Biomedical Engineering Reference
In-Depth Information
i
where
D
p N
x
is defined by (
6.4
) in this section. Using (
6.6
), (
6.13
), and the resampling step,
the posterior density of the state
(
|
)
k
k
t
can
x
given the whole path of the observed events up to time
be approximated as
N
S
å
i
i
p
(
x
|
N
)
»
p
(
D
N
|
x
)
× -
k
(
x
x
)
(6.14)
k
1:
k
k
k
k
k
i
=
1
Equation (
6.14
) shows that the posterior density of the current state given the observation is
modified by the latest probabilistic measurement of observing the spike event
D
i
p
(
N
|
x
)
, which
k
k
is the update stage in the adaptive filtering algorithm.
Without a closed form solution for state estimation, we estimate the posterior density of the
state given the observed spike event
p N
x
at every step and apply two methods to get the state
estimation
~
k
. One possible method is maximum likelihood estimation (MLE), which picks out the
sample
(
|
)
k
1:
k
i
x
with maximum posterior density, or alternatively the expectation of the posterior density can
be picked as the state. As we smooth the posterior density by convolving with a Gaussian kernel, we can
easily obtain the expectation
~
k
, and its error covariance
*
V
by a technique called collapse [18]:
N
S
∑
∆
1
~
=
p N
(
|
i
)
⋅
i
x
x
x
(6.15)
k
k
k
k
i
=
N
S
∑
(
~
~
V
=
p N
∆
x
|
i
) (
⋅ + −
σ
(
i
)(
i
−
) )
T
x
x
x
x
k
k
(6.16)
k
k
k
k
k
i
=
1
From (
6.15
) and (
6.16
), it is evident that with simple computation one can easily estimate the
next state. Hence, the expectation by collapse is simple and elegant.
The major drawback of the algorithm is computational complexity because the quality of the
solution requires many particles
{
=
L
to approximate the posterior density. Smooth-
ing the particles with kernels as in (
6.14
) alleviates the problem in particular when collapsing is
utilized, but still the computation is much higher than calculating the mean and covariance of the
PDF with a Gaussian assumption.
i
,
i
1
,
,
N
}
x
0
:
k
S
6.3 SIMUlaTIoN oF MoNTE CaRlo SEQUENTIal
ESTIMaTIoN USINg SPIKE TRaINS
Next, we will describe how the framework presented in Section
6.2
can be used to determine
neuronal receptive fields. In a conceptually simplified motor cortical neural model [
19
], the one-
