  We present the mathematical basis of a new approach to the analysis of
temporal coding. The foundation of the approach is the construction of several
families of novel distances (metrics) between neuronal impulse trains. In
contrast to most previous approaches to the analysis of temporal coding, the
present approach does not attempt to embed impulse trains in a vector space,
and does not assume a Euclidean notion of distance. Rather, the proposed
metrics formalize physiologically-based hypotheses for what aspects of the
firing pattern might be stimulus-dependent, and make essential use of the point
process nature of neural discharges. We show that these families of metrics
endow the space of impulse trains with related but inequivalent topological
structures. We show how these metrics can be used to determine whether a set of
observed responses have stimulus-dependent temporal structure without a
vector-space embedding. We show how multidimensional scaling can be used to
assess the similarity of these metrics to Euclidean distances. For two of these
families of metrics (one based on spike times and one based on spike
intervals), we present highly efficient computational algorithms for
calculating the distances. We illustrate these ideas by application to
artificial datasets and to recordings from auditory and visual cortex.
