Biomedical Engineering Reference
In-Depth Information
smoothness because of the number of free parameters (12 073 for a system with 104 neuronal inputs
and 9 outputs of 3D position, velocity, and acceleration).
5.4 PaRTIClE FIlTERS
Although the Kalman filter provides a closed-form decoding procedure for linear Gaussian models,
we have to consider the fact that the relationship between neuronal activity and behavior may be
nonlinear. Moreover, measured neuronal firing often follows Poisson distributions, and even after
binning, there is reason to believe that the Gaussian assumption is too restrictive. The consequences
for such a mismatch between model and the real system will be expressed as additional errors in the
final position estimates. To cope with this problem, we need to go beyond the linear Gaussian model
assumption. In principle, for an arbitrary nonlinear dynamical system with arbitrary known noise
distributions, the internal states (
hP
,
hV
, and
ha
) can be still estimated from the measured outputs
(neuronal activity) using the sequential estimation framework presented. In the BMI literature, re-
searchers have already implemented the most general of these models called the particle filter [
15
].
The particle filter framework alleviates the restrictions of the Kalman filter (linearity and
Gaussianity) but substantially complicates the computations because, as a result of the generality of
the model, there is no closed form solution and therefore the posterior distribution has to be esti-
mated by probing. To help create at the output an estimate of the posterior density, a set of samples
drawn from a properly determined density that is estimated at each step, is sent through the system
with the present parameters. The peak of this posterior (or another central moment) is considered
as the state estimate. Particle filters have also been applied to BMIs [
26
] where the tuning function
has been assumed exponential on linear filtered velocities [
32
].
In this most general framework, the state and output equations can include nonlinear func-
tions as given in (
5.22
) and (
5.23
).
.
(5.22)
x
=
F x
(
,
v
)
-
k
k
k
-
1
k
1
)
.
(5.23)
z
=
H x n
(
,
k
k
k
k
where
F
k
: ´Â ®Â
and
H
k
: ´Â ®Â
are known, possibly
nonlinear,
functions of
n
x
n
v
n
x
n
x
n
n
n
z
n
k
∈
are both independent and identically distributed
(i.i.d.) process noise;
n
x
,
n
v
,
n
z
, and
n
n
are dimensions of the state
x
k
, noise vector
v
k
-1,
the measure-
ment
z
k
, and measurement noise vector
n
k
, respectively; and
N
is the set of natural numbers. Note
that
v
k
and
n
k
are assumed
non-Gaussian
in the particle filter.
To deal with these general distributions in a probabilistic way, the integrals have to be evalu-
ated in a numeric way. Particle filtering is implemented to propagate and update the posterior den-
sity of the state
x
k
given the measurement
z
k
recursively over time. Particle filtering uses Sequential
Importance Sampling [
33
] to discretely approximate the posterior distribution. The key idea is to
{
v
k
,
k
∈
N
}
{
,
k
N
}
the state
x
k
-1
,
, and
−
1
