Analysis of the interrelations between frequency bands of the EEG by means of the bispectrum a preliminary study

doi:10.1016/0013-4694(71)90183-0

Electroencephalography and Clinical Neurophysiology

Volume 31, Issue 2, August 1971, Pages 137-148

https://doi.org/10.1016/0013-4694(71)90183-0 Get rights and content

Abstract

The more or less random character of spontaneous EEG activity justifies the application of mathematical models developed in time series analysis. Among these methods numerical spectral analysis has proved to be a powerful tool for analysing and quantifying EEG data, especially background activity. The spectral decomposition of an EEG sample into its frequency components, giving (after appropriate smoothing) the power spectrum, provides complete information about the statistical properties of an EEG sample only under the assumption that the underlying process is stationary and Gaussian. If this assumption is violated, only partial information is obtained and higher order moments should be investigated.

Whereas by the power spectrum the second central moment is analysed in detail, the bispectrum allows a detailed analysis of the third central moment, which is influenced either by interrelations between frequency components or by non-stationarity of the signal. The bispectrum may therefore display important additional information about the properties of a stochastic signal like the EEG. In the stationary case this information is of great value in the investigation of phase-locking between different frequency bands, e.g., between alpha and beta activities. In addition, some new insights into the non-linear aspects of the EEG generating process might be expected.

Bispectra of artificial signals are shown in order to explain the basic aspects of this extension of spectral analysis and selected examples of EEG bispectra demonstrate the interesting possibilities which this method offers in the field of EEG analysis.

Résumé

Le caractère plus ou moins aléatoire de l'activité EEG spontanée justifie l'application de modèles mathématiqus développés suivant l'analyse statistique de séries temporelles. Parmi ces méthodes, l'analyse spectrale numérique s'est révélée d'une grande valeur pour l'analyse et la quantification de l'activité EEG de base. La décomposition spectrale d'un échantillon EEG en ses composantes de fréquence, aboutissant (aprés lissage appropié) au spectre de puissance ne fournit une information complète sur les propriétés statistiques de cet échantillon que dans le cas d'un processus sousjacent stationnaire et Gaussien. Si cette condition n'est pas réalisée, on n'obtient qu'une information partielle et des investigations doivent être faites sur des moments d'ordre supérieur.

Puisque par le spectre de puissance le second moment central est analysé en détail, la méthode bispectrale permet une analyse détaillée du troisième moment central qui est influencé soit par des interrelations entre les bandes de fréquence, soit par le caractère non stationnaire du signal. La méthode bispectrale peut ainsi apporter une information additionnelle importante sur les propriétés d'un signal stochastique tel que l'EEG. Dans le cas de la stationnarité, cette information est de grande valeur dans l'investigation des relations de phase entre différentes bandes de fréquence, c'est-à-dire entre les activités alpha et bêta. De plus, on peut attendre de cette méthode une contribution à l'étude des aspects non-linéaires du processus générateur de l'EEG.

Les auteurs montrent des bispectres de signaux artificiels, afin d'expliquer les aspects fondamentaux de cette extension de l'analyse spectrale ainsi que des exemples sélectionnés de bispectres d'EEG pour prouver les possibilités intéressantes qu'offre cette méthode dans le domaine de l'analyse EEG.

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In this paper, we address the problem of mixing artifacts in relation to the use of nonlinear methods for studying cross-frequency phase-synchronization between neuronal populations. Specifically we refer to bispectral measures, which were developed and applied on EEG/MEG in abundance (Darvas et al., 2009a, 2009b; Dumermuth et al., 1971; Helbig et al., 2006; Jirsa and Müller, 2013; Schwilden, 2006; Wang et al., 2007), and we examine the question of what information can be derived from such measures that estimate true functional connectivity between brain regions as opposed to mixing artifacts. Our new contribution is, essentially, the generalization to nonlinear methods of the concepts based on the imaginary part of coherency to solve the problem of volume conduction (Marzetti et al., 2008; Nolte et al., 2004, 2008, 2009).
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^☆: Paper presented in part as a free communication at the VIIth International Congress of EEG and Clinical Neurophysiology, San Diego, U.S.A., September, 1969 (Kleiner et al. 1969).

^☆☆: Supported in part by the “Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung”, The “Emil Barell Stiftung der F. Hoffmann-La Roche zur Förderung der Medizinisch-Wissenschaftlichen Forschung”, and the “Fritz Hoffmann-La Roche-Stiftung zur Förderung Wissenschaftlicher Arbeitsgemeinshaften in der Schweiz”.

^∗: The authors thank A. Schai and H. Ammann from the Computer Centre of the Swiss Federal Institute of Technology in Zürich for their help. The assistance of O. Brunner, photographer at the Children's Hospital Zürich is gratefully acknowledged.

³: P.J. Huber and Th. Gasser are now with the Dept. of Mathematical Statistics, Princeton University, Princeton, U.S.A.

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Analysis of the interrelations between frequency bands of the EEG by means of the bispectrum a preliminary studyAnalyse des interrelations entre des differentes bandes de frequence de l'eeg par la methode de l'analyse bispectrale (etude preliminaire)☆,☆☆

Abstract

Résumé

Brain Res.

Computer programs in Biomed

Electroenceph. clin. Neurophysiol.

Electroenceph. clin. Neurophysiol.

Computer analysis of EEG data from Gemini flight GT-7

Aerospace Med.

The measurement of power spectra

Computation and interpretation of k-th order spectra

An algorithm for the machine calculation of complex Fourier series

Math. Comput.

Variance spectra of electroencephalograms in twins. A contribution to the problems of quantification of EEG background activity in childhood

Die Anwendung von Varianzspektra für einen quantitativen Vergleich von EEG bei zwillingen

Helv. paediat. Acta

Some modern aspects in numerical spectrum analysis of multichannel electroencephalographic data

Med. biol. Engng.

Gaussian behavior of the electroencephalogram: changes during performance of mental task

Science