Entropy of Entropy: Measurement of Dynamical Complexity for Biological Systems

Abstract

Healthy systems exhibit complex dynamics on the changing of information embedded in physiologic signals on multiple time scales that can be quantified by employing multiscale entropy (MSE) analysis. Here, we propose a measure of complexity, called entropy of entropy (EoE) analysis. The analysis combines the features of MSE and an alternate measure of information, called superinformation, useful for DNA sequences. In this work, we apply the hybrid analysis to the cardiac interbeat interval time series. We find that the EoE value is significantly higher for the healthy than the pathologic groups. Particularly, short time series of 70 heart beats is sufficient for EoE analysis with an accuracy of 81% and longer series of 500 beats results in an accuracy of 90%. In addition, the EoE versus Shannon entropy plot of heart rate time series exhibits an inverted U relationship with the maximal EoE value appearing in the middle of extreme order and disorder.

Is an Entropy-Based Approach Suitable for an Understanding of the Metabolic Pathways of Fermentation and Respiration?

“…The achievement of a stationary phase of lower dissipation and thermodynamic stability could be a key interpretation to understand the robustness and redundancy of neoplastic systems associated with greater gene modification and Shannon entropy [16,17]. As a result, there is an increase of the possible number of responses of the system to environment changes or equivalently the concept of Darwinian fitness [43,44].…”

“…Ratios between the rate of internal entropy associated to mass transport for lactic fermentation and the corresponding one associated to respiration. (a) ri D ferm/ri D resp expressed by Equation (16); (b) ri r ferm/ri r resp given in Equation (17). Figure 10 displays the ratios re Q ferm/re Q resp and re exch ferm/re exch resp derived from the REEDP contributions.…”

“…As a result, there is a strict relationship between thermodynamic irreversibility and information expressed as the dialectic between Clausius entropy, Boltzmann entropy and informational Shannon entropy [16][17][18][19].…”

State Change Probability: A Measure of the Complexity of Cardiac RR Interval Time Series Using Physiological State Change with Statistical Hypothesis Testing

“…Nevertheless, most of these measures show higher values for AF than NSR RR interval time series, which is closer to a measure of irregularity than complexity, and, thus, fail to reflect their adaptability. Comparatively, the multiscale entropy (MSE) [2,4], MSE-based method [21][22][23], and entropy of entropy (EoE) [24] show that the RR interval time series from NSR subjects exhibit a higher complexity than those from CHF and AF subjects. The MSE analysis evaluates the information content of a time series across multiple temporal scales, [2][3][4] while the EoE method quantifies the variation of information within a time series.…”

“…The MSE analysis evaluates the information content of a time series across multiple temporal scales, [2][3][4] while the EoE method quantifies the variation of information within a time series. [24] The MSE analysis computes the sample entropy of a coarse-grained time series over multiple temporal scales. Conceptually, sample entropy evaluates the difference between the number of repeating patterns of m length and the number of repeating patterns of m + 1 length [25], where m represents the embedding dimensions-usually 2.…”