Biomedical Engineering Reference
In-Depth Information
C H A P T E R 6
adaptive algorithms for Point Processes
additional Contributor: yiwen wang
In this chapter, we review the design of adaptive filters for
point processes
under the Gaussian assump-
tion, and then introduce a Monte Carlo sequential estimation, to probabilistically reconstruct the state
from discrete (spiking) observation events. In the previous chapter, we covered a similar topic but the
observations were assumed continuous random variables (i.e., firing rates). On the other hand, here
they are point processes, that is, 0/1 signals that contain the information solely in their time structure.
So the fundamental question is how to adapt the Bayesian sequential estimation models described in
Chapter
5
to point processes. The answer is provided by working with the probability of neural firing
(which is a continuous random variable). We will use the well-accepted Poisson model of spike gen-
eration introduced in Chapter
2
, but this time making the firing rate dependent on the system state.
Methods for analyzing spike trains have been applied primarily to understand how neurons
encode information [
1
]. In motor control BMIs, the problem is actually the reverse, where a process
called
decoding
[
2
] identifies how a spike train in motor cortex can explain the movement of a limb.
However, the primary methodologies are still inspired by the
encoding
methods. For example, the
population vector method of Georgopoulos [
3
] is a generative model of the spike activity based on
the tuning curve concept (preferential firing for a given hand position/speed) that has been exten-
sively utilized in encoding methods. In BMIs, the population vector technique has been championed
by Schwartz et al. [
4
]. All the encoding methods effectively model the PDF of the spike firings. An
alternative methodology that effectively bypasses this requirement is the use of maximum likeli-
hood methods, assuming a specific PDF. In neuroscience, the Poisson distribution assumption is
very common because it has been validated in numerous experimental setups, but it cannot account
for multimodal firing histograms that are often found in neurons. The Poisson model has been im-
proved with a time varying mean to yield what is called the inhomogeneous Poisson model.
A general adaptive filtering paradigm for point processes was recently proposed in [
5
] to
reconstruct the hand position from the discrete observation of neural firings. This algorithm mod-
eled the neural spike train as an inhomogeneous Poisson process feeding a kinematic model through
a nonlinear tuning function. The point process counterparts of the Kalman filter, recursive least
squares, and steepest descent algorithms were derived and recently compared in the decoding
of tuning parameters and states from the ensemble neural spiking activity [
6
]. The point process
