Improved simulation techniques for first exit time of neural diffusion models

We consider the fixed and exponential time-stepping Euler algorithms, with boundary tests, to calculate the mean first exit times (MFET) of two one-dimensional neural diffusion models, represented by the Ornstein-Uhlenbeck (OU) process and a stochastic space-clamped FitzHugh-Nagumo (FHN) system. The numerical methods are described and the convergence rates for the MFET analyzed. A boundary test improves the rate of convergence from order one-half to order 1. We show how to apply the multi-level Monte Carlo (MLMC) method to an Euler time-stepping method with boundary test and this improves the Monte Carlo computation of the MFET.