Likelihood Methods for Neural Spike Train Data Analysis - Computational Neuroscience: A Comprehensive Approach - page 271

1

SPATIAL
IP
IG
IIG

0.5

0

0 0.5 1

Model Quantiles

Figure 9.7
Kolmogorov-Smirnov plots of the spatial smoothing model (dotted line), and the
maximum likelihood fits of the inhomogeneous Poisson (IP) (dashed line), inhomo-
geneous gamma (IG) (thin solid line), and inhomogeneous inverse Gaussian (IIG)
(thick solid) models. As in Figure 9.4, the parallel diagonal lines are the 95% con-
fidence bounds for the degree of agreement between the models and the spike train
data.

conditional intensity function, is computed from Equation (9.2) using the maximum
likelihood estimate of q. For both of these models this equation defines a spatio-
temporal rate function whose spatial component is defined by Equation (9.34). This
is why the spatial components of these models are not the spatial rate function. For
the IP model Equation (9.2) simplifies to Equation (9.34). The smoothed spatial rate
model makes no assumption about temporal dependence, and therefore, it implicitly
states that its estimated rate function is the rate function of a Poisson process.
The KS plot goodness-of-fit comparisons are shown in Figure 9.7. The IG model
overestimates at lower quantiles, underestimates at intermediate quantiles, and lies
within the 95% confidence bounds at the upper quantiles (Figure 9.7). The IP model
underestimates the lower and intermediate quantiles, and like the IG model, lies
within the 95% confidence bounds in the upper quantiles. The KS plot of the spatial
rate model is similar to that of the IP model, yet closer to the confidence bounds.
This analysis suggests that the IG, IP and spatial rate models are most likely over-
smoothing this spike train because underestimating the lower quantiles corresponds
to underestimating the occurrence of short ISIs [3]. The fact that the IP and IG mod-