: This paper reports on the application to field measurements of time series methods developed on the basis of the theory of deterministic chaos. The major difficulties are pointed out that arise when the data cannot be assumed to be purely deterministic and the potential that remains in this situation is discussed. For signals with weakly nonlinear structure, the presence of nonlinearity in a general sense has to be inferred statistically. The paper reviews the relevant methods and discusses the implications for deterministic modeling. Most field measurements yield nonstationary time series, which poses a severe problem for their analysis. Recent progress in the detection and understanding of nonstationarity is reported. If a clear signature of approximate determinism is found, the notions of phase space, attractors, invariant manifolds etc. provide a convenient framework for time series analysis. Although the results have to be interpreted with great care, superior performance can be achieved for typical signal processing tasks. In particular, prediction and filtering of signals are discussed, as well as the classification of system states by means of time series recordings.

The search for chaotic patterns has occupied numerous investigators in neuroscience, as in many other fields of science. Their results and main conclusions are reviewed in the light of the most recent criteria that need to be satisfied since the first descriptions of the surrogate strategy. The methods used in each of these studies have almost invariably combined the analysis of experimental data with simulations using formal models, often based on modified Huxley and Hodgkin equations and/or of the Hindmarsh and Rose models of bursting neurons. Due to technical limitations, the results of these simulations have prevailed over experimental ones in studies on the nonlinear properties of large cortical networks and higher brain functions. Yet, and although a convincing proof of chaos (as defined mathematically) has only been obtained at the level of axons, of single and coupled cells, convergent results can be interpreted as compatible with the notion that signals in the brain are distributed according to chaotic patterns at all levels of its various forms of hierarchy. This chronological account of the main landmarks of nonlinear neurosciences follows an earlier publication [Faure, Korn, C. R. Acad. Sci. Paris, Ser. III 324 (2001) 773–793] that was focused on the basic concepts of nonlinear dynamics and methods of investigations which allow chaotic processes to be distinguished from stochastic ones and on the rationale for envisioning their control using external perturbations. Here we present the data and main arguments that support the existence of chaos at all levels from the simplest to the most complex forms of organization of the nervous system.

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

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.

Invasive electroencephalographic (EEG) recordings from depth and subdural electrodes, performed in eight patients with temporal lobe epilepsy, are analyzed using a variety of nonlinear techniques. A surrogate data technique is used to find strong evidence for nonlinearities in epileptogenic regions of the brain. Most of these nonlinearities are characterized as "spiking"' by a wavelet analysis. A small fraction of the nonlinearities are characterized as "recurrent" by a nonlinear prediction algorithm. Recurrent activity is found to occur in spatio-temporal patterns related to the location of the epileptogenic focus. Residual delay maps, used to characterize "lag-one nonlinearity", are remarkably stationary for a given electrode, and exhibit striking variations among electrodes. The clinical and theoretical implications of these results are discussed. Keywords: Epileptogenic focus, Invasive EEG; Nonlinear prediction; Surrogate data; Wavelets 1. Introduction Invasive electroencep...

This paper concerns the application of new nonlinear time-series modeling methods to recordings of infant respiratory patterns. The techniques used combine the concept of minimum description length modeling with radial basis models. Our first application of the methods produced results that were not entirely satisfactory, particularly with respect to accurately modeling long term quantitative and qualitative features of respiration patterns. This paper describes a number of modifications of the original methods and makes a comparison of the improvements the various modifications gave. The modifications made were increasing the class of basis function, broadening the range of possible embedding strategies, improving the optimization of the likelihood of the model parameters and calculating a closer approximation to description length. The criteria used in the comparisons were description length, root mean square prediction error, model size, free run behavior and amplitude size and vari...

We describe a procedure (and the motivation behind it) which rapidly and accurately tracks the onset and progress of an epileptic seizure. Roughly speaking, one monitors changes in the relative dispersion of a re-embedded time series. The results are robust with respect to variation of adjustable parameters such as embedding dimension, lag time, and critical distances. Moreover, the general method is virtually unaffected when the data is significantly corrupted by external noise. When the information computed for the individual channels is displayed in an appropriate space-time plot, the progress and geometric location of the seizure are easily seen. An interpretation of these results in terms of a cloud of particles moving in an abstract phase space is examined. 1 Introduction Epilepsy is a disease characterized by recurrent, unprovoked seizures accompanied by pathological electrical activity in the brain[1]. This activity can be monitored and recorded using electrodes attached to the...

this article. However, certain properties of chaotic systems can be described qualitatively. For example, chaotic systems exhibit strong dependence on initial conditions. In the can of the logistic equation, small differences in the initial value x 1 , will result in big differences in the subsequent values x n over time. This strong dependence on initial conditions means that predicting the long-term behavior of chaotic systems is difficult. Another important property of chaotic systems is the ability to show self-organization - to evolve toward ordered temporal and spatial patterns (11). The transition from chaotic to ordered behavior, or the reverse, can occur as an abrupt phase transition with a minute change in the control parameters. As we shall see subsequently, abrupt phase transitions and self-organizing behavior have been demonstrated in electroencephalographs (EEGs) from the epileptogenic foci in humans.

Epilepsy surgergy outcome strongly depends on the localization of epileptic focus. The analysis of ictal EEG (scalp or intracranial) is a gold standard for definition of localization of epileptic focus. In order to automate visual analysis of large amounts of EEG data, we examine the correlations among electrodes captured by linear, nonlinear and multilinear data analysis techniques. We study the performance of these statistical tools to understand the complex structure of epilepsy seizure and localize seizure origin. Our analysis results on four patients with temporal lobe epilepsy reveal that multiway (Tucker3 [1]) and nonlinear multiway (Kernelized Tucker3) analysis techniques are capable of capturing epileptic focus precisely when validated with clinical findings whereas linear and nonlinear analysis techniques (SVD, Kernel PCA) fail to localize seizure origin. KEY WORDS biomedical computing, data mining, unsupervised learning, multiway analysis, epileptic focus 1

The technique of surrogate data has been used as a method to test for membership of particular classes of linear systems. We suggest an obvious extension of this to classes of nonlinear parametric models and demonstrate our methods with respiratory data from sleeping human infants. Although our data are clearly distinct from the different classes of linear systems we are unable to distinguish between our data and surrogates generated by nonlinear models. Hence we conclude that human respiration is likely to be a nonlinear system with more than 2 degrees of freedom with a limit cycle that is driven by high dimensional dynamics or noise.