First passage times of univariate and bivariate diffusion processes to timevarying and constant boundaries: analytical, statistical and numerical results. Application to neuronal spiking activity
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2nd International Conference on Mathematical NeuroScience
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In neuroscience, stochastic processes and their hitting times are used to describe the membrane potential dynamics of single neurons and to reproduce temporal patterns of nerve impulses, spikes, respectively. For this reason, the first passage time (FPT) problem of univariate diffusion processes through constant boundaries has been extensively studied in the literature. Less results are available in presence of time-varying boundaries or for multivariate diffusion processes, which can be used to reproduce biological features such as the afterhyperpolarization in neurons or to provide a preliminary understanding of neural networks, respectively. In this talk we tackle both problems, investigating the FPT problem of: a) a Wiener process in presence of an exponentially decaying threshold; b) a two-dimensional correlated diffusion process in presence of some constant boundaries. We provide probabilistic, statistical and numerical methods to handle these problems, highlighting how to use them in neuroscience.