Multivariate time series analysis is extensively used in neurophysiology with the aim of studying the relationship between simultaneously recorded signals. Recently, advances on information theory and nonlinear dynamical systems theory have allowed the study of various types of synchronization from time series. In this work, we first describe the multivariate linear methods most commonly used in neurophysiology and show that they can be extended to assess the existence of nonlinear interdependences between signals. We then review the concepts of entropy and mutual information followed by a detailed description of nonlinear methods based on the concepts of phase synchronization, generalized synchronization and event synchronization. In all cases, we show how to apply these methods to study different kinds of neurophysiological data. Finally, we illustrate the use of multivariate surrogate data test for the assessment of the strength (strong or weak) and the type (linear or nonlinear) of interdependence between neurophysiological signals.

Many complex and interesting phenomena in nature are due to nonlinear phenomena. The theory of nonlinear dynamical systems, also called ‘chaos theory’, has now progressed to a stage, where it becomes possible to study self-organization and pattern formation in the complex neuronal networks of the brain. One approach to nonlinear time series analysis consists of reconstructing, from time series of EEG or MEG, an attractor of the underlying dynamical system, and characterizing it in terms of its dimension (an estimate of the degrees of freedom of the system), or its Lyapunov exponents and entropy (reflecting unpredictability of the dynamics due to the sensitive dependence on initial conditions). More recently developed nonlinear measures characterize other features of local brain dynamics (forecasting, time asymmetry, determinism) or the nonlinear synchronization between recordings from different brain regions. Nonlinear time series has been applied to EEG and MEG of healthy subjects during no-task resting states, perceptual processing, performance of cognitive tasks and different sleep stages. Many pathologic states have been examined as well, ranging from toxic states, seizures, and psychiatric disorders to Alzheimer’s, Parkinson’s and Cre1utzfeldt-Jakob’s disease. Interpretation of these results in terms of ‘functional sources ’ and ‘functional networks ’ allows the identification of three basic patterns of brain dynamics: (i) normal, ongoing dynamics during a no-task, resting state in healthy subjects; this state is characterized by a high dimensional complexity and a relatively low and fluctuating level of synchronization of the neuronal networks; (ii) hypersynchronous, highly nonlinear dynamics of epileptic seizures; (iii) dynamics of degenerative encephalopathies with an abnormally low level of between area synchronization. Only intermediate levels of rapidly fluctuating synchronization, possibly due to critical dynamics near a phase transition, are associated with normal information

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Papers published in Hydrology and Earth System Sciences Discussions are under open-access review for the journal Hydrology and Earth System Sciences Participatory scenario development for integrated assessment of nutrient flows in a Catalan river catchment

Over the last few years there has been a surge of activity within the physics community in the emerging field of Econophysics—the study of economic systems from a physicist’s perspective. Physicists tend to take a different view than economists and other social scientists, being interested in such topics as phase transitions and fluctuations. In this dissertation two simple models of stock exchange are developed and simulated numerically. The first is characterized by centralized trading with a market maker. Fluctuations are driven by a stochastic component in the agents ’ forecasts. As the scale of the fluctuations is varied a critical phase transition is discovered. Unfortunately, this model is unable to generate realistic market dynamics. The second model discards the requirement of centralized trading. In this case the stochastic driving force is Gaussian-distributed “news events ” which are public knowledge. Under variation of the control parameter the model exhibits two phase transitions: both a first- and a second-order (critical). The decentralized model is able to capture many of the interesting properties

. The maintenance of balance while sitting or standing requires a control mechanism which can maintain upright posture as well as adapt quickly and flexibly to changes in the environment. Some sort of dynamical control must link visual, auditory, vestibular, and proprioceptive perceptual input to the motoric responses required to activate appropriate muscle groups in order to maintain balance. This dynamical control mechanism needs to use perceptual input to predict the future state of posture with respect to the environment if adaptive balance is to be maintained under changing conditions. These constraints suggest that a purely stochastic random--walk postural control system is unlikely, although others have been unable to reject a linear stochastic model for postural control of quiet standing. The data presented in this chapter are drawn from an experiment that measures center of pressure in a sample of sitting infants who are exposed to a "moving room" stimulus paradigm. Three cate...

In this study we applied both linear and nonlinear spatial correlation measures to charac-terize epicardial activation patterns of sustained atrial fibrillation in instrumented conscious goats. It was investigated if nonlinearity was involved in the spatial coupling of atrial regions and to what extent fibrillation was organized in the experimental model of sustained atrial fibrillation (AF) in instrumented goats. Data were collected in five goats during experiments to convert AF by continuous infu-sion of cibenzoline. Spatial organization during AF was quantified with the linear spatial cross correlation function and the nonlinear spatial cross redundancy which was calculated using the Grassberger–Procaccia correlation integral. Two different types of correlation were evaluated to distinguish simultaneous interaction from non–simultaneous interaction, for in-stance resulting from propagation of fibrillation waves. The nonlinear association length and the linear correlation length were estimated along the principal axes of iso–correlation contours in two–dimensional correlation maps of the nonlinear spatial redundancy and the linear spatial correlation function, respectively.

This work presents a methodology for analyzing developmental and physiological time series from the perspective of dynamical systems. An overview of recent advances in nonlinear techniques for time series analysis is presented. Methods for generating a nonlinear dynamical systems analog to a covariance matrix are proposed. A novel application of structural equation modeling is proposed in which structural expectations can be fit to these nonlinear dependency matrices. A data set has been selected to demonstrate an application of some of these linear and nonlinear descriptive analyses, a suurogate data null hypothesis test, and nonlinear dependency analysis. The dynamical systems methods are evaluated in the light of (a) whether the techniques can be successfully applied to the example data and if so, (b) whether the results of these analyses provide insight into the processes under study which was not provided by other analyses.

Since perceptual and motor processes in the brain are the result of interactions between neurons, layers and areas, a lot of attention has been directed towards the development of techniques to unveil these interactions both in terms of connectivity and direction of interaction. Several techniques are derived from the Granger causality principle, and are based on multivariate autoregressive modeling, so that they can only account for the linear aspect of these interactions. We propose here a technique based on conditional mutual information which enables us not only to describe the directions of nonlinear connections, but also their time delays. We compare our technique with others using ground truth data, thus, for which we know the connectivity. As an application, we consider local field potentials (LFPs) recorded with the 96 micro-electrode UTAH array implanted in area V4 of the macaque monkey’s visual cortex. 1 Causality analysis in neural systems Understanding the connections between different recording sites in the brain,

Abstract: The main objective of the present work is to develop a framework for modelling and controlling of a real world multi-input and multi-output (MIMO) continuously drifting metallurgical process, which is shown to be a complex system. A small change in the properties of the charge composition may lead to entirely different outcome of the process. The newly emerging paradigm of soft-computing or Hybrid Computational Intelligence Systems approach which is based on neural networks, fuzzy sets, genetic algorithms and chaos theory has been applied to tackle this problem In this framework first a feed-forward neuro-model has been developed based on the data collected from a working Submerged Arc Furnace (SAF). Then the process is analysed for the existence of the chaos with the chaos theory (calculating indices like embedding dimension, Lyapunov exponent etc). After that an effort is made to evolve a fuzzy logic controller for the dynamical process using combination of genetic algorithms and the neural networks based forward model to predict the system’s behaviour or conditions in advance and to further suggest modifications to be made to achieve the desired results. Most of the real-world dynamical systems are difficult to model or control using conventional