A Reproducing Kernel Hilbert Space Framework for Spike Train Signal Processing

Abstract

This letter presents a general framework based on reproducing kernel Hilbert spaces (RKHS) to mathematically describe and manipulate spike trains. The main idea is the definition of inner products to allow spike train signal processing from basic principles while incorporating their statistical description as point processes. Moreover, because many inner products can be formulated, a particular definition can be crafted to best fit an application. These ideas are illustrated by the definition of a number of spike train inner products. To further elicit the advantages of the RKHS framework, a family of these inner products, the cross-intensity (CI) kernels, is analyzed in detail. This inner product family encapsulates the statistical description from the conditional intensity functions of spike trains. The problem of their estimation is also addressed. The simplest of the spike train kernels in this family provide an interesting perspective to others' work, as will be demonstrated in terms of spike train distance measures. Finally, as an application example, the RKHS framework is used to derive a clustering algorithm for spike trains from simple principles.

An information-geometric framework for statistical inferences in the neural spike train space

“…has an inner-product, while the others are only Banach spaces. We conjecture that for p = 2 one can particularize our framework into a Riemannian framework with full inner products, geodesics, and geodesic-based statistics, it does not seem possible for other values of p. Once we have a Hilbert manifold, other extensions, including the use of reproducing Kernel Hilbert spaces (Paiva et al 2009b), can be applied to spike train data.…”

What can spike train distances tell us about the neural code?

“…The measure S corr corresponds to an angle in the space of the spike trains (Paiva et al, 2009). Therefore, the dependence of S corr on τ S differs significantly from the one of D R on τ R (Table 1).…”

A comparison of binless spike train measures

“…Furthermore, the computation burden would increase proportional to the length of the spike trains and inversely proportional to the simulation step. However, as shown by Paiva et al [34], and utilized in Paiva et al [25], the van Rossum's distance can be evaluated in terms of a computationally effective estimator with order OðN i N j Þ, given as…”

“…So, in this sense, the two measures are directly related. Nevertheless, this measure is non-Euclidean like the VP distance, since it is an angular metric [34].…”

“…If this naïve approach is taken, then the computational complexity in evaluating the measure would suffer from the same limitations as the direct implementation of van Rossum's distance. But, like the latter, a data-effective method can be obtained in the same way to compute the distance [34],…”