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PubMed2002

Detecting determinism in short time series, with an application to the analysis of a stationary EEG recording.

We have developed a new method for detecting determinism in a short time series and used this method to examine whether a stationary EEG is deterministic or stochastic. The method is based on the observation that the trajectory of a time series generated from a differentiable dynamical system behaves smoothly in an embedded phase space. The angles between two successive directional vectors in the trajectory reconstructed from a time series at a minimum embedding dimension were calculated as a function of time. We measured the irregularity of the angle variations obtained from the time series using second-order difference plots and central tendency measures, and compared these values with those from surrogate data. The ability of the proposed method to distinguish between chaotic and stochastic dynamics is demonstrated through a number of simulated time series, including data from Lorenz, Rössler, and Van der Pol attractors, high-dimensional equations, and 1/f noise. We then applied this method to the analysis of stationary segments of EEG recordings consisting of 750 data points (6-s segments) from five normal subjects. The stationary EEG segments were not found to exhibit deterministic components. This method can be used to analyze determinism in short time series, such as those from physiological recordings, that can be modeled using differentiable dynamical processes.

Algorithms↗
PubMed2000

A first-passage-time analysis of the periodically forced noisy leaky integrate-and-fire model.

We present a general method for the analysis of the discharge trains of periodically forced noisy leaky integrate-and-fire neuron models. This approach relies on the iterations of a stochastic phase transition operator that generalizes the phase transition function used for the study of periodically forced deterministic oscillators to noisy systems. The kernel of this operator is defined in terms of the the first passage time probability density function of the Ornstein Uhlenbeck process through a suitable threshold. Numerically, it is computed as the solution of a singular integral equation. It is shown that, for the noisy system, quantities such as the phase distribution (cycle histogram), the interspike interval distribution, the autocorrelation function of the intervals, the autocorrelogram and the power spectrum density of the spike train, as well as the input-output cross-correlation and cross-spectral density can all be computed using the stochastic phase transition operator. A detailed description of the numerical implementation of the method, together with examples, is provided.

Action Potentials↗
PubMed2000

Connectance in Sorghum development: beyond the genotype-phenotype duality.

Connectance, the level of linkage between organs, was measured in different lines of Sorghum bicolor during their reproductive development. It was compared with expression of characters, their level of variability and their heritability. A negative relationship is observed between connectance and heritability. Further results indicate that connectance does not simply introduce a noise factor in expression of a pre-existing information, but that it is directly involved in phenotypic expression and plasticity. Connectance appears as partly determined by the nature and dynamics of the network of relationships. It is concluded that the phenotype is not restricted to the deterministic expression of a pre-existing program, the genotype. Morphogenesis also involves another dimension, self-organized, which confers reliability, stability and adaptability to the developmental processes. The complex interactions between these two dimensions and their evolutionary consequences are discussed.

Edible Grain↗
PubMed2001

Extremal principle for the steady-state selection in driven lattice gases with open boundaries.

This paper investigates the steady states of one-dimensional driven lattice gases with open boundary conditions. It shows how the extremal principle proposed recently by Popkov and Schütz can be modified to apply to more general cases. Monte Carlo simulations are presented for a one-dimensional totally asymmetric simple exclusion process with nearest neighbor repulsion under parallel update as an example. The simulations enable one to guess the exact phase diagram for this particular lattice gas with deterministic bulk dynamics, by fitting the data to analytic formulas, which appear to be exact in the thermodynamic limit.

Journal Article↗
PubMedDate not supplied

Cytosolic calcium oscillators: critical discussion and stochastic modelling.

In the last few years, an immense amount of experimental data on agonist-induced cytosolic Ca2+ oscillations has emphasized the necessity of theoretical models accounting for these phenomena. In the first part of the paper, a critical analysis of different minimal Ca2+ oscillator models is presented, which reveals that not all of the so far proposed mechanisms are capable to reproduce experimental data adequately. The second part of the paper is devoted to a computational method using a stochastic simulation algorithm which describes the time evolution of Ca2+ oscillations at the molecular level. In contrast to the deterministic formulation of the models presented so far, the stochastic treatment takes account of the inherent fluctuations of cytosolic Ca2+ in cellular subcompartments. In the macroscopic limit, the stochastic models display dynamics analogous to the deterministic ones.

Animals↗
PubMedDate not supplied

Statistical methods of detection of a periodic phenomenon in a short series. Example of application: a density series of nematode eggs - II.

Using the example of the statistical treatment of a series of density of nematode eggs, we give here a survey of the main methods of detection of a periodic phenomenon in a short series. The purpose of these methods, which may be ranked into 3 groups: smoothing methods, regression methods, autocorrelations, is to study non random characteristics of the process. We detected in this series and estimated a rhythm of period 12 h. We concluded that the regression methods are the most resourceful for detecting a deterministic periodic phenomenon but that it is useful to confirm the results using the other methods.

Animals↗
PubMed1989

Saltatory transitions are a naturally occurring property of evolving systems.

On the basis of paleological evidence, it has been suggested that biological evolution need not necessarily be characterized by gradual change. Rather, evolutionary history may display saltatory periods of rapid speciation alternating with periods of relative quiescence, the whole dynamic being called punctuated equilibria. The empirical evidence that has been presented in support of this hypothesis has been the object of a vigorous dispute. Mathematical investigations of complex models of biological evolution that contain random elements have demonstrated that these systems can display saltatory behavior. In this paper we address a more abstract question: can saltations occur in the evolution of very simple, deterministic mathematical systems that function in a constant environment? The answer appears to be yes. Saltations appear as a natural dynamical behavior in the evolution of simplistic information processing networks. We stress that these networks do not constitute a model of biological evolution. However, the appearance of saltations in such simple systems suggests that their appearance in a process as complex as biological evolution is not surprising.

Animals↗
PubMed2003

A stochastic model of an ozonation reactor.

Disinfection of some microorganisms is characterized by a lag-phase (a minimum required ozone exposure until disinfection occurs). This phenomenon is easy to model in laboratory batch reactors but not in continuous flow mixed reactors. This paper introduces a stochastic disinfection model where individual microorganisms are followed on their paths through full-scale reactors. Combining exponentially distributed transport processes with delayed exponential disinfection kinetics for large populations of microorganisms (up to 10,000 individuals) yields predictions which can be evaluated statistically. It could be shown that deterministic models work well for systems with good disinfection performance (more than 2 log units reduction of active microorganisms), for reactors with poor performance stochastic models have to be applied. It could be demonstrated for real reactors that Bacillus subtilis spores are poor surrogates for Cryptosporidium parvum oocysts. The differences between the two microorganisms are large for reactors that deviate significantly from plug-flow behaviour.

Animals↗
PubMed1994

Dynamics of local neuronal networks: control parameters and state bifurcations in epileptogenesis.

The aim of this overview is to present evidence that local neuronal networks (LNNs) are functionally organized in such a way that they behave as dynamic non-linear systems that can exhibit multiple types of attractor and can present bifurcations between different attractors, depending on control parameters. To begin with, some of the theoretical concepts of non-linear dynamics and chaos are briefly presented. As a case study, we described the CA1 area of the hippocampus and the changes that the corresponding LNNs undergo during kindling epileptogenesis. During epileptic seizures, evidence exists for the presence of low-dimensional chaos, since the correlation dimension estimated from the corresponding EEG signals decreases dramatically from a large value, characteristic of the resting state, to a low value typical of deterministic chaos. We propose that, among other things, an important control parameter of the dynamics of this brain area is the balance between excitatory (E) and inhibitory (I) processes. We assume that this balance can be experimentally estimated by using a paired-pulse paradigm. Accordingly, we demonstrate that the paired-pulse response changes during kindling epileptogenesis in the sense that the E/I ratio increases in the course of the establishment of a kindled epileptogenic focus. This change in E/I leads to a shift in the operating point of the LNN moving it close to a bifurcation where a rapid state change takes place. In this way, the LNN dynamics can change more readily to the basin of attraction of a chaotic attractor than under normal conditions. This is in essence what makes the behavior of the LNN more sensitive to tetanus, and predicts the facilitated occurrence of epileptic seizures during kindling.

Animals↗
PubMedDate not supplied

Input-output behaviour of a model neuron with alternating drift.

The input-output behaviour of the Wiener neuronal model subject to alternating input is studied under the assumption that the effect of such an input is to make the drift itself of an alternating type. Firing densities and related statistics are obtained via simulations of the sample-paths of the process in the following three cases: the drift changes occur during random periods characterised by (i) exponential distribution, (ii) Erlang distribution with a preassigned shape parameter, and (iii) deterministic distribution. The obtained results are compared with those holding for the Wiener neuronal model subject to sinusoidal input.

Models, Neurological↗
PubMed1994

Scientific methodology in temporomandibular disorders. Part III: Diagnostic reasoning.

Temporomandibular disorders (TMD), as a cluster of individual diseases and disorders, pose new intellectual challenges to the diagnostic skills of dentists. New technologies enable dentists to avail themselves of paraclinical data such that diagnosis can and should be disease specific or etiology specific. The importance of logic in diagnostic reasoning is discussed. Studies of the reasoning process of doctors with reputations for having good clinical judgement have resulted in protocols of diagnostic reasoning. Three specific strategies are presented and discussed-probabilistic, causal and deterministic. The difference between intellectual and managerial decisions are explained relative to utility of the strategies.

Algorithms↗
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↗
PubMed1995

Active mutation in self-reproducing networks of machines and tapes.

Self-reproduction via description is discussed in a network model of machines and description tapes. Tapes consist of bit strings, which encode the machines' function A tape is replicated when it is read by adequate machines. Generally, a machine rewrites a tape without doing correct replication. The variation in a reproduced tape is taken as mutation. Because this mutation is caused by a machine's program, we call it active mutation. Which machine is translated from a given tape is dependent on what kind of a machine reads the tape. External noise is introduced in a machine's reading process to make errors. A new reaction pathway is induced by external noise via a machine's error action. We find that the induced pathways will be mimicked deterministically in an emerging core structure. This core structure will remain stable after turning off external noise. Low external noise develops a core structure of a minimal self-replicative loop. When external noise is elevated, a more complex network evolves. Machines containing a complex core network, which has been bred in high external noise, will actively rewrite tapes rather than just replicate them. Self-replication not as an individual but as a network now becomes important.

Biological Evolution↗
PubMed2000

Stochastic modelling of environmental variation for biological populations.

We examine stochastic effects, in particular environmental variability, in population models of biological systems. Some simple models of environmental stochasticity are suggested, and we demonstrate a number of analytic approximations and simulation-based approaches that can usefully be applied to them. Initially, these techniques, including moment-closure approximations and local linearization, are explored in the context of a simple and relatively tractable process. Our presentation seeks to introduce these techniques to a broad-based audience of applied modellers. Therefore, as a test case, we study a natural stochastic formulation of a non-linear deterministic model for nematode infections in ruminants, proposed by Roberts and Grenfell (1991). This system is particularly suitable for our purposes, since it captures the essence of more complicated formulations of parasite demography and herd immunity found in the literature. We explore two modes of behaviour. In the endemic regime the stochastic dynamic fluctuates widely around the non-zero fixed points of the deterministic model. Enhancement of these fluctuations in the presence of environmental stochasticity can lead to extinction events. Using a simple model of environmental fluctuations we show that the magnitude of this system response reflects not only the variance of environmental noise, but also its autocorrelation structure. In the managed regime host-replacement is modelled via periodic perturbation of the population variables. In the absence of environmental variation stochastic effects are negligible, and we examine the system response to a realistic environmental perturbation based on the effect of micro-climatic fluctuations on the contact rate. The resultant stochastic effects and the relevance of analytic approximations based on simple models of environmental stochasticity are discussed.

Animals↗
PubMed1998

Nonlinear feedforward networks with stochastic outputs: infomax implies redundancy reduction.

We prove that maximization of mutual information between the output and the input of a feedforward neural network leads to full redundancy reduction under the following sufficient conditions: (i) the input signal is a (possibly nonlinear) invertible mixture of independent components; (ii) there is no input noise; (iii) the activity of each output neuron is a (possibly) stochastic variable with a probability distribution depending on the stimulus through a deterministic function of the inputs (where both the probability distributions and the functions can be different from neuron to neuron); (iv) optimization of the mutual information is performed over all these deterministic functions. This result extends that obtained by Nadal and Parga (1994) who considered the case of deterministic outputs.

Feedback↗
PubMed1998

Stability analysis of the FitzHugh-Nagumo differential equations driven by impulses: applied to the electrical firing of magnocellular neurons.

A stability analysis is carried out for a mathematical model which describes the electrical firing of a single vasopressin neuron. The model used in a FitzHugh-Nagumo-type system which is driven by impulses. The analysis is based on recent developments in the stability theory of impulsive differential equations. Conditions are derived under which the system of differential equations is stable at two of its equilibrium points. Biologically this bistability represents the cell alternating between periods of electrical activity and silence. The conditions for stability are specified in terms of the amplitude and frequency of the impulses perturbing the system. Both stochastic and deterministic impulses are considered.

Action Potentials↗
PubMed1985

Decision making under uncertainty: a comparison of simple scalability, fixed-sample, and sequential-sampling models.

The purpose of this article is to investigate the learning and memory processes involved in decision making under uncertainty. In two different experiments, subjects were given a choice between a certain alternative that produced a single known payoff and an uncertain alternative that produced a normal distribution of payoffs. Initially this distribution was unknown, and in the first experiment it was learned through feedback from past decisions, whereas in the second experiment it was learned by observing sample outcomes. In the first experiment, a response deadline was used to limit the amount of time available for making a decision. In the second experiment, an observation cost was used to limit the number of samples that could be purchased. The mean and variance of the uncertain alternative and the value of the certain alternative were factorially manipulated to study their joint effects on choice probability, choice response time (Experiment 1), and number of observations purchased (Experiment 2). Algebraic-deterministic theories developed for decision making with simple gambles fail to explain the present results. Two new models are developed and tested--fixed- and sequential-sampling models--that attempt to describe the learning and memory processes involved in decision making under uncertainty.

Adult↗
PubMed1977

A mechanistic model of the aerobic growth of Saccharomyces cerevisiae.

A two-stage deterministic model of the growth of Saccharomyces cerevisiae is presented. The cell cycle of this organism was used to suggest the basic model structure. The model represents the preparatory processes of substrate uptake and conversion separately from replication and division. The regulation of the fraction of the culture devoted to each of these broad areas of metabolism, and the overall growth rate, is related to the nature and availability of the energy substrate. The simulation of respiration and glycolysis is achieved by including two alternative energy producing pathways. The regulation of these pathways is described in terms of the postulated primary regulation of the proportion of the culture required for substrate uptake and conversion, and the overall kinetic constants for each pathway. This regulation is dictated primarily by the growth rate rather than the nature or concentration of the energy substrate. The model successfully describes both batch and continuous growth of S. cerevisiae under conditons of glucose limitation and oxygen excess. A preliminary assessment indicates that adjustment of the relevant parameters will allow the model to describe the growth of S. cerevisiae on other sugars and under oxygen limitation. Similarly the model could be expected to describe the growth characteristics of other yeast species.

Aerobiosis↗