  Frequency-specific patterns of neural activity are traditionally interpreted
as sustained rhythmic oscillations, and related to cognitive mechanisms such as
attention, high level visual processing or motor control. While alpha waves
(8-12 Hz) are known to closely resemble short sinusoids, and thus are revealed
by Fourier analysis or wavelet transforms, there is an evolving debate that
electromagnetic neural signals are composed of more complex waveforms that
cannot be analyzed by linear filters and traditional signal representations. In
this paper, we propose to learn dedicated representations of such recordings
using a multivariate convolutional sparse coding (CSC) algorithm. Applied to
electroencephalography (EEG) or magnetoencephalography (MEG) data, this method
is able to learn not only prototypical temporal waveforms, but also associated
spatial patterns so their origin can be localized in the brain. Our algorithm
is based on alternated minimization and a greedy coordinate descent solver that
leads to state-of-the-art running time on long time series. To demonstrate the
implications of this method, we apply it to MEG data and show that it is able
to recover biological artifacts. More remarkably, our approach also reveals the
presence of non-sinusoidal mu-shaped patterns, along with their topographic
maps related to the somatosensory cortex.
