33,841 research outputs found
Large Deviations for Nonlocal Stochastic Neural Fields
We study the effect of additive noise on integro-differential neural field
equations. In particular, we analyze an Amari-type model driven by a -Wiener
process and focus on noise-induced transitions and escape. We argue that
proving a sharp Kramers' law for neural fields poses substanial difficulties
but that one may transfer techniques from stochastic partial differential
equations to establish a large deviation principle (LDP). Then we demonstrate
that an efficient finite-dimensional approximation of the stochastic neural
field equation can be achieved using a Galerkin method and that the resulting
finite-dimensional rate function for the LDP can have a multi-scale structure
in certain cases. These results form the starting point for an efficient
practical computation of the LDP. Our approach also provides the technical
basis for further rigorous study of noise-induced transitions in neural fields
based on Galerkin approximations.Comment: 29 page
Spatially partitioned embedded Runge-Kutta Methods
We study spatially partitioned embedded Runge–Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient for problems in which the smoothness of the solution or the magnitudes of the PDE coefficients vary strongly in space. We focus on embedded partitioned methods as they offer greater efficiency and avoid the order reduction that may occur in non-embedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to non-physical effects and propose conservative additive schemes based on partitioning the fluxes rather than the ordinary differential equations. A variety of SPERK schemes are presented, including an embedded pair suitable for the time evolution of fifth-order weighted non-oscillatory (WENO) spatial discretizations. Numerical experiments are provided to support the theory
Onsager’s variational principle in soft matter : introduction and application to the dynamics of adsorption of proteins onto fluid membranes
This book is the first collection of lipid-membrane research conducted by leading mechanicians and experts in continuum mechanics. It brings the overall intellectual framework afforded by modern continuum mechanics to bear on a host of challenging problems in lipid membrane physics. These include unique and authoritative treatments of differential geometry, shape elasticity, surface flow and diffusion, interleaf membrane friction, phase transitions, electroelasticity and flexoelectricity, and computational modelling.
[Chapter] Lipid bilayers are unique soft materials operating in general in the low Reynolds limit. While their shape is predominantly dominated by curvature elasticity as in a solid shell, their in-plane behavior is that of a largely inextensible viscous fluid. Furthermore, lipid membranes are extremely responsive to chemical stimuli. Because in their biological context they are continuously brought out-of-equilibrium mechanically or chemically, it is important to understand their dynamics. Here, we introduce Onsager’s variational principle as a general and transparent modeling tool for lipid bilayer dynamics. We introduce this principle with elementary examples, and then use it to study the sorption of curved proteins on lipid membranes.Peer ReviewedPostprint (author's final draft
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