Abstract
Numerous applications all the way from biology and physics to economics depend on the density of first crossings over a boundary. Motivated by the lack of general purpose analytical tools for computing first-passage time densities (FPTDs) for complex problems, we propose a new simple method based on the independent interval approximation (IIA). We generalise previous formulations of the IIA to include arbitrary initial conditions as well as to deal with discrete time and non-smooth continuous time processes. We derive a closed form expression for the FPTD in z and Laplace-transform space to a boundary in one dimension. Two classes of problems are analysed in detail: discrete time symmetric random walks (Markovian) and continuous time Gaussian stationary processes (Markovian and non-Markovian). Our results are in good agreement with Langevin dynamics simulations.
Highlights
When the electric potential between the interior and exterior of a neurone exceed a certain threshold, the neurone fires
Discrete time symmetric random walks A prominent example of a discrete time process is the Markovian symmetric random walk that evolves via x (n) = x (n - 1) + h (n) for n 1 with the fixed initial condition x0 = x (n = 0)
We generalised the interval approximation (IIA) to discrete time and continuous time processes with non-smooth trajectories that start from fixed initial conditions
Summary
When the electric potential between the interior and exterior of a neurone exceed a certain threshold, the neurone fires. The interior potential is abruptly reset to its rest value and the process starts over. How often it starts over depends on external stimuli (e.g. light and touch) and firing frequencies of neighbouring neurones. To better understand neurone firing, and how neurones work, researchers in the field [1, 2] use stochastic models to calculate how long it takes for the interior potential to pass the firing threshold for the first time. Neurone dynamics is not the only case where first-passage problems arise. Despite enormous interest there are surprisingly few cases where we know the probability density of first-passage times analytically
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