Likelihood Methods for Neural Spike Train Data Analysis - Computational Neuroscience: A Comprehensive Approach - page 275

intensity of spiking activity (number of spikes per pass) increases from the start of
the experiment to the end.
We used the spline model of the conditional intensity function (Equation (9.41))
in the adaptive filter algorithm (Equation (9.40)) to estimate the dynamics of the
receptive field of the neuron whose spiking activity is shown in Figure 9.8. The pa-
rameter updates were computed every 2 msec and the learning rate parameters were
chosen based on the sensitivity analysis described in [13]. Examples of the spatial
and temporal components of the conditional intensity function are shown in Figure
9.9. The migration of the spatial component during the course of the experiment is
evidenced by the difference between these functions on the first pass compared with
the last pass (Figure 9.9A). On the first pass the spatial function has a height of 12,
is centered at approximately 40 cm and extends from 15 to 55 cms. By the last pass,
the center of the spatial function has migrated to 52 cm, its height has increased to
almost 20 and the range of the field extends from 15 to 70 cms. The migration of
this spatial function is an exception. Typically, the direction of field migration is in
the direction opposite the one in which the cell fires relative to the animal's motion
[24, 25]. This place field migrates in the direction that the neuron fires relative to the
animal's motion.
The temporal component of the intensity function characterizes history depen-
dence as a function of the amount of time that has elapsed since the last spike. The
temporal function shows increased values between 2 to 10 msec and around 100
msec. The former corresponds to the bursting activity of the neuron whereas the
latter is the modulation of the place specific firing of the neuron by the theta rhythm
[13]. For this neuron the modulation of the spiking activity due to the bursting ac-
tivity is stronger than the modulation due to the approximately 6 to 14 Hz theta
rhythm. Between the first and last pass the temporal component of the conditional
intensity function increases slightly in the burst range and decreases slightly in the
theta rhythm range. By definition, the rate function, i.e., the conditional intensity
function based on the model in Equation (9.41) is the product of the spatial and tem-
poral components at a given time. This is the reason why the units on the spatial and
temporal components ( Figure 9.9) are not spikes/sec. However, the product of the
spatial and temporal components at a given time gives the rate function with units of
spikes/sec. A similar issue arose in the interpretation of the spatial components of
the conditional intensity functions for the IG and IIG models in Section 9.3.2 ( Figure
9.6) .
As in the previous two examples we used the KS plots based on the time-rescaling
theorem to assess the goodness-of-fit of the adaptive point process filter estimate of
the conditional intensity function ( Figure 9.10) . We compared the estimate of the
conditional intensity function with and without the temporal component. The model
without the temporal component is an implicit inhomogeneous Poisson process. The
impact of including the temporal component is clear from the KS plots. For the
model without the temporal component the KS plot does not lie within the 95%
confidence bounds, whereas with the temporal component the plot is completely
within the bounds. The improvement of the model fit with the temporal component
is not surprising given that this component is capturing the effect of theta rhythm and