An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution.
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.
Department of Mathematics, Maxwell Institute, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom. e.buckwar@hw.ac.uk
Evelyn
E
Buckwar
Martin G
MG
Riedler
D000200:Action Potentials / Q000502:physiology*; D000465:Algorithms; D000818:Animals; D001369:Axons / Q000502:physiology; D002462:Cell Membrane / Q000502:physiology*; D049832:Decapodiformes; D007700:Kinetics; D008390:Markov Chains; D008959:Models, Neurological; D009474:Neurons / Q000502:physiology*; D015221:Potassium Channels / Q000378:metabolism; D015222:Sodium Channels / Q000378:metabolism; D013269:Stochastic Processes
D015221:Potassium Channels; D015222:Sodium Channels
Biophys J. 1970 Nov;10(11):1090-101
5471700
Chaos. 1995 Mar;5(1):174-183
12780171
Nat Rev Neurosci. 2008 Apr;9(4):292-303
18319728
Biophys J. 1997 Oct;73(4):1815-29
9336177
Biophys J. 1983 May;42(2):151-7
6305436
IET Syst Biol. 2008 May;2(3):113-35
18537453
Ann Biomed Eng. 2002 Apr;30(4):578-87
12086008
Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 1994 Apr;49(4):3421-3431
9961610
J Neurosci. 2005 Oct 26;25(43):10025-8
16251451
Biophys J. 1995 Mar;68(3):779-85
7756544
IEEE Trans Biomed Eng. 2004 Jan;51(1):13-20
14723489
IEEE Trans Biomed Eng. 2009 Oct;56(10):2493-501
19304476
IEEE Trans Biomed Eng. 2001 Apr;48(4):416-24
11322529
Cell Calcium. 1998 May;23(5):291-302
9681192
Biophys J. 2009 Jul 22;97(2):636-55
19619479
Biophys J. 1996 Dec;71(6):3013-21
8968572
Biophys J. 2003 Jan;84(1):42-56
12524264
Acta Physiol Scand. 1979 Dec;107(4):343-63
543428
PLoS Comput Biol. 2007 May;3(5):e79
17480115