  Statistical analysis and inferences on spike trains are one of the central
topics in neural coding. It is of great interest to understand the underlying
distribution and geometric structure of given spike train data. However, a
fundamental obstacle is that the space of all spike trains is not an Euclidean
space, and non-Euclidean metrics have been commonly used in the literature to
characterize the variability and pattern in neural observations. Over the past
few years, two Euclidean-like metrics were independently developed to measure
distance in the spike train space. An important benefit of these metrics is
that the spike train space will be suitable for embedding in Euclidean spaces
due to their Euclidean properties. In this paper, we systematically compare
these two metrics on theory, properties, and applications. Because of its
Euclidean properties, one of these metrics has been further used in defining
summary statistics (i.e. mean and variance) and conducting statistical
inferences in the spike train space. Here we provide equivalent definitions
using the other metric and show that consistent statistical inferences can be
conducted. We then apply both inference frameworks in a neural coding problem
for a recording in geniculate ganglion stimulated by different tastes. It is
found that both frameworks achieve desirable results and provide useful new
tools in statistical inferences in neural spike train space.
