clinical research.

Pathophysiology of the neuroregulation of growth hormone secretion in experimental animals and the human

Techniques to determine changing system complexity from data are evaluated. Convergence of a frequently used correlation dimension algorithm to a finite value does not necessarily imply an underlying deterministic model or chaos. Analysis of a recently developed family of formulas and statistics, approximate entropy (ApEn), suggests that ApEn can classify complex systems, given at least 1000 data values in diverse settings that include both deterministic chaotic and stochastic processes. The capability to discern changing complexity from such a relatively small amount of data holds promise for applications of ApEn in a variety of contexts.

https://www.pnas.org/content/pnas/88/6/2297.full.pdf

Approximate entropy as a measure of system complexity.

Aging and the complexity of cardiovascular dynamics.

Gender- and age-related differences in heart rate dynamics: are women more complex than men?

Approximate entropy (ApEn) is a recently formulated family of parameters and statistics quantifying regularity (orderliness) in serial data, with developments within theoretical mathematics as well as numerous applications to multiple biological contexts. We discuss the motivation for ApEn development, from the study of inappropriate application of dynamical systems (complexity) algorithms to general time-series settings. ApEn is scale invariant and model independent, evaluates both dominant and subordinant patterns in data, and discriminates series for which clear feature recognition is difficult. ApEn is applicable to systems with at least 50 data points and to broad classes of models: it can be applied to discriminate both general classes of correlated stochastic processes, as well as noisy deterministic systems. Moreover, ApEn is complementary to spectral and autocorrelation analyses, providing effective discriminatory capability in instances in which the aforementioned measures exhibit minimal distinctions. Representative ApEn applications to human aging studies, based on both heart rate and endocrinologic (hormonal secretory) time series, are featured. Heart rate (HR) studies include gender- and age-related changes in HR dynamics in older subjects, and analyses of “near-SIDS” infants. Endocrinologic applications establish clear quantitative changes in joint LH-testosterone secretory dynamics in older versus younger men (a “partial male menopause”), via cross-ApEn, a related two-variable asynchrony formulation; a disruption in LH-FSH-NPT (penile tumescence) synchrony in older subjects; and changes in LH-FSH secretory dynamics across menopause. The capability of ApEn to assess relatively subtle disruptions, typically found earlier in the history of a subject than mean and variance changes, holds the potential for enhanced preventative and earlier interventionist strategies.

Steven M. Pincus

Papers

Approximate entropy as a measure of system complexity.

Physiological time-series analysis: What does regularity quantify?

Aging and the complexity of cardiovascular dynamics.

Gender- and age-related differences in heart rate dynamics: are women more complex than men?

Assessing serial irregularity and its implications for health.