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PubMed2004

Modelling the movement of a soil insect.

We use a linear autoregressive model to describe the movement of a soil-living insect, Protaphorura armata (Collembola). Models of this kind can be viewed as extensions of a random walk, but unlike a correlated random walk, in which the speed and turning angles are independent, our model identifies and expresses the correlations between the turning angles and a variable speed. Our model uses data in x- and y-coordinates rather than in polar coordinates, which is useful for situations in which the resolution of the observations is limited. The movement of the insect was characterized by (i) looping behaviour due to autocorrelation and cross correlation in the velocity process and (ii) occurrence of periods of inactivity, which we describe with a Poisson random effects model. We also introduce obstacles to the environment to add structural heterogeneity to the movement process. We compare aspects such as loop shape, inter-loop time, holding angles at obstacles, net squared displacement, number, and duration of inactive periods between observed and predicted movement. The comparison demonstrates that our approach is relevant as a starting-point to predict behaviourally complex moving, e.g. systematic searching, in a heterogeneous landscape.

Animals↗
PubMed1996

Role of chaos in trial-and-error problem solving by an artificial neural network.

One role of chaotic neural activity is illustrated by means of computer simulations of an imaginary agent's goal-oriented behavior. The agent has a simplified neural network with seven neurons and three legs. The neuronal network consists of one photosensory neuron and three pairs of inter- and motor neurons. The three legs whose movements are governed by the three motor neurons allow the agent to walk in six concentric radial directions on a plane. It is intended that the neural network causes the agent to walk in a direction of greater brightness, to reach finally the most brightly lit place on the plane. The presence of only one sensory neuron has an important meaning. That is, no immediate information on directions of greater brightness is sensed by the agent. In other words, random walking in the manner of trial- and error problem solving must be involved in the agent's walking. Chaotic firing of the motor neurons is intended to play a crucial role in generating the random walking. Brief random walking and rapid straight walking in a direction of greater brightness were observed to occur alternately in the computer simulation. Controlled chaos in naturally occurring neural networks may play a similar role.

Animals↗
PubMed2006

Quantitative analysis of bottlenose dolphin movement patterns and their relationship with foraging.

1. Broad-scale telemetry studies have greatly improved our understanding of the ranging patterns and habitat-use of many large vertebrates. However, there often remains considerable uncertainty over the function of different areas or the factors influencing habitat selection. Further insights into these processes can be obtained through analyses of finer scale movement patterns. For example, search behaviour may be modified in response to prey distribution and abundance. 2. In this study, quantitative analysis techniques are applied to the movements of bottlenose dolphins, recorded from land using a theodolite, to increase our understanding of their foraging strategies. Movements were modelled as a correlated random walk (CRW) and a biased random walk (BRW) to identify movement types and using a first-passage time (FPT) approach, which quantifies the time allocated to different areas and identifies the location and spatial scale of intensive search effort. 3. Only a quarter of the tracks were classed as CRW movement. Turning angle and directionality appeared to be key factors in determining the type of movement adopted. A high degree of overlap in search effort between separate movement paths indicated that there were small key sites (0.3 km radius) within the study area (4 km(2)). Foraging behaviour occurred mainly within these intensive search areas, indicating that they were feeding sites. 4. This approach provides a quantitative method of identifying important foraging areas and their spatial scale. Such techniques could be applied to movement paths for a variety of species derived from telemetry studies and increase our understanding of their foraging strategies.

Animals↗
PubMed2005

Hierarchical structure of the energy landscape of proteins revisited by time series analysis. I. Mimicking protein dynamics in different time scales.

Time series models, which are constructed from the projections of the molecular-dynamics (MD) runs on principal components (modes), are used to mimic the dynamics of two proteins: tendamistat and immunity protein of colicin E7 (ImmE7). Four independent MD runs of tendamistat and three independent runs of ImmE7 protein in vacuum are used to investigate the energy landscapes of these proteins. It is found that mean-square displacements of residues along the modes in different time scales can be mimicked by time series models, which are utilized in dividing protein dynamics into different regimes with respect to the dominating motion type. The first two regimes constitute the dominance of intraminimum motions during the first 5 ps and the random walk motion in a hierarchically higher-level energy minimum, which comprise the initial time period of the trajectories up to 20-40 ps for tendamistat and 80-120 ps for ImmE7. These are also the time ranges within which the linear nonstationary time series are completely satisfactory in explaining protein dynamics. Encountering energy barriers enclosing higher-level energy minima constrains the random walk motion of the proteins, and pseudorelaxation processes at different levels of minima are detected in tendamistat, depending on the sampling window size. Correlation (relaxation) times of 30-40 ps and 150-200 ps are detected for two energy envelopes of successive levels for tendamistat, which gives an overall idea about the hierarchical structure of the energy landscape. However, it should be stressed that correlation times of the modes are highly variable with respect to conformational subspaces and sampling window sizes, indicating the absence of an actual relaxation. The random-walk step sizes and the time length of the second regime are used to illuminate an important difference between the dynamics of the two proteins, which cannot be clarified by the investigation of relaxation times alone: ImmE7 has lower-energy barriers enclosing the higher-level energy minimum, preventing the protein to relax and letting it move in a random-walk fashion for a longer period of time.

Colicins↗
PubMed2004

Time series analysis of collective motions in proteins.

The dynamics of alpha-amylase inhibitor tendamistat around its native state is investigated using time series analysis of the principal components of the C(alpha) atomic displacements obtained from molecular dynamics trajectories. Collective motion along a principal component is modeled as a homogeneous nonstationary process, which is the result of the damped oscillations in local minima superimposed on a random walk. The motion in local minima is described by a stationary autoregressive moving average model, consisting of the frequency, damping factor, moving average parameters and random shock terms. Frequencies for the first 50 principal components are found to be in the 3-25 cm(-1) range, which are well correlated with the principal component indices and also with atomistic normal mode analysis results. Damping factors, though their correlation is less pronounced, decrease as principal component indices increase, indicating that low frequency motions are less affected by friction. The existence of a positive moving average parameter indicates that the stochastic force term is likely to disturb the mode in opposite directions for two successive sampling times, showing the modes tendency to stay close to minimum. All these four parameters affect the mean square fluctuations of a principal mode within a single minimum. The inter-minima transitions are described by a random walk model, which is driven by a random shock term considerably smaller than that for the intra-minimum motion. The principal modes are classified into three subspaces based on their dynamics: essential, semiconstrained, and constrained, at least in partial consistency with previous studies. The Gaussian-type distributions of the intermediate modes, called "semiconstrained" modes, are explained by asserting that this random walk behavior is not completely free but between energy barriers.

Binding Sites↗
PubMed2004

Lévy walks in random scattering and growth of waves.

Random spatial wave scattering and stochastic wave growth are studied where one or both of the random processes can be described by a Lévy walk. This analysis extends previous work on randomly growing and scattering waves where both the random processes are modeled by Gaussian diffusive statistics. Both random spatial scattering and stochastic wave growth modeled by Lévy walks are studied separately, together, and in combination with Gaussian processes. Transmission coefficients, lasing thresholds, and energy densities in the medium are obtained for the different permutations.

Journal Article↗
PubMed2001

Directed self-avoiding walks in random media.

Two types of directed self-avoiding walks (SAW's), namely, three-choice directed SAW and outwardly directed SAW, have been studied on infinite percolation clusters on the square lattice in two dimensions. The walks on the percolation clusters are generated via a Monte Carlo technique. The longitudinal extension R(N) and the transverse fluctuation W(N) have been measured as a function of the number of steps N. Slight swelling is observed in the longitudinal direction on the random lattices. A crossover from shrinking to swelling of the transverse fluctuations is found at a certain length N(c) of the walks. The exponents related to the transverse fluctuations are seen to be unchanged in the random media even as the percolation threshold is reached. The scaling function form of the extensions are verified.

Journal Article↗
PubMed2005

Calculating spatial statistics for velocity jump processes with experimentally observed reorientation parameters.

Mathematical modelling of the directed movement of animals, microorganisms and cells is of great relevance in the fields of biology and medicine. Simple diffusive models of movement assume a random walk in the position, while more realistic models include the direction of movement by assuming a random walk in the velocity. These velocity jump processes, although more realistic, are much harder to analyse and an equation that describes the underlying spatial distribution only exists in one dimension. In this communication we set up a realistic reorientation model in two dimensions, where the mean turning angle is dependent on the previous direction of movement and bias is implicitly introduced in the probability distribution for the direction of movement. This model, and the associated reorientation parameters, is based on data from experiments on swimming microorganisms. Assuming a transport equation to describe the motion of a population of random walkers using a velocity jump process, together with this realistic reorientation model, we use a moment closure method to derive and solve a system of equations for the spatial statistics. These asymptotic equations are a very good match to simulated random walks for realistic parameter values.

Animals↗
PubMed2006

Simulation and analysis of solute transport in 2D fracture/pipe networks: the SOLFRAC program.

The Time Domain Random Walk (TDRW) method has been recently developed by Delay and Bodin [Delay, F. and Bodin, J., 2001. Time domain random walk method to simulate transport by advection-dispersion and matrix diffusion in fracture networks. Geophys. Res. Lett., 28(21): 4051-4054.] and Bodin et al. [Bodin, J., Porel, G. and Delay, F., 2003c. Simulation of solute transport in discrete fracture networks using the time domain random walk method. Earth Planet. Sci. Lett., 6566: 1-8.] for simulating solute transport in discrete fracture networks. It is assumed that the fracture network can reasonably be represented by a network of interconnected one-dimensional pipes (i.e. flow channels). Processes accounted for are: (1) advection and hydrodynamic dispersion in the channels, (2) matrix diffusion, (3) diffusion into stagnant zones within the fracture planes, (4) sorption reactions onto the fracture walls and in the matrix, (5) linear decay, and (6) mass sharing at fracture intersections. The TDRW method is handy and very efficient in terms of computation costs since it allows for the one-step calculation of the particle residence time in each bond of the network. This method has been programmed in C++, and efforts have been made to develop an efficient and user-friendly software, called SOLFRAC. This program is freely downloadable at the URL (labo.univ-poitiers.fr/hydrasa/intranet/telechargement.htm). It calculates solute transport into 2D pipe networks, while considering different types of injections and different concepts of local dispersion within each flow channel. Post-simulation analyses are also available, such as the mean velocity or the macroscopic dispersion at the scale of the entire network. The program may be used to evaluate how a given transport mechanism influences the macroscopic transport behaviour of fracture networks. It may also be used, as is the case, e.g., with analytical solutions, to interpret laboratory or field tracer test experiments performed in single fractures.

Computer Simulation↗
PubMed2004

Statistical properties of a discrete version of the Ornstein-Uhlenbeck process.

A discrete version of the Ornstein-Uhlenbeck process is discussed which arises from a simple generalization of the master equation of the random walk. The calculation of the statistical properties of the free propagator for this process can be obtained using essentially the same formalism as for simple random walks. These calculations are carried out in some detail for the one-dimensional case. The usual equation for the evolution of the probability distribution of the Ornstein-Uhlenbeck process is recovered in the continuum limit if the jump distribution has a finite variance. However, the discrete process is also well defined for long tailed jump distributions and, thus, can be used to describe a Lèvy walk under the effect of a harmonic potential. Finally, a brief discussion of the generalization of this process to describe random walks in general potentials is presented and briefly compared with results arising from the fractional diffusion approach.

Journal Article↗
PubMed1996

Evolution of simple sequence repeats.

Simple Sequence Repeats (SSRs) are common and frequently polymorphic in eukaryote DNA. Many are subject to high rates of length mutation in which a gain or loss of one repeat unit is most often observed. Can the observed abundances and their length distributions be explained as the result of an unbiased random walk, starting from some initial repeat length? In order to address this question, we have considered two models for an unbiased random walk on the integers, n (n0 < or = n). The first is a continuous time process (Birth and Death Model or BDM) in which the probability of a transition to n + 1 or n - 1 is lambda k, with k = n - n0 + 1 per unit time. The second is a discrete time model (Random Walk Model or RWM), in which a transition is made at each time step, either to n - 1 or to n + 1. In each case the walks start at length n0, with new walks being generated at a steady rate, S, the source rate, determined by a base substitution rate of mutation from neighboring sequences. Each walk terminates whenever n reaches n0 - 1 or at some time, T, which reflects the contamination of pure repeat sequences by other mutations that remove them from consideration, either because they fail to satisfy the criteria for repeat selection from some database or because they can no longer undergo efficient length mutations. For infinite T, the results are particularly simple for N(k), the expected number of repeats of length n = k + n0 - 1, being, for BDM, N(k) = S/k lambda, and for RWM, N(k) = 2S. In each case, there is a cut-off value of k for finite T, namely k = T lambda ln2 for BDM and k = 0.57 square root of T for RWM; for larger values of k, N(k) becomes rapidly smaller than the infinite time limit. We argue that these results may be compared with SSR length distributions averaged over many loci, but not for a particular locus, for which founder effects are important. For the data of Beckmann & Weber [(1992), Genomics 12, 627] on GT.AC repeats in the human, each model gives a reasonable fit to the data, with the source at two repeat units (n0 = 2). Both the absolute number of loci and their length distribution are well represented.

DNA↗
PubMed1996

Dynamics of root growth in microgravity.

An experiment to study the growth of garden cress roots in microgravity is described. The experiment, denoted RANDOM, was an ESA Biorack experiment in the IML-2 flight in July 1994. In the absence of gravity, it can be anticipated that the roots would show random growth, changing their direction randomly. The hypothesis that such random growth movements occur according to random walk theory, leads to predictions as to the detailed manner in which deviations increase with time. The experiment was designed to test this random walk hypothesis. The paper concentrates on the technological aspects of studying the roots in microgravity. The development of suitable plant chambers, fitting containers developed by ESA, is described as well as the techniques used to grow the seeds between agar slices. hardware was developed to record photographically root movements between the agar slices. Photos were taken once per hour. Some plant chambers were designed to allow fixation of plant material in space. The practical solutions found using glutaraldehyde for prefixation in the Spacelab, within the restrictions given, are described. The experimental results show that the growth pattern in fact followed the prediction from the random walk approach. The average changes in the growth direction stayed constant and equal to zero during the experiment while the squared angular deviations increased proportional to time. Furthermore, plant material prefixed in orbit was permanently fixed after the flight. Light microscopy and electron microscopy pictures are shown as examples of the results achieved. The long prefixation period meant a drawback for the quality of the fixation process. However, sections suitable for study were achieved. The main goals of the RANDOM experiment were therefore achieved.

Environment, Controlled↗
PubMed1970

The locomotion of mouse fibroblasts in tissue culture.

Time-lapse cinematography was used to investigate the motion of mouse fibroblasts in tissue culture. Observations over successive short time intervals revealed a tendency for the cells to persist in their direction of motion from one 2.5 hr time interval to the next. Over 5.0-hr time intervals, however, the direction of motion appeared random. This fact suggested that D, the diffusion constant of a random walk model, might serve to characterize cellular motility if suitably long observation times were used. We therefore investigated the effect of "persistence" on the pure random walk model, and we found theoretically and confirmed experimentally that the motility of a persisting cell could indeed be characterized by an augmented diffusion constant, D*. A method for determining confidence limits on D* was also developed. Thus a random walk model, modified to comprehend the persistence effect, was found to describe the motion of fibroblasts in tissue culture and to provide a numerical measure of cellular motility.

Animals↗
PubMed2005

Chemotaxis and random motility in unsteady chemoattractant fields: a computational study.

We discuss a generic computational model which captures the effects of transient chemoattractant concentration on the chemotactic motility of individual cells. The model solves the appropriate unsteady chemoattractant transport equation using finite differences, while simultaneously executing biased random walks representing individual cells. The simulations were implemented for a 2D homogeneous domain, and two case studies were considered. In the first case study, we consider a single-point source at the origin of the domain which produces chemoattractant, while other cells execute biased random walks toward this point source. We observe that for continuous chemoattractant production, chemoattractant diffusivity has no effect on cell motility, as measured by the mean of time to reach the source. However, in the case of pulsed random production with a specific average duty cycle, the mean time-to-contact is generally minimal with respect to chemoattractant diffusivity over a moderate range of diffusivities. In the second case study, two mobile cells which simultaneously secrete chemoattractant are initially placed a certain distance apart and are then allowed to execute biased random walks. Our model shows that a pulsed random protocol for chemoattractant production facilitates the two cells "finding" one another compared to continuous production. From this case study we also learn that there exists a range of moderate chemoattractant diffusivities for which the mean time-to-contact is minimal when cells both produce/detect chemoattractant and chemotactically migrate. Using these case studies, we discuss how transience in chemoattractant concentration becomes important in characterizing the effectiveness of chemotaxis.

Algorithms↗
PubMed1997

The effects of common peroneal stimulation on the effort and speed of walking: a randomized controlled trial with chronic hemiplegic patients.

OBJECTIVE: To measure the effect of the Odstock Dropped Foot Stimulator (ODFS), a common peroneal stimulator, on the effort and speed of walking. DESIGN: A randomized controlled trial. SUBJECTS: Hemiplegic patients who had suffered a single stroke at least six months prior to the start of the trial whose walking was impaired by a drop-foot. INTERVENTIONS: The treatment, functional electrical stimulation (FES) group, used the stimulator and received a course of physiotherapy; the control group received physiotherapy alone. MAIN OUTCOME MEASURES: Changes in walking speed measured over 10 m and the effort of walking measured by physiological cost index (PCI). RESULTS: Thirty-two subjects completed the trial, 16 in the FES group and 16 in the control group. Mean increase in walking speed between the beginning and end of the trial was 20.5% in the FES group (when the stimulator was used), and 5.2% in the control group. Improvement was also measured in PCI with a reduction of 24.9% in the FES group (when the stimulator was used) and 1% in the control group. No improvement in these parameters was measured in the FES group when the stimulator was not used. CONCLUSION: Walking was statistically significantly improved when the ODFS was worn but no 'carry-over' was measured. Physiotherapy alone, in this group of subjects with established stroke, did not improve walking.

Aged↗
PubMed2006

Analysis of cell growth and diffusion in a scaffold for cartilage tissue engineering.

Developments in tissue engineering over the past decade have offered promising future for the repair and reconstruction of damaged tissues. To regenerate three dimensional and weight-bearing implants, advances in biomaterials and manufacturing technologies prompted cell cultivations with natural or artificial scaffolds, in which cells are allowed to proliferate, migrate, and differentiate in vitro. In this article, we develop a mathematical model for cell growth in a porous scaffold. By treating the cell-scaffold construct as a porous medium, a continuum model is set up based on basic principles of mass conservation. In addition to cell growth kinetics, we incorporate cell diffusion in the model to describe the effects of cell random walks. Computational results are compared to experimental data found in the literature. With this model, we are able to investigate cell motility, heterogeneous cell distributions, and non-uniform seeding for tissue engineering applications. Results show that random walks tend to enhance uniform cell spreads in space, which in turn increases the probabilities for cells to acquire nutrients; therefore random walks are likely to be a positive contribution to the overall cell growth on scaffolds.

Animals↗
PubMed1992

Extending the stochastic two-stage model of carcinogenesis to include self-regulation of the nonmalignant cell population.

One of the challenges of introducing greater biological realism into stochastic models of cancer induction is to find a way to represent the homeostatic control of the normal cell population over its own size without complicating the analysis too much to obtain useful results. Current two-stage models of carcinogenesis typically ignore homeostatic control. Instead, a deterministic growth path is specified for the population of "normal" cells, while the population of "initiated" cells is assumed to grow randomly according to a birth-death process with random immigrations from the normal population. This paper introduces a simple model of homeostatically controlled cell division for mature tissues, in which the size of the nonmalignant population remains essentially constant over time. Growth of the nonmalignant cell population (normal and initiated cells) is restricted by allowing cells to divide only to fill the "openings" left by cells that die or differentiate, thus maintaining the constant size of the nonmalignant cell population. The fundamental technical insight from this model is that random walks, rather than birth-and-death processes, are the appropriate stochastic processes for describing the kinetics of the initiated cell population. Qualitative and analytic results are presented, drawn from the mathematical theories of random walks and diffusion processes, that describe the probability of spontaneous extinction and the size distribution of surviving initiated populations when the death/differentiation rates of normal and initiated cells are known. The constraint that the nonmalignant population size must remain approximately constant leads to much simpler analytic formulas and approximations, flowing directly from random walk theory, than in previous birth-death models.(ABSTRACT TRUNCATED AT 250 WORDS)

Cell Death↗
PubMed2006

Assessment of long-range correlation in time series: how to avoid pitfalls.

Due to the ubiquity of time series with long-range correlation in many areas of science and engineering, analysis and modeling of such data is an important problem. While the field seems to be mature, three major issues have not been satisfactorily resolved. (i) Many methods have been proposed to assess long-range correlation in time series. Under what circumstances do they yield consistent results? (ii) The mathematical theory of long-range correlation concerns the behavior of the correlation of the time series for very large times. A measured time series is finite, however. How can we relate the fractal scaling break at a specific time scale to important parameters of the data? (iii) An important technique in assessing long-range correlation in a time series is to construct a random walk process from the data, under the assumption that the data are like a stationary noise process. Due to the difficulty in determining whether a time series is stationary or not, however, one cannot be 100% sure whether the data should be treated as a noise or a random walk process. Is there any penalty if the data are interpreted as a noise process while in fact they are a random walk process, and vice versa? In this paper, we seek to gain important insights into these issues by examining three model systems, the autoregressive process of order 1, on-off intermittency, and Lévy motions, and considering an important engineering problem, target detection within sea-clutter radar returns. We also provide a few rules of thumb to safeguard against misinterpretations of long-range correlation in a time series, and discuss relevance of this study to pattern recognition.

Journal Article↗