First-passage-time statistics of a Brownian particle driven by an arbitrary unidimensional potential with a superimposed exponential time-dependent drift

Abstract

In one-dimensional systems, the dynamics of a Brownian particle are governed by the force derived from a potential as well as by diffusion properties. In this work, we obtain the first-passage-time statistics of a Brownian particle driven by an arbitrary potential with an exponential temporally decaying superimposed field up to a prescribed threshold. The general system analyzed here describes the sub-threshold signal integration of integrate-and-fire neuron models, of any kind, supplemented by an adaptation-like current, whereas the first-passage-time corresponds to the declaration of a spike. Following our previous studies, we base our analysis on the backward Fokker Planck equation and study the survival probability and the first-passage-time density function in the space of the initial condition. By proposing a series solution we obtain a system of recurrence equations, which given the specific structure of the exponential time-dependent drift, easily admit a simpler Laplace representation. Naturally, the present general derivation agrees with the explicit solution we found previously for the Wiener process in (2012 JPhysA 45 185001). However, to demonstrate the generality of the approach, we further explicitly evaluate the first-passage-time statistics of the underlying Ornstein Uhlenbeck process. To test the validity of the series solution, we extensively compare theoretical expressions with the data obtained from numerical simulations in different regimes. As shown, agreement is precise whenever the series is truncated at an appropriate order. Beyond the fact that both the Wiener and Ornstein Uhlenbeck processes have a direct interpretation in the context of neuronal models, given their ubiquity in different fields, our present results will be of interest in other settings where an additive state-independent temporal relaxation process is being developed as the particle diffuses.

22 pages (20 pages in the journal version), 3 figures, published in J. Phys. A

References in corpus (14)

• Intermittent search strategies
• Non-Markovian polymer reaction kinetics
• First Exit Times of Harmonically Trapped Particles: A Didactic Review
• Patterns of interval correlations in neural oscillators with adaptation
• Exit time distribution in spherically symmetric two-dimensional domains
• First passage times in homogeneous nucleation and self-assembly
• Translocation through environments with time dependent mobility
• Onset of negative interspike interval correlations in adapting neurons
• Brownian motion in time-dependent logarithmic potential: Exact results for dynamics and first-passage properties
• Survival probability and first-passage-time statistics of a Wiener process driven by an exponential time-dependent drift
• Large fluctuations in diffusion-controlled absorption
• First Invader Dynamics in Diffusion-Controlled Absorption
• Reaction-Diffusion Process Driven by a Localized Source: First Passage Properties
• Series solution to the first-passage-time problem of a Brownian motion with an exponential time-dependent drift

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