Individual neurons often produce highly variable responses over nominally identical trials, reflecting a mixture of intrinsic noise and systematic changes in the animals 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.
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