First Passage Time Distribution Research Articles

Rosetta stone for the population dynamics of spiking neuron networks.

Populations of spiking neuron models have densities of their microscopic variables (e.g., single-cell membrane potentials) whose evolution fully capture the collective dynamics of biological networks, even outside equilibrium. Despite its general applicability, the Fokker-Planck equationgoverning such evolution is mainly studied within the borders of the linear response theory, although alternative spectral expansion approaches offer some advantages in the study of the out-of-equilibrium dynamics. This is mainly due to the difficulty in computing the state-dependent coefficients of the expanded system of differential equations. Here, we address this issue by deriving analytic expressions for such coefficients by pairing perturbative solutions of the Fokker-Planck approach with their counterparts from the spectral expansion. A tight relationship emerges between several of these coefficients and the Laplace transform of the interspike interval density (i.e., the distribution of first-passage times). "Coefficients" like the current-to-rate gain function, the eigenvalues of the Fokker-Planck operator and its eigenfunctions at the boundaries are derived without resorting to integral expressions. For the leaky integrate-and-fire neurons, the coupling terms between stationary and nonstationary modes are also worked out paving the way to accurately characterize the critical points and the relaxation timescales in networks of interacting populations.

Vibrational Energy Landscapes and Energy Flow in GPCRs.

We construct and analyze disconnectivity graphs to provide the first graphical representation of the vibrational energy landscape of a protein, in this study β2AR, a G-protein coupled receptor (GPCR), in active and inactive states. The graphs, which indicate the relative free energy of each residue and the minimum free energy barriers for energy transfer between them, reveal important composition, structural and dynamic properties that mediate the flow of energy. Prolines and glycines, which contribute to GPCR plasticity and function, are identified as bottlenecks to energy transport along the backbone from which alternative pathways for energy transport via nearby noncovalent contacts emerge, seen also in the analysis of first passage time (FPT) distributions presented here. Striking differences between the disconnectivity graphs and FPT distributions for the inactive and active states of β2AR are found where structural and dynamic changes occur upon activation, contributing to allosteric regulation.

First-passage time distribution of a Brownian motion: two unexpected journeys

The distribution of the first-passage time (FPT) T a for a Brownian particle with drift μ subject to hitting an absorber at a level a > 0 is well-known and given by its density γ ( t ) = a 2 π t 3 e − ( a − μ t ) 2 2 t , t > 0 , which is normalized only if μ≥0. This article demonstrates the existence of two additional diffusion process categories (one with one parameter and the other with two) that have the same first passage-time distributions when μ < 0. For both, we identify the transition densities and thoroughly investigate the processes. A substantial implication is that the first-passage time distribution does not indicate whether the process originates from a drifted Brownian motion or from one of the new processes presented.

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.