First Passage Time Density Research Articles

First passage time densities in resonate-and-fire models

Motivated by the dynamics of resonant neurons we discuss the properties of the first passage time (FPT) densities for non-Markovian differentiable random processes. We start from an exact expression for the FPT density in terms of an infinite series of integrals over joint densities of level crossings, and consider different approximations based on truncation or on approximate summation of this series. Thus the first few terms of the series give good approximations for the FPT density on short times. For rapidly decaying correlations the decoupling approximations perform well in the whole time domain. As an example we consider resonate-and-fire neurons representing stochastic underdamped or moderately damped harmonic oscillators driven by white Gaussian or by Ornstein-Uhlenbeck noise. We show that approximations reproduce all qualitatively different structures of the FPT densities: from monomodal to multimodal densities with decaying peaks. The approximations work for the systems of whatever dimension and are especially effective for the processes with narrow spectral density, exactly when Markovian approximations fail.

First passage and arrival time densities for Lévy flights and the failure of the method of images

We discuss the first passage time problem in the semi-infinite interval, for homogeneous stochastic Markov processes with L{\'e}vy stable jump length distributions $\lambda(x)\sim\ell^{\alpha}/|x|^{1+\alpha}$ ($|x|\gg\ell$), namely, L{\'e}vy flights (LFs). In particular, we demonstrate that the method of images leads to a result, which violates a theorem due to Sparre Andersen, according to which an arbitrary continuous and symmetric jump length distribution produces a first passage time density (FPTD) governed by the universal long-time decay $\sim t^{-3/2}$. Conversely, we show that for LFs the direct definition known from Gaussian processes in fact defines the probability density of first arrival, which for LFs differs from the FPTD. Our findings are corroborated by numerical results.

FIRST PASSAGE TIME PROBLEM FOR BIASED CONTINUOUS-TIME RANDOM WALKS

We study the first passage time (FPT) problem for biased continuous time random walks. Using the recently formulated framework of fractional Fokker-Planck equations, we obtain the Laplace transform of the FPT density function when the bias is constant. When the bias depends linearly on the position, the full FPT density function is derived in terms of Hermite polynomials and generalized Mittag-Leffler functions.

First-passage time, survival probability and propagator on deterministic fractals

The first-passage time density, psi (r, t) (defined as the probability density for the time spent by a random walker to travel (for the first time) the distance r that separates the starting site from its nearest neighbours), and the survival probability S(r, t) (i.e. the probability that a random walker who starts at a site has not been absorbed by traps located on its nearest neighbours at distance r in the time interval (0, t)), were calculated for the class of deterministic fractals in which sites are isolated from the rest of the lattice by their nearest neighbours. The large xi identical to r/( square root (2Dt)1dw/) asymptotic expressions for these quantities are psi (r, t) approximately=A xi nu 2+dw/ exp(-C xi nu ) and h(r, t)=1-S(r, t) approximately=(A/C)(dw-1) xi - nu 2/exp(-C xi nu ) with v=dw/(dw-1), A and C being characteristic constants for each fractal. The asymptotic expression for S(r, t) is used to justify that, for this class of deterministic fractals, the propagator or Green function is given asymptotically by P(r, t)~t-ds/2 xi alpha exp(-C xi nu ) for large xi , with alpha = nu /2-df. This value of alpha differs from others proposed recently.

Hitting-Time Densities of a Two-Dimensional Markov Process

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.

On the asymptotic behaviour of first-passage-time densities for one-dimensional diffusion processes and varying boundaries

Making use of the integral equations given in [1], [2] and [3], the asymptotic behaviour of the first-passage time (FPT) p.d.f.'s through certain time-varying boundaries, including periodic boundaries, is determined for a class of one-dimensional diffusion processes with steady-state density. Sufficient conditions are given for the cases both of single and of pairs of asymptotically constant and asymptotically periodic boundaries, under which the FPT densities asymptotically exhibit an exponential behaviour. Explicit expressions are then worked out for the processes that can be obtained from the Ornstein–Uhlenbeck process by spatial transformations. Some new asymptotic results for the FPT density of the Wiener process are finally proved, together with a few miscellaneous results.

On the asymptotic behaviour of first-passage-time densities for one-dimensional diffusion processes and varying boundaries

Making use of the integral equations given in [1], [2] and [3], the asymptotic behaviour of the first-passage time (FPT) p.d.f.'s through certain time-varying boundaries, including periodic boundaries, is determined for a class of one-dimensional diffusion processes with steady-state density. Sufficient conditions are given for the cases both of single and of pairs of asymptotically constant and asymptotically periodic boundaries, under which the FPT densities asymptotically exhibit an exponential behaviour. Explicit expressions are then worked out for the processes that can be obtained from the Ornstein–Uhlenbeck process by spatial transformations. Some new asymptotic results for the FPT density of the Wiener process are finally proved, together with a few miscellaneous results.

Structural reliability under evolutionary seismic excitation

Two approximate methods of obtaining survival probability estimates of a structure initially at rest, and subjected to a non-stationary seismic excitation have been presented. Both methods are based on approximating the response amplitude by a Markov process. In the first one, a discrete extreme point process has been utilized for the derivation of reliability estimates of the absolute response | x( t)|. In the second approach, similar estimates for the amplitude a( t) have been derived by solving by discretization an integral equation. The seismic excitations which have been considered are stationary processes modulated by exponential type deterministic functions. It has been found that, for this particular type of excitation, the structure experiences an early dramatic reduction of its survival probability as compared to that for the unmodulated excitation. Characteristic of the structural behavior is, also, the fact that for higher barriers the survival probability does not eventually tend to zero, as is the case for the stationary excitations, but it attains a constant non-zero value. Consequently for this case the concept of first-passage time density function is meaningless. Based on numerical data obtained by conducting digital Monte Carlo simulations it has been shown that the accuracies of the proposed methods are quite acceptable for common engineering applications.

Time-dependent barriers and first-passage times--different types of neuronal models.

To study the phenomenon of spike discharge in a single neuron, first-passage time density of the membrane potential to cross or reach the threshold value in different situations is analyzed. The neuron is taken as a leaky integrator and the inputs are due to Poisson events occurring independently. Closed solutions using the compensation method and imbedding techniques are arrived at. Using Keilson's limiting procedure, diffusion processes with boundaries are solved. To simulate the refractory period of the neuron, thresholds moving linearly or exponentially with time are considered. Very interesting Wald identities are obtained in the case of jump processes which go over to the corresponding identities for diffusion processes obtained by a limiting procedure.