Escape Through a Small Opening: Receptor Trafficking in a Synaptic Membrane

We model the motion of a receptor on the membrane surface of a synapse as free Brownian motion in a planar domain with intermittent trappings in and escapes out of corrals with narrow openings. We compute the mean confinement time of the Brownian particle in the asymptotic limit of a narrow opening and calculate the probability to exit through a given small opening, when the boundary contains more than one. Using this approach, it is possible to describe the Brownian motion of a random particle in an environment containing domains with small openings by a coarse grained diffusion process. We use the results to estimate the confinement time as a function of the parameters and also the time it takes for a diffusing receptor to be anchored at its final destination on the postsynaptic membrane, after it is inserted in the membrane. This approach provides a framework for the theoretical study of receptor trafficking on membranes. This process underlies synaptic plasticity, which relates to learning and memory. In particular, it is believed that the memory state in the brain is stored primarily in the pattern of synaptic weight values, which are controlled by neuronal activity. At a molecular level, the synaptic weight is determined by the number and properties of protein channels (receptors) on the synapse. The synaptic receptors are trafficked in and out of synapses by a diffusion process. Following their synthesis in the endoplasmic reticulum, receptors are trafficked to their postsynaptic sites on dendrites and axons. In this model the receptors are first inserted into the extrasynaptic plasma membrane and then random walk in and out of corrals through narrow openings on their way to their final destination.     

Stochastic (white noise) analysis of resonant absorption in laser generated plasma

We suggest to use stochastic differential equations for laser-plasma problems. Using this technique we solve analytically the motion of the electrons at the critical layer and calculate the density profile scaling length (L). For an Nd laser flux of 10¹⁷ W/cm² we estimate L ≈ 0.15 μm, while for a CO₂ laser flux of 10¹⁵ W/cm² we get L ≈ 1.5 μm.     

Tempered wave functions: Schrödinger's equation within the light cone

The standard derivation of Schrödinger's equation from a Lorentz-invariant Feynman path integral consists in taking first the limit of infinite speed of light and then the limit of short time slice. In this order of limits the light cone of the path integral disappears, giving rise to an instantaneous spread of the wave function to the entire space. We ascribe the failure of the propagation in time according to Schrödinger's equation to retain the light cone of the path integral to the very nature of the limiting process: it is a regular expansion of a singular approximation problem, because the time-dependent boundary conditions of the path integral on the light cone are lost in this limit. We propose a distinguished limit of the time-sliced relativistic path integral, which produces Schrödinger's equation and preserves the zero boundary conditions on and outside the original light cone of the path integral. This produces an intermediate model between non-relativistic and relativistic mechanics of a single particle quantum particle. These boundary conditions relieve the solutions of Schrödinger's equation in the entire space of several annoying, seemingly unrelated unphysical artifacts, including non-analytic wave functions, spontaneous appearance of discontinuities, non-existence of moments when the initial wave function has a jump discontinuity (e.g., a collapsed wave function after a measurement), and so on. The practical implications of the present formulation are yet to be seen.     

Schrödinger propagation of initial discontinuities leads to divergence of moments

We show that the large phase expansion of the Schrödinger propagation of an initially discontinuous wave function leads to the divergence of average energy, momentum, and displacement, rendering them unphysical states. If initially discontinuous wave functions are considered to be approximations to continuous ones, the determinant of the spreading rate of these averages is the maximal gradient of the initial wave function. Therefore a dilemma arises between the inclusion of discontinuous wave functions in quantum mechanics and the requirement of finite moments.     

Boundary behavior of diffusion approximations to Markov jump processes

We show that diffusion approximations, including modified diffusion approximations, can be problematic since the proper choice of local boundary conditions (if any exist) is not obvious. For a class of Markov processes in one dimension, we show that to leading order it is proper to use a diffusion (Fokker-Planck) approximation to compute mean exit times with a simple absorbing boundary condition. However, this is only true for the leading term in the asymptotic expansion of the mean exit time. Higher order correction terms do not, in general, satisfy simple absorbing boundary conditions. In addition, the diffusion approximation for the calculation of mean exit times is shown to break down as the initial point approaches the boundary, and leads to an increasing relative error. By introducing a boundary layer, we show how to correct the diffusion approximation to obtain a uniform approximation of the mean exit time. We illustrate these considerations with a number of examples, including a jump process which leads to Kramers' diffusion model. This example represents an extension to a multivariate process.     

Narrow Escape, Part I

A Brownian particle with diffusion coefficient D is confined to a bounded domain Ω by a reflecting boundary, except for a small absorbing window . The mean time to absorption diverges as the window shrinks, thus rendering the calculation of the mean escape time a singular perturbation problem. In the three-dimensional case, we construct an asymptotic approximation when the window is an ellipse, assuming the large semi axis a is much smaller than ( is the volume), and show that the mean escape time is , where e is the eccentricity and is the complete elliptic integral of the first kind. In the special case of a circular hole the result reduces to Lord Rayleigh's formula , which was derived by heuristic considerations. For the special case of a spherical domain, we obtain the asymptotic expansion . This result is important in understanding the flow of ions in and out of narrow valves that control a wide range of biological and technological function. If Ω is a two-dimensional bounded Riemannian manifold with metric g and , we show that . This result is applicable to diffusion in membrane surfaces.     

Busy period distribution in state-dependent queues

We consider a state-dependent M/M/1 queue in which the arrival rate is a function of the instantaneous unfinished work (work backlog) in the system, and the customer's exponential service time distribution is allowed to depend on the unfinished work in the system at the instant that customer arrived. We obtain asymptotic approximations to both the busy period distributions as well as the residual busy period distribution. Our approximations are valid for systems with a rapid arrival rate and small mean service times.This research was supported in part by NSF Grants DMS-84-06110 and DMS-86-20267 and grants from the U.S. Israel Binational Science Foundation and the Israel Academy of Sciences. C. Knessl was partially supported by an I.B.M. Graduate Fellowship.