Metric-space analysis of spike trains: theory, algorithms and application

TLDR

Unlike most prior temporal coding analyses, this approach avoids embedding impulse trains in a vector space and does not assume Euclidean distance. The study introduces the mathematical foundation of a new method for analyzing temporal coding. The method constructs novel metrics between neuronal impulse trains based on physiologically grounded hypotheses, uses point‑process properties, enables detection of stimulus‑dependent temporal structure without vector‑space embedding, compares metric similarity to Euclidean distances via multidimensional scaling, and provides efficient algorithms for two metric families (spike‑time and spike‑interval based). These metrics reveal distinct yet related topological structures of impulse trains and, as demonstrated on artificial and cortical recordings, can identify stimulus‑dependent temporal patterns.

Abstract

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 those aspects of the firing pattern that 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 demonstrate how these metrics can be used to determine whether a set of observed responses has a 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 data sets and to recordings from auditory and visual cortex.