Decomposing EEG data into space–time–frequency components using Parallel Factor Analysis
Introduction
The electroencephalogram (EEG) is the reflection upon the scalp of the summed synaptic potentials of millions of neurons (Lopes da Silva, 1987). Most investigators agree Lachaux et al., 1999, Varela et al., 2001 that these neurons self-organize into transient networks (“neural masses”) that synchronize in time and space to produce a mixture of short bursts of oscillations that are observable in the EEG record. A statistical description of the oscillatory phenomena of the EEG was carried out first in the frequency domain (Lopes da Silva, 1987) by estimation of the power spectrum for quasi-stationary segments of data. More recent characterizations of transient oscillations are carried out by estimation of the time-varying (or evolutionary) spectrum in the frequency/time domain (Dahlhaus, 1997). These evolutionary spectra of EEG oscillations will have a topographic distribution on the sensors that is contingent on the spatial configuration of the neural sources that generate them as well as the properties of the head as a volume conductor (Nunez, 1993).
The purpose of the present study was to attempt the decomposition of multichannel time-varying EEG spectrum into a series of distinct components or modes. In the parlance of modern harmonic analysis (Chen and Donoho, 2001), we performed a space/frequency/time “atomic decomposition” of multidimensional data. In other words, we assume that each neural mass contributes a distinctive atom to the topographic frequency/time description of the EEG, so that the estimation of these atoms is possible by means of signal-processing techniques. Each atom will be defined by its topography, spectral content, and time profile; in other words, by its spatial, spectral, and temporal signatures. We expect that these extracted atoms ultimately will allow the identification of the corresponding neural masses that produce them.
There is a long history of atomic decompositions for the EEG. However, to date, atoms have not been defined by the triplet spatial, spectral, and temporal signatures but rather pairwise combinations of these components. Some of the current procedures for these analyses are reviewed below.
Space/time atoms are the basis of both Principal Component Analysis (PCA) and Independent Component Analysis (ICA) as applied to multichannel EEG. PCA has been used for artifact removal and to extract significant activities in the EEG Lagerlund et al., 1997, Soong and Koles, 1995. A basic problem is that atoms defined by only two signatures (space and time) are not determined uniquely. In PCA, orthogonality is therefore imposed between the corresponding signatures of different atoms. This, however, is a rather nonphysiological constraint. Even with this restriction, there is the well-known nonuniqueness of PCA that allows the arbitrary choice of rotation of axes (e.g., Varimax and Quartimax rotations). More recently, ICA has become a popular tool for space/time atomic decomposition Cichocki and Amari, 2002, Hyvarinen et al., 2001. It avoids the arbitrary choice of rotation (Jung et al., 2001). Uniqueness, however, is achieved at the price of imposing a constraint even stronger than orthogonality, namely, statistical independence. In both PCA and ICA, the frequency information may be obtained from the temporal signature of the extracted atoms in a separate step.
There are many papers on the decomposition of single-channel EEG into frequency/time atoms. For this purpose, the Fast Fourier Transformation (FFT) with sliding window (Makeig, 1993) or the wavelet transformation Bertrand et al., 1994, Tallon-Baudry et al., 1997 have been employed. In fact, any of the frequency/time atomic decompositions currently available (Chen and Donoho, 2001) could, in principle, be used for the EEG. However, these methods do not address the topographic aspects of the EEG time/frequency analysis.
Gonzalez Andino et al. (2001) improved previous analyses by analyzing regions of the frequency/time plane where a single dipole model is an adequate spatial description of the signal, thus incorporating topographic information. Topographic frequency/time decomposition of the EEG was introduced by Koenig et al. (2001), which is the first work to estimate atoms characterized simultaneously by a frequency/time and spatial signature. In their analyses, it was possible to extract physiologically significant activity in the EEG. However, in order to achieve a unique decomposition, they imposed the mathematical constraints that the combined frequency/time signatures of all atoms were required to be of minimum norm and the spatial or topographic signatures were required to have maximal smoothness. These constraints have been found to be unnecessary for unique topographic time/frequency decomposition, a fact that has motivated the work described in this paper.
It has long been known, especially in the chemometrics literature, that unique multi-linear decompositions of multi-way arrays of data (more than two dimensions) are possible under very weak conditions (Sidiropoulos and Bro, 2000). In fact, this is the basic argument for Parallel Factor Analysis (PARAFAC). This technique was proposed independently by Harshman, 1970, Harshman, 1972 and by Carroll and Chang (1970) who named the model Canonical Decomposition, and recently has been improved by Bro (1998) who also provided a Matlab toolbox (available as of this writing at: http://www.models.kvl.dk/users/rasmus/). In PARAFAC, for three-way arrays, the data is decomposed as a sum of components (corresponding to our “atoms”), each of which is the tri-linear product of one score vector and two loading vectors (corresponding to our “signatures”). The important difference between PARAFAC and techniques such as PCA or ICA is that the decomposition of multi-way data is unique even without additional orthogonality or independence constraints.
Thus, PARAFAC can be employed for a space/frequency/time atomic decomposition of the EEG. This makes use of the fact that multichannel evolutionary spectra are multi-way arrays, indexed by electrode, frequency, and time. The inherent uniqueness of the PARAFAC solution leads to a topographic time/frequency decomposition with a minimum of a priori assumptions.
Here, we use PARAFAC for the purpose of simultaneous space/frequency/time decompositions. Previous applications of PARAFAC to EEG data have analyzed only space/time, and some additional external dimensions provided by subject and drug dose, among other factors Achim and Bouchard, 1997, Estienne et al., 2001, Field and Graupe, 1991. A special interpretation of this model is also the Topographic Components Model (TCM) Möcks, 1988a, Möcks, 1988b, which gives justification for the PARAFAC model in the context of evoked potentials analysis, based on biophysical considerations (Möcks, 1988b). In this field, a relevant proof of the use of TCM over PCA using only synthetic noiseless data was given in Achim and Bouchard (1997).
To illustrate the usefulness of PARAFAC, we applied the decomposition of time-varying EEG spectrum to the comparison of resting EEG to that recorded while the subject performed mental arithmetic. Mental arithmetic produces theta activity in the frontal area and a suppression of alpha activity in the occipital area, while the converse occurs when the eyes are closed in the resting condition Harmony et al., 1999, Ishihara and Yoshii, 1972, Sasaki et al., 1996. The PARAFAC atomic decomposition should be able to extract these components, localize them correctly, and detect the corresponding level of activity in these bands in each physiological state. Once estimated, the spatial and spectral signatures of the identified atoms may be used to search for similar types of activity in new data sets. Here, this procedure will be called “screening” for the presence of an atom.
Our focus is on space/time/frequency decompositions tailored to the description of oscillatory phenomena. These are not the only interesting phenomena in the EEG, transient activity being another example. The methods described in this paper may be generalized to this application by exchanging the basic dictionary that describes oscillations.
This paper is organized as follows. We first describe the experimental methods. Then, we consider the basic theoretical development of the space/frequency/time atomic decomposition and the use of estimated factors to screen for activity in new data segments. The results and discussion follow.
Section snippets
Data acquisition
Five male right-handed subjects (mean age 25.8 ± 3.96 years) that produced clear theta activities during a mental task were studied in this work. All subjects signed an informed consent form approved by the RIKEN Human Subject Protection Committee before EEG recording. All subjects were required to concentrate, for 3 min, on mental arithmetic (subtraction by serial 7 from 1000) with closed eyes. They were asked the final residual number at the end of the task. The resting EEG with closed eyes
Parallel Factor Analysis
To evaluate the performance of PARAFAC for extracting alpha and theta activities in EEG, two different states were prepared in a benchmark data set. For this purpose, 10 segments of 1 s each were selected from the wavelet-transformed data (after wavelet transformation the time-varying EEG spectrum data set was subsampled to 100 Hz to reduce the computational cost of PARAFAC). Clear alpha activity is observed continuously during resting and task condition; however, strong theta activity appears
Discussion
This paper introduces a new type of space/frequency/time atomic decomposition of the EEG. It takes advantage of the fact that three-way arrays of data may be decomposed into a sum of atoms of which is a trilinear combination of factors or signatures. This decomposition will be unique if the number of atoms is less than half the sum of the ranks of the three matrices formed by concatenating the signatures. The application of this concept to obtain unique space/frequency/time decomposition for
Acknowledgements
The authors want to thank Prof. Mark S. Cohen, Director of Functional MR Imaging, Ahmanson-Lovelace Brain Mapping Center, UCLA School of Medicine, for his very helpful advice and suggestion for this work.
References (41)
- et al.
Toward a dynamic topographic components model
Electroencephalogr. Clin. Neurophysiol.
(1997) - et al.
Do specific EEG frequencies indicate different processes during mental calculation?
Neurosci. Lett.
(1999) - et al.
Multivariate analytic study of EEG and mental activity in juvenile delinquents
Electroencephalogr. Clin. Neurophysiol.
(1972) - et al.
Topographic time-frequency decomposition of the EEG
NeuroImage
(2001) Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics
Linear Algebra Appl.
(1977)Auditory event-related dynamics of the EEG spectrum and effects of exposure to tones
Electroencephalogr. Clin. Neurophysiol.
(1993)Decomposing event-related potential: a new topographic components model
Biol. Psychol.
(1988)- et al.
Low resolution electromagnetic tomography: a new method for localizing electrical activity in the brain
Int. J. Psychophysiol.
(1994) - et al.
Frontal mental theta wave recorded simultaneously with megnetroencephalography and electroencephalography
Neuroscience
(1996) - et al.
Time–frequency digital filtering based on an invertible wavelet transform: an application to evoked potential
IEEE Trans. Biomed. Eng.
(1994)
Analysis of individual differences in multidimensional scaling via an N-way generalization of ‘Eckart-Young’ decomposition
Psychometrika
Frequency domain distributed inverse solution
Atomic decomposition by basis pursuit
SIAM Rev.
Adaptive Blind Signal and Image Processing
Fitting time series models to non-stationary processes
Ann. Stat.
Multi-way modelling of high-dimensionality electroencephalographic data
Chemom. Intell. Lab. Syst.
Topographic component (Parallel Factor) analysis of multichannel evoked potentials: practical issues in trilinear spatiotemporal decomposition
Brain Topogr.
Linear and nonlinear current density reconstructions
J. Clin. Neurophysiol.
Non-stationary distributed source approximation: an alternative to improve localization procedures
Hum. Brain Mapp.
Cited by (324)
Harmonized-Multinational qEEG norms (HarMNqEEG)
2022, NeuroImageTensor decomposition of human narrowband oscillatory brain activity in frequency, space and time
2022, Biological PsychologyExploration of the 2021 Mw 7.3 Maduo Earthquake by Fusing the Electron Density and Magnetic Field Data of Swarm Satellites
2024, IEEE Transactions on Geoscience and Remote SensingSecond Order Blind Identification of Event Related Potentials Sources
2023, Brain Topography