Likelihood Methods for Neural Spike Train Data Analysis - Computational Neuroscience: A Comprehensive Approach - page 260

unit rate. Berman and Ogata derived their transformations by applying the general
form of the theorem. An elementary proof of the time-rescaling theorem is given in
[8].
We use the time-rescaling theorem to construct goodness-of-fit tests for a neural
spike data model. Once a model has been fit to a spike train data series we can
compute from its estimated conditional intensity the rescaled times

u j

t j = l ( u | H u ) du , (9.21)

u j − 1

for j J . If the model is correct then, according to the theorem, the t j s

are independent exponential random variables with mean 1. If we make the further
transformation

= 1 , ··· ,

= − ( − ) ,

(9.22)
then z j s are independent uniform random variables on the interval [0,1). Because
the transformations in Eqs. (9.21) and (9.22) are both one-to-one, any statistical
assessment that measures agreement between the z j s and a uniform distribution di-
rectly evaluates how well the original model agrees with the spike train data. We use
Kolmogorov-Smirnov tests to make this evaluation [8].
To construct the Kolmogorov-Smirnov test we order the z j s from smallest to
largest, denoting the ordered values as z j s and then plot the values of the cumu-
lative distribution function of the uniform density defined as b j =(

z j 1 exp t j

j − 1 / 2 ) / J for

J against the z j s. If the model is correct, then the points should lie on a
45 o line. Confidence bounds for the degree of agreement between the models and the
data may be constructed using the distribution of the Kolmogorov-Smirnov statistic
[19]. For moderate to large sample sizes the 95% confidence bounds are well ap-
proximated as b j ±

j = 1 , ··· ,

J 1 / 2 [19]. We term these plots Kolmogorov-Smirnov (KS)

1 . 36 /

plots.

9.3 Applications

9.3.1 An analysis of the spiking activity of a retinal neuron

In this first example we study a spike train data series from a goldfish retinal ganglion
cell neuron recorded in vitro ( Figure 9.2) . The data are 975 spikes recorded over 30
seconds from neuron 78 in [18]. They were provided by Dr. Satish Iyengar from
experiments originally conducted by Dr. Michael Levine at the University of Illinois
[22, 23]. The retinae were removed from the goldfish and maintained in a flow of
moist oxygen. Recordings of retina ganglion cells were made with an extracellular
microelectrode under constant illumination.
The plot of the spikes from this neuron (Figure 9.2) reveals a collection of short
and long interspike intervals (ISI). To analyze these data we consider three ISI prob-
ability models: the gamma, exponential and inverse Gaussian probability densities.