Wald's identity and random walk models for neuron firing

A class of probability models for the firing of neurons is introduced and treated analytically. The cell membrane potential is assumed to be a one-dimensional random walk on the continuum; the first passage of the moving boundary triggers a nerve spike, an all-or-none event. The interspike interval distribution of first-passage time is shown to satisfy a Volterra integral equation suitable for numerical evaluation. Explicit solutions as well as their Laplace (Mellin) transforms are obtained for some special cases. The main technique used in this paper is the extension of Wald's fundamental identity of sequential analysis to a wide range of additive stochastic processes with an (eventually) linear boundary function. This identity is also useful in evaluating model parameters in terms of observed firing times, as well as providing a unified exposition of many earlier results.

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First-passage time asymptotics over moving boundaries for random walk bridges

Journal of Applied Probability ◽

10.1017/jpr.2018.39 ◽

2018 ◽

Vol 55 (2) ◽

pp. 627-651 ◽

Cited By ~ 1

Author(s):

Fiona Sloothaak ◽

Vitali Wachtel ◽

Bert Zwart

Keyword(s):

Random Walk ◽

Moving Boundary ◽

First Passage Time ◽

Passage Time ◽

Finite Variance ◽

Tail Behavior ◽

First Passage ◽

The Impact ◽

Return Point ◽

Asymptotic Tail

Abstract We study the asymptotic tail behavior of the first-passage time over a moving boundary for a random walk conditioned to return to zero, where the increments of the random walk have finite variance. Typically, the asymptotic tail behavior may be described through a regularly varying function with exponent -½, where the impact of the boundary is captured by the slowly varying function. Yet, the moving boundary may have a stronger effect when the tail is considered at a time close to the return point of the random walk bridge, leading to a possible phase transition depending on the order of the distance between zero and the moving boundary.

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Oscillating Brownian motion

Journal of Applied Probability ◽

10.1017/s0021900200045599 ◽

1978 ◽

Vol 15 (02) ◽

pp. 300-310 ◽

Cited By ~ 3

Author(s):

Julian Keilson ◽

Jon A. Wellner

Keyword(s):

Brownian Motion ◽

Random Walk ◽

First Passage Time ◽

Passage Time ◽

Brownian Motion Process ◽

First Passage ◽

Special Cases ◽

Motion Process ◽

Transition Densities ◽

Oscillating Random Walk

An ‘oscillating' version of Brownian motion is defined and studied. ‘Ordinary' Brownian motion and ‘reflecting' Brownian motion are shown to arise as special cases. Transition densities, first-passage time distributions, and occupation time distributions for the process are obtained explicitly. Convergence of a simple oscillating random walk to an oscillating Brownian motion process is established by using results of Stone (1963).

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Oscillating Brownian motion

Journal of Applied Probability ◽

10.2307/3213403 ◽

1978 ◽

Vol 15 (2) ◽

pp. 300-310 ◽

Cited By ~ 12

Author(s):

Julian Keilson ◽

Jon A. Wellner

Keyword(s):

Brownian Motion ◽

Random Walk ◽

First Passage Time ◽

Passage Time ◽

Brownian Motion Process ◽

First Passage ◽

Special Cases ◽

Motion Process ◽

Transition Densities ◽

Oscillating Random Walk

An ‘oscillating' version of Brownian motion is defined and studied. ‘Ordinary' Brownian motion and ‘reflecting' Brownian motion are shown to arise as special cases. Transition densities, first-passage time distributions, and occupation time distributions for the process are obtained explicitly. Convergence of a simple oscillating random walk to an oscillating Brownian motion process is established by using results of Stone (1963).

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Large deviations of the first passage time for a random walk with semiexponentially distributed jumps

Siberian Mathematical Journal ◽

10.1007/s11202-006-0117-3 ◽

2006 ◽

Vol 47 (6) ◽

pp. 1084-1101 ◽

Cited By ~ 2

Author(s):

A. A. Mogul’skii

Keyword(s):

Random Walk ◽

Large Deviations ◽

First Passage Time ◽

Passage Time ◽

First Passage

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Tracking a random walk first-passage time through noisy observations

The Annals of Applied Probability ◽

10.1214/11-aap815 ◽

2012 ◽

Vol 22 (5) ◽

pp. 1860-1879 ◽

Cited By ~ 1

Author(s):

Marat V. Burnashev ◽

Aslan Tchamkerten

Keyword(s):

Random Walk ◽

First Passage Time ◽

Passage Time ◽

First Passage

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Hitting-Time Densities of a Two-Dimensional Markov Process

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800002734 ◽

1992 ◽

Vol 6 (4) ◽

pp. 561-580

Author(s):

C. H. Hesse

Keyword(s):

Global Analysis ◽

First Passage Time ◽

Type Equation ◽

Passage Time ◽

Two Dimensional ◽

First Passage ◽

Wide Range ◽

The Laplace Transform ◽

Mean And Variance

This paper deals with the two-dimensional stochastic process (X(t), V(t)) where dX(t) = V(t)dt, V(t) = W(t) + ν for some constant ν and W(t) is a one-dimensional Wiener process with zero mean and variance parameter σ2= 1. We are interested in the first-passage time of (X(t), V(t)) to the plane X = 0 for a process starting from (X(0) = −x, V(0) = ν) with x > 0. The partial differential equation for the Laplace transform of the first-passage time density is transformed into a Schrödinger-type equation and, using methods of global analysis, such as the method of dominant balance, an approximation to the first-passage density is obtained. In a series of simulations, the quality of this approximation is checked. Over a wide range of x and ν it is found to perform well, globally in t. Some applications are mentioned.

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Fractional compound Poisson processes with multiple internal states

Mathematical Modelling of Natural Phenomena ◽

10.1051/mmnp/2018001 ◽

2018 ◽

Vol 13 (1) ◽

pp. 10 ◽

Cited By ~ 8

Author(s):

Pengbo Xu ◽

Weihua Deng

Keyword(s):

Stochastic Process ◽

Random Walk ◽

Waiting Time ◽

Anomalous Diffusion ◽

First Passage Time ◽

Passage Time ◽

Poisson Processes ◽

First Passage ◽

Internal States ◽

Functional Distribution

For the particles undergoing the anomalous diffusion with different waiting time distributions for different internal states, we derive the Fokker-Planck and Feymann-Kac equations, respectively, describing positions of the particles and functional distributions of the trajectories of particles; in particular, the equations governing the functional distribution of internal states are also obtained. The dynamics of the stochastic processes are analyzed and the applications, calculating the distribution of the first passage time and the distribution of the fraction of the occupation time, of the equations are given. For the further application of the newly built models, we make very detailed discussions on the none-immediately-repeated stochastic process, e.g., the random walk of smart animals.

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Random walk and first passage time on a weighted hierarchical network

International Journal of Modern Physics C ◽

10.1142/s0129183114500375 ◽

2014 ◽

Vol 25 (09) ◽

pp. 1450037 ◽

Cited By ~ 6

Author(s):

Feng Zhu ◽

Meifeng Dai ◽

Yujuan Dong ◽

Jie Liu

Keyword(s):

Random Walk ◽

First Passage Time ◽

Passage Time ◽

Scale Factor ◽

Hierarchical Network ◽

Self Similarity ◽

First Passage ◽

Hierarchical Networks ◽

Average Value ◽

Peripheral Node

This paper reports a weighted hierarchical network generated on the basis of self-similarity, in which each edge is assigned a different weight in the same scale. We studied two substantial properties of random walk: the first-passage time (FPT) between a hub node and a peripheral node and the FPT from a peripheral node to a local hub node over the network. Meanwhile, an analytical expression of the average sending time (AST) is deduced, which reflects the average value of FPT from a hub node to any other node. Our result shows that the AST from a hub node to any other node is related to the scale factor and the number of modules. We found that the AST grows sublinearly, linearly and superlinearly respectively with the network order, depending on the range of the scale factor. Our work may shed some light on revealing the diffusion process in hierarchical networks.

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Some approximate results in renewal and dam theories

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700010284 ◽

1971 ◽

Vol 12 (4) ◽

pp. 425-432 ◽

Cited By ~ 21

Author(s):

R. M. Phatarfod

Keyword(s):

Random Walk ◽

Sequential Analysis ◽

Random Variables ◽

Renewal Theory ◽

Slight Modification ◽

Exact Results ◽

Fundamental Identity ◽

Gambler’S Ruin ◽

Gambler's Ruin Problem ◽

Two Barriers

It is well known that Wald's Fundamental Identity (F.I.) in sequential analysis can be used to derive approximate (and, sometimes exact) results in most situations wherein we have essentially a random walk phenomenon. Bartlett [2] used it for the gambler's ruin problem and also for a simple renewal problem. Phatarfod [18] used it for a problem in dam theory. It is the purpose of this paper to show how a generalization of the Fundamental Identity to Markovian variables, (Phatarfod [19]) can be used to derive approximate results in some problems in dam and renewal theories where the random variables involved have Markovian dependence. The reason for considering both the theories together is that the models usually proposed for both the theories — input distribution for dam theory, and lifedistribution for renewal theory — are similar, and only a slight modification (to account for the ‘release rules’ in dam theory, plus the fact that we have two barriers) is necessary to derive results in dam theory from those of renewal theory.

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