A characterization of the time-rescaled gamma process as a model for spike trains

The occurrence of neuronal spikes may be characterized by not only the rate but also the irregularity of firing. We have recently developed a Bayes method for characterizing a sequence of spikes in terms of instantaneous rate and irregularity, assuming that interspike intervals (ISIs) are drawn from a distribution whose shape may vary in time. Though any parameterized family of ISI distribution can be installed in the Bayes method, the ability to detect firing characteristics may depend on the choice of a family of distribution. Here, we select a set of ISI metrics that may effectively characterize spike patterns and determine the distribution that may extract these characteristics. The set of the mean ISI and the mean log ISI are uniquely selected based on the statistical orthogonality, and accordingly the corresponding distribution is the gamma distribution. By applying the Bayes method equipped with the gamma distribution to spike sequences derived from different ISI distributions such as the log-normal and inverse-Gaussian distribution, we confirm that the gamma distribution effectively extracts the rate and the shape factor.