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.

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.

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

We present different tests for phase synchronization which improve the procedures currently used in the literature. This is accomplished by using a twosamples test setup and by utilizing insights and methods from directional statistics and bootstrap theory. The tests differ in the generality of the situation in which they can be applied as well as in their complexity, including computational cost. A modification of the resampling technique of the bootstrap is introduced, making it possible to fully utilize data from time series. Contents 1

Epilepsy is characterized by the spontaneous and unforeseeable occurrence of seizures, during which the perception or behavior of patients is disturbed. The predictability of these seizures would render novel therapeutic approaches possible. Several prediction methods have claimed to be able to predict seizures based on EEG recordings minutes in advance. However, the term seizure prediction is not unequivocally defined, different criteria to assess prediction methods exist, and only little attention has been paid to issues of sensitivity and false prediction rate. We introduce an assessment criterion called the seizure prediction characteristic that incorporates the assessment of sensitivity and false prediction rate. Within this framework, three nonlinear seizure prediction methods were evaluated on a large EEG data pool of 21 patients. Altogether, 582 h intracranial EEG data and 88 seizures were examined. With a rate of 1–3.6 false predictions per day, the “dynamical similarity index” achieves a sensitivity between 21 and 42%, which was the best result of the three methods. Sensitivity was between 18 and 31 % for the extended, prospective version of the “accumulated energy ” and between 13 and 30 % for the “effective correlation dimension”. These results still are not sufficient for clinical applications.

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

Assessment and comparison of three methods by means of the seizure prediction characteristic