Complex independent component analysis of frequency-domain electroencephalographic data

Abstract

Independent component analysis (ICA) has proven useful for modeling brain and electroencephalographic (EEG) data. Here, we present a new, generalized method to better capture the dynamics of brain signals than previous ICA algorithms. We regard EEG sources as eliciting spatio-temporal activity patterns, corresponding to, e.g. trajectories of activation propagating across cortex. This leads to a model of convolutive signal superposition, in contrast with the commonly used instantaneous mixing model. In the frequency-domain, convolutive mixing is equivalent to multiplicative mixing of complex signal sources within distinct spectral bands. We decompose the recorded spectral-domain signals into independent components by a complex infomax ICA algorithm. First results from a visual attention EEG experiment exhibit: (1). sources of spatio-temporal dynamics in the data, (2). links to subject behavior, (3). sources with a limited spectral extent, and (4). a higher degree of independence compared to sources derived by standard ICA.

Fig. 1

Schematic representation of the processing stages of the complex spectral-domain ICA algorithm. Left (`spec'): the recorded electrode signals are decomposed into different spectral bands. Center (`cICA'): complex ICA decomposition is performed within each spectral band. Right: iteration steps performed by complex ICA for estimation of each separating matrix W(f).

Fig. 2

The circular symmetric super-Gaussian probability density function P(s) of the complex sources s.

Fig. 3

The distribution P_|s|(|s|) of super-Gaussian source magnitude (solid) versus the distribution of the magnitude of a two-dimensional Gaussian process with the same variance (dashed). The latter is the well-known Rayleigh distribution. The super-Gaussian source distribution is characterized by its stronger peak at small magnitudes and its longer (high-magnitude) tails.

Fig. 4

Histograms for estimated kurtosis of complex spectral-domain electrode signals (thin line) and independent component activations (thick line). Each histogram based on 3131 kurtosis estimates (see text), 44 bins of width 0.05 in the interval from 0 to 3.

Fig. 5

Residual statistical dependencies evaluated using second order (left panel) and fourth order (right panel) measures at frequency bands between 0 and 50 Hz. Residuals for the recorded electrode signals (dotted), signal separation obtained from real time-domain infomax ICA (dash-dotted), real-map constrained-complex spectral-domain ICA (dashed), and fully complex spectral-domain ICA (solid).

Fig. 6

Mean distance between the component maps obtained by time-domain infomax ICA and best-matching frequency-specific component maps of real-map constrained-complex ICA. Abscissa: frequency of spectral-domain component. Ordinate: mean distance to time-domain ICA map.

Fig. 7

Minimal mean distances D_act(f₁, f₂) computed from component activation functions obtained with the fully complex ICA algorithm in 101 frequency bands of width 5.12 Hz, spaced equidistantly between 0 and 50 Hz in 0.5-Hz increments. Right: distances for all best-matching component pairs of different frequencies. Left: enlarged view of the 0–20-Hz range.

Fig. 8

Independent component at 5 Hz obtained from standard time-domain infomax ICA. Left: scalp map. Middle: ERP-image of 5-Hz power. Right: ERP-image of complex-demodulated 5-Hz phase. Response times superimposed on data. Lower panels: mean time-courses of event-related 5-Hz power (middle) and 5-Hz intertrial coherence (ITC, right).

Fig. 9

Independent component at 5 Hz obtained from real-map constrained-complex spectral-domain ICA. Same dataset as Fig. 8. Left: scalp map. Middle: ERP-image of 5-Hz power. Right: ERP-image of complex-demodulated 5-Hz phase. Response time and lower panels analogous to Fig. 8.

Fig. 10

Independent component at 5 Hz obtained from fully complex spectral-domain ICA. Same dataset as Figs. 8 and 9. From left to right: real and imaginary part of the complex scalp map, respectively; ERP-images of 5-Hz power and complex-demodulated 5-Hz phase of the complex IC activation time-courses, respectively. Response time and lower panels analogous to Fig. 8.

Fig. 11

Magnitude maps of complex independent components obtained using the fully complex spectral-domain ICA algorithm at five frequency bands, same dataset as Figs. 8–10.