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PubMed2005

Scaling of spreading in models unidirectionally coupled to source particles.

We investigate the spreading behavior of evolving clusters using unidirectionally coupled two-level hierarchies in one spatial dimension. In the hierarchy, while only two source particles (A) hop away from each other without branching its offspring on the bottom level, different species of particles (B) evolve according to given dynamics belonging to one of known universality classes on the top level. Two levels are unidirectionally coupled from the bottom to the top level by the branching A-->A+2B. We derive the spreading exponent zU of the uncoupled region of size RU(t) approximately tzU up to the first order correction in terms of the spreading exponent of source particles (zA) and that of given dynamics of the top level (zo) as zU=(1--zA)zo/(1-zo). From the relation, zA and zU always satisfy the inequality zU< or =zA for zA> or =zo. The inequality confirms that the scaling of the spreading in the slave level should follow the scaling of the source in unidirectionally coupled systems. We numerically confirm the relation for three different B-particle dynamics; annihilating random walks, branching annihilating random walks with one and two offspring which belong to the directed percolation, and the parity conserving universality class respectively.

Journal Article↗
PubMed2005

A kinetic model of the formation of organic monolayers on hydrogen-terminated silicon by hydrosilation of alkenes.

We have analyzed a kinetic model for the formation of organic monolayers based on a previously suggested free radical chain mechanism for the reaction of unsaturated molecules with hydrogen-terminated silicon surfaces (Linford, M. R.; Fenter, P. M.; Chidsey, C. E. D. J. Am. Chem. Soc 1995, 117, 3145). A direct consequence of this mechanism is the nonexponential growth of the monolayer, and this has been observed spectroscopically. In the model, the initiation of silyl radicals on the surface is pseudo first order with rate constant, ki, and the rate of propagation is determined by the concentration of radicals and unreacted Si-H nearest neighbor sites with a rate constant, kp. This propagation step determines the rate at which the monolayer forms by addition of alkene molecules to form a track of molecules that constitute a self-avoiding random walk on the surface. The initiation step describes how frequently new random walks commence. A termination step by which the radicals are destroyed is also included. The solution of the kinetic equations yields the fraction of alkylated surface sites and the mean length of the random walks as a function of time. In mean-field approximation we show that (1) the average length of the random walk is proportional to (kp/ki)1/2, (2) the monolayer surface coverage grows exponentially only after an induction period, (3) the effective first-order rate constant describing the growth of the monolayer and the induction period (kt) is k = (2ki kp)1/2, (4) at long times the effective first-order rate constant drops to ki, and (5) the overall activation energy for the growth kinetics is the mean of the activation energies for the initiation and propagation steps. Monte Carlo simulations of the mechanism produce qualitatively similar kinetic plots, but the mean random walk length (and effective rate constant) is overestimated by the mean field approximation and when kp >> ki, we find k approximately ki0.7kp0.3 and Ea = (0.7Ei+ 0.3Ep). However the most striking prediction of the Monte Carlo simulations is that at long times, t >> 1/k, the effective first-order rate constant decreases to ki even in the absence of a chemical termination step. Experimental kinetic data for the reaction of undec-1-ene with hydrogen-terminated porous silicon under thermal reflux in toluene and ethylbenzene gave a value of k = 0.06 min(-1) and an activation energy of 107 kJ mol(-1). The activation energy is in reasonable agreement with density functional calculations of the transition state energies for the initiation and propagation steps.

Journal Article↗
PubMed2006

Dynamic properties in a family of competitive growing models.

The properties of a wide variety of growing models, generically called X-RD, involving the deposition of particles according to competitive processes, such that a particle is attached to the aggregate with probability p following the mechanisms of a generic model X that provides the correlations and at random [random deposition (RD)] with probability (1-p), are studied by means of numerical simulations and analytic developments. The study comprises the following X models: Ballistic deposition, random deposition with surface relaxation, Das Sarma-Tamboronea, Kim-Kosterlitz, Lai-Das Sarma, Wolf-Villain, large curvature, and three additional models that are variants of the ballistic deposition model. It is shown that after a growing regime, the interface width becomes saturated at a crossover time (tx2) that, by fixing the sample size, scales with p according to tx2(p) proportional variant p-y (P>0), where is an exponent. Also, the interface width at saturation (Wsat) scales as Wsat(p) proportional variant p-delta (p>0), where delta is another exponent. It is proved that, in any dimension, the exponents delta and y obey the following relationship: delta=y beta RD, where beta RD=1/2 is the growing exponent for RD. Furthermore, both exponents exhibit universality in the p --> 0 limit. By mapping the behavior of the average height difference of two neighboring sites in discrete models of type X-RD and two kinds of random walks, we have determined the exact value of the exponent delta. When the height difference between two neighbouring sites corresponds to a random walk that after walking steps returns to a distance from its initial position that is proportional to the maximum distance reached (random walk of type A), one has delta=1/2. On the other hand, when the height difference between two neighboring sites corresponds to a random walk that after steps moves steps towards the initial position (random walk of type B), one has delta=1. Finally, by linking four well-established universality classes (namely Edwards-Wilkinson, Kardar-Parisi-Zhang, linear [molecular beam epitaxy (MBE)] and nonlinear MBE) with the properties of type A and B of random walks, eight different stochastic equations for all the competitive models studied are derived.

Animals↗
PubMed2005

A computational method to detect epistatic effects contributing to a quantitative trait.

We develop a new computational method to detect epistatic effects that contribute to a complex quantitative trait. Rather than looking for epistatic effects that show statistical significance when considered in isolation, we search for a close approximation to the quantitative trait by a sum of epistatic effects. Our search algorithm consists of a sequence of random walks around the space of sums of epistatic effects. An important feature of our approach is that there is learning between random walks, i.e. the control mechanism that chooses steps in our random walks adapts to the experiences of earlier random walks. We test the effectiveness of our algorithms by applying them to synthetic datasets where the phenotype is a sum of epistatic effects plus normally distributed noise. Our test statistic is the rate of success that our methods achieve in identifying the underlying epistatic effects. We report on the effectiveness of our methods as we vary parameters that are intrinsic to the computation (length of random walks and degree of learning) as well as parameters that are extrinsic to the computation (number of markers, number of individuals, noise level, architecture of the epistatic effects).

Algorithms↗
PubMed2003

System identification applied to a visuomotor task: near-optimal human performance in a noisy changing task.

Sensory-motor integration has frequently been studied using a single-step change in a control variable such as prismatic lens angle and has revealed human visuomotor adaptation to often be partial and inefficient. We propose that the changes occurring in everyday life are better represented as the accumulation of many smaller perturbations contaminated by measurement noise. We have therefore tested human performance to random walk variations in the visual feedback of hand movements during a pointing task. Subjects made discrete targeted pointing movements to a visual target and received terminal feedback via a cursor the position of which was offset from the actual movement endpoint by a random walk element and a random observation element. By applying ideal observer analysis, which for this task compares human performance against that of a Kalman filter, we show that the subjects' performance was highly efficient with Fisher efficiencies reaching 73%. We then used system identification techniques to characterize the control strategy used. A "modified" delta-rule algorithm best modeled the human data, which suggests that they estimated the random walk perturbation of feedback in this task using an exponential weighting of recent errors. The time constant of the exponential weighting of the best-fitting model varied with the rate of random walk drift. Because human efficiency levels were high and did not vary greatly across three levels of observation noise, these results suggest that the algorithm the subjects used exponentially weighted recent errors with a weighting that varied with the level of drift in the task to maintain efficient performance.

Adult↗
PubMed1997

Stochastic processes in postural center-of-pressure profiles.

The stochastic processes of postural center-of-pressure profiles were examined in 3- and 5-year-old children, young adult students (mean 20 years), and an elderly age group (mean 67 years). Subjects stood still in an upright bipedal stance on a force platform under vision and nonvision conditions. The time evolutionary properties of the center-of-pressure dynamic were examined using basic stochastic process models. The amount of motion of the center of pressure decreased with increments of age from 3 to 5 years to young adult but increased again in the elderly age group. The availability of vision decreased the amount of motion of the center of pressure in all groups except the 3-year-old group, where there was less motion of the center of pressure with no vision. The stochastic properties of the center-of-pressure dynamic were assessed using both a two-process, random-walk model of Collins and De Luca and an Ornstein-Uhlenbeck model that is linear and has displacement governed only by a single stiffness term in the random walk. The two-process open- and closed-loop model accounted for about 96% and the Ornstein-Uhlenbeck model 92% of the variance of the diffusion term. Diffusion parameters in both models showed that the data were correlated and that they varied with age in a fashion consistent with developmental accounts of the changing regulation of the degrees of freedom in action. The findings suggest that it is premature to consider the trajectory of the center-of-pressure as a two-process, open- and closed-loop random-walk model given that: (a) the linear Ornstein-Uhlenbeck dynamic equation with only two parameters accommodates almost as much of the variance of the random walk; and (b) the linkage of a discontinuity in the diffusion process with the transition of open- to closed-loop processes is poorly founded. It appears that the nature of the stochastic properties of the random walk of the center-of-pressure trajectory in quiet, upright standing remains to be elucidated.

Adult↗
PubMed1999

Heterogeneous tube model for the study of small intestinal transit flow.

PURPOSE: A Monte-Carlo computer simulation technique was employed to study the details of the small intestinal transit flow in the gastrointestinal (GI) tract. METHODS: A heterogeneous tube model was constructed using a numerical computer simulation technique. The model was built from first principles and included several heterogeneous characteristics of the GI tract structure. We used a random, dendritic-type internal structure representing the villi of the GI tract. The small intestinal transit flow was simulated using two diffusion models, namely, the blind ant and the myopic ant models, which are different models to account the elapse of time, and which are both based on statistical properties of random walks. For each one of the models we utilize two types of biased random walk, placing different emphasis in the motion towards the output of the tube. We monitored the flow of the drug in terms of Monte-Carlo time steps (MCS) through the tube walls and dendritic villi present. RESULTS: The frequency of the transit times was dependent on the structure of the dendritic villi and on the type of biased random walk. The small intestinal flow profile of literature data for a large number of drugs was well characterized by the heterogeneous model using, as parameters, a certain number of villi per unit length of the tube and specific characteristics for both types of the biased random walk. A correspondence between the MCS and real time units was achieved. CONCLUSIONS: The transit process of the oral dosage forms in the GI tract can be reproduced with the heterogeneous model developed. This model can be used to study GI absorption phenomena.

Computer Simulation↗
PubMedDate not supplied

Diffusion tensor fiber tracking of human brain connectivity: aquisition methods, reliability analysis and biological results.

We present a description, biological results and a reliability analysis for the method of diffusion tensor tracking (DTT) of white matter fiber pathways. In DTT, diffusion-tensor MRI (DT-MRI) data are collected and processed to visualize the line trajectories of fiber bundles within white matter (WM) pathways of living humans. A detailed description of the data acquisition is given. Technical aspects and experimental results are illustrated for the geniculo-calcarine tract with broad projections to visual cortex, occipital and parietal U-fibers, and the temporo-calcarine ventral pathway. To better understand sources of error and to optimize the method, accuracy and precision were analyzed by computer simulations. In the simulations, noisy DT-MRI data were computed that would be obtained for a WM pathway having a helical trajectory passing through gray matter. The error vector between the real and ideal track was computed, and random errors accumulated with the square root of track length consistent with a random-walk process. Random error was most dependent on signal-to-noise ratio, followed by number of averages, pathway anisotropy and voxel size, in decreasing order. Systematic error only occurred for a few conditions, and was most dependent on the stepping algorithm, anisotropy of the surrounding tissue, and non-equal voxel dimensions. Both random and systematic errors were typically below the voxel dimension. Other effects such as track rebound and track recovery also depended on experimental conditions. The methods, biological results and error analysis herein may improve the understanding and optimization of DTT for use in various applications in neuroscience and medicine.

Adult↗
PubMed2006

Optimizing detection of tissue anisotropy by fluorescence recovery after photobleaching.

Fluorescence recovery after photobleaching (FRAP) has been widely used to measure fluid flow and diffusion in gels and tissues. It has not been widely used in detection of tissue anisotropy. This may be due to a lack of applicable theory, or due to inherent limitations of the method. We discuss theoretical aspects of the relationship between anisotropy of tissue structure and anisotropy of diffusion coefficients, with special regard to the size of the tracer molecule used. We derive a semi-mechanistic formula relating the fiber volume fraction and ratio of fiber and tracer molecule diameters to the expected anisotropy of the diffusion coefficients. This formula and others are tested on simulated random walks through random simulated and natural media. We determine bounds on the applicability of FRAP for detection of tissue anisotropy, and suggest minimum tracer sizes for detection of anisotropy in tissues of different composition (fiber volume fraction and fiber diameter). We find that it will be easier to detect anisotropy in monodisperse materials than in polydisperse materials. To detect mild anisotropy in a tissue, such as cartilage, which has a low fiber fraction would require a tracer molecule so large that it would be difficult to deliver to the tissue. We conclude that FRAP can be used to detect tissue anisotropy when the tracer molecule is sufficiently large relative to the fiber diameter, volume fraction, and degree of polydispersivity, and when the anisotropy is sufficiently pronounced.

Anisotropy↗
PubMed2004

DNA-protein interactions under random jump conditions.

We model the site-specific association of a protein molecule with DNA as a random walk with random jumps. Results show that the simultaneous occurrence of processes such as sliding, hopping, and intersegmental transfer can facilitate the diffusion-controlled site-specific association rate. We have also shown that sliding would dominate at lower DNA length, whereas at higher lengths hopping and intersegmental transfer would dominate. Apart from this, we predict that the association rate is directly proportional to the size of nonspecific DNA that flanks the specific site. These results are consistent with the experimental observations.

Binding Sites↗
PubMed2002

Long time correlations in lagrangian dynamics: a key to intermittency in turbulence.

Using a new experimental technique, based on the scattering of ultrasounds, we perform a direct measurement of particle velocities, in a fully turbulent flow. This allows us to approach intermittency in turbulence from a dynamical point of view and to analyze the Lagrangian velocity fluctuations in the framework of random walks. We find experimentally that the elementary steps in the walk have random uncorrelated directions but a magnitude that is extremely long range correlated in time. Theoretically, a Langevin equation is proposed and shown to account for the observed one- and two-point statistics. This approach connects intermittency to the dynamics of the flow.

Journal Article↗
PubMed2006

Reverse mapping of normal tissue complication probabilities onto dose volume histogram space: the problem of randomness of the dose volume histogram sampling.

A very important issue in contemporary inverse treatment radiotherapy planning is the specification of proper dose-volume constraints limiting the treatment planning algorithm from delivering high doses to the normal tissue surrounding the tumor. Recently we have proposed a method called reverse mapping of normal tissue complication probabilities (NTCP) onto dose-volume histogram (DVH) space, which allows the calculation of appropriate biologically based dose-volume constraints to be used in the inverse treatment planning. The method of reverse mapping requires random sampling from the functional space of all monotonically decreasing functions in the unit square. We develop, in this paper, a random function generator for the purpose of the reverse mapping. Since the proposed generator is based on the theory of random walk, it is therefore designated in this work, as a random walk DVH generator. It is theoretically determined that the distribution of the number of monotonically decreasing functions passing through a point in the dose volume histogram space follows the hypergeometric distribution. The proposed random walk DVH generator thus simulates, in a random fashion, trajectories of monotonically decreasing functions (finite series) that are situated in the unit square [0, 1] X [1,0] using the hypergeometric distribution. The DVH generator is an important tool in the study of reverse NTCP mapping for the calculation of biologically based dose-volume constraints for inverse treatment planning.

Computer Simulation↗
PubMed2005

Improving physical function and blood pressure in older adults through cobblestone mat walking: a randomized trial.

OBJECTIVES: To determine the relative effects of cobblestone mat walking, in comparison with regular walking, on physical function and blood pressure in older adults. DESIGN: Randomized trial with allocation to cobblestone mat walking or conventional walking. SETTING: General community in Eugene, Oregon. PARTICIPANTS: One hundred eight physically inactive community-dwelling adults aged 60 to 92 (mean age+/-standard deviation=77.5+/-5.0) free of neurological and mobility-limiting orthopedic conditions. INTERVENTION: Participants were randomized to a cobblestone mat walking condition (n=54) or regular walking comparison condition (n=54) and participated in 60-minute group exercise sessions three times per week for 16 consecutive weeks. MEASUREMENTS: Primary endpoint measures were balance (functional reach, static standing), physical performance (chair stands, 50-foot walk, Up and Go), and blood pressure (systolic, diastolic). Secondary endpoint measures were Short Form-12 physical and mental health scores and perceptions of health-related benefits from exercise. RESULTS: At the 16-week posttest, differences between the two exercise groups were found for balance measures (P=.01), chair stands (P<.001), 50-foot walk (P=.01), and blood pressure (P=.01) but not for the Up and Go test (P=.14). Although significant within-group changes were observed in both groups for the secondary outcome measures, there were no differences between intervention groups. CONCLUSION: Cobblestone mat walking improved physical function and reduced blood pressure to a greater extent than conventional walking in older adults. Additional benefits of this walking program included improved health-related quality of life. This new physical activity may provide a therapeutic and health-enhancing exercise alternative for older adults.

Aged↗
PubMed1989

The motile response of alveolar macrophages. An experimental study using single-cell and cell population approaches.

In this report, we studied the applicability of a random walk model of individual cell motility in predicting the motile behavior of alveolar macrophage populations under agarose. The migration of a population of cells in the absence of a chemotactic or chemokinetic gradient can be characterized by the random motility coefficient, mu, which is analogous to a particle diffusion coefficient. Random walk theory relates this latter coefficient to particle speed and collision time (equivalent to the time between changes in particle direction). By analogy, according to a similar random walk theory for cell migration, mu for a cell population is a function of the speed and persistence time (with direction changes governed by cell behavioral processes rather than by collisions) of individual cells. To test the model, normal guinea pig alveolar macrophages were incubated in the presence or absence of uniform concentrations of the chemotactic tripeptide formyl-norleucyl leucyl phenylalanine (FNLLP) to elicit different levels of motile activity. Mu was calculated from cell population density profiles obtained by fixing and staining cultures after 2, 3, or 4 days. In parallel experiments, individual cell speeds and persistence times were measured from 1-h, time-lapse video microscopy recordings. The value of mu calculated from single-cell measurements was in good agreement with that from population studies for stimulated random migration (at 10(-7) to 10(-11) M FNLLP), but not in the absence of stimulant. Overall, these results support the applicability of the random walk model of individual cell migration to randomly migrating alveolar macrophage populations.

Animals↗
PubMed2001

Characterization of structural reorganization in rice piles.

Diagnostics applied to a rice-pile cellular automaton reveal different mechanisms producing power-law behaviors of statistical attributes of grains which are germane to self organised critical phenomena. The probability distributions for these quantities can be derived from two distinct random walk models that account for correlated clustered behavior through incorporating fluctuations in the number of steps in the walk. The first model describes the distribution for a spatial quantity, the resultant flight length of grains. This has a power-law tail caused by grains moving through a discrete, power-law distributed number of random steps of finite length. Developing this model into a random walk obtains distributions for the resultant flight length with characteristics similar to Lévy distributions. The second random walk model is devised to explain a temporal quantity, the distribution of "trapping" or "residence" times of grains at single locations in the pile. Diagnostics reveal that the trapping time can be constructed as a sum of "subtrapping times," which are described by a Lévy distribution where the number of terms in the sum is a discrete random variable accurately described by a negative binomial distribution. The infinitely divisible, two-parameter, limit distribution for the resultant of such a random walk is discussed, and describes a dual-scale power-law behavior if the number fluctuations are strongly clustered. The form for the distribution of transit times of grains results as a corollary.

Journal Article↗
PubMed2000

Occupancy of a single site by many random walkers

We consider an infinite number of noninteracting lattice random walkers with the goal of determining statistical properties of the time, out of a total time T, that a single site has been occupied by n random walkers. Initially the random walkers are assumed uniformly distributed on the lattice except for the target site at the origin, which is unoccupied. The random-walk model is taken to be a continuous-time random walk and the pausing-time density at the target site is allowed to differ from the pausing-time density at other sites. We calculate the dependence of the mean time of occupancy by n random walkers as a function of n and the observation time T. We also find the variance for the cumulative time during which the site is unoccupied. The large-T behavior of the variance differs according as the random walk is transient or recurrent. It is shown that the variance is proportional to T at large T in three or more dimensions, it is proportional to T(3/2) in one dimension and to T ln T in two dimensions.

Journal Article↗
PubMed2003

Pascal principle for diffusion-controlled trapping reactions.

In this paper, we analyze the long-time behavior of the survival probability P(A)(t) of an A particle, that performs lattice random walk in the presence of randomly moving traps B. We show that for both perfect and imperfect trapping reactions, for arbitrary spatial dimension d and for a rather general class of random walks, P(A)(t) is less than or equal to the survival probability of an immobile target A in the presence of randomly moving traps.

Journal Article↗
PubMed1997

Stochastic completion fields; a neural model of illusory contour shape and salience.

We describe an algorithm- and representation-level theory of illusory contour shape and salience. Unlike previous theories, our model is derived from a single assumption: that the prior probability distribution of boundary completion shape can be modeled by a random walk in a lattice whose points are positions and orientations in the image plane (i.e., the space that one can reasonably assume is represented by neurons of the mammalian visual cortex). Our model does not employ numerical relaxation or other explicit minimization, but instead relies on the fact that the probability that a particle following a random walk will pass through a given position and orientation on a path joining two boundary fragments can be computed directly as the product of two vector-field convolutions. We show that for the random walk we define, the maximum likelihood paths are curves of least energy, that is, on average, random walks follow paths commonly assumed to model the shape of illusory contours. A computer model is demonstrated on numerous illusory contour stimuli from the literature.

Algorithms↗