First Passage Time Distribution Research Articles

Analytical expressions for the first passage time distribution and hit distribution in two and three dimensions

The distribution of the time elapsed before a random variable reaches a threshold value for the first time, called the first passage time (FPT) distribution, is a fundamental characteristic of stochastic processes. Here, by solving the standard macroscopic diffusion equation, we derive analytical expressions for the FPT distribution of a diffusing particle hitting a spherical object in two dimensions (2D) and three dimensions (3D) in the course of unrestricted diffusion in open space. In addition, we calculate, analytically, the angular dependence of the FPT, known as the hit distribution. The analytical results are also compared to simulations of the motions of a random walker on a discrete lattice. This topic could be of wide pedagogical interest because the FPT is important not only in physics but also in chemistry, biology, medicine, agriculture, engineering, and finance. Additionally, the central equations often appear in physics and engineering with only trivial variations, making the solution techniques widely applicable.

First passage on disordered intervals.

We derive unexpected first-passage properties for nearest-neighbor hopping on finite intervals with disordered hopping rates, including (a) a highly variable spatial dependence of the first-passage time, (b) huge disparities in first-passage times for different realizations of hopping rates, (c) significant discrepancies between the first moment and the square root of the second moment of the first-passage time, and (d) bimodal first-passage time distributions. Our approach relies on the backward equation, in conjunction with probability generating functions, to obtain all moments, as well as the distribution of first-passage times. Our approach is simpler than previous approaches based on the forward equation, in which computing the mth moment of the first-passage time requires all preceding moments.

Bernoulli trial under restarts: A comparative study of resetting transitions.

A Bernoulli trial describing the escape behavior of a lamb to a safe haven in pursuit by a lion is studied under restarts. The process ends in two ways: either the lamb makes it to the safe haven (success) or is captured by the lion (failure). We study the first passage properties of this Bernoulli trial and find that only mean first passage time exists. Considering Poisson and sharp resetting, we find that the success probability is a monotonically decreasing function of the restart rate. The mean time, however, exhibits a nonmonotonic dependence on the restart rate taking a minimal value at an optimal restart rate. Furthermore, for sharp restart, the mean time possesses a local and a global minima. As a result, the optimal restart rate exhibits a continuous transition for Poisson resetting while it exhibits a discontinuous transition for sharp resetting as a function of the relative separation of the lion and the lamb. We also find that the distribution of first passage times under sharp resetting exhibits a periodic behavior.

Survival probabilities and first-passage distributions of self-propelled particles in spherical cavities.

A model of self-propelled motion in a closed compartment containing simple or complex fluids is formulated in this paper in terms of the dynamics of a point particle moving in a spherical cavity under the action of random thermal forces and exponentially correlated noise. The particle's time evolution is governed by a generalized Langevin equation (GLE) in which the memory function, connected to the thermal forces by a fluctuation-dissipation relation, is described by Jeffrey's model of viscoelasticity (which reduces to a model of ordinary viscous dynamics in a suitable limit). The GLE is transformed exactly to a Fokker-Planck equation that in spherical polar coordinates is in turn found to admit of an exact solution for the particle's probability density function under absorbing boundary conditions at the surface of the sphere. The solution is used to derive an expression (that is also exact) for the survival probability of the particle in the sphere, starting from its center, which is then used to calculate the distribution of the particle's first-passage times to the boundary. The behavior of these quantities is investigated as a function of the Péclet number and the persistence time of the athermal forces, providing insight into the effects of nonequilibrium fluctuations on confined particle motion in three dimensions.

First-passage time statistics for non-linear diffusion

Most processes examined from a standpoint of the first-passage problem are modeled by linear diffusion equations. Here, we consider the non-linear diffusion equation in which diffusivity is power-law dependent on the concentration/probability density and explore its fundamental first-passage properties. Depending on the value of the power-law exponent, we demonstrate the exact and approximate expressions for the survival probability and the first-passage time distribution along with their asymptotic representations. These results refer to the freely and harmonically trapped diffusing particle. While in the former case the mean first-passage time is divergent, albeit the first-passage time distribution is normalized to unity, it is finite in the latter. To support this result, we derive the exact formula for the mean first-passage time to the target prescribed in the minimum of the harmonic potential.

The linear framework II: using graph theory to analyse the transient regime of Markov processes.

The linear framework uses finite, directed graphs with labelled edges to model biomolecular systems. Graph vertices represent chemical species or molecular states, edges represent reactions or transitions and edge labels represent rates that also describe how the system is interacting with its environment. The present paper is a sequel to a recent review of the framework that focussed on how graph-theoretic methods give insight into steady states as rational algebraic functions of the edge labels. Here, we focus on the transient regime for systems that correspond to continuous-time Markov processes. In this case, the graph specifies the infinitesimal generator of the process. We show how the moments of the first-passage time distribution, and related quantities, such as splitting probabilities and conditional first-passage times, can also be expressed as rational algebraic functions of the labels. This capability is timely, as new experimental methods are finally giving access to the transient dynamic regime and revealing the computations and information processing that occur before a steady state is reached. We illustrate the concepts, methods and formulas through examples and show how the results may be used to illuminate previous findings in the literature.

Efficient network exploration by means of resetting self-avoiding random walkers

The self-avoiding random walk (SARW) is a stochastic process whose state variable avoids returning to previously visited states. This non-Markovian feature has turned SARWs a powerful tool for modeling a plethora of relevant aspects in network science, such as network navigability, robustness and resilience. We analytically characterize self-avoiding random walkers that evolve on complex networks and whose memory suffers stochastic resetting, that is, at each step, with a certain probability, they forget their previous trajectory and start free diffusion anew. Several out-of-equilibrium properties are addressed, such as the time-dependent position of the walker, the time-dependent degree distribution of the non-visited network and the first-passage time distribution, and its moments, to target nodes. We examine these metrics for different resetting parameters and network topologies, both synthetic and empirical, and find a good agreement with simulations in all cases. We also explore the role of resetting on network exploration and report a non-monotonic behavior of the cover time: frequent memory resets induce a global minimum in the cover time, significantly outperforming the well-known case of the pure random walk, while reset events that are too spaced apart become detrimental for the network discovery. Our results provide new insights into the profound interplay between topology and dynamics in complex networks, and shed light on the fundamental properties of SARWs in nontrivial environments.

Obtaining Stable Predicted Distributions of Response Times and Decision Outcomes for the Circular Diffusion Model

The circular diffusion model represents continuous outcome decision making as evidence accumulation by a two-dimensional Wiener process with drift on the interior of a disk, whose radius represents the decision criterion for the task. The hitting point on the circumference of the disk represents the decision outcome and the hitting time represents the decision time. The Girsanov change-of-measure theorem applied to the first-passage time distribution for the Euclidean distance Bessel process yields an explicit expression for the joint distribution of decision outcomes and decision times for the model. A problem with the expression for the joint distribution obtained in this way is that the change-of-measure calculation magnifies numerical noise in the series expression for the Bessel process, which can make the expression unstable at small times when the drift rate or decision criterion is large. We introduce a new method that uses an asymptotic approximation to characterize the Bessel process at short times and the series expression for the large times. The resulting expressions are stable across all parts of the parameter space likely to be of interest in experiments, which greatly simplifies the task of fitting the model to data. The new method applies to the spherical and hyperspherical generalizations of the model and to versions of it in which the drift rates are normally distributed across trials with independent or correlated components.

Active Brownian particles in a circular disk with an absorbing boundary.

We solve the time-dependent Fokker-Planck equationfor a two-dimensional active Brownian particle exploring a circular region with an absorbing boundary. Using the passive Brownian particle as basis states and dealing with the activity as a perturbation, we provide a matrix representation of the Fokker-Planck operator and we express the propagator in terms of the perturbed eigenvalues and eigenfunctions. Alternatively, we show that the propagator can be expressed as a combination of the equilibrium eigenstates with weights depending only on time and on the initial conditions, and obeying exact iterative relations. Our solution allows also obtaining the survival probability and the first-passage time distribution. These latter quantities exhibit peculiarities induced by the nonequilibrium character of the dynamics; in particular, they display a strong dependence on the activity of the particle and, to a less extent, also on its rotational diffusivity.

Geometrical optics of first-passage functionals of random acceleration.

Random acceleration is a fundamental stochastic process encountered in many applications. In the one-dimensional version of the process a particle is randomly accelerated according to the Langevin equationx[over ̈](t)=sqrt[2D]ξ(t), where x(t) is the particle's coordinate, ξ(t) is Gaussian white noise with zero mean, and D is the particle velocity diffusion constant. Here, we evaluate the A→0 tail of the distribution P_{n}(A|L) of the functional I[x(t)]=∫_{0}^{T}x^{n}(t)dt=A, where T is the first-passage time of the particle from a specified point x=L to the origin, and n≥0. We employ the optimal fluctuation method akin to geometrical optics. Its crucial element is determination of the optimal path-the most probable realization of the random acceleration process x(t), conditioned on specified A,n, and L. The optimal path dominates the A→0 tail of P_{n}(A|L). We show that this tail has a universal essential singularity, P_{n}(A→0|L)∼exp(-α_{n}L^{3n+2}/DA^{3}), where α_{n} is an n-dependent number which we calculate analytically for n=0, 1, and 2 and numerically for other n. For n=0 our result agrees with the asymptotic of the previously found first-passage time distribution.

Slowest first passage times, redundancy, and menopause timing.

Biological events are often initiated when a random "searcher" finds a "target," which is called a first passage time (FPT). In some biological systems involving multiple searchers, an important timescale is the time it takes the slowest searcher(s) to find a target. For example, of the hundreds of thousands of primordial follicles in a woman's ovarian reserve, it is the slowest to leave that trigger the onset of menopause. Such slowest FPTs may also contribute to the reliability of cell signaling pathways and influence the ability of a cell to locate an external stimulus. In this paper, we use extreme value theory and asymptotic analysis to obtain rigorous approximations to the full probability distribution and moments of slowest FPTs. Though the results are proven in the limit of many searchers, numerical simulations reveal that the approximations are accurate for any number of searchers in typical scenarios of interest. We apply these general mathematical results to models of ovarian aging and menopause timing, which reveals the role of slowest FPTs for understanding redundancy in biological systems. We also apply the theory to several popular models of stochastic search, including search by diffusive, subdiffusive, and mortal searchers.

An accurate and efficient approach to probabilistic conflict prediction

Conflict prediction is a vital component of path planning for autonomous vehicles. Prediction methods must be accurate for reliable navigation, but also computationally efficient to enable online path planning. Efficient prediction methods are especially crucial when testing large sets of candidate trajectories. We present a prediction method that has the same accuracy as existing methods, but up to an order of magnitude faster. This is achieved by rewriting the conflict prediction problem in terms of the first-passage time distribution using a dimension-reduction transform. First-passage time distributions are analytically derived for a subset of Gaussian processes describing vehicle motion. The proposed method is applicable to 2-D stochastic processes where the mean can be approximated by line segments, and the conflict boundary can be approximated by piece-wise straight lines. The proposed method was tested in simulation and compared to two probability flow methods, as well as a recent instantaneous conflict probability method. The results demonstrate a significant decrease of computation time.

Diffusion models with time-dependent parameters: An analysis of computational effort and accuracy of different numerical methods

Drift-diffusion models have become valuable tools in many fields of contemporary psychology and the neurosciences. The present study compares and analyzes different methods (i.e., stochastic differential equation, integral method, Kolmogorov equations, and matrix method) to derive the first-passage time distribution predicted by these models. First, these methods are compared in their accuracy and efficiency. In particular, we address non-standard problems, for example, models with time-dependent drift rates or time-dependent thresholds. Second, a mathematical analysis and a classification of these methods is provided. Finally, we discuss their strengths and caveats.

Survival in a nanoforest of absorbing pillars

We investigate the survival probability of a particle diffusing between two parallel reflecting planes toward a periodic array of absorbing pillars. We approximate the periodic cell of this system by a cylindrical tube containing a single pillar. Using a mode matching method, we obtain an exact solution of the modified Helmholtz equation in this domain that determines the Laplace transform of the survival probability and the associated distribution of first-passage times (FPTs). This solution reveals the respective roles of several geometric parameters: the height and radius of the pillar, the inter-pillar distance, and the distance between confining planes. This model allows us to explore different asymptotic regimes in the probability density of the FPT. In the practically relevant case of a large distance between confining planes, we argue that the mean FPT is much larger than the typical time and thus uninformative. We also illustrate the failure of the capacitance approximation for the principal eigenvalue of the Laplace operator. Some practical implications and future perspectives are discussed.

Numerical approximation of the first-passage time distribution of time-varying diffusion decision models: A mesh-free approach

Sequential sampling theory is the dominant theoretical framework for explaining human decision making behavior. In this theory, the decision process is determined by a stochastic process in a bounded domain, with the likelihood function of the model being the first passage-time distribution of this process. However, many sequential sampling models have no analytic formulation for the first passage-time distribution, making these models difficult and computationally taxing to fit. Thus, developing an efficient and general approximation procedure for the first-passage time distribution of sequential sampling models is crucial for the field of mathematical psychology. Here, we have developed a numerical algorithm based on the mesh-free techniques for numerically solving the Fokker–Planck differential equation, which allows us to approximate the first passage-time distribution of some important classes of sequential sampling models, including the urgency signal model, Ornstein–Uhlenbeck model, and collapsing threshold model. Since the considered equation is a singular partial differential equation in a moving boundary domain, the proposed numerical method first transforms the equation to a regular domain, and then the solution is approximated based on the collocation method. However, since satisfying the boundary conditions in the collocation algorithms can be very tricky, we have imposed the boundary conditions in the basis functions. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. The performance of the proposed method is illustrated by solving several examples.

G-Fractional Diffusion on Bounded Domains in Rd

In this paper, we study g-fractional diffusion on bounded domains in Rd with absorbing boundary conditions. A new general and explicit representation of the solution is obtained. We study the first-passage time distribution, showing the dependence on the particular choice of the function g. Then, we specialize the analysis to the interesting case of a rectangular domain. Finally, we briefly discuss the connection of this general theory with the physical application to the so-called fractional Dodson diffusion model, recently discussed in the literature.

G-fractional diffusion models in bounded domains.

In the recent literature, the g-subdiffusion equationinvolving Caputo fractional derivatives with respect to another function has been studied in relation to anomalous diffusions with a continuous transition between different subdiffusive regimes. In this paper we study the problem of g-fractional diffusion in a bounded domain with absorbing boundaries. We find the explicit solution for the initial boundary value problem, and we study the first-passage time distribution and the mean first-passage time (MFPT). The main outcome is the proof that with a particular choice of the function g it is possible to obtain a finite MFPT, differently from the anomalous diffusion described by a fractional heat equationinvolving the classical Caputo derivative.

First-Passage-Time Analysis of the Pulse-Timing Statistics in a Two-Section Semiconductor Laser under Excitable and Noisy Conditions

A two-section semiconductor laser can exhibit excitability for certain parameter settings. When used as a photonic spiking neuron, it is relevant to investigate its sensitivity to noise due to, e.g., spontaneous emission. Under excitable conditions, the system emits irregularly timed noise-triggered pulses. Their statistics is analyzed in terms of a first-passage time distribution for the fluctuating intensity to reach the threshold for excitable response. Two analytic approximations valid for short and long times, respectively, are derived which very well explain measured and simulated pulse-repetition time distributions. This provides physical insight into the noise-triggered spiking mechanism.

Analytical results for the distribution of first-passage times of random walks on random regular graphs

We present analytical results for the distribution of first-passage (FP) times of random walks (RWs) on random regular graphs that consist of N nodes of degree c ⩾ 3. Starting from a random initial node at time t = 0, at each time step t ⩾ 1 an RW hops into a random neighbor of its previous node. In some of the time steps the RW may hop into a yet-unvisited node while in other time steps it may revisit a node that has already been visited before. We calculate the distribution P(T FP = t) of first-passage times from a random initial node i to a random target node j, where j ≠ i. We distinguish between FP trajectories whose backbone follows the shortest path (SPATH) from the initial node i to the target node j and FP trajectories whose backbone does not follow the shortest path (¬SPATH). More precisely, the SPATH trajectories from the initial node i to the target node j are defined as trajectories in which the subnetwork that consists of the nodes and edges along the trajectory is a tree network. Moreover, the shortest path between i and j on this subnetwork is the same as in the whole network. The SPATH scenario is probable mainly when the length ℓ ij of the shortest path between the initial node i and the target node j is small. The analytical results are found to be in very good agreement with the results obtained from computer simulations.

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.