Biomedical Engineering Reference
In-Depth Information
response variability due to the unreliable release of neurotransmitters, stochasticity
9
in channel gat-
ing, and fluctuations in the membrane potential activity? If the irregularity arises from stochastic
forces, then the irregular interspike interval may reflect a random process. If we assume that the
generation of each spike depends only on an underlying driving signal of instantaneous firing rate,
then it follows that the generation of each spike is independent of all the other spikes; hence, we
refer to this as the
independent spike hypothesis
. If the independent spike hypothesis is true, then the
spike train would be completely described by a particular kind of random process called a Pois-
son process. Knowledge of this probability density function (PDF) would uniquely define all the
statistical measures (mean, variance, etc.) of the spike train. Although the statistical properties of
neural recordings can vary depending on the sample area, the animal, and the behavior paradigm, in
general, spike trains modeled as Poisson distribution perform reasonably is well-constrained condi-
tions [
35
]. The interspike interval from an example neuron of an animal performing a BMI task is
presented in Figure
2.5
a for comparison. The Poisson model has the great appeal of being defined
by a single constant λ, the firing rate, assumed known
x
e
−
λ
P
Poisson
(
x;
λ)
=
λ
∑
(2.2)
i
i
!
i
=
1
Certain features of neuronal firing, however, violate the independent spike hypothesis. Fol-
lowing the generation of an action potential, there is an interval of time known as the absolute
refractory period during which the neuron cannot fire another spike. For a longer interval known
as the relative refractory period, the likelihood of a spike being fired is much reduced. Bursting is
another non-Poisson feature of neuronal spiking.
Perhaps more realistically, the firing rate λ may change over time, yielding a nonhomoge-
neous Poisson process [
36
]. One of the simplest neural models that still captures a great deal of the
biological detail is the linear-nonlinear-Poisson (LNP) model shown in Figure
2.5
b. This model
consists of a single linear filter,10
10
followed by an instantaneous nonlinear function that accounts for
response nonlinearities such as rectification and saturation, followed by a Poisson generative model.
The firing rate of the Poisson process is totally determined by the parameters of the linear filter
and the nonlinearity. Recently, this model has been used by Simoncelli et al. [
37
] to estimate better
the receptive field, which characterizes the computation being performed by the neuron when it
responds to an input stimulus. Later, we are going to delve into more detail on this example to show
9
A stochastic process is one whose behavior is nondeterministic. Specifically, the next state of the system is not fully
determined by the previous state.
10
The linear filter refers to a linear receptive field. This will be described in the next section.
