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with two spike times at a time limits its practical use. In particular, spike trains are
sets
of spike times but we have not yet addressed the problem of how to combine the
kernel for all spike times. One immediate approach is to utilize the linearity of the
RKHS [4]. If the
m
th spike time is represented in the RKHS by
Φ
m
, then the spike
train can be represented in the RKHS as the sum of the transformed spike times,
N
m
=
1
Φ
m
.
Ψ
=
(1.9)
Notice that if a spike time is represented by a given function, say, an impulse, the
spike train will be a sum of time-shifted impulses centered at the spike times. Then
Equation (1.9) implies that the mapping of the spike train into the RKHS induced
by the spike time kernel is linear. Using the linearity of the RKHS it results that the
inner product of spike trains is
N
j
N
j
n
=
1
κ
(
t
i
m
,
t
n
)
.
N
i
m
=
1
N
i
m
=
1
Ψ
s
i
,
Ψ
s
j
n
=
1
Φ
n
i
m
j
H
κ
=
,
Φ
H
κ
=
(1.10)
It must be remarked that Equation (1.10) is only one example of a spike train
kernel from inner products on spike times. Indeed, as is commonly done in kernel
methods, more complex spike train kernels can be defined utilizing the kernel on
spike times as a building block equating the nonlinear relationship between the spike
times. On the other hand, the main disadvantage in this approach toward spike train
analysis is that the underlying model assumed for the spike train is not clearly stated.
This is important in determining and understanding the potential limitations of a
given spike train kernel for data analysis.
Rather than utilizing this direct approach, an alternative construction is to define
first a general inner product for the spike trains from the fundamental statistical
descriptors. In fact, it will be seen that the inner product for spike trains builds upon
the kernel on single spike times. This bottom-up construction of the kernel for spike
trains is unlike the previous approach and is rarely taken in machine learning, but it
exposes additional insight on the properties of the kernel and the RKHS it induces
for optimization and data analysis.
A spike train is a realization of an underlying stochastic point process [33]. In
general, to completely characterize a point process, the conditional intensity func-
tion must be used. The Poisson process is a special case because it is memoryless
and, therefore, the intensity function (or rate function) is sufficient [33, Chapter 2].
Spike trains in particular have been found to be reasonably well modeled as realiza-
tions of Poisson processes [28, Chapter 2]. Hence, for the remaining of this study
only Poisson spike trains are considered.
Consider two spike trains,
s
i
,
s
j
∈S
(
T
)
, with
i
,
j
∈
N
. Denote the intensity of
the underlying Poisson processes by
λ
s
i
(
t
)
and
λ
s
j
(
t
)
, respectively, where
t
∈T
=
[
denotes the time coordinate. Note that the dependence of the intensity function
on
t
indicates that the Poisson processes considered may be inhomogeneous (i.e.,
nonstationary). For any practical spike train and for finite
T
, we have that
0
,
T
]
