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We introduce a generalization of multiscale entropy (MSE) analysis. The method is termed MSE_{n}_{μ}_{σ}_{σ}

The trajectories of the human heartbeat constitute an object of scientific, as well as literary, inquiry. Fluctuations in cardiac interbeat intervals (often termed heart rate variability) are regulated by the autonomic (involuntary) nervous system, which plays a fundamental role in integrative physiologic control. Heart rate time series, therefore, provide a unique window into the status of the cardiovascular system in health and disease and, more broadly, into the entirety of physiologic function [

From a dynamical perspective, cardiac interbeat interval time series pose major challenges to quantitative analysis. These signals are typically non-stationary, non-linear and exhibit complex fluctuation patterns over a wide range of time scales. The multiscale entropy method (MSE) [

The MSE method employs an entropy measure to quantify the degree of unpredictability of time series derived from the original signal by an operation called coarse-graining. This operation consists of dividing the original signal ({_{i}

Intuitively, by quantifying the level of disorder at all relevant levels of resolution of a signal, the MSE method yields a measure of its complexity. An unaddressed question is whether the coarse-graining procedure, itself, which uses a single property of the data, _{n}_{μ}

Here, we implement the MSE_{σ}

We test the hypothesis that, under baseline (“free-running”) conditions, the heartbeat volatility time series from healthy young subjects are more complex than those of healthy older subjects, which, in turn, are more complex than those from patients with heart failure.

Consider a time series ({_{i}_{n}

We analyzed cardiac interval (RR) time series derived from approximately 24 h continuous electrocardiographic (ECG) Holter monitor recordings of 26 ostensibly healthy young subjects (13 men and 13 women, aged (mean

Datasets were filtered to exclude artifacts, premature ventricular complexes, and missed beat detections. The algorithm is available at _{μ}

_{σ}^{2}) for the computation of sample entropy as described in [

This paper introduces a generalization of the multiscale entropy (MSE_{n}_{μ}_{σ}

Traditional fractal analysis [_{σ}

A key physiologic finding is that the multiscale complexity of the volatility, not only of the mean heart rate [

Mathematical models purporting to capture the dynamics of healthy heartbeat variability should account for the observed multiscale volatility and for its degradation with aging and disease. Our method also opens up inquiries into the use of MSE methodology to probe properties of the signals related to higher moments (e.g., the third moment, _{n}_{n}_{n}

We gratefully acknowledge support from G. Harold and Leila Y. Mathers Charitable Foundation; the James S. McDonnell Foundation; the National Institutes of Health grants K99/R00 AG030677, R01GM104987 and R24HL114473; and the Wyss Institute for Biologically Inspired Engineering.

The authors contributed equally to this manuscript. Both authors have read and approved the final manuscript.

The authors declare no conflict of interest.

Top: Cardiac interbeat interval (RR) time series from a healthy 20 year-old subject (left) and a 53 year-old patient with congestive heart failure (right). Middle and bottom: Variance of the RR interval time series calculated in a 20 (middle) and 40 (bottom) data point moving window. The horizontal axes are the same for all plots.

Multiscale entropy (MSE)_{σ}_{μ}