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PubMed2000

Modeling and analysis of stochastic invasion processes.

In this paper we derive spatially explicit equations to describe a stochastic invasion process. Parents are assumed to produce a random number of offspring which then disperse according to a spatial redistribution kernel. Equations for population moments, such as expected density and covariance averaged over an ensemble of identical stochastic processes, take the form of deterministic integro-difference equations. These equations describe the spatial spread of population moments as the invasion progresses. We use the second order moments to analyse two basic properties of the invasion. The first property is 'permanence of form' in the correlation structure of the wave. Analysis of the asymptotic form of the invasion wave shows that either (i) the covariance in the leading edge of the wave of invasion asymptotically achieves a permanence of form with a characteristic structure described by an unchanging spatial correlation function, or (ii) the leading edge of the wave has no asymptotic permanence of form with the length scales of spatial correlations continually increasing over time. Which of these two outcomes pertains is governed by a single statistic, phi which depends upon the shape of the dispersal kernel and the net reproductive number. The second property of the invasion is its patchy structure. Patchiness, defined in terms of spatial correlations on separate short (within patch) and long (between patch) spatial scales, is linked to the dispersal kernel. Analysis shows how a leptokurtic dispersal kernel gives rise to patchiness in spread of a population.

Animals↗
PubMed1981

Some notes on the foundations and scope of intensive design.

IN THE application of single-case and intensive design to the study of psychopathology and interpersonal processes the primary focus is upon the data of the individual subject as distinct from that of a sample of subjects. Such models begin with the notion that personality processes are stochastic processes. That is, one's moods, affects, signs and symptoms, and interactions with others out of the past or from the present fluctuate over time and with respect to various contingencies in a probability space of a very high order of dimensionality. In practice--that is, in relation to application to a given study or set of studies--we are of course constrained to consider just a very small subspace of this huge hyperspace.

Ethics, Professional↗
PubMed1997

Stochastic threshold characterization of the intensity of active channel dynamical action potential generation.

Stochastic threshold characterization of the intensity of active channel dynamical action potential generation. J. Neurophysiol. 78: 2616-2630, 1997. This paper develops a stochastic intensity description for action potential generation formulated in terms of stochastic processes, which are direct analogues of the physiological processes of the pre- and postsynaptic complex of the cochlear nerve: 1) neurotransmitter release is modeled as an inhomogeneous Poisson counting process with release intensity mu t, 2) the excitatory postsynaptic conductance (EPSC) process is modeled as a marked, linearly filtered Poisson process resulting from the linear superposition of standard shaped postsynaptic conductances of size G, and 3) action potential generation is modeled as resulting from the EPSC exceeding a random threshold determined by active channel dynamics of the Hodgkin-Huxley type. The random threshold is defined to be the least upper bound in the size of a standard-shaped neurotransmitter release injected at time t given the previous action potential time and the number of releases occurring in a short preconditioning time increment. The action potential process is modeled as a self-exciting point process with stochastic intensity resulting from the probability that the random threshold process crosses the threshold in some small time increment that is a function of time since previous action potential, release intensity, and the probability that a single synaptic event exceeds the stochastic threshold. The stochastic intensity model is consistent with a direct simulation of the nonlinear Hodgkin-Huxley differential equations over a variety of parameters for the vesicle release intensity, vesicle size, vesicle duration, and temperatures. Results are presented showing that the regularity properties seen in the vestibular primary afferent in the lizard, Calotes versicolor, associated with a slow-to-activate potassium channel resulting in a long afterhyperpolarization can be accommodated directly by the stochastic intensity description. The stimulus dependence of the model is attributed to synaptic transmission and the probabilistic nature to the threshold conductance process, which is dependent upon the EPSC process. The stochastic intensity is seen to have a form consistent with the phenomenologically based Siebert-Gaumond model, a stimulus-related function of time multiplied by a refractory-related function of time since previous action potential.

Action Potentials↗
PubMed2004

First passage time problem for a drifted Ornstein-Uhlenbeck process.

We consider a continuous stochastic process defined as a drifted Ornstein-Uhlenbeck, for which the first passage time is of interest. The process being non-homogeneous, the first passage time probability density function cannot be found analytically, but numerical methods enable to find its estimate. Estimating the first passage time implies solving an unsteady convection-diffusion equation, with variable coefficients, and we use an implicit Euler scheme to solve it. This work is applied to simulated data, and the continuous process is inspired from recent work on biological marker modelling for HIV-positive patients. The first passage time probability density function can be useful to compare the marker progression in different groups. Numerical results show that the first passage time is highly dependent from the process perturbation, and is then more relevant than methods not considering the stochastic process directly to compare the progression.

Biomarkers↗
PubMed1967

Neuronal spike trains and stochastic point processes. I. The single spike train.

In a growing class of neurophysiological experiments, the train of impulses ("spikes") produced by a nerve cell is subjected to statistical treatment involving the time intervals between spikes. The statistical techniques available for the analysis of single spike trains are described and related to the underlying mathematical theory, that of stochastic point processes, i.e., of stochastic processes whose realizations may be described as series of point events occurring in time, separated by random intervals. For single stationary spike trains, several orders of complexity of statistical treatment are described; the major distinction is that between statistical measures that depend in an essential way on the serial order of interspike intervals and those that are order-independent. The interrelations among the several types of calculations are shown, and an attempt is made to ameliorate the current nomenclatural confusion in this field. Applications, interpretations, and potential difficulties of the statistical techniques are discussed, with special reference to types of spike trains encountered experimentally. Next, the related types of analysis are described for experiments which involve repeated presentations of a brief, isolated stimulus. Finally, the effects of nonstationarity, e.g. long-term changes in firing rate, on the various statistical measures are discussed. Several commonly observed patterns of spike activity are shown to be differentially sensitive to such changes. A companion paper covers the analysis of simultaneously observed spike trains.

Action Potentials↗
PubMed1999

Stochastic multiplicative processes with reset events.

We study a stochastic multiplicative process with reset events. It is shown that the model develops a stationary power-law probability distribution for the relevant variable, whose exponent depends on the model parameters. Two qualitatively different regimes are observed, corresponding to intermittent and regular behavior. In the boundary between them, the mean value of the relevant variable is time independent, and the exponent of the stationary distribution equals -2. The addition of diffusion to the system modifies in a nontrivial way the profile of the stationary distribution. Numerical and analytical results are presented.

Journal Article↗
PubMed1994

First-passage-time problem for simulated stochastic diffusion processes.

Solving the first-passage-time problem for one-dimensional stochastic diffusion processes is a task with many applications in biomedical research. It has been noted (Musila and Lánský, Int. J. Biomed. Comput. 31, 233-245, 1992) that the first-passage time is overestimated if computed as the time when the simulated trajectory of the process crosses the threshold. It is studied in this paper how the error depends on the simulation step and on the parameters of the process. We propose an adaptive algorithm to make the simulation faster. The presented examples are related to neuronal modelling, but application in other fields is straightforward.

Algorithms↗
PubMed1984

Identification and estimation algorithm for stochastic neural system.

An algorithm for the estimation of stochastic processes in a neural system is presented. This process is defined here as the continuous stochastic process reflecting the dynamics of the neural system which has some inputs and generates output spike trains. The algorithm proposed here is to identify the system parameters and then estimate the stochastic process called neural system process here. These procedures carried out on the basis of the output spike trains which are supposed to be the data observed in the randomly missing way by the threshold time function in the neural system. The algorithm is constructed with the well-known Kalman filters and realizes the estimation of the neural system process by cooperating with the algorithm for the parameter estimation of the threshold time function presented previously (Nakao et al., 1983). The performance of the algorithm is examined by applying it to the various spike trains simulated by some artificial models and also to the neural spike trains recorded in cat's optic tract fibers. The results in these applications are thought to prove the effectiveness of the algorithm proposed here to some extent. Such attempts, we think, will serve to improve the characterizing and modelling techniques of the stochastic neural systems.

Animals↗
PubMed2005

Refractory pulse counting processes in stochastic neural computers.

This letter quantitiatively investigates the effect of a temporary refractory period or dead time in the ability of a stochastic Bernoulli processor to record subsequent pulse events, following the arrival of a pulse. These effects can arise in either the input detectors of a stochastic neural network or in subsequent processing. A transient period is observed, which increases with both the dead time and the Bernoulli probability of the dead-time free system, during which the system reaches equilibrium. Unless the Bernoulli probability is small compared to the inverse of the dead time, the mean and variance of the pulse count distributions are both appreciably reduced.

Neural Networks, Computer↗
PubMed1999

Locations of radiation-produced DNA double strand breaks along chromosomes: a stochastic cluster process formalism.

Ionizing radiation produces DNA double strand breaks (DSBs) in chromosomes. For densely ionizing radiation, the DSBs are not spaced randomly along a chromosome: recent data for size distributions of DNA fragments indicate break clustering on kbp-Mbp scales. Different DSB clusters on a chromosome are typically made by different, statistically independent, stochastically structured radiation tracks, and the average number of tracks involved can be small. We therefore model DSB positions along a chromosome as a stationary Poisson cluster process, i.e. a stochastic process consisting of secondary point processes whose locations are determined by a primary point process that is Poisson. Each secondary process represents a break cluster, typically consisting of 1-10 DSBs in a comparatively localized stochastic pattern determined by chromatin geometry and radiation track structure. Using this Poisson cluster process model, which we call the randomly located clusters (RLC) formalism, theorems are derived for how the DNA fragment-size distribution depends on radiation dose. The RLC dose-response relations become non-linear when the dose becomes so high that DSB clusters from different tracks overlap or adjoin closely. The RLC formalism generalizes previous models, fits current data adequately and facilitates mechanistically based extrapolations from high-dose experiments to the much lower doses of interest for most applications.

Chromosome Aberrations↗
PubMed2003

Queuing transitions in the asymmetric simple exclusion process.

Stochastic driven flow along a channel can be modeled by the asymmetric simple exclusion process. We confirm numerically the presence of a dynamic queuing phase transition at a nonzero obstruction strength, and establish its scaling properties. Below the transition, the traffic jam is macroscopic in the sense that the length of the queue scales linearly with system size. Above the transition, only a power-law shaped queue remains. Its density profile scales as deltarho approximately x(-nu) with nu=1/3, and x is the distance from the obstacle. We construct a heuristic argument, indicating that the exponent nu=1/3 is universal and independent of the dynamic exponent of the underlying dynamic process. Fast bonds create only power-law shaped depletion queues, and with an exponent that could be equal to nu=2/3, but the numerical results yield consistently somewhat smaller values nu approximately 0.63(3). The implications of these results to faceting of growing interfaces and localization of directed polymers in random media, both in the presence of a columnar defect are pointed out as well.

Journal Article↗
PubMed1991

Development of skin tumors in hairless mice after discontinuation of ultraviolet irradiation.

The development of skin tumors (mainly squamous cell carcinomas) in hairless Skh-HR1 mice after discontinuation of a course of daily UV irradiations (wavelengths, 280-370 nm) is compared to that when the daily irradiations are continued. Under conditions of continued daily exposures 50% of 22 animals contracted tumors with diameters of at least 4 mm in 135 days. With exposures stopped after 35 or 19 days (2 groups with 24 and 23 mice) this time interval increased to 280 and 645 days, respectively; the rate at which multiple tumors developed on the mice was correspondingly lower. A mathematical model, derived from a larger experiment (223 mice) with different levels of chronic UV exposure, successfully predicts the tumor development after discontinuation of UV exposure. This model is similar to those used in risk assessments for skin cancers in human populations, e.g., in relation to stratospheric ozone depletion, sunbeds, etc. The model separates UV-driven processes from purely time-dependent processes. These stochastic processes, described by Weibull statistics, form stages in the tumorigenesis. This interpretation of the data indicates that a late, UV-independent stage occurs between the smallest observable tumors and larger ones with diameters of over 4 mm. This could be a simple growth stage, but histopathology suggests that it may also entail a transition from actinic keratosis to squamous cell carcinoma.

Animals↗
PubMed1996

The dynamical theory of coevolution: a derivation from stochastic ecological processes.

In this paper we develop a dynamical theory of coevolution in ecological communities. The derivation explicitly accounts for the stochastic components of evolutionary change and is based on ecological processes at the level of the individual. We show that the coevolutionary dynamic can be envisaged as a directed random walk in the community's trait space. A quantitative description of this stochastic process in terms of a master equation is derived. By determining the first jump moment of this process we abstract the dynamic of the mean evolutionary path. To first order the resulting equation coincides with a dynamic that has frequently been assumed in evolutionary game theory. Apart from recovering this canonical equation we systematically establish the underlying assumptions. We provide higher order corrections and show that these can give rise to new, unexpected evolutionary effects including shifting evolutionary isoclines and evolutionary slowing down of mean paths as they approach evolutionary equilibria. Extensions of the derivation to more general ecological settings are discussed. In particular we allow for multi-trait coevolution and analyze coevolution under nonequilibrium population dynamics.

Animals↗
PubMed1996

Computation of first passage time moments for stochastic diffusion processes modelling nerve membrane depolarization.

For further understanding of neural coding, stochastic variability of interspike intervals has been investigated by both experimental and theoretical neuroscientists. In stochastic neuronal models, the interspike interval corresponds to the time period during which the process imitating the membrane potential reaches a threshold for the first time from a reset depolarization. For neurons belonging to complex networks in the brain, stochastic diffusion processes are often used to approximate the time course of the membrane potential. The interspike interval is then viewed as the first passage time for the employed diffusion process. Due to a lack of analytical solution for the related first passage time problem for most diffusion neuronal models, a numerical integration method, which serves to compute first passage time moments on the basis of the Siegert recursive formula, is presented in this paper. For their neurobiological plausibility, the method here is associated with diffusion processes whose state spaces are restricted to finite intervals, but it can also be applied to other diffusion processes and in other (non-neuronal) contexts. The capability of the method is demonstrated in numerical examples and the relation between the integration step, accuracy of calculation and amount of computing time required is discussed.

Action Potentials↗
PubMed1995

Hominoid phylogeny estimated by model selection using goodness of fit significance tests.

Phylogeny estimation from nucleotide sequence data may be thought of as a problem of choosing between different evolutionary models that vary with the branching pattern of the phylogeny and with the stochastic process of nucleotide sequence change occurring on the branches of the phylogenetic tree. Thus, each evolutionary model consists of both a particular stochastic process and a particular phylogeny. Such models produce multinomial distributions of nucleotide character patterns. As first suggested by Cavalli-Sforza and Edwards [Evolution 21: 550-570 (1967)] the distribution of patterns expected under each model can be compared to the actual observed distribution of patterns by a goodness of fit statistic such as the loglikelihood ratio G2 or Pearson's X2 after the numerical parameters for the model have been chosen to minimize the respective statistic. For each evolutionary model, the probability P of getting a value of the goodness of fit statistic greater than the observed value is computed. A very small P value means that either a rare event has occurred or that the model is false. Employing for each of 16 models a stochastic process which has 12 parameters to describe the mode of nucleotide change on each branch of each putative phylogenetic tree, we examined all 15 unrooted dichotomously branching arrangements of orthologous noncoding sequences from the gamma hemoglobin genomic region of the five hominoids (gibbon, orangutan, gorilla, chimpanzee, and human) plus the branching arrangement with a trichotomous separation of gorilla, chimpanzee, and human. Of these 16 models, all had P values less than 0.01, except for the arrangement of human joined by chimpanzee, in turn joined by gorilla, and then orangutan and gibbon. This analysis allows convincing claims to be made about hominoid phylogenetic relationships by testing the applicability of the assumed stochastic process for nucleotide sequence evolution at the same time as testing the inferred phylogenetic branching arrangement.

Animals↗
PubMed2002

Additive noise, Weibull functions and the approximation of psychometric functions.

The Weibull function is frequently chosen to define psychometric functions. Tyler and Chen (Vis. Res. 40 (2000) 3121) criticised the high-threshold postulate implied by the Weibull function and argued that this function implies the assumption of multiplicative noise. It will be shown in this paper that in fact the Weibull function is compatible with the assumption of additive noise, and that the Weibull function may be generalised to the case of detection not being high threshold. The derivations rest, however, on a representation of sensory activity lacking a satisfying degree of generality. Therefore, a more general representation of sensory activity in terms of stochastic processes will be suggested, with detection being defined as a level-crossing process, containing the original representation as a special case. Two classes of stochastic processes will be considered: one where the noise is assumed to be additive, stationary Gaussian, and another resulting from cascaded Poisson processes, representing a form of multiplicative noise. While Weibull functions turn out to approximate well psychometric functions generated by both types of stochastic processes, it also becomes obvious that there is no simple interpretation of the parameters of the fitted Weibull functions. Moreover, corresponding to Tyler and Chen's discussion of the role of multiplicative noise particular sources of this type of noise will be considered and shown to be compatible with the Weibull. It is indicated how multiplicative noise may be defined in general; however, it will be argued that in the light of certain empirical data the role of this type of noise may be negligible in most detection tasks.

Contrast Sensitivity↗
PubMed2004

Extinction times and moment closure in the stochastic logistic process.

We investigate the statistics of extinction times for an isolated population, with an initially modest number M of individuals, whose dynamics are controlled by a stochastic logistic process (SLP). The coefficient of variation in the extinction time V is found to have a maximum value when the death and birth rates are close in value. For large habitat size K we find that Vmax is of order K1/4 / M1/2, which is much larger than unity so long as M is small compared to K1/2. We also present a study of the SLP using the moment closure approximation (MCA), and discuss the successes and failures of this method. Regarding the former, the MCA yields a steady-state distribution for the population when the death rate is low. Although not correct for the SLP model, the first three moments of this distribution coincide with those calculated exactly for an adjusted SLP in which extinction is forbidden. These exact calculations also pinpoint the breakdown of the MCA as the death rate is increased.

Animals↗