How Sample Paths of Leaky Integrate-and-Fire Models Are Influenced by the Presence of a Firing Threshold

Neural membrane potential data is necessarily conditional on observation being prior to a firing time. In a stochastic Leaky Integrate and Fire model this corresponds to conditioning the process on not crossing a boundary. In the literature simulation and estimation has almost always been done using unconditioned processes. In this paper we determine the stochastic differential equations of a diffusion process conditioned to stay below a level S up to a fixed time t1 and of a diffusion process conditioned to cross the boundary for the first time at t1. This allows simulation of sample paths and identification of the corresponding mean process. Differences between the mean of free and conditioned processes are illustrated as well as the role of the noise in increasing these differences.