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PubMed1995

Nonlinear forecasting and the dynamics of cardiac rhythm.

Since the initial development of the electrocardiogram, cardiologists have made dramatic advances in the description and understanding of cardiac arrhythmias. Despite these successes, the analysis of cardiac rhythm has remained largely descriptive. Recently, the principles of nonlinear dynamics, or chaos theory, have been applied to the quantitative analysis of cardiac rhythm in a variety of diverse situations. In chaos theory, three types of signals can be defined: periodic signals, which repeat themselves over some finite time interval, chaotic signals, which, while deterministic, demonstrate complex behavior and do not repeat themselves, and random signals, which are unpredictable and nondeterministic. The technique of nonlinear forecasting defines trajectories in a suitably defined phase space and uses the future evolution of trajectories that are close to each other over short distances to make predictions for times further into the future. The ability to reliably predict the future evolution of the trajectories derived from any signal is an important characteristic of the underlying dynamics of the signal and can therefore used to determine those dynamics. The foundation of nonlinear forecasting is reviewed, and an algorithm is described that can be used to determine the underlying dynamics of a signal and has been applied to the analysis of R-R interval data.

Algorithms↗
PubMed1993

A method of screening for genes of major effect.

This paper describes a method for screening animal populations on an index of calculated probabilities of genotype status at an unknown single locus. Animals selected by such a method might then be candidates in test matings and genetic marker analyses for major gene detection. The method relies on phenotypic measures for a continuous trait plus identification of sire and dam. Some missing phenotypes and missing pedigree information are permitted. The method is an iterative two-step procedure, the first step estimates genotype probabilities and the second step estimates genotypic effects by regressing phenotypes on genotype probabilities, modeled as true genotype status plus error. Prior knowledge or choice of major locus-free heritability for the trait of interest is required, plus initial starting estimates of the effect on phenotype of carrying one and two copies of the unknown gene. Gene frequency can be estimated by this method, but it is demonstrated that the consequences of using an incorrect fixed prior for gene frequency are not particularly adverse where true frequency of the allele with major effect is low. Simulations involving deterministic sampling from the normal distribution lead to convergence for estimates of genotype effects at the true values, for a reasonable range of starting values, illustrating that estimation of major gene effects has a rational basis. In the absence of polygenic effects, stochastic simulations of 600 animals in five generations resulted in estimates of genotypic effects close to the true values. However, stochastic simulations involving generation and fitting of both major genotype and animal polygenic effects showed upward bias in estimates of major genotype effects. This can be partially overcome by not using information from relatives when calculating genotype probabilities-a result which suggests a route to a modified method which is unbiased and yet does use this information.

Alleles↗
PubMed2001

Analysis of potential radiobiological effects related to a unified skin dose limit.

A unified skin dose limit of 0.5 Sv at a depth of 70 microm averaged over the highest 10 cm2 of skin exposed was evaluated to replace the existing limit of 0.5 Sv averaged over 1 cm2. This limit would apply to all exposures including non-uniform exposures such as from hot particles on or off skin, skin contamination, or beams of charged particles or photons. The probabilities and severity of both stochastic and deterministic risks were estimated for a wide range of worst-case exposure scenarios using published radiobiological data and calculations of radial- and depth-dose distributions. Results indicate that exposures at the unified dose limit have the potential to cause effective doses of about 17 microSv (1.7 mrem), estimated stochastic risks of <3.3 x 10(-7) fatal skin cancers, and <1.6 x 10(-4) non-fatal skin cancers. The worst deterministic effects were estimated to be (a) based on a 2 Gy threshold, transient erythema induction to an area of 2.5 cm2 for uniform skin contamination over this same area and 0.65 cm2 for a 60Co hot particle 3 mm off of skin, (b) based on data for pig skin, 50% probability that 0.5 cm2 of skin would suffer 20% dermal thinning for uniform contamination with 106Rh spread over the same area, and (c) 10% probability of barely detectable transient acute necrosis or ulceration for 60Co or activated fuel particles 0.4 mm off of skin. It was concluded that the unified limit would provide a more logical system of dose control with possible savings of whole-body dose and other benefits.

Animals↗
PubMed2006

Spectral methods for parametric sensitivity in stochastic dynamical systems.

Stochastic dynamical systems governed by the chemical master equation find use in the modeling of biological phenomena in cells, where they provide more accurate representations than their deterministic counterparts, particularly when the levels of molecular population are small. The analysis of parametric sensitivity in such systems requires appropriate methods to capture the sensitivity of the system dynamics with respect to variations of the parameters amid the noise from inherent internal stochastic effects. We use spectral polynomial chaos expansions to represent statistics of the system dynamics as polynomial functions of the model parameters. These expansions capture the nonlinear behavior of the system statistics as a result of finite-sized parametric perturbations. We obtain the normalized sensitivity coefficients by taking the derivative of this functional representation with respect to the parameters. We apply this method in two stochastic dynamical systems exhibiting bimodal behavior, including a biologically relevant viral infection model.

Algorithms↗
PubMed2005

Monte Carlo simulations of VEGF binding to cell surface receptors in vitro.

The vascular endothelial growth factor (VEGF) family binds multiple endothelial cell surface receptors. Our goal is to build comprehensive models of these interactions for the purpose of simulating angiogenesis. In view of low concentrations of growth factors in vivo and in vitro, stochastic modeling of molecular interactions may be necessary. Here, we compare Monte Carlo simulations of the stochastic binding of VEGF and two of its major receptors on cells in vitro to equivalent deterministic simulations. In the range of typical VEGF concentrations, the stochastic and deterministic models are in agreement. However, we observe significant variability in receptor binding, which may be linked to biological stochastic events, e.g., blood vessel sprout initiation. We study patches of cell surface of varying sizes to investigate spatial integration of the signal by the cell, which impacts directly the variability of binding, and find significant variability up to the single-cell level. Dimerization of VEGF receptors does not significantly alter the variability in ligand binding. A 'sliding window' approach demonstrated no reduction in the variability of binding by temporal integration. The variability is expected to be more prominent in in vivo situations where the number of ligand molecules available for binding is less.

Algorithms↗
PubMed2004

Periodic local disturbance in host-parasitoid metapopulations: host suppression and parasitoid persistence.

Within many agricultural systems, insect pests and their natural enemies are forced to persist as a metapopulation, continuously recolonizing patches following disturbance through harvesting or spraying with insecticides. Despite the need to understand factors influencing biocontrol success, few theoretical studies of host-parasitoid interactions have examined the potential impact of local disturbance within a metapopulation framework. Here, we add periodic local mortality to series of classical host-parasitoid models to examine its effect on host suppression and parasitoid persistence. Using a deterministic lattice model, we show that despite the wide range of complex dynamics generated at the patch level, the region wide pattern of disturbance is the key factor influencing host suppression. The level of host suppression achieved can be understood in terms of both the strength of density dependent parasitism, and the relative amounts of host and parasitoid mixing amongst patches of different ages. Local dispersal among patches is sufficient to ensure coexistence of the host and parasitoid, though persistence is not necessarily associated with the formation of self-organized spatial structures reported in previous studies. Finally, a stochastic version of the model is developed, in order to highlight how the effects of demographic stochasticity may influence biocontrol success in highly disturbed agricultural systems.

Animals↗
PubMed2005

Stochastic amplification and signaling in enzymatic futile cycles through noise-induced bistability with oscillations.

Stochastic effects in biomolecular systems have now been recognized as a major physiologically and evolutionarily important factor in the development and function of many living organisms. Nevertheless, they are often thought of as providing only moderate refinements to the behaviors otherwise predicted by the classical deterministic system description. In this work we show by using both analytical and numerical investigation that at least in one ubiquitous class of (bio)chemical-reaction mechanisms, enzymatic futile cycles, the external noise may induce a bistable oscillatory (dynamic switching) behavior that is both quantitatively and qualitatively different from what is predicted or possible deterministically. We further demonstrate that the noise required to produce these distinct properties can itself be caused by a set of auxiliary chemical reactions, making it feasible for biological systems of sufficient complexity to generate such behavior internally. This new stochastic dynamics then serves to confer additional functional modalities on the enzymatic futile cycle mechanism that include stochastic amplification and signaling, the characteristics of which could be controlled by both the type and parameters of the driving noise. Hence, such noise-induced phenomena may, among other roles, potentially offer a novel type of control mechanism in pathways that contain these cycles and the like units. In particular, observations of endogenous or externally driven noise-induced dynamics in regulatory networks may thus provide additional insight into their topology, structure, and kinetics.

Enzyme Stability↗
PubMed1997

A simple model of transmitter release and facilitation.

We describe a model of synaptic transmitter release and presynaptic facilitation that is based on activation of release sites by single Ca2+ microdomains. Facilitation is due to Ca2+ that remains bound to release sites between impulses. This model is inherently stochastic, but deterministic equations can be derived for the mean release. The number of equations required to describe the mean release is small, so it is practical to use the model with models of neuronal electrical activity to investigate the effects of different input spike patterns on presynaptic facilitation. We use it in conjunction with a model of dopamine-secreting neurons of the basal ganglia to demonstrate that transmitter release is greater when the neuron bursts than when it spikes continuously, due to the greater facilitation generated by the bursting impulse pattern. Finally, a minimal form of the model is described that is coupled to simple models of postsynaptic receptors and passive membrane to compute the postsynaptic voltage response to a train of presynaptic stimuli. This form of the model is appropriate for neural network simulations.

Basal Ganglia↗
PubMed2005

Fractal and noisy CBV dynamics in humans: influence of age and gender.

The complexity of spontaneous cerebral blood volume (CBV) fluctuations can emerge from random, fractal, or chaotic processes. Our aims were to define the contribution of these patterns to the observed complexity and to evaluate the effect of age and gender on it. The total hemoglobin content as the measure of CBV was monitored by near-infrared spectroscopy on volunteers (men n = 19, age = 20 to 78 years; women n = 23, age = 21 to 79 years). Random and fractal patterns were distinguished by the spectral index (beta). Chaos was identified by surrogate analysis of the correlation dimension (a static chaotic parameter, the dimension of the correlation integral) and the largest Lyapunov exponent (a dynamic chaotic parameter, the rate of exponential divergence of the system states from a perturbed initial condition over the chaotic attractor). In spontaneous CBV fluctuations, both fast random and slow fractal dynamics are present separately in their spectra by a cutoff frequency, f'. Below f' the pattern is fractal, in that power rises inversely with frequency as 1/f(beta). f' decreases with age in men and women alike (F1: up to 0.12+/-0.06 Hz versus F2: up to 0.05+/-0.04 Hz at P = 0.015, and M1: up to 0.16+/-0.05 Hz versus M2: up to 0.11+/-0.04 Hz at P = 0.044). Neither pre- nor postmenopausal age groups (1 and 2, respectively) showed a (low)beta gender difference. Surrogate analysis showed that CBV dynamics cannot be characterized on the grounds of deterministic chaos. Cerebral blood volume fluctuates in a complex, bimodal manner in humans, in that the fast dynamics has no structure, while the slow dynamics exhibits a self-similar, that is, fractal temporal structure. The range of fluctuation amplitudes produced by fractal dynamics is always larger than that of random fluctuations, and it shrinks with an altered structuring in aging women only.

Adult↗
PubMed1997

Fractal modeling and segmentation for the enhancement of microcalcifications in digital mammograms.

The objective of this research is to model the mammographic parenchymal, ductal patterns and enhance the microcalcifications using deterministic fractal approach. According to the theory of deterministic fractal geometry, images can be modeled by deterministic fractal objects which are attractors of sets of two-dimensional (2-D) affine transformations. The iterated functions systems and the collage theorem are the mathematical foundations of fractal image modeling. In this paper, a methodology based on fractal image modeling is developed to analyze and model breast background structures. We show that general mammographic parenchymal and ductal patterns can be well modeled by a set of parameters of affine transformations. Therefore, microcalcifications can be enhanced by taking the difference between the original image and the modeled image. Our results are compared with those of the partial wavelet reconstruction and morphological operation approaches. The results demonstrate that the fractal modeling method is an effective way to enhance microcalcifications. It may also be able to improve the detection and classification of microcalcifications in a computer-aided diagnosis system.

Algorithms↗
PubMed2006

Deviant effects in molecular reaction pathways.

In biological networks, any manifestations of behaviors substantially 'deviant' from the predictions of continuous-deterministic classical chemical kinetics (CCK) are typically ascribed to systems with complex dynamics and/or a small number of molecules. Here we show that in certain cases such restrictions are not obligatory for CCK to be largely incorrect. By systematically identifying properties that may cause significant divergences between CCK and the more accurate discrete-stochastic chemical master equation (CME) system descriptions, we comprehensively characterize potential CCK failure patterns in biological settings, including consequences of the assertion that CCK is closer to the 'mode' rather than the 'average' of stochastic reaction dynamics, as generally perceived. We demonstrate that mechanisms underlying such nonclassical effects can be very simple, are common in cellular networks and result in often unintuitive system behaviors. This highlights the importance of deviant effects in biotechnologically or biomedically relevant applications, and suggests some approaches to diagnosing them in situ.

Biotechnology↗
PubMed2002

Demographic characteristics and population dynamical patterns of solitary birds.

In birds and many other animals, there are large interspecific differences in the magnitude of annual variation in population size. Using time-series data on populations of solitary bird species, we found that fluctuations in population size of solitary birds were affected by the deterministic characteristics of the population dynamics as well as the stochastic factors. In species with highly variable populations, annual variation in recruitment was positively related to the return rate of adults between successive breeding seasons. In stable populations, more recruits were found in years with low return rates of breeding adults. This identifies a gradient, associated with the position of the species along a "slow-fast" continuum of life history variation, from highly variable populations with a recruitment-driven demography to stable, strongly density-regulated populations with a survival-restricted demography. These results suggest that patterns in avian population fluctuations can be predicted from a knowledge of life-history characteristics and/or temporal variation in certain demographic traits.

Animals↗
PubMed2001

Source-sink dynamics between transgenic and non-transgenic habitats and their role in the evolution of resistance.

The interaction of population dynamics and movement among two habitat types (toxic transgenic fields and nontoxic refuge fields) on the evolution of insecticide resistance was examined in two different simulation models. The two models were developed to test the hypothesis that increasing habitat grain from fine-grained to coarse-grained, and the resultant increase in nonrandom mating, would increase the rate of local adaptation, here the evolution of resistance. The first model, a complex, stochastic spatially explicit model, altered habitat grain by varying adult dispersal rates between habitat patches. In contrast to the expectation that increasing patch isolation and increasing the coarseness of the habitats would increase the rate of resistance evolution, intermediate levels of dispersal actually delayed resistance by as much as fivefold over the range of dispersal levels observed. Source-sink dynamics related to ovipositional patterns and the related population dynamics appear to explain the results. A simple deterministic model was developed to abstract out the separate impacts of mating and ovipositional behaviors. This model showed qualitatively the same results, although under similar assumptions it predicted much longer delays in resistance evolution. In this model, nonrandom mating alone always increased the rate at which insects adapted to transgenic crops, but nonrandom mating in combination with nonrandom oviposition could significantly delay resistance evolution. Differences between the two models may be due to the population regulation incorporated in the spatially explicit model. The models clearly suggest that resistance management programs using untreated refuges should not over-emphasize random mating at the cost of making the habitat too fine-grained.

Animals↗
PubMed2000

Nonlinear EEG analysis and its potential role in epileptology.

Deterministic chaos offers a striking explanation for apparently irregular behavior of the brain that is evidenced in the EEG. Recent developments in the physical-mathematical framework of the theory of nonlinear dynamics (colloquially often termed chaos theory) provide new concepts and powerful algorithms to analyze such time series. Because of its high versatility, nonlinear time series analysis has already gone beyond the physical sciences and, at present, is being successfully applied in a variety of disciplines, including cardiology, neurology, psychiatry, and epileptology. However, it is well known that different influencing factors limit the use of nonlinear measures to characterize EEG dynamics in a strict sense. Nevertheless, when interpreted with care, relative estimates of, e.g., the correlation dimension or the Lyapunov exponents, can reliably characterize different states of normal and pathologic brain function. In epileptology, extraction of nonlinear measures from the intracranially recorded EEG promises to be important for clinical practice. In addition to an immense reduction of information content of long-lasting EEG recordings, previous studies have shown that these measures enable (a) localization of the primary epileptogenic area in different cerebral regions during the interictal state, (b) investigations of antiepileptic drug effects, (c) analyses of spatio-temporal interactions between the epileptogenic zone and other brain areas, and (d) detection of features predictive of imminent seizure activity. Nonlinear time series analysis provides new and supplementary information about the epileptogenic process and thus contributes to an improvement in presurgical evaluation.

Brain Mapping↗
PubMed1992

Maintenance of foot and mouth disease viruses in buffalo (Syncerus caffer Sparrman, 1779) in southern Africa.

Using age-related infection rates derived from serological data in available deterministic and specially developed stochastic simulation models, it has been possible to establish that the basic reproductive rates for South African Territory (SAT) type foot and mouth disease virus in buffalo (Syncerus caffer) are high. The models predict that there is a periodicity of infection within herds and possibly the population as a whole. Thus, buffalo herds are likely to be more infectious at some times than at others. However, because most infections in buffalo are inapparent, such episodes are difficult to identify. There is wide intratypic variation within the SAT type virus populations circulating in buffalo. This was determined by sequencing part of the 1 D gene of buffalo isolates and establishing antigenic profiles with neutralising monoclonal antibodies and conventional antisera.

Age Factors↗
PubMed1983

Paradoxical fixation of deleterious alleles in two-locus systems with epistasis.

The dynamics of four deterministic models of interaction between two Mendelian loci are explored numerically. At one locus there are two hypostatic alleles: h, the wild type and H, a deleterious mutant that in either heterozygous or homozygous state (depending on the specifics of the model) produces an abnormal phenotype. At another arbitrarily linked epistatic locus there are two alleles: e, the wild type with no effect on the expression of the H locus, and E, an epistatic mutant that in heterozygous or homozygous state (again, depending on the specifics of the model) blocks the expression of H. The parameters are the initial gamete frequencies, the recombination fraction, the genotypic viabilities, and the forward and back mutation rates at each locus. The He haplotype is eventually eliminated (unless back mutation occurs) from the population. If mutation is ignored, the evolutionary outcome is determined by the initial gamete frequencies and is either (1) an edge equilibrium comprising one pair of haplotypes only (he and He, or he and hE, or hE and HE, or He and HE, or, if there is no recombination, he and HE or hE and He); or (2) a corner equilibrium consisting of a single gametic type. Given that the forward mutation rates at both loci are greater than the back mutation rates, then the outcome is always the corner equilibrium in which HE is universal (apart from transient perturbations by mutation). In the process of fixation, the phenotypic impact of the deleterious allele H becomes neutralized by the epistatic allele E. The rate at which the initially harmful gene replaces the wild type gene depends on the recombination fraction, the genotypic viabilities, and the mutation rates.

Alleles↗
PubMed1998

Analytical and simulation results for stochastic Fitzhugh-Nagumo neurons and neural networks.

An analytical approach is presented for determining the response of a neuron or of the activity in a network of connected neurons, represented by systems of nonlinear ordinary stochastic differential equations--the Fitzhugh-Nagumo system with Gaussian white noise current. For a single neuron, five equations hold for the first- and second-order central moments of the voltage and recovery variables. From this system we obtain, under certain assumptions, five differential equations for the means, variances, and covariance of the two components. One may use these quantities to estimate the probability that a neuron is emitting an action potential at any given time. The differential equations are solved by numerical methods. We also perform simulations on the stochastic Fitzugh-Nagumo system and compare the results with those obtained from the differential equations for both sustained and intermittent deterministic current inputs with superimposed noise. For intermittent currents, which mimic synaptic input, the agreement between the analytical and simulation results for the moments is excellent. For sustained input, the analytical approximations perform well for small noise as there is excellent agreement for the moments. In addition, the probability that a neuron is spiking as obtained from the empirical distribution of the potential in the simulations gives a result almost identical to that obtained using the analytical approach. However, when there is sustained large-amplitude noise, the analytical method is only accurate for short time intervals. Using the simulation method, we study the distribution of the interspike interval directly from simulated sample paths. We confirm that noise extends the range of input currents over which (nonperiodic) spike trains may exist and investigate the dependence of such firing on the magnitude of the mean input current and the noise amplitude. For networks we find the differential equations for the means, variances, and covariances of the voltage and recovery variables and show how solving them leads to an expression for the probability that a given neuron, or given set of neurons, is firing at time t. Using such expressions one may implement dynamical rules for changing synaptic strengths directly without sampling. The present analytical method applies equally well to temporally nonhomogeneous input currents and is expected to be useful for computational studies of information processing in various nervous system centers.

Action Potentials↗
PubMed1980

Extraction of objects from structured backgrounds in the cat superior colliculus. Part I.

Specific changes occur in the cells of the upper layers of the cat's superior collicules when a two dimensional noise (background) is superimposed onto a deterministic signal (spot of light). Some of the measurements can be interpreted as meaning that some cells only react to certain relative movements of object (spot) and background (noise). The movement of the visual background is interpreted as environmental movement occurring due to the animal's own movement. The results of the measurements provide all the necessary presuppositions for a distinction between the animal's own velocity and that of the object (Part I). The experimental results can be interpreted with a model. The essential factors for the interpretation is the direction specific behavior of the cells which is bound up with an asymmetrical spatial coupling of the neurons with each other. The decisive advantage of asymmetrical systems for the pattern recognition of moving objects is that they can work without distortion and spatial displacement over large ranges of velocity (Part II).

Animals↗