On the Hamiltonian structure of large deviations in stochastic hybrid systems
| Autor: | Paul C. Bressloff, Olivier Faugeras |
|---|---|
| Přispěvatelé: | Department of Mathematics - University of Utah, University of Utah, TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Mathématiques pour les Neurosciences (MATHNEURO), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), European Project: 269921,EC:FP7:ICT,FP7-ICT-2009-6,BRAINSCALES(2011), European Project: 318723,EC:FP7:ICT,FP7-ICT-2011-8,MATHEMACS(2012), European Project: 227747,EC:FP7:ERC,ERC-2008-AdG,NERVI(2009), Paul Bressloff holds an INRIA International Chair |
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Statistics and Probability
Molecular Networks (q-bio.MN) [SDV.NEU.NB]Life Sciences [q-bio]/Neurons and Cognition [q-bio.NC]/Neurobiology [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] 01 natural sciences large deviations 010305 fluids & plasmas symbols.namesake 0103 physical sciences FOS: Mathematics Ergodic theory Applied mathematics Quantitative Biology - Molecular Networks 010306 general physics Eigenvalues and eigenvectors Mathematics WKB Markov chain [SCCO.NEUR]Cognitive science/Neuroscience Probability (math.PR) Statistical and Nonlinear Physics [SDV.BBM.MN]Life Sciences [q-bio]/Biochemistry Molecular Biology/Molecular Networks [q-bio.MN] Hamiltonian [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] FOS: Biological sciences Quantitative Biology - Neurons and Cognition Hybrid system symbols Piecewise Neurons and Cognition (q-bio.NC) Large deviations theory Statistics Probability and Uncertainty Hamiltonian (quantum mechanics) Rate function Mathematics - Probability stochastic hybrid systems |
| Zdroj: | Journal of Statistical Mechanics: Theory and Experiment Journal of Statistical Mechanics: Theory and Experiment, IOP Publishing, 2017, 2017, pp.33206. ⟨10.1088/1742-5468/aa64f3⟩ Journal of Statistical Mechanics: Theory and Experiment, 2017, 2017, pp.33206. ⟨10.1088/1742-5468/aa64f3⟩ |
| ISSN: | 1742-5468 |
| Popis: | International audience; We present a new derivation of the classical action underlying a large deviation principle (LDP) for a stochastic hybrid system, which couples a piecewise deterministic dynamical system in R d with a time-homogeneous Markov chain on some discrete space Γ. We assume that the Markov chain on Γ is ergodic, and that the discrete dynamics is much faster than the piecewise deterministic dynamics (separation of timescales). Using the Perron-Frobenius theorem and the calculus-of-variations, we show that the resulting action Hamiltonian is given by the Perron eigenvalue of a |Γ|-dimensional linear equation. The corresponding linear operator depends on the transition rates of the Markov chain and the nonlinear functions of the piecewise deterministic system. We compare the Hamiltonian to one derived using WKB methods, and show that the latter is a reduction of the former. We also indicate how the analysis can be extended to a multi-scale stochastic process, in which the continuous dynamics is described by a piecewise stochastic differential equations (SDE). Finally, we illustrate the theory by considering applications to conductance-based models of membrane voltage fluctuations in the presence of stochastic ion channels. |
| Databáze: | OpenAIRE |
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