Article

Methodology and Computing in Applied Probability

, Volume 13, Issue 1, pp 29-57

The First Passage Time Problem for Gauss-Diffusion Processes: Algorithmic Approaches and Applications to LIF Neuronal Model

  • Aniello BuonocoreAffiliated withDipartimento di Matematica e Applicazioni, Università di Napoli Federico II
  • , Luigia CaputoAffiliated withDipartimento di Matematica, Università di Torino
  • , Enrica PirozziAffiliated withDipartimento di Matematica e Applicazioni, Università di Napoli Federico II
  • , Luigi M. RicciardiAffiliated withDipartimento di Matematica e Applicazioni, Università di Napoli Federico II Email author 

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Abstract

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 Mehr and McFadden (J R Stat Soc B 27:505–522, 1965). This is essentially based on a space-time transformation, originally due to Doob (Ann Math Stat 20:393–403, 1949), 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.

Keywords

Gaussian process Diffusion LIF neuronal models Numerical approximations Asymptotics

PACS

02.50Ey 02.50.Ga 02.60.Jh

Mathematics Subject Classifications (2000)

60J60 60J70 6015 92C20

AMS 2000 Subject Classifications

60J60 60G15 60J70