Sustained oscillations via coherence resonance in SIR

Sustained oscillations in a stochastic SIR model are studied using a new multiple scale analysis. It captures the interaction of the deterministic and stochastic elements together with the separation of time scales inherent in the appearance of these dynamics. The nearly regular fluctuations in the infected and susceptible populations are described via an explicit construction of a stochastic amplitude equation. The agreement between the power spectral densities of the full model and the approximation verifies that coherence resonance is driving the behavior. The validity criteria for this asymptotic approximation give explicit expressions for the parameter ranges in which one expects to observe this phenomenon.     

Genetical feedback repression. I. Single locus models

The process of genetical feedback-repression analogous to the Jacob-Monod system has been simulated by digital computer modelling, allowing inclusion of stochastic variation of the components. In the absence of stochastic variability, the model shows a damped oscillation over a wide range of specifications. Inclusion of stochastic variation results in the model maintaining oscillations which, if summed over several independent genetical feedback-repression loops, results in a regular oscillation.     

心肌单细胞兴奋模式转迁的非线性动力学机理

利用Moms-Lecar模型研究实验观察到的培养心肌单细胞自发性兴奋模式转迁规律的动力学机理,确定性模型仿真,揭示了心肌单细胞随参数由“极化”静息经规则节律到“去极化”静息的节律变化规律。随机因素扰动下的模型仿真发现在分岔序列中的分岔点附近会出现含延迟后去极化电位、旱后去极化电位的节律模式,其中,延迟后去极化节律产生于从“极化”静息到规则节律的分岔点附近,而旱后去极化节律产生于从规则节律到“去极化”静息的分岔点附近。这表明含延迟后去极化电位的节律和含旱后去极化电位的节律是系统在自动兴奋和静息之间的分岔点附近由于参数的随机扰动而产生的。     

Receptors, sparks and waves in a fire-diffuse-fire framework for calcium release

Calcium ions are an important second messenger in living cells. Indeed calcium signals in the form of waves have been the subject of much recent experimental interest. It is now well established that these waves are composed of elementary stochastic release events (calcium puffs or sparks) from spatially localised calcium stores. The aim of this paper is to analyse how the stochastic nature of individual receptors within these stores combines to create stochastic behaviour on long time-scales that may ultimately lead to waves of activity in a spatially extended cell model. Techniques from asymptotic analysis and stochastic phase–plane analysis are used to show that a large cluster of receptor channels leads to a release probability with a sigmoidal dependence on calcium density. This release probability is incorporated into a computationally inexpensive model of calcium release based upon a stochastic generalisation of the fire-diffuse-fire (FDF) threshold model. Numerical simulations of the model in one and two dimensions (with stores arranged on both regular and disordered lattices) illustrate that stochastic calcium release leads to the spontaneous production of calcium sparks that may merge to form saltatory waves. Illustrations of spreading circular waves, spirals and more irregular waves are presented. Furthermore, receptor noise is shown to generate a form of array enhanced coherence resonance whereby all calcium stores release periodically and simultaneously.     

The Effect of Loss of Immunity on Noise-Induced Sustained Oscillations in Epidemics

The effect of loss of immunity on sustained population oscillations about an endemic equilibrium is studied via a multiple scales analysis of a SIRS model. The analysis captures the key elements supporting the nearly regular oscillations of the infected and susceptible populations, namely, the interaction of the deterministic and stochastic dynamics together with the separation of time scales of the damping and the period of these oscillations. The derivation of a nonlinear stochastic amplitude equation describing the envelope of the oscillations yields two criteria providing explicit parameter ranges where they can be observed. These conditions are similar to those found for other applications in the context of coherence resonance, in which noise drives nearly regular oscillations in a system that is quiescent without noise. In this context the criteria indicate how loss of immunity and other factors can lead to a significant increase in the parameter range for prevalence of the sustained oscillations, without any external driving forces. Comparison of the power spectral densities of the full model and the approximation confirms that the multiple scales analysis captures nonlinear features of the oscillations.     

Stochastic traits of molecular evolution--acceptance of point mutations in native actin genes

A stochastic matrix of nucleotide mutation probabilities is derived by counting differences and identities in alignments of native actin genes, with the aim of obtaining a more reliable data base for regular modes of molecular evolution. The evolution of DNA sequences is thereby considered as a Markov process consisting of events (point mutations) characterized by a stochastic matrix for codon-codon interchanges. The genetic distance is set to 1 PAM (percentage of accepted point mutations). The results can be reproduced by Monte Carlo simulations which are subjected to selective constraints. The latter are observed as nonrandom codon usage and ratios of silent to recognizable point mutations. Specific patterns within the matrix of mutation probabilities attest to preferences of natural selection in the evolution of a specific protein.     

Visualization of stochastic Ca2+ signals in the formed somites during the early segmentation period in intact,normally developing zebrafish embryos

Localized Ca²⁺ signals were consistently visualized in the formed somites of intact zebrafish embryos during the early segmentation period. Unlike the regular process of somitogenesis, these signals were stochastic in nature with respect to time and location. They did, however, occur predominantly at the medial and lateral boundaries within the formed somites. Embryos were treated with modulators of [Ca²⁺]_i to explore the signal generation mechanism and possible developmental function of the stochastic transients. Blocking elements in the phosphoinositol pathway eliminated the stochastic signals but had no obvious effect, stochastic or otherwise, on the formed somites. Such treatments did, however, result in the subsequently formed somites being longer in the mediolateral dimension. Targeted uncaging of buffer (diazo‐2) or Ca²⁺ (NP‐ethyleneglycoltetraacetic acid [EGTA]) in the presomitic mesoderm, resulted in a regular mediolateral lengthening and shortening, respectively, of subsequently formed somites. These data suggest a requirement for IP₃ receptor‐mediated Ca²⁺ release during convergence cell movements in the presomitic mesoderm, which appears to have a distinct function from that of the IP₃ receptor‐mediated stochastic Ca²⁺ signaling in the formed somites.     

On the comparative static properties of the expected population density in the presence of stochastic fluctuations

We consider the impact of increased stochastic fluctuations on the expected density of an unstructured population evolving according to a regular diffusion process subject to a concave expected growth rate. By relying on the flow nature of the solutions of stochastic differential equations and Girsanovs theorem, we demonstrate that typically increased volatility decreases the expected future population density. As a consequence, we are able to characterize the sensitivity of the expected population density with respect to changes in the diffusion coefficient measuring the size of the stochastic fluctuations. We provide both qualitative and quantitative information about the consequences of a mis-specified volatility structure and, especially, of a deterministic approximation to stochastic population growth. We also consider the effect of uncertainty in the initial density and demonstrate that the sign of the relationship between the expected population density and initial uncertainty is unambiguosly negative. Received: 15 February 1999 / Revised version: 29 September 1999 / Published online: 5 May 2000     

Interaction of microwave radiation undergoing stochastic phase jumps with plasmas or gases

New types of beam-plasma devices generating intense stochastic microwave radiation in the interaction of electron beams with hybrid plasma waveguides were developed and put into operation at the National Science Center Kharkov Institute of Physics and Technology (Ukraine). The objective of the paper is to discuss the results of theoretical and experimental studies and numerical simulations of the normal and oblique incidence of linearly polarized electromagnetic waves on an interface between a vacuum and an overcritical plasma. The main results of the reported investigations are as follows: (i) for the parameter values under analysis, the transmission coefficient for microwaves with a stochastically jumping phase is one order of magnitude greater than that for a broadband regular electromagnetic wave with the same spectral density; (ii) the electrons are heated most efficiently by obliquely incident waves with a stochastically jumping phase and, in addition, the electron distribution function has a high-energy tail; and (iii) necessary conditions for gas breakdown and for the initiation of a microwave discharge in stochastic fields in a light source are determined. The anomalously large transmission coefficient for microwaves, the anomalous character of the breakdown conditions, the anomalous behavior of microwave gas discharges, and the anomalous nature of collisionless electron heating, are attributed to stochastic jumps in the phase of microwave radiation.     

Regularity and Synchrony in Motor Proteins

We investigate the origin of the regularity and synchrony which have been observed in numerical experiments of two realistic models of molecular motors, namely Fisher–Kolomeisky’s model of myosin V for vesicle transport in cells and Duke’s model of myosin II for sarcomere shortening in muscle contraction. We show that there is a generic organizing principle behind the emergence of regular gait for a motor pulling a large cargo and synchrony of action of many motors pulling a single cargo. These results are surprising in that the models used are inherently stochastic, and yet they display regular behaviors in the parameter range relevant to experiments. Our results also show that these behaviors are not tied to the particular models used in these experiments, but rather are generic to a wide class of motor protein models.     

An ecological framework linking scales across space and time based on self-thinning

Scaling up from measurements made at small spatial and short temporal scales is a central challenge in the ecological and related sciences, where predictions at larger scales and over long time periods are required. It involves two quite distinct aspects: a formulation of a theoretical framework for calculating space-time averages, and an acquisition of data to support that framework. In this paper, we address the theoretical part of the question, and although our primary motivation was an understanding of carbon accounting our formulation is more general. To that end, we adopt a dynamical systems approach, and incorporate a new dynamical formulation of self-thinning. We show how to calculate rates of change for total (and average) plant dry mass, volume, and carbon, in terms of the properties of the individual plants. The results emphasize how local scale statistics (such as, variation in the size of individuals) lead to nonlinear variation at larger scales. Further, we describe how regular and stochastic disturbance can be readily incorporated into this framework. It is shown that stochastic disturbance at patch-scales, results in (to first approximation) regular disturbance at ecosystem scales, and hence can be formulated as such. We conclude that a dynamical formulation of self-thinning can be used as a generic framework for scaling ecological processes in space and time.     

Intrinsic coherence resonance in excitable membrane patches

The influence of intrinsic channel noise on the spiking activity of excitable membrane patches is studied by use of a stochastic generalization of the Hodgkin-Huxley model. Internal noise stemming from the stochastic dynamics of individual ion channels does affect the electric properties of the cell-membrane patches. There exists an optimal size of the membrane patch for which the internal noise alone can cause a nearly regular spontaneous generation of action potentials. We consider the influence of intrinsic channel noise in presence of a constant and an oscillatory current driving for both, the mean interspike interval and the phenomenon of coherence resonance for neuronal spiking. Given small membrane patches, implying that channel noise dominates the excitable dynamics, we find the phenomenon of intrinsic coherence resonance. In this case, the relatively regular spiking behavior becomes essentially independent of an applied stimulus. We observed, however, the occurrence of a skipping of supra-threshold input events due to channel noise for intermediate patch sizes. This effect consequently reduces the overall coherence of the spiking.     

Emergence of organized bursting in clusters of pancreatic beta-cells by channel sharing.

Pancreatic beta-cells in an intact Islet of Langerhans exhibit bursting electrical behavior. The Chay-Keizer model describes this using a calcium-activated potassium (K-Ca) channel, but cannot account for the irregular spiking of isolated beta-cells. Atwater I., L. Rosario, and E. Rojas, Cell Calcium. 4:451-461, proposed that the K-Ca channels, which are rarely open, are shared by several cells. This suggests that the chaotic behavior of isolated cells is stochastic. We have revised the Chay-Keizer model to incorporate voltage clamp data of Rorsman and Trube and extended it to include stochastic K-Ca channels. This model can describe the behavior of single cells, as well as that of clusters of cells tightly coupled by gap junctions. As the size of the clusters is increased, the electrical activity shows a transition from chaotic spiking to regular bursting. Although the model of coupling is over-simplified, the simulations lend support to the hypothesis that bursting is the result of channel sharing.     

A stochastic attractor participates in chymotrypsin catalysis. A new facet of enzyme catalysis

The previously observed discrete levels of vibrational parameters of chymotrypsin and tosyl-chymotrypsin were analyzed by the methods of non-linear dynamics in order to investigate their origin. The fractal dimensionality of the step sequence was determined using the correlation function and the Farey-tree method. These methods yielded the same value and indicated the presence of a "Devil's Staircase", i.e. the existence of a stochastic attractor. The latter was assigned to the substrate-mobilizable conformation. The attractor dimension of the catalytic site in the acylated state was found to be indistinguishable from that of the substrate mobilizable conformation. A "Devil's Staircase" signals the transition from the regular to the stochastic regime. This transition is shared by critical phenomena and may be a prerequisite for catalysis.     

A new approach for the assessment of stochastic variation: analysis of behavioural response in blue mussel (Mytilus edulis L.)

One of the most effective techniques for evaluating stress is the analysis of developmental stability, measured by stochastic variation based particularly on fluctuating asymmetry, i.e. a variance in random deviations from perfect bilateral symmetry. However, the application of morphological methods is only possible when an organism lives under testing conditions during a significant part of its ontogenesis. Contrary to morphological characters, behavior can change very fast. Consequently, methods based on behavioural characters may have advantages over more traditional approaches. In this study we describe the technique of assessing stochastic variation, using not morphological, but behavioural characters. To measure stochastic variation of behavioural response, we assessed the stability of the isolation reaction of blue musselMytilus edulis at regular changes of salinity. With increasing temperature from +12°C to +20°C stochastic variation of the isolation reaction increased, which is a common response to change of environmental conditions. In this way, we have developed a method of assessing stochastic variation of behavioural response in molluscs. This method may find a great range of applications, because its usage does not require keeping animals in tested conditions for a long time.     

Temporal clustering of tropical cyclones on the Great Barrier Reef and its ecological importance

Tropical cyclones have been a major cause of reef coral decline during recent decades, including on the Great Barrier Reef (GBR). While cyclones are a natural element of the disturbance regime of coral reefs, the role of temporal clustering has previously been overlooked. Here, we examine the consequences of different types of cyclone temporal distributions (clustered, stochastic or regular) on reef ecosystems. We subdivided the GBR into 14 adjoining regions, each spanning roughly 300 km, and quantified both the rate and clustering of cyclones using dispersion statistics. To interpret the consequences of such cyclone variability for coral reef health, we used a model of observed coral population dynamics. Results showed that clustering occurs on the margins of the cyclone belt, being strongest in the southern reefs and the far northern GBR, which also has the lowest cyclone rate. In the central GBR, where rates were greatest, cyclones had a relatively regular temporal pattern. Modelled dynamics of the dominant coral genus, Acropora, suggest that the long-term average cover might be more than 13 % greater (in absolute cover units) under a clustered cyclone regime compared to stochastic or regular regimes. Thus, not only does cyclone clustering vary significantly along the GBR but such clustering is predicted to have a marked, and management-relevant, impact on the status of coral populations. Additionally, we use our regional clustering and rate results to sample from a library of over 7000 synthetic cyclone tracks for the GBR. This allowed us to provide robust reef-scale maps of annual cyclone frequency and cyclone impacts on Acropora. We conclude that assessments of coral reef vulnerability need to account for both spatial and temporal cyclone distributions.

     

The probability that beneficial mutations are lost in populations with periodic bottlenecks

Population bottlenecks affect the dynamics of evolution, increasing the probability that beneficial mutations will be lost. Recent protocols for the experimental study of evolution involve repeated bottlenecks-when fresh media are inoculated during serial transfer or when chemostat tubes are changed. Unlike population reductions caused by stochastic environmental factors, these bottlenecks occur at known, regular intervals and with a fixed dilution ratio. We derive the ultimate probability of extinction for a beneficial mutation in a periodically bottlenecked population, using both discrete and continuous approaches. We show that both approaches yield the same approximation for extinction probability. From this, we derive an approximate expression for an effective population size.     

The stochastic resonance of magnetosomes fixed in the cytoskeleton

The rotation of microscopic magnetic particles, magnetosomes, embedded into the cytoskeleton and subjected to a magnetic field and thermal noise was considered. The dynamics of magnetosome is shown to comply with the conditions of the stochastic resonance under not too tight constraints on the character of particle fastening. The excursion of regular rotations attains the value of the order of radian, which facilitates explaining the biological effects of low-frequency weak magnetic fields and geomagnetic fluctuations.     

Stochastic differential equation model for cerebellar granule cell excitability

Neurons in the brain express intrinsic dynamic behavior which is known to be stochastic in nature. A crucial question in building models of neuronal excitability is how to be able to mimic the dynamic behavior of the biological counterpart accurately and how to perform simulations in the fastest possible way. The well-established Hodgkin-Huxley formalism has formed to a large extent the basis for building biophysically and anatomically detailed models of neurons. However, the deterministic Hodgkin-Huxley formalism does not take into account the stochastic behavior of voltage-dependent ion channels. Ion channel stochasticity is shown to be important in adjusting the transmembrane voltage dynamics at or close to the threshold of action potential firing, at the very least in small neurons. In order to achieve a better understanding of the dynamic behavior of a neuron, a new modeling and simulation approach based on stochastic differential equations and Brownian motion is developed. The basis of the work is a deterministic one-compartmental multi-conductance model of the cerebellar granule cell. This model includes six different types of voltage-dependent conductances described by Hodgkin-Huxley formalism and simple calcium dynamics. A new model for the granule cell is developed by incorporating stochasticity inherently present in the ion channel function into the gating variables of conductances. With the new stochastic model, the irregular electrophysiological activity of an in vitro granule cell is reproduced accurately, with the same parameter values for which the membrane potential of the original deterministic model exhibits regular behavior. The irregular electrophysiological activity includes experimentally observed random subthreshold oscillations, occasional spontaneous spikes, and clusters of action potentials. As a conclusion, the new stochastic differential equation model of the cerebellar granule cell excitability is found to expand the range of dynamics in comparison to the original deterministic model. Inclusion of stochastic elements in the operation of voltage-dependent conductances should thus be emphasized more in modeling the dynamic behavior of small neurons. Furthermore, the presented approach is valuable in providing faster computation times compared to the Markov chain type of modeling approaches and more sophisticated theoretical analysis tools compared to previously presented stochastic modeling approaches.     

Stochastic simulation of processive and oscillatory sliding using a two-headed model for axonemal dynein

Computer simulations have been used in an attempt to understand experimental observations of processive and oscillatory sliding by one or a few axonemal dyneins. A simple two-headed model has been examined using stochastic simulation methods. To explain the experimental observations, the model must be capable of taking backward steps, as well as forward steps, and there must be hysteresis in switching between forward or backward stepping. When the effects of Brownian movement on motor strain are included, it is not possible to obtain oscillations as regular as the experimental records by using motor strain to regulate switching between forward or backward stepping.