First Passage Time Analysis Research Articles

Clustering of primordial black holes from quantum diffusion during inflation

We study how large fluctuations are spatially correlated in the presence of quantum diffusion during inflation. This is done by computing real-space correlation functions in the stochastic-δ N formalism. We first derive an exact description of physical distances as measured by a local observer at the end of inflation, improving on previous works. Our approach is based on recursive algorithmic methods that consistently include volume-weighting effects. We then propose a “large-volume” approximation under which calculations can be done using first-passage time analysis only, and from which a new formula for the power spectrum in stochastic inflation is derived. We then study the full two-point statistics of the curvature perturbation. Due to the presence of exponential tails, we find that the joint distribution of large fluctuations is of the form P(ζ R 1, ζ R 2) = F(R 1,R 2, r) P(ζ R 1)P( ζ R 2), where ζ R 1 and ζ R 2 denote the curvature perturbation coarse-grained at radii R 1 and R 2, around two spatial points distant by r. This implies that, on the tail, the reduced correlation function, defined as P(ζ R 1 > ζc, ζ R 2 > ζc)/[P(ζ R 1 > ζc) P(ζ R 2 > ζc)]-1, is independent of the threshold value ζc. This contrasts with Gaussian statistics where the same quantity strongly decays with ζc, and shows the existence of a universal clustering profile for all structures forming in the exponential tails. Structures forming in the intermediate (i.e. not yet exponential) tails may feature different, model-dependent behaviours.

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.

Dissection and integration of bursty transcriptional dynamics for complex systems

RNA velocity estimation is a potentially powerful tool to reveal the directionality of transcriptional changes in single-cell RNA-sequencing data, but it lacks accuracy, absent advanced metabolic labeling techniques. We developed an approach, TopicVelo, that disentangles simultaneous, yet distinct, dynamics by using a probabilistic topic model, a highly interpretable form of latent space factorization, to infer cells and genes associated with individual processes, thereby capturing cellular pluripotency or multifaceted functionality. Focusing on process-associated cells and genes enables accurate estimation of process-specific velocities via a master equation for a transcriptional burst model accounting for intrinsic stochasticity. The method obtains a global transition matrix by leveraging cell topic weights to integrate process-specific signals. In challenging systems, this method accurately recovers complex transitions and terminal states, while our use of first-passage time analysis provides insights into transient transitions. These results expand the limits of RNA velocity, empowering future studies of cell fate and functional responses.

First-Passage Time Analysis Based on GPS Data Offers a New Approach to Estimate Restricted Zones for the Management of Infectious Diseases in Wildlife: A Case Study Using the Example of African Swine Fever

An essential part of any disease containment and eradication policy is the implementation of restricted zones, but determining the appropriate size of these zones can be challenging for managers. We designed a new method, based on animal movement, to help assess how large restricted zones should be after a spontaneous outbreak to successfully control infectious diseases in wildlife. Our approach uses first-passage time (FPT) analysis and Cox proportional hazard (CPH) models to calculate and compare the risk of an animal leaving different-sized areas. We illustrate our approach using the example of the African swine fever (ASF) virus and its wild pig reservoir host species, the wild boar (Sus scrofa), and we investigate the feasibility of applying this method to other systems. Using GPS data from 57 wild boar living in the Hainich National Park, Germany, we calculate the time spent by each individual in areas of different sizes using FPT analysis. We apply CPH models on the derived data to compare the risk of leaving areas of different sizes and to assess the effects of season and the sex of the wild boar on the risk of leaving. We conduct survival analyses to estimate the risk of leaving an area over time. Our results indicate that the risk of leaving an area decreases exponentially by 10% for each 100 m increase in radius size so that the differences were more pronounced for small sizes. Furthermore, the probability of leaving increases exponentially with time. Wild boar had a similar risk of leaving an area of a given size throughout the year, except in spring and winter, when females had a much lower risk of leaving. Our findings are in agreement with the literature on wild boar movement, further validating our method, and repeated analyses with location data resampled at different rates gave similar results. Our results may be applicable only to our study area, but they demonstrate the applicability of the proposed method to any ecosystem where wild boar populations are likely to be infected with ASF and where restricted zones should be established accordingly. The outlined approach relies solely on the analysis of movement data and provides a useful tool to determine the optimal size of restricted zones. It can also be applied to future outbreaks of other diseases.

Discrete and Continuous Operational Calculus in N-Critical Shocks Reliability Systems with Aging under Delayed Information

We study a reliability system subject to occasional random shocks of random magnitudes W0,W1,W2,… occurring at times τ0,τ1,τ2,…. Any such shock is harmless or critical dependent on Wk≤H or Wk>H, given a fixed threshold H. It takes a total of N critical shocks to knock the system down. In addition, the system ages in accordance with a monotone increasing continuous function δ, so that when δT crosses some sustainability threshold D at time T, the system becomes essentially inoperational. However, it can still function for a while undetected. The most common way to do the checking is at one of the moments τ1,τ2,… when the shocks are registered. Thus, if crossing of D by δ occurs at time T∈τk,τk+1, only at time τk+1, can one identify the system’s failure. The age-related failure is detected with some random delay. The objective is to predict when the system fails, through the Nth critical shock or by the observed aging moment, whichever of the two events comes first. We use and embellish tools of discrete and continuous operational calculus (D-operator and Laplace–Carson transform), combined with first-passage time analysis of random walk processes, to arrive at fully explicit functionals of joint distributions for the observed lifetime of the system and cumulative damage to the system. We discuss various special cases and modifications including the assumption that D is random (and so is T). A number of examples and numerically drawn figures demonstrate the analytic tractability of the results.

First-passage time analysis of diffusion-controlled reactions in single-molecule detection

Single-molecule detection (SMD) aims to achieve the ultimate limit-of-detection (LOD) in biosensing. This method detects a countable number of targeted analyte molecules in solution, where the dynamics of molecule diffusion, capturing, identification and delivery greatly impact the SMD's efficiency and accuracy. In this study, we adopt the first-passage time method to investigate the diffusion-controlled reaction process in SMD. We analyze the influence of detection conditions on incubation time and the expected coefficient of variation (CV) under three SMD molecule capturing strategies, including solid-phase capturing (one-dimensional solid-liquid interface fixation), liquid-phase magnetic bead (MB) capturing, and liquid-phase direct fluorescence pair labeling. We find that inside a finite-sized reaction chamber, a finite average reaction time exists in all three capturing strategies, while the liquid-phase strategies are in general more efficient than the solid-phase approaches. CV can be estimated by averaging first-passage time solely in all three strategies, and the CV reduction is achievable given an extended reaction time. To further enable zeptomolar detection, extra treatments, such as adopting liquid-phase fluorescence pairs with high diffusion rates to label the molecule, or designing specific sensing devices with large effective sensing areas would be required. This framework provides solid theoretical support to guide the design of SMD sensing strategies and sensor structures to achieve desired measurement time and CV.

Reproduction drives changes in space use and habitat selection in a highly adaptable invasive mammal

Abstract For ungulates, it is not well understood how the interaction between habitat and reproduction affects movement behavior, space use, and habitat selection. We used known parturition (farrowing) data to validate First Passage Time (FPT) movement analysis to identify wild pig (Sus scrofa) parturition events from data collected with GPS collars. We examined home range and habitat selection during three physiologically distinct time periods in the reproductive cycle: late-gestation; farrowing; and neonate-care period. Sows exhibited a behavioral change 1-2 days prior to farrowing, suggesting FPT analysis can reliably identify parturition events for wide-ranging species that have a focused birthing area. Home range was smallest during the farrowing period, likely reflective of nest building, parturition, and protection of neonates. Home range size during the neonate-care period was intermediate between the late-gestation and farrowing periods, indicative of offspring care that may restrict maternal movement. Across all periods, sows avoided developed areas that have sparse canopy and ground cover and are associated with human activities. During late-gestation, sows avoided pine forests that have an open understory and less vegetative cover. During late-gestation and neonate-care, sows selected bottomland hardwood forests, habitat associated with ample food, cover, and water. During farrowing and neonate-care periods, sows selected upland hardwood forests, habitat with high quality food and cover for neonates. The physiological requirements of reproduction drive female habitat selection and spatial scale of movement patterns associated with parturition. Our study contributes to delineation of the appropriate scale at which to analyze movement data to provide insight about where individuals chose to place their home range, how much space to use, and how they use resources on the landscape to maximize reproductive success and fitness.

Cavity nucleation in single-component homogeneous amorphous solids under negative pressure

Understanding the cavity formation and cavity growth mechanisms in solids has fundamental and applied importance for the correct determination of their exploitation capabilities and mechanical characteristics. In this work, we present the molecular dynamics simulation results for the process of homogeneous formation of nanosized cavities in a single-component amorphous metallic alloy. To identify cavities of various shapes and sizes, an original method has been developed, which is based on filling cavities by virtual particles (balls) of the same diameter. By means of the mean first-passage time analysis, it was shown that the cavity formation in an amorphous metallic melt is the activation-type process. This process can be described in terms of the classical nucleation theory, which is usually applied to the case of first order phase transitions. Activation energy, critical size and nucleation rate of cavities are calculated, the values of which are comparable with those for the case of crystal nucleation in amorphous systems.

Non-volatile memory characteristics of a Ti/HfO2/Pt synaptic device with a crossbar array structure

The resistive switching and synaptic behavior of a fabricated Ti/HfO2/Pt crossbar array device are investigated. The results demonstrated that TiOx layers are created by the movement of oxygen ions during the positive SET process, thereby improving the endurance and multilevel switching behavior of the device. The random properties of SET process were described with the help of stochastic model of memristor based on the length of conductive filament. The analysis of the mean first passage time allows estimating the parameters of the dielectric switching layer such as the activation energy of the diffusive defects, its variation under the influence of the driving voltage and the value of the working temperature. Furthermore, the conductive mechanism of the Ti/HfO2/Pt device with temperature variations and Schottky emission fitting was studied. Furthermore, the potentiation, depression, and the spike-time dependent plasticity behavior of the Ti/HfO2/Pt devices have been demonstrated by well-designed pulse inputs. The observed gradual change in conductance is suitable for a synaptic device in a neuromorphic system.

Relating cellular signaling timescales to single-molecule kinetics: A first-passage time analysis of Ras activation by SOS

Son of Sevenless (SOS) is a Ras guanine nucleotide exchange factor (GEF) that plays a central role in numerous cellular signaling pathways. Like many other signaling molecules, SOS is autoinhibited in the cytosol and activates only after recruitment to the membrane. The mean activation time of individual SOS molecules has recently been measured to be ∼60 s, which is unexpectedly long and seemingly contradictory with cellular signaling timescales, which have been measured to be as fast as several seconds. Here, we rectify this discrepancy using a first-passage time analysis to reconstruct the effective signaling timescale of multiple SOS molecules from their single-molecule activation kinetics. Along with corresponding experimental measurements, this analysis reveals how the functional response time, comprised of many slowly activating molecules, can become substantially faster than the average molecular kinetics. This consequence stems from the enzymatic processivity of SOS in a highly out-of-equilibrium reaction cycle during receptor triggering. Ultimately, rare, early activation events dominate the macroscopic reaction dynamics.

Physics of free climbing.

The theory of stochastic processes provides theoretical tools which can be efficiently used to explore the properties of noise-induced escape kinetics. Since noise-facilitated escape over the potential barrier resembles free climbing, one can use the first-passage time theory in an analysis of rock climbing. We perform the analysis of the mean first-passage time in order to answer the question regarding the optimal, i.e., resulting in the fastest climbing, rope length. It is demonstrated that there is a discrete set of favorable rope lengths assuring the shortest climbing times, as they correspond to local minima of mean first-passage time. Within the set of favorable rope lengths there is the optimal rope giving rise to the shortest climbing time. In particular, more experienced climbers can decrease their climbing time by using longer ropes.

Tipping point and noise-induced transients in ecological networks.

A challenging and outstanding problem in interdisciplinary research is to understand the interplay between transients and stochasticity in high-dimensional dynamical systems. Focusing on the tipping-point dynamics in complex mutualistic networks in ecology constructed from empirical data, we investigate the phenomena of noise-induced collapse and noise-induced recovery. Two types of noise are studied: environmental (Gaussian white) noise and state-dependent demographic noise. The dynamical mechanism responsible for both phenomena is a transition from one stable steady state to another driven by stochastic forcing, mediated by an unstable steady state. Exploiting a generic and effective two-dimensional reduced model for real-world mutualistic networks, we find that the average transient lifetime scales algebraically with the noise amplitude, for both environmental and demographic noise. We develop a physical understanding of the scaling laws through an analysis of the mean first passage time from one steady state to another. The phenomena of noise-induced collapse and recovery and the associated scaling laws have implications for managing high-dimensional ecological systems.

Habitat preferences of Southern Ground-hornbills in the Kruger National Park: implications for future conservation measures

Understanding how a species utilises its habitat, and the processes that give rise to its movements and patterns of space use, is critical for its conservation. Southern Ground-hornbills Bucorvus leadbeateri are listed as Endangered in South Africa, as a result of habitat loss and persecution. The National Species Recovery Plan lists reintroductions as a suitable conservation action, but highlights “understanding the exact habitat requirements of Southern Ground-hornbills” as a knowledge gap. In this study, we used tracking data from six Southern Ground-hornbill groups (a total of 37,060 GPS locations) in the Kruger National Park to investigate their seasonal home range differences and habitat preferences. We used first-passage time analysis to determine the scale at which Southern Ground-hornbills concentrate their foraging efforts and whether specific movement behaviours were linked to habitat types. We found marked differences in seasonal home ranges, with all groups showing a range contraction during the breeding season. Grassland and open woodland habitat types were used throughout the year in accordance with their availability within the territory, with grassland, open woodland and dense thicket being favoured habitats for foraging. Our habitat preference results, based on longitudinal GPS data, allowed us to determine ideal habitat ratios (grassland:open woodland:low shrubland of 1.00:6.10:0.09 ha) to assist with the selection of suitable reintroduction sites for Southern Ground-hornbills. With an increasing number of species being threatened with extinction, reintroductions into suitable habitats may be a useful conservation mitigation measure. However, our findings highlight the importance of a thorough understanding of a species’ movement and space use prior to the selection of areas for reintroduction to ensure the establishment and sustainability of these species at these sites.

Bridging Dynamics of Telechelic Polymers between Solid Surfaces

We employ Brownian dynamics simulations combined with forward flux sampling and theoretical first-passage time analyses to describe the rates of transitions between loops and bridges of telechelic polymers between solid surfaces representing e.g. latex colloids in the limit that the telechelic stickers bind strongly enough to the surfaces to make free chains very rare. It is shown that the bridge formation rate can be expressed by combining times for two processes, namely, the escape of one end sticker from the narrow-but-deep association well near the colloidal surface and the longer-range motion of the chain end to the other surface inhibited by stretching free energy. We find that when using multibead chains to represent the telechelic polymers, the longer-range motion requires use of a multidimensional first passage time analysis that we borrow from the work of Likhtman and co-workers, which was originally developed to describe polymer end fluctuations in a one-dimensional reptation tube. From these i...

Stochastic Spikes and Strong Noise Limits of Stochastic Differential Equations

Motivated by studies of indirect measurements in quantum mechanics, we investigate stochastic differential equations with a fixed point subject to an additional infinitesimal repulsive perturbation. We conjecture, and prove for an important class, that the solutions exhibit a universal behavior when time is rescaled appropriately: by fine-tuning of the time scale with the infinitesimal repulsive perturbation, the trajectories converge in a precise sense to spiky trajectories that can be reconstructed from an auxiliary time-homogeneous Poisson process. Our results are based on two main tools. The first is a time change followed by an application of Skorokhod's lemma. We prove an effective approximate version of this lemma of independent interest. The second is an analysis of first passage times, which shows a deep interplay between scale functions and invariant measures. We conclude with some speculations of possible applications of the same techniques in other areas.

A framework for testing assumptions about foraging scales, body mass, and niche separation using telemetry data.

Ecological theory predicts that if animals with very similar dietary requirements inhabit the same landscape, then they should avoid niche overlap by either exploiting food resources at different times or foraging at different spatial scales. Similarly, it is often assumed that animals that fall in different body mass modes and share the same body plan will use landscapes at different spatial scales. We developed a new methodological framework for understanding the scaling of foraging (i.e. the range and distribution of scales at which animals use their landscapes) by applying a combination of three well‐established methods to satellite telemetry data to quantify foraging patch size distributions: (1) first‐passage time analysis; (2) a movement‐based kernel density estimator; and (3) statistical comparison of resulting histograms and tests for multimodality. We demonstrate our approach using two sympatric, ecologically similar species of African ducks with quite different body masses: Egyptian Geese (actually a shelduck), and Red‐billed Teal. Contrary to theoretical predictions, the two species, which are sympatric throughout the year, foraged at almost identical spatial scales. Our results show how ecologists can use GPS tracking data to explicitly quantify and compare the scales of foraging by different organisms within an animal community. Our analysis demonstrates both a novel approach to foraging data analysis and the need for caution when making assumptions about the relationships among niche separation, diet, and foraging scale.

Performance analysis of queue with two-stage vacation policy

ABSTRACTIn this paper, we investigate an M/M/1 queue with a two-stage vacation policy which comprises of single working vacation and single vacation. Using the matrix-analytic method, we obtain the distribution of stationary system size, and then the decomposition structures of the stationary system size and the sojourn time are demonstrated. Furthermore, we study the waiting time by first-passage time analysis. Meanwhile, the busy-cycle analysis is provided by the limiting theorem of alternative renewal process. Finally, several numerical examples are presented in the paper.

A martingale analysis of first passage times of time-dependent Wiener diffusion models

Research in psychology and neuroscience has successfully modeled decision making as a process of noisy evidence accumulation to a decision bound. While there are several variants and implementations of this idea, the majority of these models make use of a noisy accumulation between two absorbing boundaries. A common assumption of these models is that decision parameters, e.g., the rate of accumulation (drift rate), remain fixed over the course of a decision, allowing the derivation of analytic formulas for the probabilities of hitting the upper or lower decision threshold, and the mean decision time. There is reason to believe, however, that many types of behavior would be better described by a model in which the parameters were allowed to vary over the course of the decision process.In this paper, we use martingale theory to derive formulas for the mean decision time, hitting probabilities, and first passage time (FPT) densities of a Wiener process with time-varying drift between two time-varying absorbing boundaries. This model was first studied by Ratcliff (1980) in the two-stage form, and here we consider the same model for an arbitrary number of stages (i.e. intervals of time during which parameters are constant). Our calculations enable direct computation of mean decision times and hitting probabilities for the associated multistage process. We also provide a review of how martingale theory may be used to analyze similar models employing Wiener processes by re-deriving some classical results. In concert with a variety of numerical tools already available, the current derivations should encourage mathematical analysis of more complex models of decision making with time-varying evidence.

Performance Analysis of the GI/M/1 Queue with Single Working Vacation and Vacations

In this paper, we consider a new class of the GI/M/1 queue with single working vacation and vacations. When the system become empty at the end of each regular service period, the server first enters a working vacation during which the server continues to serve the possible arriving customers with a slower rate, after that, the server may resume to the regular service rate if there are customers left in the system, or enter a vacation during which the server stops the service completely if the system is empty. Using matrix geometric solution method, we derive the stationary distribution of the system size at arrival epochs. The stochastic decompositions of system size and conditional system size given that the server is in the regular service period are also obtained. Moreover, using the method of semi-Markov process (SMP), we gain the stationary distribution of system size at arbitrary epochs. We acquire the waiting time and sojourn time of an arbitrary customer by the first-passage time analysis. Furthermore, we analyze the busy period by the theory of limiting theorem of alternative renewal process. Finally, some numerical results are presented.

Exploring the environmental drivers of waterfowl movement in arid landscapes using first-passage time analysis.

BackgroundThe movement patterns of many southern African waterfowl are typified by nomadism, which is thought to be a response to unpredictable changes in resource distributions. Nomadism and the related movement choices that waterfowl make in arid environments are, however, poorly understood. Tracking multiple individuals across wide spatiotemporal gradients offers one approach to elucidating the cues and mechanisms underpinning movement decisions. We used first-passage time (FPT) to analyse high spatial and temporal resolution telemetry data for Red-billed Teal and Egyptian Geese across a 1500 km geographical gradient between 2008 and 2014. We tested the importance of several environmental variables in structuring movement patterns, focusing on two competing hypotheses: (1) whether movements are driven by resource conditions during the current period of habitat occupation (reactive movement hypothesis), or (2) whether movements are structured by shifts in the magnitude and direction of environmental variables at locations prior to occupation (prescient movement hypothesis).ResultsAn increase in rainfall at a 32 day lag (i.e., prior to wetland occupancy), along with tagging site, were significant predictors of FPT in both waterfowl species. There was a positive relationship between NDVI and FPT for Egyptian Geese during this 32 day period; the relationship was negative for Red-billed Teal. Consistent with findings for migratory grazing geese, Egyptian Geese prioritised food quality over food biomass. Red-billed Teal showed few immediate responses to wetland filling, contrary to what one would predict for a dabbling duck, suggesting high dietary flexibility. Our results were consistent with the prescient movement hypothesis.ConclusionsUsing FPT analysis we showed that the proximate drivers of southern African waterfowl movement are the dynamics of rainfall and primary productivity. Waterfowl appeared to be able to perceive and respond to temporal shifts in resource conditions prior to habitat patch occupation. This in turn suggests that their movements in semi-arid landscapes may be underpinned by intimate knowledge of the local environment; waterfowl pursue a complex behavioural strategy, locating suitable habitat patches proactively, rather than acting as passive respondents.