Likelihood Methods for Neural Spike Train Data Analysis - Computational Neuroscience: A Comprehensive Approach - page 272

els estimate a different temporal structure for this spike train data series is evidenced
by the fact that while they both have the same spatial model components ( Figure
9.6) , Their KS plots differ significantly ( Figure 9.7 ). This difference is due entirely
to the fact that ˆ y

= . =

1byas-
sumption. The IP, the IG, and the smoothed spatial rate function have the greatest
lack of fit in that order. Of the 4 models, the IIG is the one that is closest to the
confidence bounds. Except for an interval around the 0.30 quantile where this model
underestimates these quantiles, and a second interval around the 0.80 quantile where
it overestimates these quantiles, the KS plot of the IIG model lies within the 95%
confidence bounds.
The findings from the analysis of the spatial model fits (Figure 9.6) appear to
contradict the findings of the overall goodness-of-fit analysis (Figure 9.7). The IIG
gives the poorest description of the spatial structure in the data yet, the best overall
description in terms of the KS plot. The smoothed spatial rate function model seems
to give the best description of the data's spatial structure however, its overall fit is one
of the poorest. To reconcile the findings, we first note that the better overall fit of the
smoothed spatial rate function relative to the IP (Figure 9.7) is to be expected because
the IP estimates the spatial component of a Poisson model with a three parameter
model that must have a Gaussian shape. The smoothed spatial rate function model,
on the other hand, uses a smoothed histogram that has many more parameters to
estimate the spatial component of the same Poisson model. The greater flexibility
of the smoothed spatial rate function allows it to estimate a bimodal structure in
the rate function. Both the IP and smoothed spatial rate models use an elementary
temporal model in that both assume that the temporal structure in the data is Poisson.
For an inhomogeneous Poisson model the counts in non-overlapping intervals are
independent whereas the interspike interval probability density is Markov (Equation
(9.35)). The importance of correctly estimating the temporal structure is also seen
in comparing the IP and IG model fits. These two models have identical spatial
components yet, different KS plots because ˆy

0 61 for the IG model whereas for the IP model y

1forthe
IP model by assumption. The KS plots suggest that while the IIG model does not
describe the spatial component of the data well, its better overall fit comes because
it does a better job at describing the temporal structure in the spike train. In contrast,
the smoothed spatial rate function fits exclusively the spatial structure in the data to
the exclusion of the temporal structure.
In summary, developing an accurate model of the place-specific firing activity of
this hippocampal neuron requires specifying correctly both its spatial and temporal
components. Our results suggest that combining a flexible spatial model, such as in
the smoothed spatial rate function model with non-Poisson temporal structure as in
the IG and IIG models, should be a way of developing a more accurate description.
Another important consideration for hippocampal pyramidal neurons is that place-
specific firing does not remain static. The current models would not capture this
dynamic behavior in the data. In the next example we analyze a place cell from
this same experiment using a point process adaptive filter algorithm to estimate the
dynamics of the place cell spatial receptive fields using a model with flexible spatial
and temporal structures.

= 0 . 61 for the IG and y =