On approximations of Stein's neuronal model

Abstract

Stein's model represents a commonly-used description of spontaneous neuronal activity. Substituting Stein's model by the Ornstein-Uhlenbeck diffusion process increases the model tractability. A diffusion approximation of Stein's model is summarized in the present paper. It is proved that the cumulative distribution functions of interspike intervals under Stein's model converge to the cumulative distribution function of interspike intervals which are generated in accordance with the limiting Ornstein-Uhlenbeck diffusion model. The approach used allows us to determine to what extent Stein's model modifications and generalizations affect the possibility of diffusion approximation. It can be seen that non-diffusion approximations exist and they are also studied here. The results achieved can be considered as complementary to the numerical study published recently.