Foundations of Neuronal Representations - Brain-Machine Interface Engineering - page 45

In conjunction with these techniques, a second class of nonlinear models for BMIs was de-
rived from system theory using the concept of Volterra series expansions in Hilbert spaces. Volterra
series have been proposed to model neuronal receptive fields using the reverse correlation or white
noise approach [ 83 ], and more recently they have been applied to cognitive BMIs to model the
CA3 region of the hippocampus [ 84 ]. A Volterra series is similar to a Taylor series expansion in
functional spaces, where the terms are convolution integrals of products of impulse responses called
kernels. The Volterra models are universal approximators, but they require the estimation of many
parameters. Therefore, when applied in practice, one of the design difficulties is how to minimize
the parameters, either by limiting the expansion to the second or third term or by using polynomial
approximations, such as Laguerre polynomials [ 85 ] or, preferably, Kautz functions, to decrease the
number of free parameters for the impulse responses.
However, because the neural response tends to be nonlinear, and higher order terms of the
Volterra expansion have many parameters, lots of experimental data are needed to estimate the
parameters well. The LNP (as shown in Figure 2.5 ) can reduce the collection of large data sets by
using a more efficient estimation of the linear filter term, followed by the instantaneous nonlinear-
ity. The assumptions for the method to work are mild: basically, one assumes that the stimuli com-
ponents are uncorrelated and they are Gaussian distributed with the same variance (i.e., that the
D -dimensional distribution is spherically symmetric).
In the LNP model, the linear filter collapses the multidimensional input space to a line.
Spike-triggered averaging provides an estimate of this axis, under the assumption that the raw
stimulus distribution is spherically symmetric [ 86 ]. Once the linear filter has been estimated, one
may compute its response and then examine the relationship between the histograms of the raw and
spike-triggered ensembles within this one-dimensional space to estimate the nonlinearity that now
can be done with histograms without requiring large data sets. The model can be generalized to take
into consideration several directions at the same time by using the covariance. Such approaches have
been proposed by Moran and Schwartz introduced an exponential velocity and direction tuned mo-
tor cortical model [ 67 ]. Eden et al. used a Gaussian tuning function for the hippocampal pyramidal
neurons [ 87 ]. These nonlinear mathematical models can encounter difficulties when dealing with
the real data because the neurons can have very different tuning properties and probably change
over time. The accuracy of the tuning function estimation can directly affect the prior knowledge of
the Bayesian approach and, therefore, the results of the kinematic estimation. Simoncelli et al. built
upon the approach of Marmarelis and proposed a statistical method to model the neural responses
with stochastic stimuli [ 85 ]. Through parametric model identification, the nonlinear property be-
tween the neural spikes and the stimuli was directly estimated from data, which is more reliable. For
motor BMIs, one is seeking sequential state estimation of the point process algorithm to infer the
kinematic vectors from the neural spike train, which is the opposite of sensory neurons. However,