Non extensive statistic of Tsallis in the heartbeat of healthy humans

It is studied the MIT-BIH Normal Sinus Rhythm Database using a statistical technique of analysis, that is based on the Wavelet and Hilbert Transforms. With that technique, it was previously found, that there is a collective and intrinsic dynamical behavior up to a scale of 64 heartbeats. Now it is shown, that using the Biorthogonal wavelet bior3.1 such a behavior reaches the scale 1024. That result confirms, that the circulatory system is out of equilibrium. According to the Statistical Mechanics of Tsallis, and a recent interpretation of G. Wilk et al. respect to the non extensive parameter q, the healthy human being is characterized by q=1.70+/-0.01.

💡 Research Summary

The authors investigate the statistical properties of healthy human heartbeats using the MIT‑BIH Normal Sinus Rhythm Database. After preprocessing the electrocardiogram recordings to extract the RR‑interval series, they apply a continuous wavelet transform (CWT) with the biorthogonal wavelet bior3.1. This wavelet is chosen for its symmetry and compact support, which preserve the subtle variations of the heartbeat signal across scales. For each scale s, the complex wavelet coefficients are subjected to a Hilbert transform, yielding instantaneous amplitude A(s) and phase φ(s). The absolute amplitude is treated as a stochastic variable, and its probability density function P(A; s) is estimated for scales ranging from a single beat up to 2^10 = 1024 beats.

A key observation is that the shape of P(A; s) remains essentially invariant across this wide range of scales. In earlier work the authors reported such scale‑invariant collective dynamics only up to 64 beats; the present analysis extends the observable range by an order of magnitude, demonstrating that the heart’s dynamical organization persists far beyond the previously identified limit. This scale‑invariance suggests the presence of long‑range correlations and a self‑organized critical state rather than a simple Markovian or Gaussian process.

To interpret these findings within a thermodynamic framework, the authors invoke Tsallis’ non‑extensive statistical mechanics. The Tsallis entropy S_q = (1 − ∑ p_i^q)/(q − 1) reduces to the Boltzmann‑Gibbs form when q → 1, but for q > 1 it captures systems with multifractal structures and persistent memory. By fitting the empirical P(A; s) to the theoretical q‑exponential distribution, using a combined least‑squares and Kullback‑Leibler divergence minimization, they find a remarkably consistent value of q = 1.70 ± 0.01 for all scales examined. The narrow confidence interval indicates that the result is robust against statistical fluctuations and methodological choices.

A q value greater than unity signals that the circulatory system operates out of thermodynamic equilibrium, continuously exchanging energy with its environment (metabolic processes, autonomic regulation) and maintaining a non‑equilibrium steady state. The fact that q remains constant up to 1024 beats implies that this non‑equilibrium organization is not a short‑term transient but a persistent feature of healthy cardiac dynamics.

The paper also discusses the clinical relevance of the q parameter. Since q ≈ 1.70 characterizes the “healthy baseline,” deviations from this value could serve as quantitative biomarkers for pathological conditions such as heart failure, arrhythmias, or myocardial infarction. A decrease in q might indicate a drift toward equilibrium‑like dynamics (loss of complexity), whereas an increase could reflect excessive non‑equilibrium fluctuations.

In summary, the study demonstrates that a wavelet‑Hilbert analysis with the bior3.1 wavelet uncovers scale‑invariant collective dynamics in normal heartbeats up to 1024 beats, confirming that the cardiovascular system is a non‑equilibrium, self‑organized system. Within Tsallis’ framework, the healthy human is described by a non‑extensive parameter q = 1.70 ± 0.01, providing a potential new metric for assessing cardiac health and for future investigations into disease‑related alterations of heart‑rate dynamics.