Entropy of Entropy: Measurement of Dynamical Complexity for Biological Systems

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.

[1]  A. N. Mamelak,et al.  Long-range temporal correlations in the spontaneous spiking of neurons in the hippocampal-amygdala complex of humans , 2005, Neuroscience.

[2]  Chengyu Liu,et al.  Multiscale Entropy Analysis of the Differential RR Interval Time Series Signal and Its Application in Detecting Congestive Heart Failure , 2017, Entropy.

[3]  Stephan Hartmann,et al.  Probabilities in physics , 2011 .

[4]  J. Shieh,et al.  Complexity of intracranial pressure correlates with outcome after traumatic brain injury. , 2012, Brain : a journal of neurology.

[5]  Koichi Takahashi,et al.  Antipsychotics reverse abnormal EEG complexity in drug-naive schizophrenia: A multiscale entropy analysis , 2010, NeuroImage.

[6]  Tad Hogg,et al.  Complexity and adaptation , 1986 .

[7]  Murray Gell-Mann,et al.  What Is Complexity , 2002 .

[8]  C. Peng,et al.  Frailty and the degradation of complex balance dynamics during a dual-task protocol. , 2009, The journals of gerontology. Series A, Biological sciences and medical sciences.

[9]  Men-Tzung Lo,et al.  Revealing the brain's adaptability and the transcranial direct current stimulation facilitating effect in inhibitory control by multiscale entropy , 2014, NeuroImage.

[10]  Gustavo Deco,et al.  Bottom up modeling of the connectome: Linking structure and function in the resting brain and their changes in aging , 2013, NeuroImage.

[11]  Chung-Kang Peng,et al.  Adaptive Data Analysis of Complex Fluctuations in physiologic Time Series , 2009, Adv. Data Sci. Adapt. Anal..

[12]  S. MacDonald,et al.  Neuroscience and Biobehavioral Reviews Review Moment-to-moment Brain Signal Variability: a next Frontier in Human Brain Mapping? , 2022 .

[13]  B. Pompe,et al.  Permutation entropy: a natural complexity measure for time series. , 2002, Physical review letters.

[14]  Yi-Cheng Zhang.  Complexity and 1/f noise. A phase space approach , 1991 .

[15]  Ranjan Bose,et al.  Alternate measure of information useful for DNA sequences. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  S M Pincus,et al.  Approximate entropy as a measure of system complexity. , 1991, Proceedings of the National Academy of Sciences of the United States of America.

[17]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[18]  Weiting Chen,et al.  Measuring complexity using FuzzyEn, ApEn, and SampEn. , 2009, Medical engineering & physics.

[19]  A. Porta,et al.  K-nearest-neighbor conditional entropy approach for the assessment of the short-term complexity of cardiovascular control , 2013, Physiological measurement.

[20]  P. Tu,et al.  Complexity of spontaneous BOLD activity in default mode network is correlated with cognitive function in normal male elderly: a multiscale entropy analysis , 2013, Neurobiology of Aging.

[21]  Murray Gell-Mann,et al.  What is complexity? Remarks on simplicity and complexity by the Nobel Prize-winning author of The Quark and the Jaguar , 1995, Complex..

[22]  Melanie Mitchell,et al.  Complexity - A Guided Tour , 2009 .

[23]  Yi-Lwun Ho,et al.  Multi-scale symbolic entropy analysis provides prognostic prediction in patients receiving extracorporeal life support , 2014, Critical Care.

[24]  Claude E. Shannon,et al.  Prediction and Entropy of Printed English , 1951 .

[25]  Vinzenz von Tscharner,et al.  Multi-scale transitions of fuzzy sample entropy of RR-intervals and their phase-randomized surrogates: A possibility to diagnose congestive heart failure , 2017, Biomed. Signal Process. Control..

[26]  L. Lenert,et al.  Utility of B-type natriuretic peptide in the diagnosis of congestive heart failure in an urgent-care setting. , 2001, Journal of the American College of Cardiology.

[27]  Luiz Otavio Murta Junior,et al.  Multiscale entropy-based methods for heart rate variability complexity analysis , 2015 .

[28]  Qin Wei,et al.  Multivariate Multiscale Entropy Applied to Center of Pressure Signals Analysis: An Effect of Vibration Stimulation of Shoes , 2012, Entropy.

[29]  J. Richman,et al.  Physiological time-series analysis using approximate entropy and sample entropy. , 2000, American journal of physiology. Heart and circulatory physiology.

[30]  Madalena Costa,et al.  Multiscale entropy analysis of biological signals. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  Madalena Costa,et al.  Broken asymmetry of the human heartbeat: loss of time irreversibility in aging and disease. , 2005, Physical review letters.

[32]  Madalena Costa,et al.  Multiscale entropy analysis of complex physiologic time series. , 2002, Physical review letters.

[33]  Jun Wang,et al.  A dynamic marker of very short-term heartbeat under pathological states via network analysis , 2014 .