Abstract.We give a short overview of recent results on a specific class of Markov process: the Piecewise Deterministic Markov Processes (PDMPs). We first recall the definition of these processes and give some general results. On more specific cases such as the TCP model or a model of switched vector fields, better results can be proved, especially as regards long time behaviour. We continue our review with an infinite dimensional example of neuronal activity. From the statistical point of view, these models provide specific challenges: we illustrate this point with the example of the estimation of the distribution of the inter-jumping times. We conclude with a short overview on numerical methods used for simulating PDMPs. General introductionThe piecewise deterministic Markov processes (denoted PDMPs) were first introduced in the literature by Davis ( [Dav84,Dav93]). Already at this time, the theory of diffusions had such powerful tools as the theory of Itō calculus and stochastic differential equations at its disposal. Davis's goal was to endow the PDMP with rather general tools. The main reason for that was to provide a general framework, since up to then only very particular cases had been dealt with, which turned out not to be easily generalizable.PDMPs form a family of càdlàg Markov processes involving a deterministic motion punctuated by random jumps. The motion of the PDMP {X(t)} t≥0 depends on three local characteristics, namely the jump rate λ, the flow φ and the transition measure Q according to which the location of the process at the jump time is chosen. The process starts from x and follows the flow φ(x, t) until the first jump time T 1 which occurs either spontaneously in a Poisson-like fashion with rate λ (φ(x, t)) or when the flow φ(x, t) hits the boundary of the state-space. In both cases, the location of the process at the jump time T 1 , denoted by Z 1 = X(T 1 ), is selected by the transition measure Q(φ(x, T 1 ), ·) and the motion restarts from this new point as before. This fully describes a piecewise continuous trajectory for {X(t)} with jump times {T k } and post jump locations {Z k }, and which evolves according to the flow φ between two jumps. Article published online by EDP Sciences and available at http://www.esaim-proc.org or http://dx
Abstract. We present recent results on Piecewise Deterministic Markov Processes (PDMPs), involved in biological modeling. PDMPs, first introduced in the probabilistic literature by [30], are a very general class of Markov processes and are being increasingly popular in biological applications. They also give new interesting challenges from the theoretical point of view. We give here different examples on the long time behavior of switching Markov models applied to population dynamics, on uniform sampling in general branching models applied to structured population dynamic, on time scale separation in integrate-and-fire models used in neuroscience, and, finally, on moment calculus in stochastic models of gene expression. Résumé. Nous présentons des résultats récents sur les Processus de Markov Déterministes parMorceaux (PDMPs) utilisés en modélisation en biologie. Les PDMPs, introduits pour la première fois dans la littérature probabiliste par [30], forment une classe générale de processus de Markov et sont de plus en plus populaires dans les applications en biologie. Ils fournissent également de nouveaux défis intéressant du point de vue théorique. Nous donnons ici différents exemples sur le comportement en temps long de modèles de Markov modulés appliqués à la dynamique des populations, sur le tirage uniforme dans des modèles génériques de branchement appliqués à la dynamique de populations structurées, sur les séparations d'échelles de temps dans des modèles intègre-et-tire utilisés en neuroscience, et, finalement, sur le calcul de moments dans des modèles stochastiques d'expression des gènes.
In this paper, we consider the generalized Hodgkin-Huxley model introduced by Austin in [1]. This model describes the propagation of an action potential along the axon of a neuron at the scale of ion channels. Mathematically, this model is a fully-coupled Piecewise Deterministic Markov Process (PDMP) in infinite dimensions. We introduce two time scales in this model in considering that some ion channels open and close at faster jump rates than others. We perform a slow-fast analysis of this model and prove that asymptotically this 'two-time-scales' model reduces to the so called averaged model which is still a PDMP in infinite dimensions for which we provide effective evolution equations and jump rates.
In this paper we consider the generalized Hodgkin-Huxley model introduced in Austin (2008). This model describes the propagation of an action potential along the axon of a neuron at the scale of ion channels. Mathematically, this model is a fully coupled piecewise-deterministic Markov process (PDMP) in infinite dimensions. We introduce two time scales in this model in considering that some ion channels open and close at faster jump rates than others. We perform a slow-fast analysis of this model and prove that, asymptotically, this ‘two-time-scale’ model reduces to the so-called averaged model, which is still a PDMP in infinite dimensions, for which we provide effective evolution equations and jump rates.
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