5,594 research outputs found
Piecewise deterministic Markov processes in biological models
We present a short introduction into the framework of piecewise deterministic
Markov processes. We illustrate the abstract mathematical setting with a series
of examples related to dispersal of biological systems, cell cycle models, gene
expression, physiologically structured populations, as well as neural activity.
General results concerning asymptotic properties of stochastic semigroups
induced by such Markov processes are applied to specific examples.Comment: in: Semigroup of Operators - Theory and Applications, J. Banasiak et
al. (eds.), Springer Proceedings in Mathematics & Statistics 113, (2015), pp.
235-25
Weak convergence of marked point processes generated by crossings of multivariate jump processes. Applications to neural network modeling
We consider the multivariate point process determined by the crossing times
of the components of a multivariate jump process through a multivariate
boundary, assuming to reset each component to an initial value after its
boundary crossing. We prove that this point process converges weakly to the
point process determined by the crossing times of the limit process. This holds
for both diffusion and deterministic limit processes. The almost sure
convergence of the first passage times under the almost sure convergence of the
processes is also proved. The particular case of a multivariate Stein process
converging to a multivariate Ornstein-Uhlenbeck process is discussed as a
guideline for applying diffusion limits for jump processes. We apply our
theoretical findings to neural network modeling. The proposed model gives a
mathematical foundation to the generalization of the class of Leaky
Integrate-and-Fire models for single neural dynamics to the case of a firing
network of neurons. This will help future study of dependent spike trains.Comment: 20 pages, 1 figur
The Hitchhiker's Guide to Nonlinear Filtering
Nonlinear filtering is the problem of online estimation of a dynamic hidden
variable from incoming data and has vast applications in different fields,
ranging from engineering, machine learning, economic science and natural
sciences. We start our review of the theory on nonlinear filtering from the
simplest `filtering' task we can think of, namely static Bayesian inference.
From there we continue our journey through discrete-time models, which is
usually encountered in machine learning, and generalize to and further
emphasize continuous-time filtering theory. The idea of changing the
probability measure connects and elucidates several aspects of the theory, such
as the parallels between the discrete- and continuous-time problems and between
different observation models. Furthermore, it gives insight into the
construction of particle filtering algorithms. This tutorial is targeted at
scientists and engineers and should serve as an introduction to the main ideas
of nonlinear filtering, and as a segway to more advanced and specialized
literature.Comment: 64 page
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