Generating Quantitatively Accurate, but Computationally Concise, Models of Single Neurons - Computational Neuroscience: A Comprehensive Approach - page 203

What would we expect if spikes were generated wholly independently of the previ-
ous incidence of spikes? For a renewal process, the probability of an event occurring
at any given time is independent of the timing of preceding events. In particular:

1. The tail of the inter-spike interval distribution should be well described by a
single negative exponential (i.e. except where the refractory period of the cell
prevents firing).

2. Data should show invariant statistical characteristics when shuffled randomly.

3. As the variance of the event frequency (s 2 ) equals the mean of the event fre-
quency (m) for a Poisson process, the index of dispersion (s 2

m) should be
close to 1 (if the relative refractory period is relatively small), and should be
independent of bin-width.

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We analysed recordings from oxytocin cells to investigate how they deviate from
randomness by each of these criteria. For each cell, the sampled firing rates (sam-
pled in successive short time intervals, e.g., 1s bins) were expressed as a distribution,
showing the relative frequency of the occurrence of particular firing rates. These dis-
tributions are typically bell-shaped, with the peak around the mean firing rate, and a
rather symmetrical spread. In the next step the interspike intervals were shuffled ran-
domly creating a new recording . We recalculated the firing rates - again for 1s bins
- and compared the new distribution of firing rates to the original distribution. If the
original distribution is based on randomly occurring intervals, than further randomi-
sation should have no effect. However, in the overwhelming majority of cases, the
distribution of randomised firing rates was wider than the distribution of observed
firing rates ( Figure 7.6) . In other words, the observed firing rate distribution was
more uniform than would be expected if there were no serial dependence between
interspike intervals.
If randomisation did not consistently affect the shape of the distribution we might
have reasonably concluded that there were no activity-dependent influences that had
a significant influence on spike patterning beyond those that influence the interspike
interval distribution. Since it does, we can conclude that activity-dependent mech-
anisms underlie the spontaneous activity of oxytocin cells. The mechanisms are
weak in the sense that their effects are not readily apparent in the interspike interval
distribution, but are apparent on a time scale of 1s. In other words, variations in
average activity of the order observed from second to second are enough to produce
discernible feedback effects on spike activity.

7.3.7 Index of dispersion

We stated above what we expect from the index of dispersion if spike arrival would
be generated independently of previous spikes. The index of dispersion is the ratio
of the variance of firing rate to the mean firing rate. For a Poisson distribution, the
variance of the event frequency equals the mean, and the index of dispersion = 1, and
is independent of bin width. The closer to 0 the index of dispersion, the more ordered