Biomedical Engineering Reference
In-Depth Information
aspikein
[
t
,
t
+
)
given
H
t
and that there has been no spike in
(
0
,
t
)
.Thatis,
Pr
(
u
∈
[
t
,
t
+
)
∩
u
>
t
|
H
t
)
Pr
(
u
∈
[
t
,
t
+
)
|
u
>
t
,
H
t
)=
Pr
(
u
>
t
|
H
t
)
Pr
(
u
∈
[
t
,
t
+
)
|
H
t
)
=
Pr
(
u
>
t
|
H
t
)
t
+
p
(
u
|
H
u
)
du
t
=
(9.3)
t
1
−
p
(
u
|
H
u
)
du
0
p
(
t
|
H
t
)
=
du
+
o
(
)
t
−
(
|
)
1
p
u
H
u
0
=
l
(
t
|
H
t
)
+
o
(
)
(
)
where
o
, such as two or more events
(spikes) occurring in an arbitrarily small interval. This establishes Equation (9.2).
The power of the conditional intensity function is that if it can be defined as Equation
(9.3) suggests then, it completely characterizes the stochastic structure of the spike
train. In any time interval
refers to all events of order smaller than
defines the probability of a spike given
the history up to time
t
. If the spike train is an inhomogeneous Poisson process then,
l
[
t
,
t
+
)
,
l
(
t
|
H
t
)
becomes the Poisson rate function. Thus, the conditional intensity
function (Equation (9.1)) is a history-dependent rate function that generalizes the
definition of the Poisson rate. Similarly, Equation (9.1) is also a generalization of the
hazard function for renewal processes [15, 20].
We can write
(
t
|
H
t
)=
l
(
t
)
d
log
t
[
1
−
p
(
u
|
H
u
)
du
]
0
l
(
t
|
H
t
)=
−
(9.4)
dt
or on integrating we have
log
1
du
t
t
−
l
(
u
|
H
u
)
du
=
−
p
(
u
|
H
u
)
.
(9.5)
0
0
Finally, exponentiating yields
exp
du
t
t
−
l
(
u
|
H
u
)
=
1
−
p
(
u
|
H
u
)
du
.
(9.6)
0
0
Therefore, by Equations. (9.2) and (9.6) we have
exp
du
t
(
|
)=
(
|
)
−
(
|
)
.
p
t
H
t
l
t
H
t
l
u
H
u
(9.7)
0
Together Equations (9.2) and (9.7) show that given the conditional intensity func-
tion the interspike interval probability density is specified and vice versa. Hence,
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