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The first passage time problem for Gauss-diffusion processes: algorithmic approaches and applications to LIF neuronal model. (English) Zbl 1211.60037
Summary: Motivated by some unsolved problems of biological interest, such as the description of firing probability densities for Leaky Integrate-and-Fire neuronal models, we consider the first-passage-time problem for Gauss-diffusion processes along the line of C. B. Mehr and J. A. McFadden [J. R. Stat. Soc., Ser. B 27, 505–522 (1965; Zbl 0234.60050)]. This is essentially based on a space-time transformation, originally due to J. L. Doob [Ann. Math. Stat. 20, 393–403 (1949; Zbl 0035.08901)], by which any Gauss-Markov process can expressed in terms of the standard Wiener process. Starting with an analysis that pinpoints certain properties of mean and autocovariance of a Gauss-Markov process, we are led to the formulation of some numerical and time-asymptotically analytical methods for evaluating first-passage-time probability density functions for Gauss-diffusion processes. Implementations for neuronal models under various parameter choices of biological significance confirm the expected excellent accuracy of our methods.

60J60 Diffusion processes
60G15 Gaussian processes
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
92C20 Neural biology
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