non-Poisson Statistics Research Articles

Intermittent Kac's flights and the generalized telegrapher's equation.

A generalized one-dimensional telegrapher equationassociated with an intermittent change of sign in the velocity of a Kac's flight has been proposed. To solve this random differential equation, we used the enlarged master equationapproach to obtain the exact differential equationfor the evolution of a normalized positive distribution. This distribution is associated with a generalized finite-velocity diffusionlike process. We studied the robustness of the ballistic regime, the cutoff of its domain, and the time-dependent Gaussian convergence. The second moment for the evolution of the profile has been studied as a function of non-Poisson statistics (possibly intermittent) for the time intervals Δ_{ij} in the Kac's flight. Numerical results for the evolution of sharp and wide initial profiles have also been presented. In addition, for comparison with a non-Gaussian process at all times, we have revisited the non-Markov Poisson's flight with exponential pulses. A theory for generalized random flights with intermittent stochastic velocity and in the presence of a force is also presented, and the stationary distribution for two classes of potential has been obtained.

Effect of Ergodic and Non-Ergodic Fluctuations on a Charge Diffusing in a Stochastic Magnetic Field.

In this paper, we study the basic problem of a charged particle in a stochastic magnetic field. We consider dichotomous fluctuations of the magnetic field where the sojourn time in one of the two states are distributed according to a given waiting-time distribution either with Poisson or non-Poisson statistics, including as well the case of distributions with diverging mean time between changes of the field, corresponding to an ergodicity breaking condition. We provide analytical and numerical results for all cases evaluating the average and the second moment of the position and velocity of the particle. We show that the field fluctuations induce diffusion of the charge with either normal or anomalous properties, depending on the statistics of the fluctuations, with distinct regimes from those observed, e.g., in standard Continuous-Time Random Walk models.

Revealing Multiple Pathways in T4 Lysozyme Substep Conformational Motions by Single-Molecule Enzymology and Modeling

Enzyme conformational dynamics play crucial roles in catalytic functions. Obtaining molecular level insights into the conformational transition dynamics of enzyme-substrate complex from the inactive state to the active state is fundamental for understanding enzymatic function and dynamics. Here, we report our progress on the real-time conformational transition dynamics of T4 lysozyme under enzymatic reactions using single-molecule fluorescence resonance energy transfer. The time duration in forming the active enzyme-substrate complex state (ES*) shows distinctive Poisson and non-Poisson statistics, including exponential and nonexponential, convoluted Poisson distributions, and Gaussian-like distributions. These complex dynamic behaviors of T4 lysozyme are in excellent agreement with a Markov dynamic simulation and a transition steps modeling. Specifically, we are able to obtain mechanistic understandings: (1) Transiting from enzyme (E) to ES*, T4 lysozyme hinge-bending conformational changes undergo multiple steps following multiple pathways. In the case of shortest pathway, this transition only requires one elementary transition or reaction step. (2) Substep conformational motions, associating with multiple nuclear coordinates and a common projected FRET-sensitive nuclear coordinate, can give rise to multiple conformational intermediate states. (3) The consequence of the multiple pathways, intermediate states, and nuclear coordinates is the time bunching effect; i.e., time durations of conformational changes tend to bunch in a narrowly distributed time window. The physical picture of multiple intermediate states along with bunching effect suggests that the conformational dynamics of T4 lysozyme shows a complementary characteristic behavior of convoluted enzyme conformation selection and induced-fit dynamics driven by substrate-enzyme interactions.

Analyzing and modeling heterogeneous behavior

Recently, it was pointed out that the non-Poisson statistics with heavy tail existed in many scenarios of human behaviors. But most of these studies claimed that power-law characterized diverse aspects of human mobility patterns. In this paper, we suggest that human behavior may not be driven by identical mechanisms and can be modeled as a Semi-Markov Modulated Process. To verify our suggestion and model, we analyzed a total of 1,619,934 records of library visitations (including undergraduate and graduate students). It is found that the distribution of visitation intervals is well fitted with three sections of lines instead of the traditional power law distribution in log–log scale. The results confirm that some human behaviors cannot be simply expressed as power law or any other simple functions. At the same time, we divided the data into groups and extracted period bursty events. Through careful analysis in different groups, we drew a conclusion that aggregate behavior might be composed of heterogeneous behaviors, and even the behaviors of the same type tended to be different in different period. The aggregate behavior is supposed to be formed by “heterogeneous groups”. We performed a series of experiments. Simulation results showed that we just needed to set up two states Semi-Markov Modulated Process to construct proper representation of heterogeneous behavior.

Contact formation in random networks of elongated objects.

The effect of steric hindrance is an important aspect of granular packings as it gives rise to, e.g., limitations on the densities of ordered and disordered packings, both of which are essentially defined by the geometry of the constituents. Here we focus on the random packing of rods via deposition and their distributions of contact number and segment length. Such statistical properties are relevant for mechanical properties of the structures, but the (quite large) steric effects on them have not been addressed in previous studies. We therefore develop a theory that describes the statistical properties of rod packings, while taking into account that the deposited rods cannot overlap and thus induce steric hindrances. The distributions derived from the theory are compared with experimental results and numerical simulations of networks constructed via deposition. The results explain the non-Poisson statistics observed in the experiments and show that the induced steric range of the rods can be large compared to their diameter and decreases with compactification of the pile, implying local orientational ordering of the structure.

A longitudinal study of Caenorhabditis elegans larvae reveals a novel locomotion switch, regulated by Gαs signaling

Despite their simplicity, longitudinal studies of invertebrate models are rare. We thus sought to characterize behavioral trends of Caenorhabditis elegans, from the mid fourth larval stage through the mid young adult stage. We found that, outside of lethargus, animals exhibited abrupt switching between two distinct behavioral states: active wakefulness and quiet wakefulness. The durations of epochs of active wakefulness exhibited non-Poisson statistics. Increased Gαs signaling stabilized the active wakefulness state before, during and after lethargus. In contrast, decreased Gαs signaling, decreased neuropeptide release, or decreased CREB activity destabilized active wakefulness outside of, but not during, lethargus. Taken together, our findings support a model in which protein kinase A (PKA) stabilizes active wakefulness, at least in part through two of its downstream targets: neuropeptide release and CREB. However, during lethargus, when active wakefulness is strongly suppressed, the native role of PKA signaling in modulating locomotion and quiescence may be minor. DOI:http://dx.doi.org/10.7554/eLife.00782.001.

Diffusion Scaling in Event-Driven Random Walks: An Application to Turbulence

Scaling laws for the diffusion generated by three different random walk models are reviewed. The random walks, defined on a one-dimensional lattice, are driven by renewal intermittent events with non-Poisson statistics and inverse power-law tail in the distribution of the inter-event or waiting times, so that the event sequences are characterized by self-similarity. Intermittency is a ubiquitous phenomenon in many complex systems and the power exponent of the waiting time distribution, denoted as complexity index, is a crucial parameter characterizing the system's complexity. It is shown that different scaling exponents emerge from the different random walks, even if the self-similarity, i.e. the complexity index, of the underlying event sequence remains the same. The direct evaluation of the complexity index from the time distribution is affected by the presence of added noise and secondary or spurious events. It is possible to minimize the effect of spurious events by exploiting the scaling relationships of the random walk models. This allows to get a reliable estimation of the complexity index and, at the same time, a confirmation of the renewal assumption. An application to turbulence data is shown to explain the basic ideas of this approach.

Counting statistics for genetic switches based on effective interaction approximation

Applicability of counting statistics for a system with an infinite number of states is investigated. The counting statistics has been studied a lot for a system with a finite number of states. While it is possible to use the scheme in order to count specific transitions in a system with an infinite number of states in principle, we have non-closed equations in general. A simple genetic switch can be described by a master equation with an infinite number of states, and we use the counting statistics in order to count the number of transitions from inactive to active states in the gene. To avoid having the non-closed equations, an effective interaction approximation is employed. As a result, it is shown that the switching problem can be treated as a simple two-state model approximately, which immediately indicates that the switching obeys non-Poisson statistics.

Empirical Analysis on the Human Dynamics of a Large-Scale Short Message Communication System

Research on human behavior has attracted increasing attention recently because of its scientific significance and potential applications. Some empirical results have indicated that the timing of many human activities follows non-Poisson statistics. We analyze a real-life huge dataset of short message communication with 6326713 users and 37577781 records during the 2006 Chinese New Year. The results show that the number of short message sendings, the interevent time between two consecutive short message sendings and the response time all follow heavy-tailed distribution. We further observe a strongly positive correlation between the activity and the power-law exponent of the interevent time distribution. In addition, the short message communication system displays a bursty property yet no memory effects, which is in particular different from some well-studied human-activited systems such as email-sending, library-loaning and file printing.

Non-Poisson statistics of settling spheres

Direct tracking of the particle positions in a sedimenting suspension indicates that the particles are not simply randomly distributed. The initial mixing of the suspension leads to a microstructure which consists of regions devoid of particles surrounded by regions where particles have an excess of close neighbors and which is maintained during sedimentation.

Superstatistics and renewal critical events

Abstract An approach to intermittent systems based on renewal processes is reviewed. The Waiting Times (WTs) between events are the main variables of interest in intermittent systems. A crucial role is played by the class of critical events, characterized by Non-Poisson statistics and non-exponential WT distribution. A particular important case is given by WT distributions with power tail. Critical events play a crucial role in the behavior of a property known as Renewal Aging. Focusing on the role of critical events, the relation between superstatistics and non-homogeneous Poisson processes is discussed, and the role of Renewal Aging is illustrated by comparing a Non-Poisson model with a Poisson one, both of them modulated by a periodic forcing. It is shown that the analysis of Renewal Aging is sensitive to the presence of critical events and that this property can be exploited to detect Non-Poisson statistics in a time series. As a consequence, it is claimed that, apart from the characterization of superstatistical features such as the distribution of the intensive parameter or the separation of the time scales, the Renewal Aging property can give some effort to better determine the role of Non-Poisson critical events in intermittent systems.

RENEWAL AGING IN NON-HOMOGENEOUS POISSON PROCESSES WITH PERIODIC RATE MODULATION

The statistical properties of sequences of Waiting Times generated by Non-Homogeneous Poisson Processes are investigated by means of Renewal Aging, i.e. a statistical analysis being able to detect the presence of genuine renewal events with non-Poisson statistics. The Renewal Aging features of two non-homogeneous Poisson models with different prescriptions of the rate modulation are compared. Both prescriptions are essentially periodic. In the first model, the rate is a totally deterministic and smooth harmonic function. In the second model, the rate modulation is deterministic and smooth almost everywhere, except in some singular points occurring with a periodicity affected by weak random fluctuations. The main parameter of both models is the modulation speed, defined as the ratio between the modulation period and the average Waiting Time. This allows to distinguish between slow, intermediate and fast modulation of the rate. As expected, in the very slow modulation case both models have a strongly reduced Renewal Aging, practically zero, because the analysis is affected by local clusters of Waiting Times with the same Poisson rate. In the intermediate range of modulation speeds, both models show non-zero Renewal Aging and the behavior of the two models tends to be more different as the modulation speed decreases. Going towards the fast modulation regime, the behavior of the two models becomes completely different from one another. Model I does not show a definite global Renewal Aging. On the time scale of the modulation period, local oscillations arise in the Probability Density Function and an apparent oscillating aging is also observed. However, this form of aging is not the manifestation of genuine renewal non-Poisson events, but it is related to the local oscillations. On the contrary, Model II displays Renewal Aging features in agreement with a homogeneous Renewal non-Poisson Process. It is argued that this is related to the singular points in the rate modulation of Model II, which, due to their random occurrence, could be identified with genuine critical non-Poisson events. The approach presented here could have interesting applications in problems involving dichotomous noise and, in particular, in the statistical characterization of ON–OFF fluorescence intermittency experimentally observed in complex systems, such as nano-crystals and single bio-molecules stimulated by laser fields.

Fluctuations, Pauses, and Backtracking in DNA Transcription

Transcription is a vital stage in the process of gene expression and a major contributor to fluctuations in gene expression levels for which it is typically modeled as a single-step process with Poisson statistics. However, recent single molecule experiments raise questions about the validity of such a simple single-step picture. We present a molecular multistep model of transcription elongation that demonstrates that transcription times are in general non-Poisson-distributed. In particular, we model transcriptional pauses due to backtracking of the RNA polymerase as a first passage process. By including such pauses, we obtain a broad, heavy-tailed distribution of transcription elongation times, which can be significantly longer than would be otherwise. When transcriptional pauses result in long transcription times, we demonstrate that this naturally leads to bursts of mRNA production and non-Poisson statistics of mRNA levels. These results suggest that transcriptional pauses may be a significant contributor to the variability in transcription rates with direct implications for noise in cellular processes as well as variability between cells.

Responses of neurons in the inferior colliculus to binaural disparities: Insights from the use of Fisher information and mutual information

The minimal change in a stimulus property that is detectable by neurons has been often quantified using the receiver operating characteristic (ROC) curve, but recent studies introduced the use of the related Fisher information (FI). Whereas ROC analysis and FI quantify the information available for discriminating between two stimuli, global aspects of the information carried by a neuron are quantified by the mutual information (MI) between stimuli and responses. FI and MI have been shown to be related to each other when FI is large. Here the responses of neurons recorded in the inferior colliculus of anesthetized guinea pigs in response to ensembles of sounds differing in their interaural time differences (ITDs) or binaural correlation (BC) were analyzed. Although the FI is not uniformly large, there are strong relationships between MI and FI. Information-theoretic measures are used to demonstrate the importance of the non-Poisson statistics of these responses. These neurons may reflect the maximization of the MI between stimuli and responses under constraints on the coded stimulus range and the range of firing rates. Remarkably, whereas the maximization of MI, in conjunction with the non-Poisson statistics of the spike trains, is enough to create neurons whose ITD discrimination capabilities are close to the behavioral limits, the same rule does not achieve single-neuron BC discrimination that is as close to behavioral performance.

Recruitment variability in exploited aquatic populations

While lognormal distributions have been proposed as useful descriptors of recruitment variability, the very nature of the recruitment distributions is still debated. To account quantitatively for recruitment distributions, I here propose a Weibull exponential model; it derives from a simple and natural hypothesis for uncorrelated recruitment processes of spawning, hatching, growth and survival through the early life stages to the point of vulnerability to the fishery. The quantification of Weibull exponentials is particularly important with regards to extrapolations to low recruits that have not yet been observed. To test the Weibull exponential null-hypothesis, I examine annual time-series of recruitment in major aquatic stocks. The Weibull exponential quite describes the bulk (95%) of the recruitment distributions of widely differing stocks, while the remaining 5% of the largest recruits are occurring with a much larger rate than predicted by the Weibull exponential. Further, I study the inter-event times between unusually high numbers in recruitment time-series data and find that intermittent pulses of strong recruitment follow non-Poisson statistics, which arises from year-to-year persistence of the magnitude of recruitment: large (or small) recruits are more likely to be followed by large (or small) recruits. This recruitment clustering effect is confirmed by the rescaled range analysis method. The empirical results imply that individual survivals on recruitment levels are independent of initial cohort sizes but year-to-year recruiting events exhibit long-term correlations.

Modeling bursts and heavy tails in human dynamics

The dynamics of many social, technological and economic phenomena are driven by individual human actions, turning the quantitative understanding of human behavior into a central question of modern science. Current models of human dynamics, used from risk assessment to communications, assume that human actions are randomly distributed in time and thus well approximated by Poisson processes. Here we provide direct evidence that for five human activity patterns, such as email and letter based communications, web browsing, library visits and stock trading, the timing of individual human actions follow non-Poisson statistics, characterized by bursts of rapidly occurring events separated by long periods of inactivity. We show that the bursty nature of human behavior is a consequence of a decision based queuing process: when individuals execute tasks based on some perceived priority, the timing of the tasks will be heavy tailed, most tasks being rapidly executed, while a few experiencing very long waiting times. In contrast, priority blind execution is well approximated by uniform interevent statistics. We discuss two queuing models that capture human activity. The first model assumes that there are no limitations on the number of tasks an individual can handle at any time, predicting that the waiting time of the individual tasks follow a heavy tailed distribution P(tau(w)) approximately tau(w)(-alpha) with alpha=3/2. The second model imposes limitations on the queue length, resulting in a heavy tailed waiting time distribution characterized by alpha=1. We provide empirical evidence supporting the relevance of these two models to human activity patterns, showing that while emails, web browsing and library visitation display alpha=1, the surface mail based communication belongs to the alpha=3/2 universality class. Finally, we discuss possible extension of the proposed queuing models and outline some future challenges in exploring the statistical mechanics of human dynamics.

Correlation function and generalized master equation of arbitrary age

We study a two-state statistical process with a non-Poisson distribution of sojourn times. In accordance with earlier work, we find that this process is characterized by aging and we study three different ways to define the correlation function of arbitrary age of the corresponding dichotomous fluctuation. These three methods yield exact expressions, thus coinciding with the recent result by Godrèche and Luck [J. Stat. Phys. 104, 489 (2001)]. Actually, non-Poisson statistics yields infinite memory at the probability level, thereby breaking any form of Markovian approximation, including the one adopted herein, to find an approximated analytical formula. For this reason, we check the accuracy of this approximated formula by comparing it with the numerical treatment of the second of the three exact expressions. We find that, although not exact, a simple analytical expression for the correlation function of arbitrary age is very accurate. We establish a connection between the correlation function and a generalized master equation of the same age. Thus this formalism, related to models used in glassy materials, allows us to illustrate an approach to the statistical treatment of blinking quantum dots, bypassing the limitations of the conventional Liouville treatment.

Absorption and emission in the non-poisson case: the theoretical challenge posed by renewal aging

This short paper aims at clarifying the physical meaning of a previous pubblication [G. Aquino, P. Grigolini, L. Palatella, Phys. Rev. Lett. 93 050601 (2004)], using later, although very recent, results. This has to do with the challenges posed to the Kubo-Anderson (KA) theory, and more in general to any form of Liouville-like approach, by the discovery of intermittent resonant fluorescence with a non-exponential distribution of waiting times. We show that to properly address the treatment of these problems the KA theory, valid in the case of aged systems, should be extended to aging systems, aging for a very extended time period or even forever, being a crucial consequence of non-Poisson statistics. This ambitious goal can be realized if we adopt the assumption that real wave-function collapses occur.

The origin of bursts and heavy tails in human dynamics

The dynamics of many social, technological and economic phenomena are driven by individual human actions, turning the quantitative understanding of human behaviour into a central question of modern science. Current models of human dynamics, used from risk assessment to communications, assume that human actions are randomly distributed in time and thus well approximated by Poisson processes. In contrast, there is increasing evidence that the timing of many human activities, ranging from communication to entertainment and work patterns, follow non-Poisson statistics, characterized by bursts of rapidly occurring events separated by long periods of inactivity. Here I show that the bursty nature of human behaviour is a consequence of a decision-based queuing process: when individuals execute tasks based on some perceived priority, the timing of the tasks will be heavy tailed, with most tasks being rapidly executed, whereas a few experience very long waiting times. In contrast, random or priority blind execution is well approximated by uniform inter-event statistics. These finding have important implications, ranging from resource management to service allocation, in both communications and retail.

Absorption and emission in the non-Poissonian case.

This Letter addresses the challenging problems posed to the Kubo-Anderson (KA) theory by the discovery of intermittent resonant fluorescence with a nonexponential distribution of waiting times. We show how to extend the KA theory from aged to aging systems, aging for a very extended time period or even forever, being a crucial consequence of non-Poisson statistics.