An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution

Abstract

In this paper, we present a mathematical description for excitable biological membranes, in particular neuronal membranes. We aim to model the (spatio-) temporal dynamics, e.g., the travelling of an action potential along the axon, subject to noise, such as ion channel noise. Using the framework of Piecewise Deterministic Processes (PDPs) we provide an exact mathematical description-in contrast to pseudo-exact algorithms considered in the literature-of the stochastic process one obtains coupling a continuous time Markov chain model with a deterministic dynamic model of a macroscopic variable, that is coupling Markovian channel dynamics to the time-evolution of the transmembrane potential. We extend the existing framework of PDPs in finite dimensional state space to include infinite-dimensional evolution equations and thus obtain a stochastic hybrid model suitable for modelling spatio-temporal dynamics. We derive analytic results for the infinite-dimensional process, such as existence, the strong Markov property and its extended generator. Further, we exemplify modelling of spatially extended excitable membranes with PDPs by a stochastic hybrid version of the Hodgkin-Huxley model of the squid giant axon. Finally, we discuss the advantages of the PDP formulation in view of analytical and numerical investigations as well as the application of PDPs to structurally more complex models of excitable membranes.

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.…”

“…, [7]). Fix N ≥ 1 and let (Ω, F, (F t ) 0≤t≤T , P) be a filtered probability space satisfying the usual conditions.…”

“…In this paper, the author uses the Schaeffer fixed point theorem to show that when the jump process r jumps at rate 1, there exists a solution to (2) and uses the Girsanov theorem for cádlág processes with finite state space to recover the dynamic of r. Another approach has been developed in [7]. In this paper, the authors construct explicitly the process (u, r) as a piecewise deterministic Markov process generalizing in this way the theory of piecewise deterministic Markov process developed by Davis from the finite to the infinite dimension setting, see [14,15].…”

“…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].…”

“…The PDMPs, also called markovian hybrid systems, have been introduced by Davis in [14,15] for the finite dimensional setting and generalized in [7] to cover the infinite dimensional case. 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].…”