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PubMed2001

Criticality of natural absorbing states.

We study a recently introduced ladder model that undergoes a transition between an active and an infinitely degenerate absorbing phase. In some cases the critical behavior of the model is the same as that of the branching-annihilating random walk with N>/=2 species both with and without hard-core interaction. We show that certain static characteristics of the so-called natural absorbing states develop power-law singularities that signal the approach of the critical point. These results are also explained using random-walk arguments. In addition to that we show that when dynamics of our model is considered as a minimum-finding procedure, it has the best efficiency very close to the critical point.

Journal Article↗
PubMed2006

Detecting an orientation component in animal paths when the preferred direction is individual-dependent.

An orientation component leads to directionally biased paths, with major consequences in animal population redistribution. Classical orientation analyses, which focus on the overall direction of motion, are useless for detecting such a component when the preferred direction is not common to the whole population, but differs from one path to another. In-depth path analyses are required in this case. They consist of determining whether paths are more suitably represented as biased or unbiased random walks. The answer is not easy because most animals' paths show some forward persistence propensity that acts as a purely local directional bias and, hence, blurs the possible occurrence of an additional, consistent bias in a preferred direction. I highlight the key differences between biased and unbiased random walks and the different ways orientation mechanisms can generate a consistent directional bias. I then examine the strength and weakness of the available methods likely to detect it. Finally, I introduce a new procedure based on the backward evolution of the beeline distance, from the end of the path, which might correspond to a goal toward which the animal orients itself, to each of the animal's preceding locations. This new procedure proves to be very efficient, as it requires only a small sample of short paths for detecting a possible orientation component.

Animal Migration↗
PubMed2001

Lower body adynamia as a factor to reduce the risk of hypobaric decompression sickness.

BACKGROUND: We define lower body adynamia (LBA) as restricted lower body movement, particularly walking, during both the denitrogenation phase at site pressure and during the exercise phase while at altitude. HYPOTHESIS: Our null hypothesis is that subjects who are adynamic in the lower body but do upper body exercise will be at similar risk of decompression sickness (DCS) and venous gas emboli (VGE) as subjects who randomly walk but do no planned exercise while at altitude. METHODS: We selected a data set that contained 1401 altitude exposures with the following conditions: a) walking was part of the exercise at altitude; or b) there was no planned exercise done at altitude but walking was not restricted; or c) LBA was inforced, but upper body exercise was done at altitude. We used logistic regression (LR) on all 1401 exposures, a log logistic survival analysis (SA) on a subset of data from "a" and "c" (n = 234), and estimated a model for how the incidence of VGE changes through time. RESULTS: The estimated probabilities of DCS and VGE with 95% confidence intervals (Cls) from the LR with a simulation of a 3-h oxygen prebreathe, a 4-h exposure to 4.3 psia in a male, and exercise and LBA conditions as described above are: (see text). CONCLUSION: LBA that includes upper body exercise appears to be as protective against DCS and VGE as random walking by subjects who did no prescribed exercise while at altitude, and is more protective than exercise that included walking. Our conclusions are based on an assumption that we have adequately controlled, through our data selection process and the use of multivariable models, important variables in tests that were not done at the Johnson Space Center.

Cardiac Volume↗
PubMed2006

A model for bacterial colonization of sinking aggregates.

Sinking aggregates provide important nutrient-rich environments for marine bacteria. Quantifying the rate at which motile bacteria colonize such aggregations is important in understanding the microbial loop in the pelagic food web. In this paper, a simple analytical model is presented to predict the rate at which bacteria undergoing a random walk encounter a sinking aggregate. The model incorporates the flow field generated by the sinking aggregate, the swimming behavior of the bacteria, and the interaction of the flow with the swimming behavior. An expression for the encounter rate is computed in the limit of large Péclet number when the random walk can be approximated by a diffusion process. Comparison with an individual-based numerical simulation is also given.

Bacteria↗
PubMed2006

Diffusion maps and coarse-graining: A unified framework for dimensionality reduction, graph partitioning, and data set parameterization.

We provide evidence that nonlinear dimensionality reduction, clustering, and data set parameterization can be solved within one and the same framework. The main idea is to define a system of coordinates with an explicit metric that reflects the connectivity of a given data set and that is robust to noise. Our construction, which is based on a Markov random walk on the data, offers a general scheme of simultaneously reorganizing and subsampling graphs and arbitrarily shaped data sets in high dimensions using intrinsic geometry. We show that clustering in embedding spaces is equivalent to compressing operators. The objective of data partitioning and clustering is to coarse-grain the random walk on the data while at the same time preserving a diffusion operator for the intrinsic geometry or connectivity of the data set up to some accuracy. We show that the quantization distortion in diffusion space bounds the error of compression of the operator, thus giving a rigorous justification for k-means clustering in diffusion space and a precise measure of the performance of general clustering algorithms.

Algorithms↗
PubMed2003

A model of animal movements in a bounded space.

Most studies describing animal movements have been developed in the framework of population dispersion or population dynamics, and have mainly focused on movements in open spaces. During their trips, however, animals are likely to encounter physical heterogeneities that guide their movements and, as a result, influence their spatial distribution. In this paper, we develop a statistical model of individual movement in a bounded space. We introduced cockroaches in a circular arena and quantified accurately the behaviors underlying their movement in a finite space. Close to the edges, we considered that the animals exhibit a linear movement mode with a constant probability per unit time to leave the edge and enter the central zone of the arena. Far from the walls cockroaches were assumed to move according to a diffusive random walk which enabled us to overcome the inherent problem of the quantification of the turning angle distribution. A numerical model implementing the behavioral rules derived from our experiments, confirms that the pattern of the spatial distribution of animals observed can be reliably accounted for by wall-following behaviors combined with a diffusive random walk. The approach developed in this study can be applied to model the movements of animals in various environment under consideration of spatial structure.

Animals↗
PubMed2003

The effects of visual input on postural control of the lumbar spine in unstable sitting.

Postural control of the lumbar spine in unstable sitting was quantified through the analysis of the center of pressure (CoP) movement recorded by a force plate situated underneath a seat that incorporated a hemisphere. Thirteen healthy subjects were tested under conditions of increasing seat instability and elimination of visual input. The purpose of this study was to determine the relative effects of visual input and support surface instability on open and closed loop postural control mechanisms in sitting and to determine the association between traditional summary statistics and random walk analysis of CoP movement. The effects of the seat instability level and visual input on the CoP movement parameters were tested with a two-factor, repeated measures ANOVA (p<0.01). In all summary statistics CoP movement parameters increased significantly due to the seat instability level and lack of visual input. The random walk analysis identified two regions, short- and long-term, which has been postulated to represent open and closed loop control mechanism, respectively. While short-term scaling exponents were independent from visual input, CoP displacement in the short-term region was significantly increased in the eyes closed condition. Summary statistic of CoP total path length per second correlated highly with critical point coordinates and short-term diffusion coefficients. The CoP movement in the long-term region was consistent with a closed loop control mechanisms. The findings of visual influence on what is assumed as an open-loop control mechanism does not, at face value, support the hypothesis that two separate mechanisms are working to achieve postural control.

Adult↗
PubMed2006

The scaling laws of human travel.

The dynamic spatial redistribution of individuals is a key driving force of various spatiotemporal phenomena on geographical scales. It can synchronize populations of interacting species, stabilize them, and diversify gene pools. Human travel, for example, is responsible for the geographical spread of human infectious disease. In the light of increasing international trade, intensified human mobility and the imminent threat of an influenza A epidemic, the knowledge of dynamical and statistical properties of human travel is of fundamental importance. Despite its crucial role, a quantitative assessment of these properties on geographical scales remains elusive, and the assumption that humans disperse diffusively still prevails in models. Here we report on a solid and quantitative assessment of human travelling statistics by analysing the circulation of bank notes in the United States. Using a comprehensive data set of over a million individual displacements, we find that dispersal is anomalous in two ways. First, the distribution of travelling distances decays as a power law, indicating that trajectories of bank notes are reminiscent of scale-free random walks known as Lévy flights. Second, the probability of remaining in a small, spatially confined region for a time T is dominated by algebraically long tails that attenuate the superdiffusive spread. We show that human travelling behaviour can be described mathematically on many spatiotemporal scales by a two-parameter continuous-time random walk model to a surprising accuracy, and conclude that human travel on geographical scales is an ambivalent and effectively superdiffusive process.

Diffusion↗
PubMed1991

Channeling and percolation in two-dimensional chaotic dynamics.

The Hamiltonian dynamics of a particle moving in a nearly periodic two-dimensional (2-D) potential of square symmetry is analyzed. The particle undergoes two types of unbounded stochastic or random walks in such a system: a quasi-1-D motion (a "stochastic channeling") and a 2-D motion which results from a sort of stochastic percolation. A scenario for the onset of this stochastic percolation is analyzed. The threshold energy for percolation is found as a function of the perturbation parameter. Each type of random walk has the property of intermittency. The particle transport is anomalous in certain energy intervals.

Journal Article↗
PubMed1981

The average number of distinct sites visited by a one-dimensional random walker and its application to isotope exchange in polypeptides.

The average number of distinct sites visited by a random walker moving with arbitrary transition probability on a one-dimensional lattice is calculated. Asymptotic forms of this quantity for both asymmetric and symmetric random walks are determined, and an exact solution for the latter case is also given for any number of steps. The average number of sites visited is then analyzed for intermediate numbers of steps by introducing an exponent. This approach is applied to explain the results of isotope exchange experiments in polypeptides, and applications of asymmetric random walks to other biological problems are briefly discussed.

Journal Article↗
PubMed2005

From diffusion to anomalous diffusion: a century after Einstein's Brownian motion.

Einstein's explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term memory. The coarse-grained behavior of such processes is described by the diffusion equation. However, many natural processes do not possess the Markovian property and exhibit anomalous diffusion. We consider here the case of subdiffusive processes, which correspond to continuous-time random walks in which the waiting time for a step is given by a probability distribution with a diverging mean value. Such a process can be considered as a process subordinated to normal diffusion under operational time which depends on this pathological waiting-time distribution. We derive two different but equivalent forms of kinetic equations, which reduce to known fractional diffusion or Fokker-Planck equations for waiting-time distributions following a power law. For waiting time distributions which are not pure power laws one or the other form of the kinetic equation is advantageous, depending on whether the process slows down or accelerates in the course of time.

Journal Article↗
PubMed2006

Dynamics of asynchronous random Boolean networks with asynchrony generated by stochastic processes.

An asynchronous Boolean network with N nodes whose states at each time point are determined by certain parent nodes is considered. We make use of the models developed by Matache and Heidel [Matache, M.T., Heidel, J., 2005. Asynchronous random Boolean network model based on elementary cellular automata rule 126. Phys. Rev. E 71, 026232] for a constant number of parents, and Matache [Matache, M.T., 2006. Asynchronous random Boolean network model with variable number of parents based on elementary cellular automata rule 126. IJMPB 20 (8), 897-923] for a varying number of parents. In both these papers the authors consider an asynchronous updating of all nodes, with asynchrony generated by various random distributions. We supplement those results by using various stochastic processes as generators for the number of nodes to be updated at each time point. In this paper we use the following stochastic processes: Poisson process, random walk, birth and death process, Brownian motion, and fractional Brownian motion. We study the dynamics of the model through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed-point analysis. The dynamics of the system show that the number of nodes to be updated at each time point is of great importance, especially for the random walk, the birth and death, and the Brownian motion processes. Small or moderate values for the number of updated nodes generate order, while large values may generate chaos depending on the underlying parameters. The Poisson process generates order. With fractional Brownian motion, as the values of the Hurst parameter increase, the system exhibits order for a wider range of combinations of the underlying parameters.

Logistic Models↗
PubMed2006

Encounter success of free-ranging marine predator movements across a dynamic prey landscape.

Movements of wide-ranging top predators can now be studied effectively using satellite and archival telemetry. However, the motivations underlying movements remain difficult to determine because trajectories are seldom related to key biological gradients, such as changing prey distributions. Here, we use a dynamic prey landscape of zooplankton biomass in the north-east Atlantic Ocean to examine active habitat selection in the plankton-feeding basking shark Cetorhinus maximus. The relative success of shark searches across this landscape was examined by comparing prey biomass encountered by sharks with encounters by random-walk simulations of 'model' sharks. Movements of transmitter-tagged sharks monitored for 964 days (16754 km estimated minimum distance) were concentrated on the European continental shelf in areas characterized by high seasonal productivity and complex prey distributions. We show movements by adult and sub-adult sharks yielded consistently higher prey encounter rates than 90% of random-walk simulations. Behavioural patterns were consistent with basking sharks using search tactics structured across multiple scales to exploit the richest prey areas available in preferred habitats. Simple behavioural rules based on learned responses to previously encountered prey distributions may explain the high performances. This study highlights how dynamic prey landscapes enable active habitat selection in large predators to be investigated from a trophic perspective, an approach that may inform conservation by identifying critical habitat of vulnerable species.

Animals↗
PubMed2004

Dynamics of stock prices.

We show that the dynamics of stock prices can be accurately described as a continuous time random walk with a time dependent diffusion coefficient. The time evolution of the diffusion coefficient can be derived from tick by tick databases provided the stock price is characterized in terms of a couple of values describing the best ask and the best bid. We are then led to a finding and, namely, that the transition rate of the random walk process is different from the frequency of transactions. Our results allow us to obtain a fast and reliable determination of the diffusion coefficient and precisely confirm that fat tails in the distribution of price variations are due to volatility fluctuations.

Journal Article↗
PubMed1975

Number fluctuation spectroscopy of motile microorganisms.

A random-walk model of motility is used to predict the dynamics of fluctuations in the number of particles in a small observation volume. The results show that number fluctuations provide a measure of the mean swimming speed as well as the persistence length. Experimental light-scattering results are presented for three strains of Escherichia coli whose motion appears random-walk in nature. For the strain with th elongest persistence length, excellent agreement is found that theoretical predictions. For the more erratic strains, however, the shape of the measured scattered light intensity correlation functions indicates the presence of a contribution due to orientational fluctuations.

Cell Movement↗
PubMed2000

Parallel excluded volume tempering for polymer melts.

We have developed a technique to accelerate the acquisition of effectively uncorrelated configurations for off-lattice models of dense polymer melts that makes use of both parallel tempering and large-scale Monte Carlo moves. The method is based upon simulating a set of systems in parallel, each of which has a slightly different repulsive core potential, such that a thermodynamic path from full excluded volume to an ideal gas of random walks is generated. While each system is run with standard stochastic dynamics, resulting in an NVT ensemble, we implement the parallel tempering through stochastic swaps between the configurations of adjacent potentials, and the large-scale Monte Carlo moves through attempted pivot and translation moves that reach a realistic acceptance probability as the limit of the ideal gas of random walks is approached. Compared to pure stochastic dynamics, this results in an increased efficiency even for a system of chains as short as N=60 monomers, however at this chain length the large-scale Monte Carlo moves were ineffective. For even longer chains, the speedup becomes substantial, as observed from preliminary data for N=200. We also compare our scheme to the end bridging algorithm of Theodorou et al. For N=60, end bridging must allow a polydispersity of more than 10% in order to relax the end-to-end vector more quickly than our method. The comparison is, however, hampered by the fact that the end-to-end vector becomes a somewhat artificial quantity when one implements end bridging, and is perhaps no longer the slowest dynamic variable.

Algorithms↗
PubMed1983

A stochastic model for helix bending in B-DNA.

Bending in double-helical B-DNA apparently occurs only by rolling adjacent base pairs over one another along their long axes. The lifting apart of ends that would be required by tilt or wedge angle contributions is too costly in free energy and does not occur. Roll angles at base steps can be positive (compression of major groove) or negative (compression of minor groove), with the former somewhat easier. Individual steps may advance or oppose the overall direction of bend, or make lateral excursions, but the result of this series of "random roll" steps is the production of a net bending in the helix axis. Because the natural roll points for bending in a given plane occur every 5 base pairs, one would expect that double-helical DNA wrapped around a nucleosome core would exhibit bends with the same periodicity. Alternate bends might be particularly acute where the major groove faced the nucleosome core and was compressed against it. The "annealed kinking" model proposed by Fratini et al. (J. Biol. Chem. 257, 14686 (1982) was suggested from the observation that a major bend at a natural roll point is flanked by decreasing roll angles at the steps to either side, as though local strain was being minimized by somewhat blurring the bend out rather than keeping it localized. The random walk model suggested in this paper would describe this as a decreased roll angle as the helix step rotates toward a direction perpendicular to the overall bend. Bending of DNA is seen to be a more stochastic process than had been suspected. Detailed analysis of every helix step reveals both side excursions and backward or retrograde motion, as in any random walk situation. Yet these isolated steps counteract one another, to leave behind a residuum of overall bending in a specific direction.

Base Sequence↗
PubMed2000

Postural control of trunk during unstable sitting.

A method for quantifying postural control of the lumbar spine during unstable sitting was developed. The unstable seat apparatus was equipped with leg and foot supports to isolate the control of the lumbar spine and trunk from the adjustments in the lower body joints. Polyester resin hemispheres with decreasing diameters were attached to the bottom of the seat to achieve increasing levels of task difficulty. The seat was placed on a force plate at the edge of a table and the participating subjects were instructed to maintain their balance while sitting on the seat. Coordinates of center of pressure (CoP) were recorded and quantified with summary statistics and random walk analysis. The CoP movement increased significantly with increased seat instability (task difficulty) (p<0.01). Stabilogram plots of the CoP movement revealed short and long-term regions consistent with the hypothesis that the two regions reflect open and closed-loop postural control mechanisms. Repeatability of the CoP parameters was excellent for the summary statistics and the short-term random walk coefficients (0.77<R<0.96). It was fair for the long-term diffusion coefficients (0.56<R<0.57) and poor for the long-term scaling exponents (0.14<R<0.40). Summary statistics of the CoP movement were positively correlated with body weight (0.69<R<0. 73) and the T9 to L4/L5 distance (0.43<R<0.54) of the subjects. This method can be applied to study the deficits in postural control of the lumbar spine in low-back pain population.

Adult↗