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dc.contributor.advisorBuckwar, Doctor Evelyn
dc.contributor.authorRiedler, Martin Georg
dc.date.accessioned2012-05-10T09:56:49Z
dc.date.available2012-05-10T09:56:49Z
dc.date.issued2011-07
dc.identifier.urihttp://hdl.handle.net/10399/2484
dc.description.abstractA 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.en_US
dc.description.sponsorshipEngineering and Physics Research Council (EPRC) EP/E03635X/1en_US
dc.description.sponsorshipANR project MANDy (ANR-09-BLAN-0008-01)en_US
dc.description.sponsorshipBC/DAAD ARC project Nr 1349/50021880en_US
dc.description.sponsorshipARC project Nr. 1333en_US
dc.language.isoenen_US
dc.publisherHeriot-Watt Universityen_US
dc.publisherMathematical and Computer Scienceen_US
dc.rightsAll items in ROS are protected by the Creative Commons copyright license (http://creativecommons.org/licenses/by-nc-nd/2.5/scotland/), with some rights reserved.
dc.titleSpatio-temporal stochastic hybrid models of biological excitable membranesen_US
dc.typeThesisen_US


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