Biomedical Engineering Reference
In-Depth Information
p
p
11
1
C
p
p
,
21
2
C
P
=
p
p
(3.12)
M
1
MC
where
p
ic
is the cross-correlation vector between neuron
i
and the
c
coordinate of hand position. The
estimated weights
w
Wiener
are optimal based on the assumption that the error is drawn from white
Gaussian distribution, and the data are stationary. The predictor
w
T
Wiener
x
minimizes the mean
square error (MSE) cost function,
2
J
=
E
[
e
],
e
= −
d
y
,
(3.13)
Each sub-block matrix
r
ij
can be further decomposed as
r
( )
0
r
( )
1
r
(
L
−
1
)
ij
ij
ij
r
(
−
1
)
r
( )
0
r
(
L
−
2
)
,
ij
ij
ij
r
ij
=
r
(
1
−
L
)
r
(
2
−
L
)
r
( )
0
(3.14)
ij
where
r
ij
(τ) represents the correlation between neurons
i
and
j
with time lag τ. These correlations,
which are the second-order moments of discrete time random processes
x
i
(
m
) and
x
j
(
k
), are the
functions of the time difference (
m
−
k
) based on the assumption of wide sense stationarity (
m
and
k
denote discrete time instances for each process). Assuming that the random process
x
i
(
k
) is ergodic
for all
i
, we can utilize the time average operator to estimate the correlation function. In this case,
the estimate of correlation between two neurons,
r
ij
(
m
−
k
), can be obtained by
1
N
∑
r
(
m
− =
k
)
E x
[
(
m x
)
( )]
k
≈
x
(
n
−
m x
)
(
n
−
k
)
,
(3.15)
ij
i
j
i
j
N
−
1
n
=
1
The cross-correlation vector
p
ic
can be decomposed and estimated in the same way.
r
ij
(
m
−
k
) is estimated using (
3.15
) from the neuronal bin count data with
x
i
(
m
) and
x
j
(
k
) being
the bin count of neurons
i
and
j
, respectively. From (
3.15
), it can be seen that
r
ij
(
m
−
k
) is equal to
r
ji
(
k
−
m
). Because these two correlation estimates are positioned at the opposite side of the diagonal
entries of
R
, the equality leads to a symmetric
R
. The symmetric matrix
R
, then, can be inverted
effectively by using the Cholesky factorization [
28
]. This factorization reduces the computational
complexity for the inverse of
R
from
O
(
N
3
) using Gaussian elimination to
O
(
N
2
) where
N
is the
number of parameters.
In the application of this technique for BMIs, we must consider the nonstationary charac-
teristics of the input, which can be investigated through observation of the temporal change of the
