Biomedical Engineering Reference
In-Depth Information
binning at 100 msec is a reasonable value for motor BMIs when signal processing models based on
correlation are used. We can expect that there are neurons that modulate their firing rates concur-
rently with the movement, and therefore, linear or nonlinear input output models should provide
acceptable performance. But this picture also shows the difficulties between the tremendous gap in
resolution between the spike trains and the movement, that are difficult to bridge with pure data
driven operations (i.e., without involving some form of physiological modeling). Therefore, it opens
up many other interesting possibilities to improve the present state of the art in BMI modeling.
2.7.4 Using Firing Rates to Compute Tuning
Classically, a single motor neuron's physiological response to a behavioral task has been described
using directional tuning curves. As originally derived from a standard center-out task by Georgop-
oulos et al. [
63
], the tuning curve relates the mean of movement-related cell activity to movement
direction. The preferred direction of a neuron, measured in degrees, is the direction that yields the
maximal firing response over many trials. Tuning curves convey the expected value of a PDF indicat-
ing the average firing a cell will exhibit given a particular movement direction [
63
] as shown in Figure
2.10
. Cells that have cosine-shaped tuning curves participate in neural computation by forming a set
of basis functions
12
that represent the output. It has been shown in the literature that a variety of hand
trajectories can be reconstructed by simply weighting and summing the vectors indicating preferred
directions of cells in an ensemble of neurons [
56
,
64-66
]. Weighing the preferred directions with the
neural activities in the population gives a resultant direction vector called the “population vector,”
which has been shown to be correlated with the actual movement direction [
56
], and the modeling
technique will be described later. To use this measurement, the kinematic data histograms must first
be computed by using the desired behaviors from which the corresponding position, velocity, and ac-
celeration vectors (magnitude and direction) between successive points in the trajectory is computed.
The quantity that is commonly used in BMI experiments is the hand movement direction measured
as an angle between 0° and 360° [
67
]. Because cellular tuning can produce properties where the aver-
age of angle 0° and 360° is not 180° (this results from the wrap-around effect of the measurements),
the mean of each cell's hand direction tuning is computed using circular statistics as in (
2.1
)
13
[
68
],
i
θ
circular mean
= ∑
arg
r e
N
(2.1)
N
N
where
r
N
is the cell's average firing rate for angle θ
N
, where
N
is from 1 to 360.
12
A term used in vector spaces where a group of functions can be linearly combined to encode any function in a
subspace.
13
In computing the circular mean, we used the four quadrant inverse tangent.
