Estimating the time-course of coherence between single-trial brain signals: an introduction to wavelet coherence

Abstract

This paper introduces the use of wavelet analysis to follow the temporal variations in the coupling between oscillatory neural signals. Coherence, based on Fourier analysis, has been commonly used as a first approximation to track such coupling under the assumption that neural signals are stationary. Yet, stationary neural processing may be the exception rather than the rule. In this context, the recent application to physical systems of a wavelet-based coherence, which does not depend on the stationarity of the signals, is highly relevant. This paper fully develops the method of wavelet coherence and its statistical properties so that it can be practically applied to continuous neural signals. In realistic simulations, we show that, in contrast to Fourier coherence, wavelet coherence can detect short, significant episodes of coherence between non-stationary neural signals. This method can be directly applied for an ‘online’ quantification of the instantaneous coherence between two signals.

Résumé

Cet article montre comment l’analyse par ondelettes peut être utilisée pour suivre les variations temporelles rapides de la cohérence entre deux signaux neuronaux oscillatoires. Pour mesurer le couplage entre deux phénomènes oscillatoires stationaires, le calcul classique de la cohérence entre deux signaux, qui s’appuie sur l’analyse de Fourier, est suffisant et satisfaisant. Malheureusement, à de rares exceptions près, cette hypothèse de stationarité n’est pas vraie pour les signaux cérébraux. Face à cette contrainte, l’apparition récente pour l’étude de systèmes physiques non-stationaires d’une mesure de cohérence basée non pas sur des décompositions de Fourier mais sur des analyses en ondelettes est tout à fait intéressante. La méthode consiste à estimer la cohérence entre deux signaux pour chaque fréquence et chaque instant, à partir de l’amplitude et de la phase instantanées des signaux obtenues par une convolution avec une ondelette de Morlet. Nous développons précisément les propriétés statistiques de cette méthode de façon à ce qu’elle puisse être appliquée à l’étude de signaux cérébraux continus. À partir de simulations réalistes, nous montrons qu’à la différence de la cohérence de Fourier, la cohérence par ondelettes est capable de détecter des épisodes de synchronisation très brefs, de quelques centaines de millisecondes, dans des essais individuels. Il devient donc possible de quantifier en direct, alors même que le sujet est en train d’être enregistré, la cohérence instantanée entre deux mesures, qu’il s’agisse par exemple d’électroencéphalogrammes, d’électromyogrammes, de magnétoencéphalogrammes ou de mesures intracrâniennes.

Introduction

Our understanding of cerebral networks dynamics depends on our ability to follow the temporal variations in the coupling between neural signals. Most methods used so far to estimate neural coupling have relied on the assumption that brain signals are stationary. This paper introduces a general method to estimate the coupling between non-stationary neural signals, as a function of time.

The most common way to estimate phase-coupling is to compute the coherence function (see 〚1〛 for discussion). The coherence function is a direct measure of the correlation between the spectra of two random processes. If the processes are stationary, i.e. if their spectra do not change in time, Fourier analysis can be used to provide accurate estimates of these spectra and, thus, of coherence (reviewed in 〚2〛). This coherence estimation, based on Fourier analysis, has been repeatedly used in neuroscience, with the assumption that neural signals are stationary and that the amplitude and phase components of the signals need not be distinguished 3, 4, 5, 6. Yet, non-stationarity is the rule rather than the exception in neural processing (e.g. 〚7〛 or 〚8〛). Therefore, coherence needs to be studied as a function of time, a constraint that limits the application of Fourier analysis.

Fourier analysis has been used to study the time-course of coherence between non-stationary signals (following a procedure called event-related coherence) 〚9〛. This method requires repeated observations and is based on the assumption that repeated presentations of a stimulus trigger the same spectral modifications of brain processes. Therefore, it replaces an assumption of stationarity in time by an assumption of stationarity across trials, an assumption that has not, to our knowledge, been validated.

We introduce here a wavelet-based method to follow the time-course of coherence between brain signals, based on its recent use in physics to estimate the interactions among non-stationary signals 10, 11, 12. In contrast with Fourier-based coherence, wavelet coherence provides a coherence measure as a function of time, and is thus a good candidate for the measurement of dynamic neural interactions. Since previous papers on this topic have provided only a brief definition of wavelet coherence, with no mention of its statistical properties, one purpose of this paper is to develop a complete overview of this method and to demonstrate its validity for the study of brain signals.

The present article is organized as follows: in the second, tutorial, section, we briefly review the definition of coherence and its physical meaning, and try to summarize concepts that are essential for understanding wavelet coherence. In Section 3, we introduce the use of wavelet analysis for coherence estimation and show how it can be related to classical Fourier estimations. The statistical properties of wavelet coherence are detailed in Section 4. Finally, we show applications of wavelet coherence to real brain signals in Section 5.