Likelihood Methods for Neural Spike Train Data Analysis - Computational Neuroscience: A Comprehensive Approach - page 265

from 25 to 50 msec ( Figure 9.3B) and of the three models, underpredicts the long
ISIs the least (Figure 9.3C).
Because of Equation (9.2), specifying the spike time probability density is equiv-
alent to specifying the conditional intensity function. From Equation (9.2) and the
invariance of the maximum likelihood estimate discussed in Section 9.2.5, it follows
that if q denotes the maximum likelihood estimate of q then the maximum likeli-
hood estimate of the conditional intensity function for each model can be computed
from Equation (9.2) as

q

p ( t | u N ( t ) , )

q

l ( t | H t , )=

du ,

(9.33) t

q

1 − p ( u | u N ( t ) , )

u N ( t )

for t u N ( t ) where u N ( t ) is the time of the last spike prior to t . The estimated condi-

tional intensity from each model may be used in the time-rescaling theorem to assess
model goodness-of-fit as described in Section 9.2.5.
An important advantage of the KS plot is that it allows us to visualize the goodness-
of-fit of the three models without having to discretize the data into a histogram
( Figure 9.4 ). While the KS plot for neither of the three models lies entirely within
the 95% confidence bounds, the inverse Gaussian model is closer to the confidence
bounds over the entire range of the data. These plots also show that the gamma model
gives a better fit to the data than the exponential model.
The AIC and KS distance are consistent with the KS plots ( Table 1) . The inverse
Gaussian model has the smallest AIC and KS distance, followed by the gamma and
exponential models in that order for both. The approximate 95% confidence interval
for each model parameter was computed from maximum likelihood estimates of the
parameters (Equations (9.27), (9.28), (9.30), (9.31)) and the estimated Fisher infor-
mation matrices (Equations (9.29), (9.32)) using Equation (9.18). Because none of
the 95% confidence intervals includes zero, all parameter estimates for all three mod-
els are significantly different from zero. While all three models estimate the mean
ISI as 30.73 msec, the standard deviation estimate from the inverse Gaussian model
of 49.0 msec is more realistic given the large number of long ISIs ( Figure 9.2B) .
In summary, the inverse Gaussian model gives the best overall fit to the retinal
ganglion spike train series. This finding is consistent with the results of [18] who
showed that the generalized inverse Gaussian model describes the data better than
a lognormal model. Our inverse Gaussian model is the special case of their gener-
alized inverse Gaussian model in which the index parameter of the Bessel function
in their normalizing constant equals

>

5. In their analyses Iyengar and Liao es-
timated the index parameter for this neuron to be

− 0 .

76. The KS plots suggests
that the model fits can be further improved. The plot of the spike train data series
(Figure 9.2) suggests that the fit may be improved by considering an ISI model that
would specifically represent the obvious propensity of this neuron to burst as well
as produce long ISIs in a history-dependent manner. Such a model could be derived
as the mixture model that Iyengar and Liao [18] suggest as a way of improve their
generalized inverse Gaussian fits.

− 0 .