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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.

MSC:
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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