  The spectrum and coherency are useful quantities for characterizing the
temporal correlations and functional relations within and between point
processes. This paper 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 which 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 which is 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. While the material discussed is of general applicability
to point processes attention will be confined to sequences of neuronal action
potentials (spike trains) which were the motivation for this work.
