Optimization in Reproducing Kernel Hilbert Spaces of Spike Trains

António R.C. Paiva (), Il Park () and José C. Príncipe ()
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António R.C. Paiva: University of Florida
Il Park: University of Florida
José C. Príncipe: University of Florida

Chapter Chapter 1 in Computational Neuroscience, 2010, pp 3-29 from Springer

Abstract: Abstract This chapter presents a framework based on reproducing kernel Hilbert spaces (RKHS) for optimization with spike trains. To establish the RKHS for optimization we start by introducing kernels for spike trains. It is shown that spike train kernels can be built from ideas of kernel methods or from the intensity functions underlying the spike trains. However, the later approach shall be the main focus of this study. We introduce the memoryless cross-intensity (mCI) kernel as an example of an inner product of spike trains, which defines the RKHS bottom-up as an inner product of intensity functions. Being defined in terms of the intensity functions, this approach toward defining spike train kernels has the advantage that points in the RKHS incorporate a statistical description of the spike trains, and the statistical model is explicitly stated. Some properties of the mCI kernel and the RKHS it induces will be given to show that this RKHS has the necessary structure for optimization. The issue of estimation from data is also addressed. We finalize with an example of optimization in the RKHS by deriving an algorithm for principal component analysis (PCA) of spike trains.

Keywords: Point Process; Spike Train; Intensity Function; Kernel Method; Reproduce Kernel Hilbert Space (search for similar items in EconPapers)
Date: 2010
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DOI: 10.1007/978-0-387-88630-5_1

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