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## Details

Additional Physical Format: | (GyWOH)har130418244 |
---|---|

Material Type: | Manuscript |

Document Type: | Book, Archival Material |

All Authors / Contributors: |
Zeev Schuss |

ISBN: | 9781461476863 1461476860 |

OCLC Number: | 839388739 |

Description: | xx, 322 pages : illustrations (some color) ; 24 cm. |

Contents: | Mathematical Brownian motion -- Definition of Mathematical Brownian motion -- Mathematical Brownian motion in Rd -- Construction of Mathematical Brownian motions -- Analytical and statistical properties of Brownian paths -- Integration with respect to MBM : the Itô integral -- Stochastic differentials -- The chain rule and Itô's formula -- Stochastic differential equations -- The Langevin equation -- Itô stochastic differential equations -- SDEs of Itô type -- Diffusion processes -- SDEs and PDEs -- The Kolmogorov representation -- The Feynman-Kac representation and terminating Trajectories -- The Pontryagin-Andronov-Vitt equation for the MFPT -- The exit distribution -- The PDF of the FPT -- The Fokker-Planck equation -- The backward Kolmogorov equation -- The survival probability and the PDF of the FPT -- Euler's scheme and Wiener's measure -- Euler's scheme for Itô SDEs and its convergence -- The pdf of Euler's scheme in R and the FPE -- Euler's scheme in Rd -- The convergence of the pdf in Euler's scheme in Rd -- Unidirectional and net probability Flux -- Brownian dynamics at Boundaries -- Absorbing boundaries -- Unidirectional Flux and the survival probability -- Reflecting and partially reflecting boundaries -- Reflection and partial reflection in one dimension -- Partially reflected diffusion in Rd -- Partial reflection in a half-space : constant diffusion matrix -- State-dependent diffusion and partial oblique reflection -- Curved boundary -- Boundary conditions for the backward equation -- Discussion and annotations -- Brownian simulation of Langevin's -- Diffusion limit of physical Brownian motion -- The Overdamped Langevin equation -- Diffusion approximation to the Fokker-Planck equation -- The Unidirectional current in the Smoluchowski equation -- Trajectories between fixed concentrations -- Trajectories, Fluxes, and boundary concentrations -- Connecting a simulation to the continuum -- The interface between simulation and the continuum -- Brownian dynamics simulations -- Application to channel simulation -- Annotation -- The first passage time to a boundary -- The FPT and escape from a domain -- The PDF of the FPT and the density of the mean time spent at a point -- The exit density and probability Flux density -- Conditioning -- Conditioning on Trajectories that reach A before B -- Application of the FPT to diffusion theory -- Stationary absorption flux in one dimension -- The probability law of the first arrival time -- The first arrival time for steady-state diffusion in R³ -- The next arrival times -- The exponential decay of G(r, t) -- Brownian models of Chemical reactions in microdomains -- A stochastic model of a non-arrhenius reaction -- Calcium dynamics in Dendritic spines -- Dendritic spines and their function -- Modeling dendritic spine dynamics -- Biological simplifications of the model -- A simplified physical model of the spine. A schematic model of spine twitching -- Final model simplifications -- The mathematical model -- Mathematical simplifications -- The Langevin equations -- Reaction-diffusion model of binding and unbinding -- Specification of the Hydrodynamic flow -- Chemical Kinetics of binding and unbinding Reactions -- Simulation of calcium kinetics in dendritic spines -- A Langevin (Brownian) dynamics simulation -- An estimate of a decay rate -- Summary and discussion -- Annotations -- Interfacing at the Stochastic separatrix -- Transition state theory of thermal activation -- The diffusion model of activation -- The FPE and TST -- Reaction rate and the principal eigenvalue -- MFPT -- The rate k abs (D), MFPT (... (D)), an eigenvalue ... (D) -- MFPT for domains of types I and II in Rd -- Recrossing, Stochastic Separatrix, Eigenfunctions -- The Eigenvalue problem -- Can recrossings be neglected? -- Accounting for recrossings and the MFPT -- The transmission coefficient Ktr -- Summary and discussion -- Annotations -- Narrow escape in R² -- Introduction -- The NET problem in Neuroscience -- NET, Eigenvalues, and time-scale separation -- A neumann-Dirichlet boundary value problem -- The Neumann function and an integral equation -- The NET problem in two dimensions -- Brownian motion in Dire straits -- The MFPT to a bottleneck -- Exit from several bottlenecks -- Diffusion and NET on a surface of revolution -- A composite domain with a bottleneck -- The NET from domains with bottlenecks in R² and R³ -- The principal Eigenvalue and bottlenecks -- Connecting head and neck -- The principal Eigenvalue in Dumbbell-Shaped domains -- A Brownian needle in Dire straits -- The diffusion law of a Brownian needle in a planar strip -- The turnaround time ... L-R -- Applications of the NET -- Annotations -- Annotation to the NET problem -- Narrow escape in R³ -- The Neumann function in Regulär domains in R³ -- Elliptic absorbing window -- Second-order asymptotics for a circular window -- Leakage in a conductor of Brownian particles -- Activation through a narrow opening -- The Neumann function -- Narrow escape -- Deep well : a Markov chain model -- The NET in a solid funnel-shaped domain -- Selected applications in molecular biophysics -- Leakage from a cylinder -- Applications of the NET -- Annotations -- Bibliography -- Index. |

Series Title: | Applied mathematical sciences (Springer-Verlag New York Inc.), v. 186. |

Responsibility: | Zeev Schuss. |

## Reviews

*Editorial reviews*

Publisher Synopsis

From the book reviews: "This text provides an excellent entry point for applied mathematicians who would like to get a first understanding of the field of neuronal modeling, with a bold motivation and immediate application to highly relevant phenomena in the science ... . this text may serve as an excellent basis for a specialized course on neuronal modeling or biophysics at master's level and as a common reference text for interdisciplinary teams, which perfectly reflects the author's long working experience." (P. R. C. Ruffino, zbMATH, Vol. 1305, 2015) "This book uniquely combines an introduction to the mathematical theory of Brownian motion with applications to chemical kinetics, primarily in biology and physiology. ... this a unique and valuable book. ... Exercises are included throughout the book, particularly relating to the mathematical theory. The book will be extremely useful to both mathematicians and biologists/physiologists, etc., who work at the interface of these two subjects." (D. J. W. Simpson, SIAM Review, Vol. 56 (4), December, 2014) "This book will be of interest to a broad group of students and researchers. It presents a style of analysis that is typical of applications in physics and applied sciences-an explicit transition-density style based on the Fokker-Planck and Langevin equations, and the forward Kolmogorov equation, and defining solutions to stochastic differential equations through the Euler scheme of successive approximations ... ." (David R. Steinsaltz, Mathematical Reviews, November, 2014) Read more...

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