Heart rate variability (HRV) contains important information about the modulation of the cardiovascular system. Various methods of nonlinear dynamics (e.g., estimating Lyapunov exponents) and complexity measures (e.g., correlation dimension or entropies) have been applied to HRV analysis. Permutation entropy, which was proposed recently, has been widely used in many fields due to its conceptual and computational simplicity. It maps a time series onto a symbolic sequence of permutation ranks. The original permutation entropy assumes the time series under study has a continuous distribution, thus equal values are rare and can be ignored by ranking them according to their order of emergence, or broken by adding small random perturbations to ensure every symbol in a sequence is different. However, when the observed time series is digitized with lower resolution leading to a greater number of equal values, or the equalities represent certain characteristic sequential patterns of the system, it may not be rational to simply ignore or break them. In the present paper, a modified permutation entropy is proposed that, by mapping the equal value onto the same symbol (rank), allows for a more accurate characterization of system states. The application of the modified permutation entropy to the analysis of HRV is investigated using clinically collected data. Results show that modified permutation entropy can greatly improve the ability to distinguish the HRV signals under different physiological and pathological conditions. It can characterize the complexity of HRV more effectively than the original permutation entropy.

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This paper proposes that a multiscale multifractality (MSMF) method be adopted for the spatiotemporal analysis of 12-lead ECG. By using this method, the authors find that, in some frequency range, 12-lead ECG has a more complex fractal structure, and the position of the largest singularity strength range delta alpha is not relying on the data length but on the scale factor. By determining the inflexion, the MSMF proves to be more sensitive in displaying the trend that the singularity strength range delta alpha of human ECG decreases with human aging.

A broad range of complex physical and biological systems exhibits scaling laws. The human language is a complex system of words organization. Studies of written texts have revealed intriguing scaling laws that characterize the frequency of words occurrence, rank of words, and growth in the number of distinct words with text length. While studies have predominantly focused on the language system in its written form, such as books, little attention is given to the structure of spoken language. Here we investigate a database of spoken language transcripts and written texts, and we uncover that words organization in both spoken language and written texts exhibits scaling laws, although with different crossover regimes and scaling exponents. We propose a model that provides insight into words organization in spoken language and written texts, and successfully accounts for all scaling laws empirically observed in both language forms.

Traditional methods for nonlinear dynamic analysis, such as correlation dimension, Lyapunov exponent, approximate entropy, detrended fluctuation analysis, using a single parameter, cannot fully describe the extremely sophisticated behavior of electroencephalogram (EEG). The multifractal formalism reveals more "hidden" information of EEG by using singularity spectrum to characterize its nonlinear dynamics. In this paper, the zero-crossing time intervals of sleep EEG were studied using multifractal analysis. A new multifractal measure Δ as α was proposed to describe the asymmetry of singularity spectrum, and compared with the singularity strength range Δα that was normally used as a degree indicator of multifractality. One-way analysis of variance and multiple comparison tests showed that the new measure we proposed gave better discrimination of sleep stages, especially in the discrimination between sleep and awake, and between sleep stages 3 and 4.The research of nonlinear dynamics in electroencephalogram (EEG) has made much headway in recent years. Nonlinear analysis methods have been successfully applied to the studies of brain functions and pathological changes in EEG [1][2][3] . These studies also proved that EEG exhibited at least partly chaotic characteristics. The detrended fluctuation analysis (DFA) [4] , which has been widely used recently, revealed the longrange power-law correlation in EEG, indicating time scale invariant and fractal structure [5,6] . However, EEG is also rather noisy, displaying short-term decorrelation like white noise, and consequently, the EEG has been traditionally considered as a linear stationary random process. The paradoxical combination of short-term decorrelation and long-range correlation, stochastic and deterministic suggests that a single nonlinear parameter, such as largest Lyapunov exponent, correlation dimension, fractal dimension, scaling exponent, etc., may not be able to fully characterize the "stochastic chaos" (as named by Freeman [7] ) nature of EEG.The long time behavior of chaotic, nonlinear dynamic systems can often be characterized by (mono) fractal or multifractal measures. Monofractals are homogeneous in the sense that they have the same scaling properties, characterized by a single singularity exponent throughout the entire signal. In contrast, multifractals can be decomposed into many (possibly infinite) sub-sets characterized by different exponents. Multifractal signals are intrinsically more complex and inhomogeneous than monofractals. Multifractal models have been used to account for scale invariance properties of various objects in very different domains ranging from the energy dissipation or the velocity field in turbulent flows [8] to underlying hierarchical structure in proteins [9] . Physiologic signals generated by complex self-regulating systems, such as heartbeat interval, electrocardiogram, gait etc. have been proven to be multifractal, and the degree of multifractality often relates to pathological state or natural aging process [10][11][12] ....

Scaling laws characterize diverse complex systems in a broad range of fields, including physics, biology, finance, and social science. The human language is another example of a complex system of words organization. Studies on written texts have shown that scaling laws characterize the occurrence frequency of words, words rank, and the growth of distinct words with increasing text length. However, these studies have mainly concentrated on the western linguistic systems, and the laws that govern the lexical organization, structure and dynamics of the Chinese language remain not well understood. Here we study a database of Chinese and English language books. We report that three distinct scaling laws characterize words organization in the Chinese language. We find that these scaling laws have different exponents and crossover behaviors compared to English texts, indicating different words organization and dynamics of words in the process of text growth. We propose a stochastic feedback model of words organization and text growth, which successfully accounts for the empirically observed scaling laws with their corresponding scaling exponents and characteristic crossover regimes. Further, by varying key model parameters, we reproduce differences in the organization and scaling laws of words between the Chinese and English language. We also identify functional relationships between model parameters and the empirically observed scaling exponents, thus providing new insights into the words organization and growth dynamics in the Chinese and English language.

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