On the blind source separation of human electroencephalogram by approximate joint diagonalization of second order statistics

Abstract

Over the last ten years blind source separation (BSS) has become a prominent processing tool in the study of human electroencephalography (EEG). Without relying on head modeling BSS aims at estimating both the waveform and the scalp spatial pattern of the intracranial dipolar current responsible of the observed EEG. In this review we begin by placing the BSS linear instantaneous model of EEG within the framework of brain volume conduction theory. We then review the concept and current practice of BSS based on second-order statistics (SOS) and on higher-order statistics (HOS), the latter better known as independent component analysis (ICA). Using neurophysiological knowledge we consider the fitness of SOS-based and HOS-based methods for the extraction of spontaneous and induced EEG and their separation from extra-cranial artifacts. We then illustrate a general BSS scheme operating in the time-frequency domain using SOS only. The scheme readily extends to further data expansions in order to capture experimental source of variations as well. A simple and efficient implementation based on the approximate joint diagonalization of Fourier cospectral matrices is described (AJDC). We conclude discussing useful aspects of BSS analysis of EEG, including its assumptions and limitations.

Introduction

Recent studies on human electroencephalogram (EEG) are based on the theory of brain volume conduction. It is well established that the generators of brain electric fields recordable from the scalp are macroscopic post-synaptic potentials created by assemblies of pyramidal cells of the neocortex (Speckmann and Elger, 2005). Pyramidal cells are aligned and oriented perpendicularly to the cortical surface. Their synchrony is possible thanks to a dense net of local horizontal connections (mostly <1 mm). At recording distances larger than about three/four times the diameter of the synchronized assemblies the resulting potential behaves as if it were produced by electric dipoles; all higher terms of the multipole expansion vanish and we obtain the often invoked dipole approximation (Lopes da Silva and Van Rotterdam, 2005, Nunez and Srinivasan, 2006). Three physical phenomena are important for the arguments we advocate in this study. First, unless dipoles are moving there is no appreciable delay in the scalp sensor measurement (Lopes da Silva and Van Rotterdam, 2005). Second, in brain electric fields there is no appreciable electro-magnetic coupling (magnetic induction) in the frequencies up to about 1 MHz, thus the quasi-static approximation of Maxwell equations holds throughout the spectrum of interest (Nunez and Srinivasan, 2006, p. 535-540). Finally, for source oscillations below 40 Hz it has been verified experimentally that capacitive effects are also negligible, implying that potential difference is in phase with the corresponding generator (Nunez and Srinivasan, 2006, p. 61). These phenomena strongly support the superposition principle, according to which the relation between neocortical dipolar fields and scalp potentials may be approximated by a system of linear equations (Sarvas, 1987). Whether this is a great simplification, we need to keep in mind that it does not hold true for all cerebral phenomena. Rather, it does at the macroscopic spatial scale we are interested in here.

A common approach to the study of human EEG is to describe patterns in space and time and link empirical findings with anatomical and physiological knowledge. The problem is characterized by high temporal resolution (about 1 ms) and low spatial resolution (several cm³). For example, it has been estimated that without time averaging about 60 million contiguous neurons must be synchronously active as to produce observable scalp potentials (Nunez and Srinivasan, 2006, p. 21). Such a cluster would realistically extend over several cm² of cortical gyral surface, whereas disentangling fields emitted by cortical functional units may require much higher precision. Because of volume conduction, scalp EEG potentials describe a mixture of the fields emitted by several dipoles extending over large cortical areas. Practically, in order to improve the spatial resolution it is often necessary to trade in the temporal one operating some form of temporal averaging. In summary, the path followed by much of current EEG research is to “isolate” in space and time the generators of the observed EEG as much as possible, counteracting the mixing caused by volume conduction and maximizing the signal-to-noise ratio (SNR).

Over the years we have assisted to the development of several classes of methods to improve the spatial specificity. Those include, among others, surface and cortical Laplacian (Nunez and Srinivasan, 2006), equivalent dipole fitting (Mosher et al., 1992) and distributed minimum norm (model-driven) or minimum variance (data-driven) inverse solutions (Greenblatt et al., 2005, Lopes da Silva, 2004). Targeted attempts include sparsification approaches (Gorodnitsky et al., 1995, Cotter et al., 2005) and spatial filters known as beamformers (Rodrı´guez-Rivera et al., 2006, Congedo, 2006). Surface Laplacian methods apply a spatial high-pass filtering to the scalp potential by estimating their second spatial derivative. They tend to overemphasize high spatial frequency and radial (to the scalp surface) dipolar fields. Inverse solutions seek source localization in a chosen solution space and rely on geometrical models of the head tissue. Unfortunately, the accurate description of EEG volume conduction is complicated by inhomogeneity (resistivity varies with type of tissue) and anisotropy (resistivity varies in different directions); therefore source localization methods are inevitably undermined by geometrical modeling error.

Another approach that persists in EEG literature is blind source separation (BSS). First studied in our laboratory during the first half of the 80’s (Ans et al., 1985, Hérault and Jutten, 1986) BSS has enjoyed considerable interest worldwide only a decade later, inspired by the seminal papers of Jutten and Herault, 1991, Comon, 1994 and Bell and Sejnowski (1995). BSS has today greatly expanded encompassing a wide range of engineering applications such as speech enhancement, image processing, geophysical data analysis, wireless communication and biological signal analysis (Hyvärinen et al., 2001, Cichocki and Amari, 2002, Choi et al., 2005). Such ubiquity springs from the “blind” nature of the BSS problem formulation: no knowledge of volume conduction or of source waveform is assumed. The problem may be attacked from several perspectives; several hundred BSS algorithms have been proposed over the last 20 years with more added on every year. Typically, such methods are based on the cancellation of second order statistics (SOS) and/or of higher (than two) order statistics (HOS). Their commonality resides in the assumption of a certain degree of source spatial independence, which is precisely modeled by the cancellation of those statistics. Both HOS and SOS have been employed with success in EEG. They are today established for denoising/artifact rejection (Vigário, 1997, Jung et al., 2000, Vorobyov and Cichocki, 2002, Iriarte et al., 2003, Joyce et al., 2004, Kierkels et al., 2006, Fitzgibbon et al., 2007, Frank and Frishkoff, 2007, Halder et al., 2007, Phlypo et al., 2007, Romero et al., 2008, Crespo-Garcia et al., 2008), improving brain computer interfaces (Qin et al., 2004, Serby et al., 2005, Wang and James, 2007, Dat and Guan, 2007, Kachenoura et al., 2008) and for increasing the SNR of single-trial time-locked responses (Cao et al., 2002, Sander et al., 2005, Lemm et al., 2006, Tang et al., 2006, Guimaraes et al., 2007, Zeman et al., 2007). Yet, it appears that only four of the many existing algorithms have repeatedly occurred in EEG literature. They are known as FastICA (Hyvärinen, 1999), JADE (Cardoso and Souloumiac, 1993), InfoMax (Bell and Sejnowski, 1995) and SOBI (Belouchrani et al., 1997). FastICA, InfoMax and JADE are ICA (HOS) methods, while SOBI is a SOS method. JADE and SOBI are solved by approximate joint diagonalization (Cardoso and Souloumiac, 1993, Pham, 2001b, Yeredor, 2002, Ziehe et al., 2004, Vollgraf and Obermayer, 2006, Li and Zhang, 2007, Fadaili et al., 2007, Degerine and Kane, 2007), a powerful algebraic tool which allows promising extensions that we will consider in this study.