Biomedical Engineering Reference
In-Depth Information
variances, and the decision about the adequacy of the fitting is left to the experimenter. In the prac-
tical situation of BMIs, more important issues require our attention. In fact, the spike train time
series are not even stationary, and the mappings to the desired response are probably not linear. The
approaches presented thus far assume that the spike train statistics do not change over time. This is
an unrealistic assumption that can be counteracted using sample by sample adaptation [the famous
least mean square (LMS) algorithm] to track the optimal solution through time as will be shown in
3.1.2
or by partitioning the solution space into a set of local linear models that switch automatically
among themselves [
30
]. Moreover, the MIMO models have many parameters, and so generaliza-
tion is a major concern. Therefore, one must proceed with great caution.
3.1.1 linear Modeling for BMIs and the wiener Filter
Because the VAR decomposes into a parallel set of independent linear models (one for each output),
we will present here this more familiar notation. Moreover, because this solution coincides with the
Wiener-Hopf solution applied to discrete time data and finite impulse responsive filters (FIRs), it
will be called here the digital Wiener filter [
2
]. Consider a set of spike counts from
M
neurons, and
a hand position vector
d
∈ℜ
C
(
C
is the output dimension,
C
= 2 or 3). The spike count of each neu-
ron is embedded by an
L
-tap discrete time-delay line. Then, the input vector for a linear model at a
given time instance
n
is composed as
x
(
n
) = [
x
1
(
n
),
x
1
(
n
− 1) …
x
1
(
n
−
L
+1),
x
2
(
n
) …
x
M
(
n
−
L
+1)]
T
,
x
∈ℜ
L
⋅
M
, where
x
i
(
n
−
j
) denotes the spike count of neuron
i
at a time instance
n
−
j
. A linear model
estimating hand position at time instance
n
from the embedded spike counts can be described as
−
∑
(
L
1
M
y
c
=
x
n
−
j w
)
c
+
b
c
(3.8)
i
ij
i
=
0
j
=
1
where
y
c
is the
c
coordinate of the estimated hand position by the model,
w
ij
c
is a weight on the
connection from
x
i
(
n
−
j
) to
y
c
, and
b
c
is a bias for the
c
coordinate. The bias can be removed from
the model when we normalize
x
and
d
such that
E
[
x
] =
0
,
0
∈ℜ
L
⋅
M
, and
E
[
d
] =
0
,
0
∈ℜ
C
, where
E
[⋅]
denotes the mean operator. Note that this model can be regarded as a combination of three separate
linear models estimating each coordinate of hand position from identical inputs. In a matrix form,
we can rewrite (
3.8
) as
y
=
w
T
x
(3.9)
where
y
is a
C
-dimensional output vector, and
w
is a weight matrix of dimension (
L
⋅
M
+1)×
C
. Each
column of
w
consists of [
w
10
c
,
w
11
c
,
w
12
c
…,
w
1
L
− 1
c
,
w
20
c
,
w
21
c
…,
w
M
0
c
,…,
w
ML
−1
c
]
T
.
Figure
3.2
shows the topology of the MIMO linear model for the BMI application, which
will be kept basically unchanged for any linear model through this topic. The most signiicant dif-
