Sampling Properties of the Spectrum and Coherency of Sequences of Action Potentials

The spectrum and coherency are useful quantities for characterizing the temporal correlations and functional relations within and between point processes. This article begins with a review of these quantities, their interpretation, and how they may be estimated. A discussion of how to assess the statistical significance of features in these measures is included. In addition, new work is presented that builds on the framework established in the review section. This work investigates how the estimates and their error bars are modified by finite sample sizes. Finite sample corrections are derived based on a doubly stochastic inhomogeneous Poisson process model in which the rate functions are drawn from a low-variance gaussian process. It is found that in contrast to continuous processes, the variance of the estimators cannot be reduced by smoothing beyond a scale set by the number of point events in the interval. Alternatively, the degrees of freedom of the estimators can be thought of as bounded from above by the expected number of point events in the interval. Further new work describing and illustrating a method for detecting the presence of a line in a point process spectrum is also presented, corresponding to the detection of a periodic modulation of the underlying rate. This work demonstrates that a known statistical test, applicable to continuous processes, applies with little modification to point process spectra and is of utility in studying a point process driven by a continuous stimulus. Although the material discussed is of general applicability to point processes, attention will be confined to sequences of neuronal action potentials (spike trains), the motivation for this work.