First Passage Time Density Research Articles

Time-Inhomogeneous Finite Birth Processes with Applications in Epidemic Models

We consider the evolution of a finite population constituted by susceptible and infectious individuals and compare several time-inhomogeneous deterministic models with their stochastic counterpart based on finite birth processes. For these processes, we determine the explicit expressions of the transition probabilities and of the first-passage time densities. For time-homogeneous finite birth processes, the behavior of the mean and the variance of the first-passage time density is also analyzed. Moreover, the approximate duration until the entire population is infected is obtained for a large population size.

Encounter-based reaction-subdiffusion model I: surface adsorption and the local time propagator

In this paper, we develop an encounter-based model of partial surface adsorption for fractional diffusion in a bounded domain. We take the probability of adsorption to depend on the amount of particle-surface contact time, as specified by a Brownian functional known as the boundary local time . If the rate of adsorption is state dependent, then the adsorption process is non-Markovian, reflecting the fact that surface activation/deactivation proceeds progressively by repeated particle encounters. The generalized adsorption event is identified as the first time that the local time crosses a randomly generated threshold. Different models of adsorption (Markovian and non-Markovian) then correspond to different choices for the random threshold probability density . The marginal probability density for particle position prior to absorption depends on ψ and the joint probability density for the pair , also known as the local time propagator. In the case of normal diffusion one can use a Feynman–Kac formula to derive an evolution equation for the propagator. Here we derive the local time propagator equation for fractional diffusion by taking a continuum limit of a heavy-tailed continuous-time random walk (CTRW). We begin by considering a CTRW on a one-dimensional lattice with a reflecting boundary at n = 0. We derive an evolution equation for the joint probability density of the particle location and the amount of time spent at the origin. The continuum limit involves rescaling by a factor , where is the lattice spacing. In the limit , the rescaled functional becomes the Brownian local time at x = 0. We use our encounter-based model to investigate the effects of subdiffusion and non-Markovian adsorption on the long-time behavior of the first passage time (FPT) density in a finite interval with a reflecting boundary at x = L. In particular, we determine how the choice of function ψ affects the large-t power law decay of the FPT density. Finally, we indicate how to extend the model to higher spatial dimensions.

Impact of Evanescence Process on Three-Dimensional Sub-Diffusion based Molecular Communication Channel.

In most of the existing works of molecular communication (MC), the standard diffusion environment is taken into account where the mean square displacement (MSD) of an information molecule (IM) scales linearly with time. On the contrary, this work considers the sub-diffusion motion that appears in crowded and complex (porous or fractal) environments (movement of the particles in the living cells) where the particle's MSD scales as a fractional order power law in time. Moreover, we examine an additional evanescence process resulting from which the molecules can degrade before hitting the boundary of the receiver (RX). Thus, in this work, we present a 3D MC system with a point transmitter (TX) and the spherical RX with the sub-diffusive behavior of an IM along with its evanescence. Furthermore, an IM's closed-form expressions for the arrival probability and the first passage time density (FPTD) are emulated in the above context. Additionally, we investigate the performance of MC by using the concentration-based modulation technique in a sub-diffusion channel. Finally, the considered MC channel is exploited in terms of the probability of detection, probability of false alarm, and probability of error for different parameters such as the reaction rate, fractional power, and radius of the RX.

On the Estimation of the Persistence Exponent for a Fractionally Integrated Brownian Motion by Numerical Simulations

For a fractionally integrated Brownian motion (FIBM) of order α∈(0,1],Xα(t), we investigate the decaying rate of P(τSα>t) as t→+∞, where τSα=inf{t>0:Xα(t)≥S} is the first-passage time (FPT) of Xα(t) through the barrier S>0. Precisely, we study the so-called persistent exponent θ=θ(α) of the FPT tail, such that P(τSα>t)=t−θ+o(1), as t→+∞, and by means of numerical simulation of long enough trajectories of the process Xα(t), we are able to estimate θ(α) and to show that it is a non-increasing function of α∈(0,1], with 1/4≤θ(α)≤1/2. In particular, we are able to validate numerically a new conjecture about the analytical expression of the function θ=θ(α), for α∈(0,1]. Such a numerical validation is carried out in two ways: in the first one, we estimate θ(α), by using the simulated FPT density, obtained for any α∈(0,1]; in the second one, we estimate the persistent exponent by directly calculating Pmax0≤s≤tXα(s)<1. Both ways confirm our conclusions within the limit of numerical approximation. Finally, we investigate the self-similarity property of Xα(t) and we find the upper bound of its covariance function.

On the Absorbing Problems for Wiener, Ornstein–Uhlenbeck, and Feller Diffusion Processes: Similarities and Differences

For the Wiener, Ornstein–Uhlenbeck, and Feller processes, we study the transition probability density functions with an absorbing boundary in the zero state. Particular attention is paid to the proportional cases and to the time-homogeneous cases, by obtaining the first-passage time densities through the zero state. A detailed study of the asymptotic average of local time in the presence of an absorbing boundary is carried out for the time-homogeneous cases. Some relationships between the transition probability density functions in the presence of an absorbing boundary in the zero state and between the first-passage time densities through zero for Wiener, Ornstein–Uhlenbeck, and Feller processes are proven. Moreover, some asymptotic results between the first-passage time densities through zero state are derived. Various numerical computations are performed to illustrate the role played by parameters.

Close agreement between deterministic versus stochastic modeling of first-passage time to vesicle fusion

Ca2+-dependent cell processes, such as neurotransmitter or endocrine vesicle fusion, are inherently stochastic due to large fluctuations in Ca2+ channel gating, Ca2+ diffusion, and Ca2+ binding to buffers and target sensors. However, previous studies revealed closer-than-expected agreement between deterministic and stochastic simulations of Ca2+ diffusion, buffering, and sensing if Ca2+ channel gating is not Ca2+ dependent. To understand this result more fully, we present a comparative study complementing previous work, focusing on Ca2+ dynamics downstream of Ca2+ channel gating. Specifically, we compare deterministic (mean-field/mass-action) and stochastic simulations of vesicle exocytosis latency, quantified by the probability density of the first-passage time (FPT) to the Ca2+-bound state of a vesicle fusion sensor, following a brief Ca2+ current pulse. We show that under physiological constraints, the discrepancy between FPT densities obtained using the two approaches remains small even if as few as ∼50 Ca2+ ions enter per single channel-vesicle release unit. Using a reduced two-compartment model for ease of analysis, we illustrate how this close agreement arises from the smallness of correlations between fluctuations of the reactant molecule numbers, despite the large magnitude of fluctuation amplitudes. This holds if all relevant reactions are heteroreaction between molecules of different species, as is the case for bimolecular Ca2+ binding to buffers and downstream sensor targets. In this case, diffusion and buffering effectively decorrelate the state of the Ca2+ sensor from local Ca2+ fluctuations. Thus, fluctuations in the Ca2+ sensor’s state underlying the FPT distribution are only weakly affected by the fluctuations in the local Ca2+ concentration around its average, deterministically computable value.

On the First-Passage Time Problem for a Feller-Type Diffusion Process

We consider the first-passage time problem for the Feller-type diffusion process, having infinitesimal drift B1(x,t)=α(t)x+β(t) and infinitesimal variance B2(x,t)=2r(t)x, defined in the space state [0,+∞), with α(t)∈R, β(t)>0, r(t)>0 continuous functions. For the time-homogeneous case, some relations between the first-passage time densities of the Feller process and of the Wiener and the Ornstein–Uhlenbeck processes are discussed. The asymptotic behavior of the first-passage time density through a time-dependent boundary is analyzed for an asymptotically constant boundary and for an asymptotically periodic boundary. Furthermore, when β(t)=ξr(t), with ξ>0, we discuss the asymptotic behavior of the first-passage density and we obtain some closed-form results for special time-varying boundaries.

Mapping input noise to escape noise in integrate-and-fire neurons: a level-crossing approach

Noise in spiking neurons is commonly modeled by a noisy input current or by generating output spikes stochastically with a voltage-dependent hazard rate (“escape noise”). While input noise lends itself to modeling biophysical noise processes, the phenomenological escape noise is mathematically more tractable. Using the level-crossing theory for differentiable Gaussian processes, we derive an approximate mapping between colored input noise and escape noise in leaky integrate-and-fire neurons. This mapping requires the first-passage-time (FPT) density of an overdamped Brownian particle driven by colored noise with respect to an arbitrarily moving boundary. Starting from the Wiener–Rice series for the FPT density, we apply the second-order decoupling approximation of Stratonovich to the case of moving boundaries and derive a simplified hazard-rate representation that is local in time and numerically efficient. This simplification requires the calculation of the non-stationary auto-correlation function of the level-crossing process: For exponentially correlated input noise (Ornstein–Uhlenbeck process), we obtain an exact formula for the zero-lag auto-correlation as a function of noise parameters, mean membrane potential and its speed, as well as an exponential approximation of the full auto-correlation function. The theory well predicts the FPT and interspike interval densities as well as the population activities obtained from simulations with colored input noise and time-dependent stimulus or boundary. The agreement with simulations is strongly enhanced across the sub- and suprathreshold firing regime compared to a first-order decoupling approximation that neglects correlations between level crossings. The second-order approximation also improves upon a previously proposed theory in the subthreshold regime. Depending on a simplicity-accuracy trade-off, all considered approximations represent useful mappings from colored input noise to escape noise, enabling progress in the theory of neuronal population dynamics.

Time-Inhomogeneous Feller-type Diffusion Process with Absorbing Boundary Condition

A time-inhomogeneous Feller-type diffusion process with linear infinitesimal drift alpha (t)x+beta (t) and linear infinitesimal variance 2r(t)x is considered. For this process, the transition density in the presence of an absorbing boundary in the zero-state and the first-passage time density through the zero-state are obtained. Special attention is dedicated to the proportional case, in which the immigration intensity function beta (t) and the noise intensity function r(t) are connected via the relation beta (t)=xi ,r(t), with 0le xi <1. Various numerical computations are performed to illustrate the effect of the parameters on the first-passage time density, by assuming that alpha (t), beta (t) or both of these functions exhibit some kind of periodicity.

Partial derivatives for the first-passage time distribution in Wiener diffusion models

The Wiener diffusion model with two absorbing barriers is frequently used in modeling decisions and decision latencies jointly. We derive representations of the partial derivatives of the density and cumulative distribution function of first-passage times. Different representations, based on infinite series, differ in convergence for small and large times, and we provide methods that control the approximation errors. These methods and their implementation in an R package are helpful whenever gradient information is required as is the case in many frequentist and Bayesian approaches to modeling.

Generalized master equation for first-passage problems in partitioned spaces

Motivated by a range of biological applications related to the transport of molecules in cells, we present a modular framework to treat first-passage problems for diffusion in partitioned spaces. The spatial domains can differ with respect to their diffusivity, geometry, and dimensionality, but can also refer to transport modes alternating between diffusive, driven, or anomalous motion. The approach relies on a coarse-graining of the motion by dissecting the trajectories on domain boundaries or when the mode of transport changes, yielding a small set of states. The time evolution of the reduced model follows a generalized master equation (GME) for non-Markovian jump processes; the GME takes the form of a set of linear integro-differential equations in the occupation probabilities of the states and the corresponding probability fluxes. Further building blocks of the model are partial first-passage time (FPT) densities, which encode the transport behavior in each domain or state. After an outline of the general framework for multiple domains, the approach is exemplified and validated for a target search problem with two domains in one- and three-dimensional space, first by exactly reproducing known results for an artificially divided, homogeneous space, and second by considering the situation of domains with distinct diffusivities. Analytical solutions for the FPT densities are given in Laplace domain and are complemented by numerical backtransforms yielding FPT densities over many decades in time, confirming that the geometry and heterogeneity of the space can introduce additional characteristic time scales.

A physicist’s guide to explicit summation formulas involving zeros of Bessel functions and related spectral sums

In this pedagogical review, we summarize the mathematical basis and practical hints for the explicit analytical computation of spectral sums that involve the eigenvalues of the Laplace operator in simple domains such as [Formula: see text]-dimensional balls (with [Formula: see text]), an annulus, a spherical shell, right circular cylinders, rectangles and rectangular cuboids. Such sums appear as spectral expansions of heat kernels, survival probabilities, first-passage time densities, and reaction rates in many diffusion-oriented applications. As the eigenvalues are determined by zeros of an appropriate linear combination of a Bessel function and its derivative, there are powerful analytical tools for computing such spectral sums. We discuss three main strategies: representations of meromorphic functions as sums of partial fractions, Fourier–Bessel and Dini series, and direct evaluation of the Laplace-transformed heat kernels. The major emphasis is put on a pedagogic introduction, the practical aspects of these strategies, their advantages and limitations. The review gathers many summation formulas for spectral sums that are dispersed in the literature.

Molecular Communication in Fractional Diffusive Channel

The molecular communication system with anomalous/fractional diffusion inside a one-dimensional (1-D) environment is considered. The time-dependent diffusivity is incorporated in terms of the power-law diffusivity, and the expression of first passage time density (FPTD) is derived. Further, the peak pulse time and peak concentration corresponding to the derived FPTD function are obtained. Moreover, the analysis is extended in terms of the average probability of error and throughput for the anomalous diffusion channel. The analytical results are validated through simulations.

First-passage statistics under stochastic resetting in bounded domains

We investigate the first-passage problem where a diffusive searcher stochastically resets to a fixed position at a constant rate in a bounded domain. We put forward an analytical framework for this problem where the resetting rate r, the resetting position xr, the initial position x0, the domain size L, and the particle’s diffusion constant D are independent variables. From this we obtain analytical expressions for the mean-first passage time, survival probability and the first-passage time density in Laplace space in terms of the above independent variables, and closed forms in time-domain expression in some cases. For the first-passage time distributions their full-time profiles in time-domain are numerically obtained and validated by Monte Carlo simulations. We show that for the general resetting condition the first-passage process has rich nontrivial features as it combines effects from resetting and the finiteness of the domain. A phase diagram showing the regions for resetting-enhanced and -suppressed first-passages is obtained analytically. Counter-intuitively, resetting facilitates the first-passage event even if the particle resets to a position that is further away from the target than where it started. Finally, we show that averaging over resetting positions, resetting never helps the search compared to a random search, unless the resetting position is displaced from the initial position.

How does transient signaling input affect the spike timing of postsynaptic neuron near the threshold regime: an analytical study

The noisy threshold regime, where even a small set of presynaptic neurons can significantly affect postsynaptic spike-timing, is suggested as a key requisite for computation in neurons with high variability. It also has been proposed that signals under the noisy conditions are successfully transferred by a few strong synapses and/or by an assembly of nearly synchronous synaptic activities. We analytically investigate the impact of a transient signaling input on a leaky integrate-and-fire postsynaptic neuron that receives background noise near the threshold regime. The signaling input models a single strong synapse or a set of synchronous synapses, while the background noise represents a lot of weak synapses. We find an analytic solution that explains how the first-passage time (ISI) density is changed by transient signaling input. The analysis allows us to connect properties of the signaling input like spike timing and amplitude with postsynaptic first-passage time density in a noisy environment. Based on the analytic solution, we calculate the Fisher information with respect to the signaling input’s amplitude. For a wide range of amplitudes, we observe a non-monotonic behavior for the Fisher information as a function of background noise. Moreover, Fisher information non-trivially depends on the signaling input’s amplitude; changing the amplitude, we observe one maximum in the high level of the background noise. The single maximum splits into two maximums in the low noise regime. This finding demonstrates the benefit of the analytic solution in investigating signal transfer by neurons.

Stochastic Dynamics of Spike Trains of Neuron Models for Firing Times with Exponential Decay and Jump-Diffusion

Herein we derive the first passage time density (FPTD) for a spike discharge in a single neuron with inputs that produce either inhibitory or excitatory impulses. We study the stochastic dynamics using imbedding approach and derive an integro-differential equation for the FPTD. By taking the kernel in the master equation in separable form, we obtain a third order differential equation. Applying a suitable limiting procedure, we convert the differential equation into that of an O.U. process. Finally we introduce jump-diffusion to study the FPTD of spike train of a single neuron.

Universal Proximity Effect in Target Search Kinetics in the Few-Encounter Limit

When does a diffusing particle reach its target for the first time? This first-passage time (FPT) problem is central to the kinetics of molecular reactions in chemistry and molecular biology. Here we explain the behavior of smooth FPT densities, for which all moments are finite, and demonstrate universal yet generally non-Poissonian long-time asymptotics for a broad variety of transport processes. While Poisson-like asymptotics arise generically in the presence of an effective repulsion in the immediate vicinity of the target, a time-scale separation between direct and reflected indirect trajectories gives rise to a universal proximity effect: Direct paths, heading more or less straight from the point of release to the target, become typical and focused, with a narrow spread of the corresponding first passage times. Conversely, statistically dominant indirect paths exploring the system size tend to be massively dissimilar. The initial distance to the target particularly impacts gene regulatory or competitive stochastic processes, for which often few binding events determine the regulatory outcome. The proximity effect is independent of details of the transport, highlighting the robust character of the FPT features uncovered here.

A simple method to calculate first-passage time densities with arbitrary initial conditions

Numerous applications all the way from biology and physics to economics depend on the density of first crossings over a boundary. Motivated by the lack of general purpose analytical tools for computing first-passage time densities (FPTDs) for complex problems, we propose a new simple method based on the independent interval approximation (IIA). We generalise previous formulations of the IIA to include arbitrary initial conditions as well as to deal with discrete time and non-smooth continuous time processes. We derive a closed form expression for the FPTD in z and Laplace-transform space to a boundary in one dimension. Two classes of problems are analysed in detail: discrete time symmetric random walks (Markovian) and continuous time Gaussian stationary processes (Markovian and non-Markovian). Our results are in good agreement with Langevin dynamics simulations.

Fast and accurate Monte Carlo sampling of first-passage times from Wiener diffusion models.

We present a new, fast approach for drawing boundary crossing samples from Wiener diffusion models. Diffusion models are widely applied to model choices and reaction times in two-choice decisions. Samples from these models can be used to simulate the choices and reaction times they predict. These samples, in turn, can be utilized to adjust the models’ parameters to match observed behavior from humans and other animals. Usually, such samples are drawn by simulating a stochastic differential equation in discrete time steps, which is slow and leads to biases in the reaction time estimates. Our method, instead, facilitates known expressions for first-passage time densities, which results in unbiased, exact samples and a hundred to thousand-fold speed increase in typical situations. In its most basic form it is restricted to diffusion models with symmetric boundaries and non-leaky accumulation, but our approach can be extended to also handle asymmetric boundaries or to approximate leaky accumulation.

Explicit form of the first-passage-time density for accelerating subdiffusion

The first passage time (FPT) statistics play a central role in different fields. Apart of description of the underlying system state and its evolution it is essential to investigate the time at which this state reaches a certain area for the first time. In this paper we study the FPT problem for recently developed models of an accelerating subdiffusion which extends, incorporating various short and long time scalings of the mean square displacement (MSD), popular models of subdiffusion. Based on fractional-order differential equations we derive the Laplace transform of the FPT density function in the case of constant bias. For a force depending linearly on the position we provide a full analytical formula for the FPT density in terms of generalized Mittag-Leffler functions and the Hermite polynomials. We also provide a numerical evidence that FPT properties are strictly connected with MSD scalings of the accelerating subdiffusion.