Nonlinear multivariate analysis of neurophysiological signals

doi:10.1016/j.pneurobio.2005.10.003

Progress in Neurobiology

Volume 77, Issues 1–2, September–October 2005, Pages 1-37

https://doi.org/10.1016/j.pneurobio.2005.10.003 Get rights and content

Abstract

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.

Introduction

One of the most common ways of obtaining information about neurophysiological systems is to study the features of the signal(s) recorded from them by using time series analysis techniques (e.g., Galka, 2000). If one is only interested in the features of a single signal, univariate analysis can perfectly carry out this task by itself. But an increasing number of experiments are being carried out in which several neurophysiological signals are simultaneously recorded, and the assessment of the interdependence between signals can give new insights into the functioning of the systems that produce them. Therefore, univariate analysis alone cannot accomplish such a task, as it is necessary to make use of the multivariate analysis.

In spite of their different aims and scopes, univariate and multivariate time series analysis have an important point in common: they have traditionally relied on the use of linear methods in the time and frequency domains (see, e.g., Bendat and Piersol, 2000). Unfortunately, these methods cannot give any information about the nonlinear features of the signal. Due to the intrinsic nonlinearity of neuronal activity, these nonlinear features might be present in neurophysiological data, which has led researchers to try out other techniques that do not present the aforementioned limitation.

Univariate nonlinear time series analysis methods started to be applied to neurophysiological data about two decades ago (Babloyantz et al., 1985); see, e.g., Elbert et al. (1994), Faure and Korn (2001), Galka (2000), Jansen (1991), Korn and Faure (2003), Segundo (2001), Segundo et al. (1998), Stam (2005) for surveys. As an example, the EEG has been characterized in terms of its correlation dimension, a nonlinear index that has been roughly interpreted as a measure of the irregularity or complexity of a signal³ (see, e.g., Kantz and Schreiber, 2004). This index has been useful in sleep-wake research (Pereda et al., 1998, Pradhan et al., 1995), mental load research (Lamberts et al., 2000), monitoring the depth of anesthesia (van den Broek et al., 2005, Widman et al., 2000) and in studies of epilepsy (Pijn, 1990) to name but a few applications (see Stam, 2005 for a recent review).

Similarly, in the last few years several nonlinear multivariate techniques have started to be used in neurophysiology, mainly as a result of recent advances in information theory (see, e.g., Kraskov et al., 2004, Schreiber, 2000) and in the study of the synchronization between chaotic systems (Boccaletti et al., 2002, Pikovsky et al., 2001). Two relevant concepts are: generalized synchronization (GS) (Rulkov et al., 1995), a state in which a functional dependence between the systems exist, and phase synchronization (PS) (Rosenblum et al., 1996), a state in which the phases of the systems are correlated whereas their amplitudes may not be. In fact, and unlike complete synchronization (Fujisaka and Yamada, 1983) (which may exist only between identical systems and entails the exact equality of their variables), GS and PS may exist between nonindentical systems even in the presence of noise. This makes GS and PS methods appealing for the analysis of neurophysiological signals.

The multivariate nonlinear time series methods derived for the study of GS and PS, as well as those based on information theory, are theoretically useful in neurophysiology due to their ability to detect nonlinear interactions, which might not be fully captured by linear techniques. Nevertheless, the application of these methods to neurophysiological signals is not a plain subject. On the one hand, these signals are often noisy, non-stationary and of finite (sometimes quite short) size. On the other hand, the theoretical subtleties underlying the calculation of many nonlinear interdependence indexes from experimental time series must be taken into account before applying them to the data. In this work, we go through all these questions by reviewing the theoretical and practical aspects of the multivariate nonlinear methods more frequently used for the analysis of neurophysiological signals, ranging from recordings of neuronal action potentials (spikes) to the electroencephalogram (EEG) and the magnetoencephalogram (MEG) as typical examples of integrated neuronal activity.

The paper is organized as follows: we first review the traditional linear tools for the assessment of the interdependence between neurophysiological data in the time and frequency domain; the nonlinear counterparts of the time domain tools are also discussed. Then, we present methods based on information theory as a natural extension of the concept of linear statistical dependence between time series. Next, we explore the idea of PS indexes, which assess the existence of interdependence between the phases of the signals regardless of whether their amplitudes are correlated. Subsequently, methods based on a state space reconstruction are introduced, which analyze the interdependence between the amplitudes of the signals in the reconstructed state spaces and can be used for the assessment of GS. Further, we review the study of the interdependence between signals that present marked events. Finally, we conclude by comparing the performance of the main multivariate nonlinear methods and give some practical recipes.

Two appendixes are added at the end. Appendix A deals with the use of multivariate surrogate data for the assessment of the strength (strong or weak) and the character (linear or nonlinear) of the interdependence between neurophysiological signals. Appendix B is devoted to interesting Internet sites from where it is possible to gather information on how to put into practice the different nonlinear methods reviewed in the text.

Section snippets

Definition and estimation

This is one of the oldest and most classical measures of interdependence between two time series. The cross-correlation function measures the linear correlation between two variables X and Y as a function of their delay time (τ), which is of interest because such a time delay may reflect a causal relationship between the signals. In particular, if X causes Y, one may in principle get a delay from the first signal to the second one. This is, however, not necessarily always the case, since

Definition and estimation

The coherence function gives the linear correlation between two signals as a function of the frequency. Coherence, also termed as magnitude squared coherence or coherence spectrum, between two signals is their cross-spectral density function – which is in fact the Fourier transform of Eq. (1) – normalized by their individual auto-spectral density functions. These spectral quantities are usually derived via the FFT algorithm (Cooley and Tukey, 1965). However, due to finite size of the neural

Definition and estimation

This measure is primarily a nonparametric nonlinear regression coefficient, which describes the dependency of X on Y in a most general way without any direct specification of the type of relationship between them (Lopes da Silva et al., 1989, Pijn et al., 1990). The underlying idea is that if the value of X is considered as a function of the value of Y, the value of Y given X can be predicted according to a nonlinear regression curve. The variance of Y according to the regression curve is

Definition and estimation

In neurophysiology, a question of great interest is whether there exists a causal relation between two brain regions without any specific information on direction. Both the cross-correlation function and the nonlinear correlation coefficient are, in principle, able to indicate the delay in coupling, but inferring causality from the time delay is not always straightforward (Lopes da Silva et al., 1989). This encouraged the researchers to develop new methods explicitly tailored for this aim. One

Multichannel analysis

Most of the methods discussed so far are defined for two signals only: a functional relationship is obtained by pairwise analysis of bivariate signals. However, as discussed earlier, a bivariate method for each pair of signals from a multichannel set of signals does not account for all the covariance structure information from the full data set. In a simple network consisting of one driver and two responses, pairwise analysis is likely to find some correlation between the two responses due to

Information-theory based methods

It might be said that the different methods presented hitherto have a point in common: they all try to establish whether there is any common information between the time series as a sign of their relationship. Therefore, it has become usual to investigate directly the existence of such relationship by means of information-theoretic tools.

The concept of phase synchronization

It is well known at present that the phases of two coupled nonlinear (noisy or chaotic) oscillators may synchronize even if their amplitudes remain uncorrelated, a state referred to as PS (Pikovsky et al., 2001). By synchronization, it is meant here that the following phase locking condition applies for any time t: $φ_{n, m} (t) = | n ϕ_{x} (t) - m ϕ_{y} (t) | \leq constant$ where ϕ_x(t) and ϕ_y(t) are the phases of the signals associated to each system defined on the real line (unwrapped). An example of this is shown in Fig.

Assessment of synchronization in state space

Neurons are highly nonlinear devices, which in some cases show chaotic behavior (Matsumoto and Tsuda, 1988). Thus, the study of their collective activity, as measured by EEG or MEG, could profit from the use of nonlinear measures derived for the study of chaotic dynamical systems. First encouraging results claimed that macroscopic EEG signals also have chaotic structure (Babloyantz et al., 1985), but further studies dit not find any strong evidence of chaos in EEG (Pijn, 1990, Theiler et al.,

Event synchronization

All the measures covered up to now are defined for continuous signals, in which we look for linear or nonlinear correlations between amplitude values, frequencies, phases, or trajectories in phase space. As described, these measures have been very useful for different applications. However, many systems in nature express themselves as point-like processes and, in this case, the applicability of such measures may be limited. Examples of point processes in neurophysiological signals are spike

The current role of linear methods

After getting acquainted with the different multivariate nonlinear methods, one might be tempted to favor them in prejudice of the linear methods or, at least, to relegate these to the background. But this would be a serious mistake: the nonlinear tools are not intended to substitute linear ones and neither they should be claimed to be superior as such. Instead, they must be regarded as a complement of the linear approach that allows a more comprehensive picture of the analyzed data. In fact,

Conclusions

We have reviewed here the current state of the main nonlinear analysis techniques applied to multivariate neurophysiological data, a subject that is earning growing popularity to the extent that these methods have been applied to almost any kind of neurophysiological signals ranging from fMRI data to spike recordings of a neuron. Certainly, and despite its possible advantages, we have also seen that this new approach is not free of caveats. It might be even argued, as suggested in the

Acknowledgements

E. Pereda acknowledges the financial support of the grant n. POS2005/047 of the Canary Government and the grant BFI2002-01159 of the MCyT.

J. Bhattacharya acknowledges the support of JST.Shimojo ERATO project.

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Nonlinear multivariate analysis of neurophysiological signals

Abstract

Introduction

Section snippets

Definition and estimation

Definition and estimation

Definition and estimation

Definition and estimation

Multichannel analysis

Information-theory based methods

The concept of phase synchronization

Assessment of synchronization in state space

Event synchronization

The current role of linear methods

Conclusions

Acknowledgements

The cooperative behavior of neuronal populations during sleep and mental tasks

Electroencephalogr. Clin. Neurophysiol.

Analysis of brain wave records from Gemini flight GT-7 by computations to be used in a thirty day primate flight

Life Sci. Space Res.

Computer analysis of EEG data from Gemini flight GT-7

Aerosp. Med.

Dynamics of neuronal firing correlation: modulation of “effective connectivity”

J. Neurophysiol.

Wavelet entropy for subband segmentation of EEG during injury and recovery

Ann. Biomed. Eng.

Is partial coherence a viable technique for identifying generators of neural oscillations?

Biol. Cybern.

An approach to multivariate phase synchronization analysis and its application to event-related potentials: Synchronization Cluster Analysis

Int. J. Bifurcation Chaos

Testing for phase synchronization

Int. J. Bifurcation Chaos

Event-related coherence as a tool for studying dynamic interaction of brain regions

Electroencephalogr. Clin. Neurophysiol.

Bivariate surrogate techniques: necessity, strengths, and caveats

Phys. Rev. E

Steady-state visual evoked potentials and phase synchronization in migraine

Phys. Rev. Lett.

A robust method for detecting interdependences: application to intracranially recorded EEG

Physica D

Adaptive AR modeling of nonstationary time series by means of Kalman filtering

IEEE Trans. Biomed. Eng.

Causal Modeling

Evidence of chaotic dynamics of brain activity during the sleep cycle

Phys. Lett. A

Overcoming the limitations of correlation analysis for many simultaneously processed neural structures

Prog. Brain Res.

Partial directed coherence: a new concept in neural structure determination

Biol. Cybern.

Combined MEG and EEG source imaging by minimization of mutual information

IEEE Trans. Biomed. Eng.

Neural networks involving the medial temporal structures in temporal lobe epilepsy

Clin. Neurophysiol.

Random Data—Analysis and Measurement Procedure

Bi-directional interactions between visual areas in the awake behaving cat

Neuroreport

Reduced degree of long-range phase synchrony in pathological human brain

Acta Neurobiol. Exp. (Warsz.)

Effective detection of coupling in short and noisy bivariate data

IEEE Trans. Syst. Man Cybern. B

Musicians and the gamma band—a secret affair?

Neuroreport

Shadows of artistry: cortical synchrony during perception and imagery of visual art

Cogn. Brain Res.

Drawing on mind's canvas: differences in cortical integration patterns between artists and non-artists

Hum. Brain Mapp.

Phase synchrony analysis of EEG during music perception reveals changes in functional connectivity due to musical expertise

Signal Process.

Interdependencies in the spontaneous EEG while listening to music

Int. J. Psychophysiol.

Long-range synchrony in the gamma band: role in music perception

J. Neurosci.

Nonlinear dynamics of evoked neuromagnetic responses signifies potential defensive mechanisms against photosensitivity

Int. J. Bifurcation Chaos

Stationarity of the EEG time series

IEEE Eng. Med. Biol. Mag.

EEG data reduction by means of autoregressive representation and discriminant analysis procedures

Electroencephalogr. Clin. Neurophysiol.

The synchronization of chaotic systems

Phys. Rep.