The Gamma renewal process as an output of the diffusion leaky integrate-and-fire neuronal model

TL;DR

This paper explores the relationship between the diffusion leaky integrate-and-fire neuronal model and the Gamma renewal process, showing how the latter can be derived from the former under certain conditions, linking physiological mechanisms to statistical models.

Contribution

It demonstrates the conditions under which the Gamma renewal process can be derived as an output of the diffusion Ornstein-Uhlenbeck neuronal model, connecting descriptive and mechanistic models.

Findings

Gamma distribution can be derived from the Ornstein-Uhlenbeck process under specific conditions

Certain parameter regimes make the Gamma model unrealistic for the given neuronal model

The relationship clarifies how physiological mechanisms relate to statistical spike train models

Abstract

Statistical properties of spike trains as well as other neurophysiological data suggest a number of mathematical models of neurons. These models range from entirely descriptive ones to those deduced from the properties of the real neurons. One of them, the diffusion leaky integrate-and-fire neuronal model, which is based on the Ornstein-Uhlenbeck stochastic process that is restricted by an absorbing barrier, can describe a wide range of neuronal activity in terms of its parameters. These parameters are readily associated with known physiological mechanisms. The other model is descriptive, Gamma renewal process, and its parameters only reflect the observed experimental data or assumed theoretical properties. Both of these commonly used models are related here. We show under which conditions the Gamma model is an output from the diffusion Ornstein-Uhlenbeck model. In some cases we can see that the Gamma distribution is unrealistic to be achieved for the employed parameters of the Ornstein-Uhlenbeck process.