Biomedical Engineering Reference
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defining one completely defines the other. This relation between the conditional in-
tensity or hazard function and the inter-event time probability density is well known
in survival analysis and renewal theory [15, 20]. Equations (9.2) and (9.7) show
that it holds for a general point process model. This relation is exploited in the data
analysis examples we discuss.
9.2.2
The likelihood function of a point process model
The likelihood of a neural spike train, like that of any statistical model, is defined
by finding the joint probability density of the data. We show in the next proposition
that the joint probability of any point process is easy to derive from the conditional
intensity function.
Proposition 1
.Given0
T
, a set of neural spike train mea-
surements, the sample path probability density of this neural spike train, i.e.
<
u
1
<
u
2
< ···<
u
J
<
the
probability density of exactly these
J
events in
(
0
,
T
]
,is
exp
du
T
J
'
j
p
(
N
0:
T
)=
l
(
u
j
|
H
u
j
)
−
l
(
u
|
H
u
)
0
=
1
(9.8)
exp
T
0
du
T
=
log l
(
u
|
H
u
)
dN
(
u
)
−
l
(
u
|
H
u
)
.
0
K
k
=
1
Proof.
Let
{
t
k
}
be a partition of the observation interval
(
0
,
T
]
.Take
k
=
t
k
−
0. Assume that the partition is sufficiently fine so that there is
at most one spike in any
t
k
−
1
,where
t
0
=
(
t
k
−
1
,
t
k
]
. For a neural spike train choosing
k
≤
1msec
would suffice. We define
dN
(
k
)=
1 if there is a spike in
(
t
k
−
1
,
t
k
]
and 0 otherwise,
and the events
A
k
=
{
spike in
(
t
k
−
1
,
t
k
]
|
H
k
}
dN
(
k
)
A
k
}
1
−
dN
(
k
)
E
k
=
{
A
k
}
{
(9.9)
1
E
j
k
−
1
H
k
=
∩
j
=
for
k
=
1
, ··· ,
K
. In any interval
(
t
k
−
1
,
t
k
]
we have (
Figure 9.1B)
(
E
k
)=
(
t
k
|
H
k
)
k
+
(
k
)
Pr
l
o
(9.10)
E
k
)=
(
−
(
t
k
|
H
k
)
k
+
(
k
)
.
Pr
1
l
o
By construction of the partition we must have
u
j
∈
(
t
k
j
−
1
,
t
k
j
]
,
j
=
1
, ··· ,
J
for a subset
of the intervals satisfying
k
1
<
J
intervals have no
spikes. The spike events form a sequence of correlated Bernoulli trials. It follows
from Equation (9.10) and the Lemma in the Appendix, that given the partition, the
k
2
··· <
k
J
. The remaining
K
−
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