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In one-dimensional systems, the dynamics of a Brownian particle are governedby the force derived from a potential as well as by diffusion properties. Inthis work, we obtain the first-passage-time statistics of a Brownian particledriven by an arbitrary potential with an exponential temporally decayingsuperimposed field up to a prescribed threshold. The general system analyzedhere describes the sub-threshold signal integration of integrate-and-fireneuron models, of any kind, supplemented by an adaptation-like current, whereasthe first-passage-time corresponds to the declaration of a spike. Following ourprevious studies, we base our analysis on the backward Fokker Planck equationand study the survival probability and the first-passage-time density functionin the space of the initial condition. By proposing a series solution we obtaina system of recurrence equations, which given the specific structure of theexponential time-dependent drift, easily admit a simpler Laplacerepresentation. Naturally, the present general derivation agrees with theexplicit solution we found previously for the Wiener process in (2012 JPhysA 45185001). However, to demonstrate the generality of the approach, we furtherexplicitly evaluate the first-passage-time statistics of the underlyingOrnstein Uhlenbeck process. To test the validity of the series solution, weextensively compare theoretical expressions with the data obtained fromnumerical simulations in different regimes. As shown, agreement is precisewhenever the series is truncated at an appropriate order. Beyond the fact thatboth the Wiener and Ornstein Uhlenbeck processes have a direct interpretationin the context of neuronal models, given their ubiquity in different fields,our present results will be of interest in other settings where an additivestate-independent temporal relaxation process is being developed as theparticle diffuses.

journal of physics. a, mathematical and theoretical

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