Some Numerical Methods for Rare Events Simulation and Analysis

  • Gilles Wainrib
Part of the Lecture Notes in Mathematics book series (LNM, volume 2058)


We present several numerical approaches to investigate rare events in stochastic systems, with a specific focus on application to biological models. We first present several aspects concerning variance reduction of Monte-Carlo methods, with a focus on importance sampling. We show how these methods can be applied to basic biological models. We show that these techniques can be useful in dealing with multiscale continuous-time Markov chains that are important in the context of biochemical reaction networks. We treat with more detail the problem of first passage time for a linear diffusion process, arising from the integrate-and-fire neuron model, to show the kind of mathematical problems that may arise when trying to design an optimal importance sampling scheme. This leads to the observation that large deviation theory can be helpful to solve these questions. We also review a numerical method designed to estimate large deviations quantities such as the quasipotential and the optimal path and apply this method to estimate numerically the quasipotential of the Morris–Lecar neuron model perturbed by small noise.


Rare Event Neuron Model Importance Sampling Large Deviation Principle Large Deviation Theory 
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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Laboratoire Analyse Géométrie et ApplicationsInstitut GaliléeVilletaneuseFrance

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