A lower bound for the first passage time density of the suprathreshold Ornstein-Uhlenbeck process

Abstract

We prove that the first passage time density ρ(t) for an Ornstein-Uhlenbeck process X(t) obeying dX = -β Xdt + σdW to reach a fixed threshold θ from a suprathreshold initial condition x₀ > θ > 0 has a lower bound of the form ρ(t) > kexp[-pe^6βt] for positive constants k and p for times t exceeding some positive value u. We obtain explicit expressions for k, p, and u in terms of β, σ, x₀, and θ, and discuss the application of the results to the synchronization of periodically forced stochastic leaky integrate-and-fire model neurons.