Evaluating Temporal Correlations in Time Series Using Permutation Entropy, Ordinal Probabilities and Machine Learning.

Time series analysis comprises a wide repertoire of methods for extracting information from data sets. Despite great advances in time series analysis, identifying and quantifying the strength of nonlinear temporal correlations remain a challenge. We have recently proposed a new method based on training a machine learning algorithm to predict the temporal correlation parameter, α, of flicker noise (FN) time series. The algorithm is trained using as input features the probabilities of ordinal patterns computed from FN time series, xαFN(t), generated with different values of α. Then, the ordinal probabilities computed from the time series of interest, x(t), are used as input features to the trained algorithm and that returns a value, αe, that contains meaningful information about the temporal correlations present in x(t). We have also shown that the difference, Ω, of the permutation entropy (PE) of the time series of interest, x(t), and the PE of a FN time series generated with α=αe, xαeFN(t), allows the identification of the underlying determinism in x(t). Here, we apply our methodology to different datasets and analyze how αe and Ω correlate with well-known quantifiers of chaos and complexity. We also discuss the limitations for identifying determinism in highly chaotic time series and in periodic time series contaminated by noise. The open source algorithm is available on Github.

Ecology and signal structure drive the evolution of synchronous displays.

Animal synchrony is found in phylogenetically distant animal groups, indicating behavioral adaptations to different selective pressures and in different signaling modalities. A notable example of synchronous display is found in fiddler crabs in that males wave their single enlarged claw during courtship. They present species-specific signals, which are composed of distinctive movement signatures. Given that synchronous waving has been reported for several fiddler crab species, the display pattern could influence the ability of a given species to sufficiently adjust wave timing to allow for synchrony. In this study, we quantified the wave displays of fiddler crabs to predict their synchronous behavior. We combined this information with the group's phylogenetic relationships to trace the evolution of display synchrony in an animal taxon. We found no phylogenetic signal in interspecific variation in predicted wave synchrony, which mirrors the general nonphylogenetic pattern of synchrony across animal taxa. Interestingly, our analyses show that the phenomenon of synchronization stems from the peculiarities of display pattern, mating systems, and the complexity of microhabitats. This is the first study to combine mathematical simulations and phylogenetic comparative methods to reveal how ecological factors and the mechanics of animal signals affect the evolution of the synchronous phenomena.

Mathematical model of brain tumour with glia-neuron interactions and chemotherapy treatment.

In recent years, it became clear that a better understanding of the interactions among the main elements involved in the cancer network is necessary for the treatment of cancer and the suppression of cancer growth. In this work we propose a system of coupled differential equations that model brain tumour under treatment by chemotherapy, which considers interactions among the glial cells, the glioma, the neurons, and the chemotherapeutic agents. We study the conditions for the glioma growth to be eliminated, and identify values of the parameters for which the inhibition of the glioma growth is obtained with a minimal loss of healthy cells.

Using recurrences to characterize the hyperchaos-chaos transition.

Hyperchaos occurs in a dynamical system with more than one positive Lyapunov exponent. When the equations governing the time evolution of the dynamical system are known, the transition from chaos to hyperchaos can be readily obtained when the second largest Lyapunov exponent crosses zero. If the only information available on the system is a time series, however, such method is difficult to apply. We propose the use of recurrence quantification analysis of a time series to characterize the chaos-hyperchaos transition. We present results obtained from recurrence plots of coupled chaotic piecewise-linear maps and Chua-Matsumoto circuits, but the method can be applied as well to other systems, even when one does not know their dynamical equations.

Lyapunov spectrum and synchronization of piecewise linear map lattices with power-law coupling.

We study the synchronization properties of a lattice of chaotic piecewise linear maps. The coupling strength decreases with the lattice distance in a power-law fashion. We obtain the Lyapunov spectrum of the coupled map lattice and investigate the relation between spatiotemporal chaos and synchronization of amplitudes and phases, using suitable numerical diagnostics.