Spatio-temporal stochastic hybrid models of biological excitable membranes
Abstract
A large number of biological systems are intrinsically random, in particular, biological
excitable membranes, such as neuronal membranes, cardiac tissue or models for
calcium dynamics. The present thesis is concerned with hybrid stochastic models of
spatio-temporal dynamics of biological excitable membranes using Piecewise Deterministic
Markov Processes (PDMPs). This class of processes allows a precise mathematical
description of the internal noise structure of excitable membranes. Overall the
aim of the thesis is two-fold: On the one hand, we establish a general hybrid modelling
framework for biological excitable membranes and, on the other hand, we are interested
in a general advance of PDMP theory which the former necessitates. Regarding
the first aim we exemplify the modelling framework on the classical Hodgkin-Huxley
model of a squid giant axon. Regarding the latter we present a general PDMP theory
incorporating spatial dynamics and present tools for their analysis. Here we focus on
two aspects.
Firstly, we consider the approximation of PDMPs by deterministic models or continuous
stochastic processes. To this end we derive as general theoretical tools a law of
large numbers for PDMPs and martingale central limit theorems. The former establishes
a connection of stochastic hybrid models to deterministic models given, e.g., by
systems of partial differential equations. Whereas the latter connects the stochastic
fluctuations in the hybrid models to diffusion processes. Furthermore, these limit
theorems provide the basis for a general Langevin approximation to PDMPs, i.e., certain
stochastic partial differential equations that are expected to be similar in their
dynamics to PDMPs.
Secondly, we also address the question of numerical simulation of PDMPs. We present
and analyse the convergence in the pathwise sense of a class of simulation methods
for PDMPs in Euclidean space.