Biomedical Engineering Reference
In-Depth Information
Independently Coupled hMM.
One of the difficulties of the VQ-HMM is the distortion
associated with VQ, which limited the performance of the classification. In addition, because dif-
ferent neurons generally have different response properties, dynamic ranges and preferred stimuli,
synchronous neural operation is not a good assumption (but rather piecewise stationary). It is dif-
ficult, however, to train a HMM with a 104 input vector with the amount of data available in BMI
experiments. Therefore, our next implementation invoked an independence assumption among the
channels to create an ICHMM.
There is some evidence for this assumption. The literature explains that different neurons
in the brain modulate independently from other neurons [
45
] during the control of movement.
Specifically, during movement, different muscles may activate for synchronized directions and ve-
locities, yet, are controlled by independent neural masses or clusters in the motor cortex [
2
,
12
,
45
].
Conversely, within the neural clusters themselves, temporal dependencies (and coactivations) have
been shown to exist [
45
]. Therefore, for our HMM classifier, we make the assumption that enough
neurons are sampled from different neural clusters to avoid overlap or dependencies. We can further
justify this assumption by looking at the correlation coefficients (CC) between all the neural chan-
nels in our data set.
The best CCs (0.59, 0.44, 0.42, 0.36) occurred between only four out of thousands of possible
neural pairs, whereas the rest of the neural pairs were a magnitude smaller. We believe this indicates
weak dependencies between the neurons in our particular data set. In addition, despite these pos-
sible weak underlying dependencies, there is a long history of making such independence assump-
tions to create models that are tractable or computationally efficient. The factorial hidden Markov
model is one example among many [
46
].
By making an independence assumption between neurons, we can treat each neural channel
HMM independently. Therefore the joint probability
(D)
(1)
(2)
, . . . O
P
(
O
, O
,
|λ
full
)
(5.58)
T
T
T
becomes the product of the marginals
D
∏
|
1
(5.59)
( )
i
( )
i
P O
(
λ
)
T
i
=
of the observation sequences (each length
T
) for each
d
th HMM chain
λ
. Because the marginal
probabilities are independently coupled, yet try to model multiple hidden processes, we name this
classifier the ICHMM.
By using an ICHMM instead of a fully coupled HMM (FCHMM) (Figure
5.9
), the overall
complexity reduces from (
2
D
2
O TN
or
(
)
2
2
) to
O DTN
given that each HMM chain has
(
O TD N
(
)
a complexity of
O TN
. In addition, because we are using a single HMM chain to train on a
2
(
)
