Biomedical Engineering Reference
In-Depth Information
1
0 . 9
0 . 8
0 . 7
0 . 6
0 . 5
0 . 4
0 . 3
0 . 2
0 . 1
0
0
0 . 2
0 . 4
0 . 6
0 . 8
1
1 . 2
1 . 4
1 . 6
1 . 8
2
t i m e
5
x 1 0
FIgURE 6.2:
Simulated neuron spike train generated by an exponential tuning function.
With the exponential tuning function operating on the velocity, we can see that when the
velocity is negative, there are few spikes; whereas when the velocity is positive, many spikes appear.
The problem lies in obtaining from this spike train the desired velocity of Figure
6.1
, assuming the
Poisson model of (
6.17
) and of course using one of the sequential estimation techniques discussed.
To implement the Monte Carlo sequential estimation of the point process, we regard both
modulation parameter
b
k
and velocity
v
as the state
x
=
[
v
b
]
T
. One hundred samples of veloc-
k
k
k
ity
v
and modulation parameter
b
k
were initialized, respectively, with a uniform and with a Gauss-
ian distribution. The new samples are generated according to the linear state evolution (
6.6
), where
=diag[1 1]
0
F
and
Q
is the covariance matrix of the i.i.d. noise. The kernel size utilized in (
6.14
) to
estimate the maximum of the posterior density (through MLE) was the average spike interval.
To obtain realistic performance assessments of the different models (MLE and collapse), the
state estimations
v
k
~
β
~
for the duration of the trajectory are drawn 20 different times for different
runs of the noise covariance matrices
Q
for state generation. The MSE between the desired trajec-
tory and the model output is shown in Table 6.1 for the adaptive filtering and sequential estimation
with both MLE and collapse. In general, if
Q
is too large, one needs many samples to estimate
the PDF of the state appropriately and the state trajectory may not be smooth. If it is too small, the
reconstructed velocity may get stuck in the same position, whereas the simulated one moves away
by a distance much larger than
Q
.
The best velocity reconstruction, shown in the first row of Table
6.1
, by both methods is
shown in Figure
6.1
.
From Figure
6.3
and Table
6.1
, we can see that compared with the desired velocity (dash-
dotted line) the best velocity reconstruction was achieved by the sequential estimation with collapse
(solid black line). It is more sensitive than the adaptive filtering approach involving the Gaussian
k
