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

We prove that the first passage time density $\rho (t)$ for an Ornstein-Uhlenbeck process $X(t)$ obeying $dX=-\beta Xdt+\sigma dW$ to reach a fixed threshold $\theta$ from a suprathreshold initial condition $x_0>\theta >0$ has a lower bound of the form $\rho(t)>k \exp\left[-p e^{6\beta t}\right]$forpositiveconstants$k$and$p$fortimes$t$ 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.