The Shannon entropy and quantitative time irreversibility (qTIR) are statistical quantifiers that are widely used for characterizing complex processes. However, the differences and associations between them have not been subjected to detailed investigation. In this Letter, we report a comparative analysis of the Shannon entropy and qTIR using model series and real-world heartbeats. We find that the permutation-based Shannon entropy (PEn) and time irreversibility (PYs) detect nonlinearities in the model series differently according to the surrogate theory. Moreover, PEn and PYs, based on either the original or the equal-value permutation, give contradictory results for congestive heart failure cases and healthy young and elderly heartbeats. PEn quantifies the complexity by calculating the amount of mean information, whereas PYs measures the probabilistic differences among symmetric nonequilibrium distributions, and these yield different or even contradictory outcomes. Our findings demonstrate the statistical associations between the Shannon entropy and qTIR, contribute to more reliable elucidation of the nonlinear dynamics of heartbeats, and improve our understanding of the complexity and nonequilibrium nature of complex systems.

Schizophrenia is a neuropsychiatric disease that affects the nonlinear dynamics of brain activity. The primary objective of this study was to explore the complexity of magnetoencephalograms (MEG) in patients with schizophrenia. We combined a multiscale method and weighted permutation entropy to characterize MEG signals from 19 schizophrenia patients and 16 healthy controls. When the scale was larger than 42, the MEG signals of schizophrenia patients were significantly more complex than those of healthy controls ($p<0.004$). The difference in complexity between patients with schizophrenia and the controls was strongest in the frontal and occipital areas ($p<0.001$), and there was almost no difference in the central area. In addition, the results showed that the dynamic range of MEG complexity is wider in healthy individuals than in people with schizophrenia. Overall, the multiscale weighted permutation entropy method reliably quantified the complexity of MEG from schizophrenia patients, contributing to the development of potential magnetoencephalographic biomarkers for schizophrenia.

Schizophrenia is a serious mental disorder. Network analysis of magnetoencephalogram signals may help to identify potential biomarkers of schizophrenia. The goal of this investigation was to identify potential biomarkers in the magnetoencephalogram signals of patients with schizophrenia, global brain connectivity measures was used for emotion recognition in discriminating the patients from controls. First, we employed a mutual information method to explore the topological characteristics of the brain network in patients with schizophrenia among different frequency bands in response to four different stimulus conditions. Second, multidimensional cross-recurrence quantification analysis was performed to investigate the differences in dynamic coupling among different frequencies of brain magnetic waves in patients with schizophrenia in response to four different stimulus conditions, as the major novel contribution of our study. We found that the differences in topological features of the brain network appear in different frequency bands under different stimulus conditions. The differences are evident in the alpha 1 (8-10 Hz) and beta (13-30 Hz) frequency bands in response to negative stimuli, in the alpha 1 (8-10 Hz) frequency band in response to positive stimuli, and in the theta (4-8 Hz) and alpha 1 (8-10 Hz) frequency bands in response to neutral and gray-cross stimuli. In addition, differences in dynamic coupling among pairs of frequency bands were the most prominent in response to positive stimuli. The characteristics identified by our methods may be potential markers of schizophrenia present in magnetoencephalogram data, which can facilitate the clinical identification of schizophrenia patients. Our method provides a comprehensive perspective of brain networks in patients with schizophrenia and has practical applications for the clinical diagnosis of this disease.