Biomedical Engineering Reference
In-Depth Information
studied during the BMI experiment.
1
In addition, some neuron firings are correlated with each
other, which creates problems in the estimation of LS (
R
may become rank deficient); thus, it may
be advantageous to blend these inputs to improve model performance. Subspace projection, which
can at the same time reduce the noise, and blend correlated input signals together, may be very
beneficial. Moreover, it also reduces the number of degrees of freedom in the multichannel data,
and consequently decreases the variance of the model for the same training data size. Here, we
introduce a hybrid subspace projection method that is derived by combining the criteria of PCA
and PLS. Then, we will design the subspace Wiener filter based on this hybrid subspace projection
for BMIs.
PCA, which preserves maximum variance in the subspace data, has been widely adopted as
a projection method [
14
]. The projection vector
w
PCA
is determined by maximizing the variance of
the projection outputs as
2
PCA
T
T
(4.24)
w
=
argmax
J
(
w
)
=
E
x w
=
w
w
R
PCA
s
w
where
R
s
is the input covariance matrix computed over the neuronal space only (it is an
M
×
M
matrix where
M
is the number of neurons).
x
is an
M
× 1 instantaneous neuronal bin count vector.
It has been well known that
w
PCA
turns out to be the eigenvector of
R
s
corresponding to the largest
eigenvalues. Then an
M
×
S
projection matrix that constructs an
S
-dimensional subspace consists
of
S
eigenvectors corresponding to the
S
largest eigenvalues. However, PCA does not exploit in-
formation in the joint space of the input (neuronal) and desired (behavioral) response. This means
that there may be directions with large variance that are not important to describe the correlation
between input and desired response (e.g., some neuronal modulations related to the BMI patient's
anticipation of reward might be substantial, but less useful for the direct estimation of movement
parameters), but will be preserved by the PCA decomposition.
One of the subspace projection methods to construct the subspace in the joint space is PLS,
which seeks the projection maximizing the cross-correlation between the projection outputs and
desired response [
22
]. Given an input vector
x
, and a desired response
d
, a projection vector of PLS,
w
PLS
, maximizes the following criterion,
=
( )
=
=
(4.25)
w
=
argmax
J
PLS
(
w
)
E
x
T
d
E
w x
T
(
d
)
w p
T
w
PLS
w
1
At the time of surgery, the targeting of neurons is related to the broad functional specialization of each area of
cortex. The ability to target specific neurons can be achieved through sensory, motor, or electrical stimulation.
However, it is common to obtain the activity of neurons not related to the motor task of interest.
