Measuring synchronization in coupled simulation and coupled cardiovascular time series: A comparison of different cross entropy measures

Synchronization provides an insight into underlying the interaction mechanisms among the bivariate time series and has recently become an increasing focus of interest. In this study, we proposed a new cross entropy measure, named cross fuzzy measure entropy (C-FuzzyMEn), to detect the synchronization of the bivariate time series. The performances of C-FuzzyMEn, as well as two existing cross entropy measures, i.e., cross sample entropy (C-SampEn) and cross fuzzy entropy (C-FuzzyEn), were first tested and compared using three coupled simulation models (i.e., coupled Gaussian noise, coupled MIX(p) and coupled Henon model) by changing the time series length, the threshold value for entropy and the coupling degree. The results from the simulation models showed that compared with C-SampEn, C-FuzzyEn and C-FuzzyMEn had better statistical stability and compared with C-FuzzyEn, C-FuzzyMEn had better discrimination ability. These three measures were then applied to a cardiovascular coupling problem, synchronization analysis for RR and pulse transit time (PTT) series in both the normal subjects and heart failure patients. The results showed that the heart failure group had lower cross entropy values than the normal group for all three cross entropy measures, indicating that the synchronization between RR and PTT time series increases in the heart failure group. Further analysis showed that there was no significant difference between the normal and heart failure groups for C-SampEn (normal 2.13 ± 0.37 vs. heart failure 2.07 ± 0.16, P = 0.36). However, C-FuzzyEn had significant difference between two groups (normal 1.42 ± 0.25 vs. heart failure 1.31 ± 0.12, P < 0.05). The statistical difference was larger for two groups when performing C-FuzzyMEn analysis (normal 2.40 ± 0.26 vs. heart failure 2.15 ± 0.13, P < 0.01).