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.

Although MEG/EEG signals are highly variable, systematic changes in distinct frequency bands are commonly encountered. These frequency-specific changes represent robust neural correlates of cognitive or perceptual processes (for example, alpha rhythms emerge on closing the eyes). However, their functional significance remains a matter of debate. Some of the mechanisms that generate these signals are known at the cellular level and rest on a balance of excitatory and inhibitory interactions within and between populations of neurons. The kinetics of the ensuing population dynamics determine the frequency of oscillations. In this work we extended the classical nonlinear lumped-parameter model of alpha rhythms, initially developed by Lopes da Silva and colleagues [Kybernetik 15 (1974) 27], to generate more complex dynamics. We show that the whole spectrum of MEG/EEG signals can be reproduced within the oscillatory regime of this model by simply changing the population kinetics. We used the model to examine the influence of coupling strength and propagation delay on the rhythms generated by coupled cortical areas. The main findings were that (1) coupling induces phase-locked activity, with a phase shift of 0 or π when the coupling is bidirectional, and (2) both coupling and propagation delay are critical determinants of the MEG/EEG spectrum. In forthcoming articles, we will use this model to (1) estimate how neuronal interactions are expressed in MEG/EEG oscillations and establish the construct validity of various indices of nonlinear coupling, and (2) generate event-related transients to derive physiologically informed basis functions for statistical modelling of average evoked responses.

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.

This paper reviews recent progress in the diagnosis of Alzheimer’s disease (AD) from electroencephalograms (EEG). Three major effects of AD on EEG have been observed: slowing of the EEG, reduced complexity of the EEG signals, and perturbations in EEG synchrony. In recent years, a variety of sophisticated computational approaches has been proposed to detect those subtle perturbations in the EEG of AD patients. The paper first describes methods that try to detect slowing of the EEG. Next the paper deals with several measures for EEG complexity, and explains how those measures have been used to study fluctuations in EEG complexity in AD patients. Then various measures of EEG synchrony are considered in the context of AD diagnosis. Also the issue of EEG pre-processing is briefly addressed. Before one can analyze EEG, it is necessary to remove artifacts due to for example head and eye movement or interference from electronic equipment. Pre-processing of EEG has in recent years received much attention. In this paper, several state-of-the-art pre-processing techniques are outlined, for example, based on blind source separation and other non-linear filtering paradigms. In addition, the paper outlines opportunities and limitations of computational approaches for diagnosing AD based on EEG. At last, future challenges and open problems are discussed.

Abstract—In the study of complex systems, one of the primary concerns is the characterization and quantification of interdepen-dencies between different subsystems. In real-life systems, the na-ture of dependencies or coupling can be nonlinear and asymmetric, rendering the classical linear methods unsuitable for this purpose. Furthermore, experimental signals are noisy and short, which pose additional constraints for the measurement of underlying coupling. We discuss an index based on nonlinear dynamical system theory to measure the degree of coupling which can be asymmetric. The usefulness of this index has been demonstrated by several exam-ples including simulated and real-life signals. This index is found to effectively disclose the nature and the degree of interactions even when the coupling is very weak and data are noisy and of limited length; by this way, new insight into the functioning of the under-lying complex system is possible. Index Terms—Asymmetry, interdependency, noise, nonlin-earity, state space, time series.

Three synchronization measures are applied to scalp electroencephalogram (EEG) data collected from 20 patients diagnosed to have either: (1) no dementia, (2) mild cognitive impairment (MCI), or (3) Alzheimer’s disease (AD). We apply the three synchronization measures — the phase synchronization, and two measures of nonlinear interdependency — to the data collected from awake patients resting with eyes closed. We show that the synchronization in potential between electrodes near the left and right occipital lobes provides a statistically significant discriminant between the healthy and AD subjects, and the MCI and AD subjects. None of the three measures appears able to distinguish between the healthy and MCI subjects, although MCI subjects show synchronization values intermediate between healthy subjects (with high synchronization values) and AD subjects (with low synchronization values) on average. Keywords: 1.

Objective According to Friston, brain dynamics can be modelled as a large ensemble of coupled nonlinear dynamical subsystems with unstable and transient dynamics. In the present study two predictions from this model (the existence of nonlinear synchronization between macroscopic field potentials and itinerant nonlinear dynamics) were investigated. The dependence of nonlinearity on the method of measuring brain activity (EEG versus MEG) was also investigated. Methods Dataset I consisted of 10 MEG recordings in 10 healthy subjects. Dataset II consisted of simultaneously recorded MEG (126 channels) and EEG (19 channels) in 5 healthy subjects. Nonlinear coupling was assessed with the synchronization likelihood and dynamic itinerancy with the synchronization entropy. Significance was assessed with surrogate data testing (ensembles of 20 surrogates). Results Significant nonlinear synchronization was detected in 14 out of 15 subjects. The nonlinear dynamics were associated with a high index of itinerant behaviour. Nonlinear interdependence was significantly more apparent in MEG data than EEG. Conclusion Synchronous oscillations in MEG and EEG recordings contain a significant nonlinear component which exhibits characteristics of unstable and itinerant behaviour. These findings are in line with Friston’s proposal that the brain can be conceived as a large ensemble of coupled nonlinear dynamical subsystems with labile and unstable dynamics. The spatial scale and physical properties of MEG acquisition may increase the sensitivity of the data to underlying nonlinear structure. Key words MEG EEG synchronization non-linear oscillations dynamics entropy

Abstract—We made use of multivariate nonlinear analysis methods to study the interdependence between the cardiac in-terval variability and both the respiratory activity and the systolic pressure in rats. The study was carried out in basal conditions and after the application of different drugs affecting the cardiovas-cular system. The results showed that there are changes both in the extent and in the directionality of such interdependences because of the drugs. The inhibition of the NO and the parasympathetic blockade changed the cardiovascular coordination, with the latter one also modifying the interdependence between the cardiac in-terval and the respiratory signal. This suggests that the nonlinear approach might be very helpful to explore the interaction between subsystems of the cardiovascular control system. Index Terms—Cardiovascular system, nonlinear detection, syn-chronization, time series.

Abstract—Today, the concept of brain connectivity plays a central role in the neuroscience. While functional connectivity is defined as the temporal coherence between the activities of different brain areas, the effective connectivity is defined as the simplest brain circuit that would produce the same temporal relationship as observed experimentally between cortical sites. The most used method to estimate effective connectivity in neuroscience is the structural equation modeling (SEM), typically used on data related to the brain hemodynamic behavior. However, the use of hemody-namic measures limits the temporal resolution on which the brain process can be followed. The present research proposes the use of the SEM approach on the cortical waveforms estimated from the hiigh-resolution EEG data, which exhibits a good spatial resolution and a higher temporal resolution than hemodynamic measures. We performed a simulation study, in which different main factors were

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