Biomedical Engineering Reference
In-Depth Information
now a filter with coefficients (1/2, 1/2). For a given discrete time series,
x
(
k
) =
v
0
(
k
), the first resolu-
tion is obtained by convolution
v
0
(
k
) with
h
, such that
1
v
(
k
)
=
(
v
(
k
)
+
v
(
k
−
1
)),
(2.13)
1
0
0
2
and the wavelet coefficients are obtained by
(2.14)
w
( )
k
= -
v
( )
k
v
( ).
k
1
0
1
For the
j
th resolution,
1
j
−
1
v
(
k
)
=
(
v
(
k
)
+
v
(
k
−
2
)
(2.15)
j
j
−
1
j
−
1
2
w
( )
k
=
v
( )
k
-
v
( ).
k
(2.16)
j
j
-1
j
Hence, the computation in this wavelet transform at time
k
involves only information from the
previous time step
k
− 1.
The HatWT provides a set of features from the time series data, such as the wavelet coeffi-
cients [
w
1
(
k
), … ,
w
S
− 1
(
k
)] and the last convolution output [
v
S − 1
(
k
)]. However, if we seek to associate
the HatWT with the binning process a spike trains, the set [
v
0
(
k
), … ,
v
S
− 1
(
k
)] can be translated into
the bin count data with multiple bin widths. To yield the multiscale bin count data using (
2.13
), we
only have to multiply
v
j
(
k
) by 2
j
, such that
u
j
(
k
)
= 2
j
v
j
(
k
)
,
for
j = 0, … , S − 1.
(2.17)
Hence, the convolution output in the HatWT provides a feature set related with binning with dif-
ferent window widths. In the design of decoding models for BMIs, we may utilize the scaled con-
volution outputs [
u
j
(
k
)] for
j
= 0,…,
S
− 1, or equivalently, the bin count data with different window
widths as input features.
To apply the multiresolution analysis to the BMI data, we must choose a suitable set of scales.
Although it is not straightforward to determine a set of scales for the HatWT for spike trains, the fir-
ing properties of neuronal data collected from the particular BMI experimental paradigm can be used
to help guide the determination of scales. For example, the smallest scale may be chosen to be larger
than 1 msec because of the refractory period of neuronal firing. Also, the largest scale may be chosen
