Limit theorems for infinite-dimensional piecewise deterministic Markov processes. Applications to stochastic excitable membrane models

Abstract

We present limit theorems for a sequence of Piecewise Deterministic Markov Processes (PDMPs) taking values in a separable Hilbert space. This class of processes provides a rigorous framework for stochastic spatial models in which discrete random events are globally coupled with continuous space-dependent variables solving partial differential equations, e.g., stochastic hybrid models of excitable membranes. We derive a law of large numbers which establishes a connection to deterministic macroscopic models and a martingale central limit theorem which connects the stochastic fluctuations to diffusion processes. As a prerequisite we carry out a thorough discussion of Hilbert space valued martingales associated to the PDMPs. Furthermore, these limit theorems provide the basis for a general Langevin approximation to PDMPs, i.e., stochastic partial differential equations that are expected to be similar in their dynamics to PDMPs. We apply these results to compartmental-type models of spatially extended excitable membranes. Ultimately this yields a system of stochastic partial differential equations which models the internal noise of a biological excitable membrane based on a theoretical derivation from exact stochastic hybrid models.

Multiscale Piecewise Deterministic Markov Process in infinite dimension: central limit theorem and Langevin approximation

“…This model was first considered in [1], and later in [7,20,35]. Although we are interested in the multi scale stochastic Hodgkin-Huxley model, we start by describing the model that does not display different time scales, for the sake of clarity.…”

“…The authors in [7] prove that their process is markovian and moreover characterize its generator. Another approach based on the marked point process theory is also possible, see for instance [26] and the extension to our framework in [35].…”

“…Recently, the asymptotic behavior of PDMPs have been investigated in [4,5,12,37], limit theorems for infinite dimensional PDMPs in [35], control problems in [10,11,22], numerical methods in [33,6], time reversal in [27] and to end up this list with no claim of completeness, estimation of the jump rates for PDMPs in [2]. Hybrid systems are the object of great attention because they offer an accurate description of a large class of phenomena arising in various domains such as physics or biology.…”

“…For example, in mathematical neuroscience, a domain the authors are more particularly interested in, PDMP models arise naturally in the description of the propagation of the nerve impulse, see [1,7]. This mathematical description has been proved to be consistent with classical models such as the Hodgkin-Huxley model and compartment type models, see [35]. In this paper, the authors are interested in the question of averaging for a spatially extended PDMP model of propagation of the nerve impulse and corresponding fluctuations.…”

Persistence of Stability for Equilibria of Map Iterations in Banach Spaces Under Small Random Perturbations

“…[23], see also [15]) and more closely to piecewise deterministic Markov processes (PDMPs), (cf. [5,7,18,29]). The NSS models are tailored towards random switching among a family of dynamical systems, driven by a hidden noise variable, rather than the 'impulsive' change in state described in Eq.…”