In this chapter we have reviewed the most common approaches to the blind separation of neural sources underlying scalp EEG measurements. Further-more, we have proposed three novel algorithms (M-COMBI, F-COMBI and ENRICA) with attractive features for the analysis of EEG data. M-COMBI and F-COMBI combine the strengths of two complementary BSS methods (WASOBI and EFICA) in order to improve separation accuracy in hybrid mixtures of non-Gaussian and time-correlated sources. Apart from their ro-bustness, the main advantage of these two algorithms is their computational simplicity, in contrast to the computational burden imposed by non-parametric approaches like ENRICA. This is especially the case of F-COMBI that allows for almost real-time separation (e.g. in less than 1 second) of mixtures of up to 10 sources in a powerful personal computer. Moreover, parallel implemen-tations of F-COMBI are straightforward, and could easily boost separation speed in modern multicore machines. Although M-COMBI is more reliable for offline applications, F-COMBI may be more appropriate for applications in which speed is critical, like brain computer interfaces [199] or neuropros-thetics [80].
F-COMBI and M-COMBI are useful mainly for high-dimensional mix-tures for two reasons. First, because multidimensional clusters of components that cannot be separated by WASOBI and EFICA are more easily found in high-dimensional mixtures. Second, because their computational burden is affordable. With respect to similar algorithms aiming at the separation of
hy-2.7. Conclusions to the chapter 49
brid mixtures of non-Gaussian and spectrally diverse Gaussian sources (e.g.
ThinICA, JCC or JADET D), M-COMBI and F-COMBI are clearly faster.
The results shown in [215] and [67] with mixtures of 40 sources show that M-COMBI and F-COMBI can be up to 10 and 100 times faster, respectively.
The brain sources underlying EEG measurements are likely to be connected by neuroelectric pathways, especially in experimental paradigms that aim to study functional brain connectivity. Time-lagged information exchange be-tween the hidden sources can have a very negative impact on BSS algorithms that assume mutual independence, as shown in this chapter with coupled Lorenz oscillators, and as will be confirmed in chapter 4 with VAR sources.
In our experience, BSS algorithms based on temporal structure (e.g. WA-SOBI) are less affected by this problem than ICA methods. For the latter, pre-processing the mixtures using a VAR filter can be helpful, especially in the case of ICA algorithms based on brute-force global optimization of the contrast function (e.g. RADICAL and MILCA). The poor performance of parametric and semiparametric ICA algorithms with the Lorenz dataset is mainly ex-plained by the distribution of the data, which is multimodal and very far from Gaussian. We have to admit that this distribution is rather uncommon and these results do not imply poor performance in real-life applications, including the analysis of EEG data. In fact, in chapter 4 we found that the combina-tion of VAR filtering and EfICA obtains excellent results in the separacombina-tion of sources with characteristics similar to those of real EEG.
Our experiments with real EEG time-series have raised concerns on whether it is possible to accurately separate many EEG sources (e.g. more than 10) from relatively short EEG epochs (e.g. in the order of 10 to 20 seconds).
We found that the common rule-of-thumb of requiring 30M2 samples [45] to separateMsources is overoptimistic. We would recommend using of the order of ten times as many samples, assuming a standard sampling rate of 250 Hz.
Chapter 3
Measures of effective connectivity
3.1 Introduction
The intriguing ability of neuronal populations to establish oscillatorycoupling at large-scale levels has been potentially regarded as one of the brain mecha-nisms underlying cognition (e.g. [218]). These large-scale interactions are due to a complex pattern of underlying brain connectivity. In this macroscale con-text,anatomicalconnectivity refers to the specific arrangement of macroscopic fiber pathways linking different brain regions. In contrast,functional connec-tivity is a fundamentally statistical concept, which refers to the existence of synchronized patterns between the temporal activations of often spatially re-mote neural systems. By this definition, functional connectivity can be mea-sured e.g. by cross correlation, cross-spectra [164] or mutual information [201].
In highly interconnected cerebral systems it is relevant not only to iden-tify anatomical and functional links, but also to measure to what extent the individual brain networks contribute to information production, and at what rate they exchange information among each other. These directional interac-tions form a pattern ofeffectivebrain connectivity [56] that carries important information on thefunctional integration mechanisms of the brain.
There are several distinct approaches to understanding and measuring ef-fective connectivity. One approach - dynamic causal modeling (DCM) [57, 58]
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- uses state-space continuous-time models for each and every step of the phys-iological and biophysical chain of events leading to the observed functional measurements. These detailed models are then used to characterize dynam-ical and structural perturbations in the system due to known deterministic inputs. DCM is especially suitable for functional magnetic resonance imaging (fMRI) studies, due to the relatively simple models involved, and due to the fact that fMRI records responses to deterministic experimental manipulations.
However, electrophysiological DCM models (see e.g. [40, 41]) are complex and still largely speculative (even more in the case of neurological disorders), mak-ing DCM much less suitable for EEG studies. In addition, application of DCM methods to spontaneous EEG signals is not straightforward [148, 149].
An alternative to DCM is to consider that an effective connection between two neural networks exists whenever the prediction of the neural states of one of the networks can be improved by incorporating information from the other.
This so-calledGranger causality(GC) was originally proposed by Wiener [227], and later formalized by Granger [73], in the context of linear regression mod-els of stochastic processes. Since then, linear GC has also been extended to non-linear models [30, 54, 191] and to non-parametric GC indices derived from information theory [29, 55, 195, 222].
Closely related to linear GC is the concept ofdirected coherence (DC) in-troduced by Saito and Harashima [186] in the early 1980s. Like linear GC, effective connectivity indices based on DC rely heavily on vector autoregres-sive (VAR) models to infer causal relationships between temporally structured time-series. The most prominent DC-based connectivity measure is the di-rected transfer function (DTF) [104], which roughly measures to what extent a spectral component in a neural signal induces the generation of the same spectral component in another neural signal. The DTF has been used in pre-vious EEG studies (see e.g. [6, 52, 111, 119]) and we will use it in chapters 4 and 5 to characterize flows of oscillatory activity underlying the generation of the human alpha rhythm.
Lastly, Nolte et al. have recently proposed the so-called phase-slope index (PSI) [159] as a new method to estimate the direction of causal interactions between time-series. The idea behind the PSI is that a time-lagged interaction between a pair of systems induces a slope in the phase of the cross-spectra of those systems. The existence of this slope allows detecting the interaction.
The PSI is a promising tool for assessing effective connectivity, and has been tentatively applied to EEG data [159]. However, a major limitation of the PSI is that it is a bivariate measure and, therefore, might be confounded by the presence of common drivers in systems consisting of more than two variates.
In this chapter, we review the most important information theoretic