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

Part of: Markov processes Physiological, cellular and medical topics

Published online by Cambridge University Press:  14 July 2016

Peter J. Thomas
Peter J. Thomas*
Affiliation:
Case Western Reserve University
*
∗Postal address: Department of Mathematics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA. Email address: pjthomas@case.edu
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Abstract

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We prove that the first passage time density ρ(t) for an Ornstein-Uhlenbeck process X(t) obeying dX = -β Xdt + σdW to reach a fixed threshold θ from a suprathreshold initial condition x0 > θ > 0 has a lower bound of the form ρ(t) > kexp[-pe6βt] 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 β, σ, x0, and θ, and discuss the application of the results to the synchronization of periodically forced stochastic leaky integrate-and-fire model neurons.

Keywords

Leaky integrate and fire neuronOrnstein-Uhlenbeck processreliabilitysynchronizationneural modellower bound

MSC classification

Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 92C20: Neural biology
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 2 , June 2011 , pp. 420 - 434
Copyright
Copyright © Applied Probability Trust 2011 

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