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PubMed2002

Emotional behaviour as the result of stochastic interactions. A process crucial for cognition.

This paper is aimed at exhibiting two striking features of the usual approach of emotional expression in science and philosophy, suggesting a different perspective. One is the generally shared belief that emotions are a state of utter disarray, which hampers objective knowledge; the other is the search for causal explanation, along a wide range of categorized approaches (psychology, neurosciences, developmental biology) each proposing its own theoretical framework. In both cases the result is to play down emotional expression. Alternatively, we propose to view emotions as something crucial in the choice of our conceptual tools, ideas and involvements, in the genesis of which various explanations interact in a complex stochastic way. Rather than being a harmful disruption of the mind calling for identification of a definite causality, emotional behaviour appears as a necessary process in cognition, which is irreducible to a unique origin.

Journal Article↗
PubMed1999

Cyclostationarity and stochastic resonance in threshold devices.

This paper intends to show how the theory of stochastic cyclostationary processes can be used to study stochastic resonance in static nonlinearities. The statistic we use is the covariance function of the output. The covariance is a second-order cumulant and is not dependent on by the mean. Furthermore, this covariance is not averaged in time as is usually done in the stochastic resonance literature. A two-dimensional Fourier transform of the covariance gives the so-called spectral correlation. The spectral correlation depends on the usual harmonic frequency and on another frequency, called cycle frequency. The cyclostationarity of a signal makes the spectral correlation discrete in the cycle frequency. The zero cycle frequency corresponds to the usual "stationary power spectrum" used in the stochastic resonance literature. We thus exploit all the second-order statistical information. We first revisit classical stochastic resonance in threshold devices using the spectral correlation, showing that the effect is seen for nonzero cycle frequencies. The cases of additive and multiplicative noise are detailed. We then study stochastic resonance in threshold devices for communication signals. These signals are usually modeled as stochastic cyclostationary processes. We show that stochastic resonance occurs, and the phenomenon is quantified using the spectral correlation of the output: The amplitude of the spectral correlation at nonzero cycle frequencies presents a maximum as the power of the input noise is increased.

Journal Article↗
PubMed1999

The Galton-Watson branching process as a quantitative tool in parasitology.

Stochastic growth processes abound in the biology of parasitism, and one mathematical tool that is particularly well suited for describing such phenomena is the Galton-Watson branching process. Introduced more than a century ago to settle a debate over the rate of disappearance of surnames in the British peerage, branching processes are applied today in fields as diverse as quantum physics and theoretical computer science. In this article, Dale Taneyhill, Alison Dunn and Melanie Hatcher provide a simple introduction to branching processes, and demonstrate their uses in quantitative parasitology.

Animals↗
PubMed2001

Population dynamics: Poisson approximation and its relation to the Langevin process.

We discuss how to simulate a stochastic evolution process in terms of difference equations with Poisson distributions of independent events when the problem is naturally described by discrete variables. For large populations the Poisson approximation becomes a discrete integration of the Langevin approximation [T. G. Kurtz, J. Appl. Prob. 7, 49 (1970); 8, 344 (1971)]. We analyze when the latter gives a reasonable representation of the original evolution for finite size systems. A simple example of an epidemic process is used to organize the discussion and to perform statistical tests that underline the goodness of the proposed method.

Poisson Distribution↗
PubMed1984

On chiasma formation point processes having the count location property.

The count-location (C-L) chiasma formation schemes introduced by Karlin and Liberman (1979b) encompass a broad class of map functions involving positive, negative or no chiasma interference. The C-L schemes do not explicitly assume a specific mechanism of crossover formation, but rather a statistical property of the process. If viewed as a stochastic point process along the chromosome, it is shown that a crossing over mechanism having the C-L property is actually a rescaled mixture of Poisson processes. Surprisingly it turns out that these C-L point processes involve negative interference throughout the entire genome.

Crossing Over, Genetic↗
PubMed2006

A constitutive law for the failure behavior of medial collateral ligaments.

A constitutive model is proposed for the description of the tensile properties of medial collateral ligaments (MCLs). The model can reproduce the three regions -- the toe region, the linear region, and the failure region -- of the stress-stretch curve of ligamentous tissues. The collagen fibers are assumed to be the only load-bearing component of the tissues. They are all oriented along the physiological loading direction of the ligament. They are crimped in the slack configuration and are unable to sustain load. After becoming taut and before failing, each collagen fiber exhibits a linear elastic behavior. The fiber straightening and failure processes are defined stochastically by means of Weibull distributions. Published experimental data for the MCLs are employed to validate the constitutive relationship. Finally, the constitutive model is generalized in order to describe the three-dimensional mechanical behavior of the ligaments by following he structural approach.

Algorithms↗
PubMed2006

Estimation of the reaction efficiency in polymerase chain reaction.

Polymerase chain reaction (PCR) is largely used in molecular biology for increasing the copy number of a specific DNA fragment. The succession of 20 replication cycles makes it possible to multiply the quantity of the fragment of interest by a factor of 1 million. The PCR technique has revolutionized genomics research. Several quantification methodologies are available to determine the DNA replication efficiency of the reaction which is the probability of replication of a DNA molecule at a replication cycle. We elaborate a quantification procedure based on the exponential phase and the early saturation phase of PCR. The reaction efficiency is supposed to be constant in the exponential phase, and decreasing in the saturation phase. We propose to model the PCR amplification process by a branching process which starts as a Galton-Watson branching process followed by a size-dependent process. Using this stochastic modelling and the conditional least-squares estimation method, we infer the reaction efficiency from a single PCR trajectory.

Animals↗
PubMed2004

Modelling the dynamics of biosystems.

The need for a more formal handling of biological information processing with stochastic and mobile process algebras is addressed. Biology can benefit this approach, yielding a better understanding of behavioural properties of cells, and computer science can benefit this approach, obtaining new computational models inspired by nature.

Algorithms↗
PubMed2004

Testing time symmetry in time series using data compression dictionaries.

Time symmetry, often called statistical time reversibility, in a dynamical process means that any segment of time-series output has the same probability of occurrence in the process as its time reversal. A technique, based on symbolic dynamics, is proposed to distinguish such symmetrical processes from asymmetrical ones, given a time-series observation of the otherwise unknown process. Because linear stochastic Gaussian processes, and static nonlinear transformations of them, are statistically reversible, but nonlinear dynamics such as dissipative chaos are usually statistically irreversible, a test will separate large classes of hypotheses for the data. A general-purpose and robust statistical test procedure requires adapting to arbitrary dynamics which may have significant time correlation of undetermined form. Given a symbolization of the observed time series, the technology behind adaptive dictionary data compression algorithms offers a suitable estimate of reversibility, as well as a statistical likelihood test. The data compression methods create approximately independent segments permitting a simple and direct null test without resampling or surrogate data. We demonstrate the results on various time-series-reversible and irreversible systems.

Journal Article↗
PubMed1995

The Poisson process as a model for compartment digesta flow in ruminants.

The Poisson process, the simplest stochastic flow process, was used to develop a multicompartment model of ruminant digesta flow with Gamma distributed retention times. Although mathematically the model is a generalization of many previously published models, the physiological model differs substantially in asserting that the distributed delay time and the exponential rate (scale) parameters, including the scale parameter of the Gamma distribution, are determined by total digesta flow, and thus invariant with respect to the fraction marked. The shape factor of the Gamma distribution is shown to be sufficient to explain the difference between markers in rate of marker excretion. Consequently, the parameters of multiple markers can be simultaneously estimated with the constraint that the exponential scale parameters and the delay time are invariant with respect to marker. This constraint leads to a measure of pure error to strengthen statistical tests for model rejection. Steady-state digesta retention time is estimated from the transient marker retention parameters, eliminating the necessity of speculating on what fraction of digesta the marked fraction represents. Tests of various models, using simulations and animal experiments indicate that, even if a model is correct, it is not possible to obtain reliable parameter estimates by fitting to a single marker. Even with multiple markers some caution must be used in interpreting parameter estimates derived from least squares fitting.

Animals↗
PubMed2000

Spread rate for a nonlinear stochastic invasion.

Despite the recognized importance of stochastic factors, models for ecological invasions are almost exclusively formulated using deterministic equations [29]. Stochastic factors relevant to invasions can be either extrinsic (quantities such as temperature or habitat quality which vary randomly in time and space and are external to the population itself) or intrinsic (arising from a finite population of individuals each reproducing, dying, and interacting with other individuals in a probabilistic manner). It has been long conjectured [27] that intrinsic stochastic factors associated with interacting individuals can slow the spread of a population or disease, even in a uniform environment. While this conjecture has been borne out by numerical simulations, we are not aware of a thorough analytical investigation. In this paper we analyze the effect of intrinsic stochastic factors when individuals interact locally over small neighborhoods. We formulate a set of equations describing the dynamics of spatial moments of the population. Although the full equations cannot be expressed in closed form, a mixture of a moment closure and comparison methods can be used to derive upper and lower bounds for the expected density of individuals. Analysis of the upper solution gives a bound on the rate of spread of the stochastic invasion process which lies strictly below the rate of spread for the deterministic model. The slow spread is most evident when invaders occur in widely spaced high density foci. In this case spatial correlations between individuals mean that density dependent effects are significant even when expected population densities are low. Finally, we propose a heuristic formula for estimating the true rate of spread for the full nonlinear stochastic process based on a scaling argument for moments.

Animals↗
PubMed2006

A comparison of descriptive models of a single spike train by information-geometric measure.

In examining spike trains, different models are used to describe their structure. The different models often seem quite similar, but because they are cast in different formalisms, it is often difficult to compare their predictions. Here we use the information-geometric measure, an orthogonal coordinate representation of point processes, to express different models of stochastic point processes in a common coordinate system. Within such a framework, it becomes straightforward to visualize higher-order correlations of different models and thereby assess the differences between models. We apply the information-geometric measure to compare two similar but not identical models of neuronal spike trains: the inhomogeneous Markov and the mixture of Poisson models. It is shown that they differ in the second- and higher-order interaction terms. In the mixture of Poisson model, the second- and higher-order interactions are of comparable magnitude within each order, whereas in the inhomogeneous Markov model, they have alternating signs over different orders. This provides guidance about what measurements would effectively separate the two models. As newer models are proposed, they also can be compared to these models using information geometry.

Action Potentials↗
PubMed1999

Exact density profile of a stochastic reaction-diffusion process.

We calculate exactly the time dependent density profile of a one-dimensional stochastic reaction-diffusion process of hard-core particles subjected to the reactions AA <--> OO and AO <--> OA. The solution is based on the fundamental property that the evolution operator, defined over an appropriate vector space, transforms vectors with n kinks into vectors with n or n+2 kinks, only. In this space, a basis vector is represented by a string of plus and minus signs and a kink is defined as a pair of opposite signs. The exact time dependent profiles are calculated for the cases of uncorrelated initial states that are translational invariant as well as initial states that are inhomogeneous in space.

Journal Article↗
PubMed1986

A stochastic model of retinotopy: a self organizing process.

Following Kohonen and using the Hebb principle, we define a self organizing stochastic process, which is a simple modelization of the retinotopy, i.e. the establishment of well-ordered connexions between the retina and the cortex. We give some mathematical results about convergence of this process. These results are illustrated by computer simulations.

Depth Perception↗