  Individual neurons often produce highly variable responses over nominally
identical trials, reflecting a mixture of intrinsic "noise" and systematic
changes in the animal's cognitive and behavioral state. Disentangling these
sources of variability is of great scientific interest in its own right, but it
is also increasingly inescapable as neuroscientists aspire to study more
complex and naturalistic animal behaviors. In these settings, behavioral
actions never repeat themselves exactly and may rarely do so even
approximately. Thus, new statistical methods that extract reliable features of
neural activity using few, if any, repeated trials are needed. Accurate
statistical modeling in this severely trial-limited regime is challenging, but
still possible if simplifying structure in neural data can be exploited. We
review recent works that have identified different forms of simplifying
structure -- including shared gain modulations across neural subpopulations,
temporal smoothness in neural firing rates, and correlations in responses
across behavioral conditions -- and exploited them to reveal novel insights
into the trial-by-trial operation of neural circuits.
