Information Technology Reference
In-Depth Information
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(a) Projection of the spike trains in the training set.
(b) Projection of the spike trains in the testing set.
Fig. 1.7: Projection of spike trains onto the first two principal components of the
covariance matrix of binned spike trains. The different point marks differentiate
between spike trains corresponding to each one of the classes.
sufficient detail
. Since the binned cross-correlation and the mCI kernel are concep-
tually equivalent apart from the discretization introduced by binning, this shows the
ill effects of this preprocessing step for analysis and computation with spike train,
and point process realizations in general.
1.8 Conclusion
A reproducing kernel Hilbert space (RKHS) framework for optimization with spike
trains is introduced. Although the application of kernel methods to spike trains
without binning is not entirely novel [4, 31], a more general view of the problem
is presented. Instead of a top-down approach often taken in kernel methods, the
mCI kernel was built bottom-up from the concept of intensity functions which are
basic statistical descriptors of spike trains. Indeed, intensity functions are the core
concept of the statistical analysis of spike trains and is perhaps one of reasons why
binning is such a well-established technique, at any timescale of interest [28, 5].
Kernel methods applied before to spike trains seemed to have no connection to
intensity estimation. This chapter, however, bridges these two perspectives seam-
lessly. In one perspective, the mCI kernel approximates our intuitive understanding
regarding intensity functions as functional descriptors of point processes. On the
other hand, the evaluation (or estimation) of the mCI kernel for given spike trains
easily links to other methodologies in the literature. Most importantly, the approach
taken lends itself to generalization to other point process models and spike train
kernels nonlinear in the space of intensity functions taking advantage of the RKHS
mathematical structure and without sacrifice in rigor.
In addition to this enlightening connection of point of view, the rigorous yet gen-
eral mathematical approach toward the problem of optimization for manipulating
