caa a22 4500 445329874 CHVBK 20180317142733.0 cr unu---uuuuu 170323e20111201xx s 000 0 eng 10.1007/s00285-010-0395-z doi (NATIONALLICENCE)springer-10.1007/s00285-010-0395-z An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution [Elektronische Daten] [Evelyn Buckwar, Martin Riedler] 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. Springer-Verlag, 2011 Stochastic hybrid system nationallicence Spatio-temporal dynamics nationallicence Neuron model nationallicence Cable equation nationallicence Channel noise nationallicence Buckwar Evelyn Department of Mathematics, Maxwell Institute, Heriot-Watt University, EH14 4AS, Edinburgh, United Kingdom aut Riedler Martin Department of Mathematics, Maxwell Institute, Heriot-Watt University, EH14 4AS, Edinburgh, United Kingdom aut Journal of Mathematical Biology Springer-Verlag 63/6(2011-12-01), 1051-1093 0303-6812 63:6<1051 2011 63 285 https://doi.org/10.1007/s00285-010-0395-z text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00285-010-0395-z text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Buckwar Evelyn Department of Mathematics, Maxwell Institute, Heriot-Watt University, EH14 4AS, Edinburgh, United Kingdom aut NATIONALLICENCE 700 1- Riedler Martin Department of Mathematics, Maxwell Institute, Heriot-Watt University, EH14 4AS, Edinburgh, United Kingdom aut NATIONALLICENCE 773 0- Journal of Mathematical Biology Springer-Verlag 63/6(2011-12-01), 1051-1093 0303-6812 63:6<1051 2011 63 285 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer