A Lower Bound for the First Passage Time Density of the Suprathreshold Ornstein-Uhlenbeck Process

TL;DR

This paper establishes a lower bound on the first passage time density for a suprathreshold Ornstein-Uhlenbeck process, providing explicit formulas and discussing implications for neuron synchronization.

Contribution

It introduces a novel lower bound for the first passage time density of the Ornstein-Uhlenbeck process with explicit parameter expressions.

Findings

Derived a lower bound of the form k \, e^{-p e^{6eta t}} for the density

Provided explicit formulas for constants in the bound

Discussed applications to neuron synchronization models

Abstract

We prove that the first passage time density $\rho(t)$ for an Ornstein-Uhlenbeck process $X(t)$ obeying $dX=-\beta X dt + \sigma dW$ to reach a fixed threshold $\theta$ from a suprathreshold initial condition $x_0>\theta>0$ has a lower bound of the form $\rho(t)>k \exp\left[-p e^{6\beta t}\right]$ 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 $\beta$, $\sigma$, $x_0$ and $\theta$, and discuss application of the results to the synchronization of periodically forced stochastic leaky integrate-and-fire model neurons.