Information Technology Reference
In-Depth Information
1.1 Introduction
A spike train
s
∈S
(
T
)
is a sequence of ordered spike times
s
=
{
t
m
∈T
:
m
=
1
at which a
neuron fires. In a different perspective, spike trains are realizations of stochastic
point processes. Spike trains can be observed whenever studying either real or ar-
tificial neurons. In neurophysiological studies, spike trains result from the activity
of multiple neurons in single-unit recordings by ignoring the stereotypical shape of
action potentials [5]. And, more recently, there has also been a great interest in using
spike trains for biologically inspired computation paradigms such as the liquid-state
machine (LSM) [13,12] or spiking neural networks (SNN) [3,12]. Regardless of the
nature of the process giving rise to the spike trains, the ultimate goal is to filter or
classify the spike trains to manipulate or extract the encoded information.
Filtering, eigendecomposition, clustering, and classification are often formulated
in terms of a criterion to be optimized. However, formulation of a criterion and/or
optimization directly with spike trains is not a straightforward task. The most widely
used approach is to bin the spike trains, obtained by segmenting the spike train in
small intervals and counting the number of spikes within each interval [5]. The
advantage of this approach is that the randomness in time is mapped to randomness
in amplitude of a discrete-time random process, and, therefore, our usual statistical
signal processing and machine learning techniques can be applied. It is known that if
the bin size is large compared to the average inter-spike interval this transformation
provides a rough estimate of the instantaneous rate. However, the discretization of
time introduced by binning leads to low resolution.
The caveats associated with binned spike trains have motivated alternative
methodologies involving the spike times directly. For example, to deal with the
problem of classification, Victor and Purpura [36, 37] defined a distance metric be-
tween spike trains resembling the edit distance in computer science. An alternative
distance measure was proposed by van Rossum [34]. Using spike train distances for
classification simplifies the problem to that of finding a threshold value. However,
for more general problems the range of applications that can be solved directly using
distances is limited since these metrics do not lend themselves to optimization. The
reason is that although distances are useful concepts in classification and pattern
analysis they do not provide a general framework for statistical signal processing
and machine learning. Recent attempts were also made to develop a mathematical
theory from simple principles [4, 31], such as the definition of an inner product and
an associated kernel, but these developments are mainly associated with the earlier
proposed distance measures [37, 34].
The framework described in this chapter is different in the sense that it does
not attempt to propose a distance or criterion directly. Rather, we propose to de-
fine first inner product kernel functions
1
,...,
N
}
corresponding to the time instants in the interval
T
=[
0
,
T
]
for spike trains. These kernels induce
1
Throughout this document we will refer to inner products and kernels indistinguishably since
they represent the same concept. However, stated more correctly, kernels denote inner products in
a reproducing kernel Hilbert space of functions on the arguments of the kernel.
