Piecewise Homogeneous Random Walk with a Moving Boundary

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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Wald's identity and random walk models for neuron firing

Advances in Applied Probability ◽

10.1017/s0001867800042130 ◽

1976 ◽

Vol 8 (02) ◽

pp. 257-277 ◽

Cited By ~ 3

Author(s):

V. I. Kryukov

Keyword(s):

Random Walk ◽

Sequential Analysis ◽

Moving Boundary ◽

First Passage Time ◽

Model Parameters ◽

First Passage ◽

Fundamental Identity ◽

Main Technique ◽

Special Cases ◽

Wide Range

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.

Download Full-text

Wald's identity and random walk models for neuron firing

Advances in Applied Probability ◽

10.2307/1425904 ◽

1976 ◽

Vol 8 (2) ◽

pp. 257-277 ◽

Cited By ~ 11

Author(s):

V. I. Kryukov

Keyword(s):

Random Walk ◽

Sequential Analysis ◽

Moving Boundary ◽

First Passage Time ◽

Model Parameters ◽

First Passage ◽

Fundamental Identity ◽

Main Technique ◽

Special Cases ◽

Wide Range

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.

Download Full-text