  In recent years, a number of methods have been proposed to estimate the times
at which neurons spike on the basis of calcium imaging data. However,
quantifying the uncertainty associated with these estimated spikes remains an
open problem. We consider a simple and well-studied model for calcium imaging
data, which states that calcium decays exponentially in the absence of a spike,
and instantaneously increases when a spike occurs. We wish to test the null
hypothesis that the neuron did not spike -- i.e., that there was no increase in
calcium -- at a particular timepoint at which a spike was estimated. In this
setting, classical hypothesis tests lead to inflated Type I error, because the
spike was estimated on the same data. To address this problem, we propose a
selective inference approach to test the null hypothesis. We describe an
efficient algorithm to compute finite-sample p-values that control selective
Type I error, and confidence intervals with correct selective coverage, for
spikes estimated using a recent proposal from the literature. We apply our
proposal in simulation and on calcium imaging data from the spikefinder
challenge.
