Improved Simulation Techniques for First Exit Time of Neural Diffusion Models

Abstract

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.

Keywords:

• Exponential time-stepping Euler algorithm
• First exit time
• FitzHugh–Nagumo model
• Fixed time-step Euler method
• Ornstein–Uhlenbeck process

Mathematics Subject Classification:

• Primary 65
• Secondary C30