Topics in Applied Stochastic Dynamics.

[摘要] Randomness in natural systems come from various sources, for example from thediscrete nature of the underlying dynamical process when viewed on a small scale. Inthis thesis we study the effect of stochasticity on the dynamics in three applications,each with different sources and effects of randomness. In the first application we studythe Hodgkin-Huxley model of the neuron with a random ion channel mechanism vianumerical simulation. Randomness affects the nonlinear mechanism of a neuron’sfiring behavior by spike induction as well as by spike suppression. The sensitivity todifferent types of channel noise is explored and robustness of the dynamical propertiesis studied using two distinct stochastic models. In the second application we compareand contrast the effectiveness of mixing of a passive scalar by stirring using differentnotions of mixing efficiency. We explore the non-commutativity of the limits of largePeclet numbers and large spatial scale separation between the flow and sources andsinks, and propose and examine a conceptual approach that captures some compat-ible features of the different models and measures of mixing. In the last applicationwe design a stochastic dynamical system that mimics the properties of so-called ho-mogeneous Rayleigh-Benard convection and show that arbitrary small noise changesthe dynamical properties of the model. The system’s properties are further exam-ined using the first exit time problem. The three applications show that randomnessof small magnitude may play important and counterintuitive roles in determinig asystem’s properties.

[发布机构] University of Michigan

[效力级别] Hodgkin-Huxley Model

[关键词] Stochastic Dynamics;Hodgkin-Huxley Model;Rayleigh-Benard Convection;Mixing of Fluids;Stochastic Lyapunov Function;Mathematics;Science;Applied and Interdisciplinary Mathematics