  Neural time-series data contain a wide variety of prototypical signal
waveforms (atoms) that are of significant importance in clinical and cognitive
research. One of the goals for analyzing such data is hence to extract such
'shift-invariant' atoms. Even though some success has been reported with
existing algorithms, they are limited in applicability due to their heuristic
nature. Moreover, they are often vulnerable to artifacts and impulsive noise,
which are typically present in raw neural recordings. In this study, we address
these issues and propose a novel probabilistic convolutional sparse coding
(CSC) model for learning shift-invariant atoms from raw neural signals
containing potentially severe artifacts. In the core of our model, which we
call $\alpha$CSC, lies a family of heavy-tailed distributions called
$\alpha$-stable distributions. We develop a novel, computationally efficient
Monte Carlo expectation-maximization algorithm for inference. The maximization
step boils down to a weighted CSC problem, for which we develop a
computationally efficient optimization algorithm. Our results show that the
proposed algorithm achieves state-of-the-art convergence speeds. Besides,
$\alpha$CSC is significantly more robust to artifacts when compared to three
competing algorithms: it can extract spike bursts, oscillations, and even
reveal more subtle phenomena such as cross-frequency coupling when applied to
noisy neural time series.
