  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. In addition to
investigating how noise and state changes impact neural computation,
statistical models of trial-to-trial variability are becoming increasingly
important as experimentalists aspire to study naturalistic animal behaviors,
which never repeat themselves exactly and may rarely do so even approximately.
Estimating the basic features of neural response distributions may seem
impossible in this trial-limited regime. Fortunately, by identifying and
leveraging simplifying structure in neural data -- e.g. shared gain modulations
across neural subpopulations, temporal smoothness in neural firing rates, and
correlations in responses across behavioral conditions -- statistical
estimation often remains tractable in practice. We review recent advances in
statistical neuroscience that illustrate this trend and have enabled novel
insights into the trial-by-trial operation of neural circuits.
