A unified description of cerebellar inter-spike interval distributions and variabilities using summation of Gaussians

Neuronal inter-spike intervals (ISIs) have previously been described as Poisson, Gamma, inverse Gaussian or other unimodal distributions. We analyzed ISIs of rhythmic and arrhythmic neuronal spike trains in cerebellum recorded from freely behaving rats, and found that their distributions can be described as the summation or integration of multiple Gaussian distributions. The ISIs of rhythmic cerebellar Purkinje cells have a main Gaussian peak at a basic firing interval and exponentially reduced peaks at multiples of this firing period. ISIs of arrhythmic Purkinje cells can be modeled as the integration of multiple Gaussian distributions centered at continuous intervals with exponentially reduced peak amplitudes. The sources of variability are directly related to the relative timing of action potentials between neighboring cells since we show that irregularities of discharge in one cell are associated with the previous history of its discharge in time relative to another cell. Through relative phase analyses, we demonstrate that the shape and the mathematical form of the ISI distributions in cerebellum are direct result of dynamic interactions in the nearby neuronal network, in addition to intrinsic firing properties. The analysis in this paper provides a unified description of cerebellar inter-spike interval distributions which deviate from the usual Poisson assumptions. Our results suggest the existence of an intrinsic rhythmicity in cells exhibiting arrhythmic spike trains in cerebellum, and may identify an important source of variability in neuronal firing patterns that is relevant to the mechanism of neural computation in cerebellum.