Information Technology Reference
In-Depth Information
kernel in the RKHS to our intuitive understanding of the covariance function and
the associated linear statistics. In other words, due to the congruence between the
two spaces an algorithm can be derived and interpreted in any of the spaces.
1.3 Inner Product for Spike Times
Denote the
m
th spike time in a spike train indexed by
i
as
t
i
m
∈T
, with
m
∈
{
and
N
i
the number of spike times in the spike train. To simplify the
notation, however, the spike train index will be omitted if is irrelevant for the pre-
sentation or obvious from the context.
The simplest inner product that can be defined for spike trains operates with only
two spike times at a time as observed by Carnell and Richardson [4]. In the general
case, such an inner product can be defined in terms of a kernel function defined on
T×T
1
,
2
,...,
N
i
}
denote such a kernel.
Conceptually, this kernel operates in the same way as the kernels operating on data
samples in kernel methods [29] and information theoretic learning [24]. Although
it operates only with two spike times, it will play a major role whenever we operate
with complete realizations of spike trains. Indeed, as the next sections show, the
estimators for one of the kernels we define on spike trains rely on this kernel as an
elemental operation for computation.
To take advantage of the framework for statistical signal processing provided by
RKHS theory,
into the reals, with
T
the interval of spike times. Let
κ
is required to be a symmetric positive definite function. By the
Moore-Aronszajn theorem [1], this ensures that an RKHS
κ
H
κ
must exist for which
κ
is a reproducing kernel. The inner product in
H
κ
is given as
κ
(
t
m
,
t
n
)=
κ
(
t
m
, ·
)
,
κ
(
t
n
, ·
)
H
κ
=
Φ
m
,
Φ
n
H
κ
,
(1.4)
where
Φ
m
is the element in
H
κ
corresponding to
t
m
(that is, the transformed spike
time).
Since the kernel operates directly on spike times and is, typically, undesirable to
emphasize events in this space,
κ
is further required to be
shift-invariant
; that is, for
any
θ
∈
R
,
κ
(
t
m
,
t
n
)=
κ
(
t
m
+
θ
,
t
n
+
θ
)
,
∀
t
m
,
t
n
∈T.
(1.5)
In other words, the kernel is only sensitive to the difference of the arguments and,
consequently, we may also write
.
For any symmetric, shift-invariant, and positive definite kernel, it is known that
κ
(
t
m
,
t
n
)=
κ
(
t
m
−
t
n
)
.
2
This is important in establishing
κ
(
as a similarity measure between
spike times. In other words, as usual, an inner product should intuitively measure
some form of inter-dependence between spike times. However, notice that the con-
ditions posed do not restrict this study to a single kernel. Quite on the contrary, any
0
)
≥|
κ
(
θ
)
|
κ
2
This is a direct consequence of the fact that symmetric positive definite kernels denote inner
products that obey the Cauchy-Schwarz inequality.
