Information Technology Reference
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2
s
i
(
T
λ
t
)
dt
<
∞
.
(1.11)
As a consequence, the intensity functions of spike trains are valid elements of
L
2
(
T
)
⊂
L
2
. Moreover, in this space, we can define an inner product of intensity
functions as the usual inner product in
L
2
,
s
j
)=
λ
s
i
,
λ
s
j
L
2
(
T
)
=
I
(
s
i
,
T
λ
s
i
(
t
)
λ
s
j
(
t
)
dt
.
(1.12)
We shall refer to
I
as the memoryless cross-intensity (mCI) kernel. Notice that
the mCI kernel incorporates the statistics of the processes directly and treats seam-
lessly even the case of inhomogeneous Poisson processes.
Furthermore, the definition of inner product naturally induces a norm in the space
of the intensity functions,
(
·, ·
)
s
i
(
λ
s
i
(
·
)
L
2
(
T
)
=
λ
s
i
,
λ
s
i
L
2
(
T
)
=
T
λ
t
)
dt
(1.13)
which is very useful for the formulation of optimization problems.
It is insightful to compare the mCI kernel definition in Equation (1.12) with the
so-called
generalized cross-correlation
(GCC) [18],
C
AB
(
θ
)=
E
{
λ
A
(
t
)
λ
B
(
t
+
θ
)
}
T
−
T
λ
A
(
(1.14)
1
2
T
=
lim
T
t
)
λ
B
(
t
+
θ
)
dt
.
→
∞
Although the GCC was proposed directly as a more general form of cross-correlation
of spike trains, one verifies that the two ideas are fundamentally equivalent. Never-
theless, the path toward the definition of mCI is more principled. More importantly,
this path suggests alternative spike train kernel definitions which may not require a
Poisson assumption, or, if the Poisson model is assumed, extract more information
in the event of deviations from the model.
1.5 Properties and Estimation of the Memoryless Cross-Intensity
Kernel
1.5.1 Properties
In this section some relevant properties of the mCI kernel are presented. In addition
to the knowledge they provide, they are necessary for a clear understanding of the
following sections.
Property 1.1.
The mCI kernel is a symmetric, nonnegative, and linear operator in
the space of the intensity functions.
