Adaptive Algorithms for Point Processes - Brain-Machine Interface Engineering - page 183

of milliseconds to the second scale). Although at the microscale the spike timing is appropriately
modeled by a Poisson random process, behavior is exquisitely organized albeit at a much larger
time scale. Modeling methods should appropriately exploit this gap in time scales, instead of using
simple averaging of spike occurrences in windows, as we do today.
The tuning function is a good example of a model that builds a functional stochastic relation-
ship across time scales. In the estimation of the tuning function, most researchers use an exponential
nonlinearity, but for a more realistic assessment, it is necessary to estimate the tuning functions of
the neurons in a training set, and also to estimate the delays between the neural firing and the actual
movement. This will have to be done in the training data, but once completedand assuming that
the neurons do not change their characteristicsthe model is ready to be used for testing (possibly
even in other types of movements not found in the training set).
6.4.1 Spike generation From Neuronal Tuning
Many different functional forms of tuning have been proposed. Most consist of linear projec-
tions of the neural modulation on two or three dimensions of kinematic vectors and bias. Moran
and Schwartz introduced an exponential velocity and direction tuned motor cortical model [ 19 ].
Eden et al have used a Gaussian tuning function for hippocampal pyramidal neurons [ 6 ] . These
general mathematical models may not be optimal for dealing with the real data because the tun-
ing properties across the ensemble can vary significantly. Moreover, the tuning may change over
time. The accuracy of the tuning function estimation can directly affect the modeling in the
Bayesian approach and the results of kinematic estimation.
One appropriate methodology is to estimate neural tuning using the training set data obtained
in experiments. Marmarelis and Naka [ 23 ] developed a statistical method, called white noise analysis,
to model the neural responses with stochastic stimuli. This method was improved by Simoncelli et
al. [ 1 ]. By parametric model identification, the nonlinear property between the neural spikes and the
stimuli was directly estimated from data, which is more reliable than just assuming a linear or Gaussian
shape. Here we will describe how to use sequential state estimation on a point process algorithm to
infer the kinematic vectors from the neural spike train, which is the opposite of sensory representation
in spite of the fact that the kinematic vector can be regarded as the outcome of the motor cortex neu-
romodulation. The tuning function between the kinematic vector and the neural spike train is exactly
the observation model between the state and the observed data in our algorithm.
6.4.1.1 Modeling and assumptions. Recall that in Chapter 2 we show that a tuning function
can be modeled as a linear filter coupled to a static nonlinearity followed by a Poisson model, the
so-called linear-nonlinear-Poisson (LNP) model (Figure 6.5 ).