Noise-Induced Coherence and Network Oscillations in a Reduced Bursting Model

The dynamics of the Hindmarsh-Rose (HR) model of bursting thalamic neurons is reduced to a system of two linear differential equations that retains the subthreshold resonance properties of the HR model. Introducing a reset mechanism after a threshold crossing, we turn this system into a resonant integrate-and-fire (RIF) model. Using Monte-Carlo simulations and mathematical analysis, we examine the effects of noise and the subthreshold dynamic properties of the RIF model on the occurrence of coherence resonance (CR). Synchronized burst firing occurs in a network of such model neurons with excitatory pulse-coupling. The coherence level of the network oscillations shows a stochastic resonance-like dependence on the noise level. Stochastic analysis of the equations shows that the slow recovery from the spike-induced inhibition is crucial in determining the frequencies of the CR and the subthreshold resonance in the original HR model. In this particular type of CR, the oscillation frequency strongly depends on the intrinsic time scales but changes little with the noise intensity. We give analytical quantities to describe this CR mechanism and illustrate its influence on the emerging network oscillations. We discuss the profound physiological roles this kind of CR may have in information processing in neurons possessing a subthreshold resonant frequency and in generating synchronized network oscillations with a frequency that is determined by intrinsic properties of the neurons. PACS 05.45.-a, 05.40.Ca, 87.18.Sn, 87.19     

Stochastic models for circadian rhythms: effect of molecular noise on periodic and chaotic behaviour

Circadian rhythms are endogenous oscillations that occur with a period close to 24 h in nearly all living organisms. These rhythms originate from the negative autoregulation of gene expression. Deterministic models based on such genetic regulatory processes account for the occurrence of circadian rhythms in constant environmental conditions (e.g., constant darkness), for entrainment of these rhythms by light-dark cycles, and for their phase-shifting by light pulses. When the numbers of protein and mRNA molecules involved in the oscillations are small, as may occur in cellular conditions, it becomes necessary to resort to stochastic simulations to assess the influence of molecular noise on circadian oscillations. We address the effect of molecular noise by considering the stochastic version of a deterministic model previously proposed for circadian oscillations of the PER and TIM proteins and their mRNAs in Drosophila. The model is based on repression of the per and tim genes by a complex between the PER and TIM proteins. Numerical simulations of the stochastic version of the model are performed by means of the Gillespie method. The predictions of the stochastic approach compare well with those of the deterministic model with respect both to sustained oscillations of the limit cycle type and to the influence of the proximity from a bifurcation point beyond which the system evolves to stable steady state. Stochastic simulations indicate that robust circadian oscillations can emerge at the cellular level even when the maximum numbers of mRNA and protein molecules involved in the oscillations are of the order of only a few tens or hundreds. The stochastic model also reproduces the evolution to a strange attractor in conditions where the deterministic PER-TIM model admits chaotic behaviour. The difference between periodic oscillations of the limit cycle type and aperiodic oscillations (i.e. chaos) persists in the presence of molecular noise, as shown by means of Poincaré sections. The progressive obliteration of periodicity observed as the number of molecules decreases can thus be distinguished from the aperiodicity originating from chaotic dynamics. As long as the numbers of molecules involved in the oscillations remain sufficiently large (of the order of a few tens or hundreds, or more), stochastic models therefore provide good agreement with the predictions of the deterministic model for circadian rhythms.     

System-size resonance for intracellular and intercellular calcium signaling

Dynamical behaviors of unidirectionally, linearly coupled as well as isolated calcium subsystems are investigated by taking into account the internal noise resulting from finite system size and thus small numbers of interacting molecules. For an isolated calcium system, the internal noise can induce stochastic oscillations for a steady state close to the Hopf-bifurcation point, and the regularity of those stochastic oscillations depends resonantly on the system size, exhibiting system-size resonance. For the coupled system consisting of two subsystems, the system-size resonance effect observed in the subsystem subject to coupling is significantly amplified due to the nontrivial effects of coupling.     

Simulation of between Repeat Variability in Real Time PCR Reactions

While many decisions rely on real time quantitative PCR (qPCR) analysis few attempts have hitherto been made to quantify bounds of precision accounting for the various sources of variation involved in the measurement process. Besides influences of more obvious factors such as camera noise and pipetting variation, changing efficiencies within and between reactions affect PCR results to a degree which is not fully recognized. Here, we develop a statistical framework that models measurement error and other sources of variation as they contribute to fluorescence observations during the amplification process and to derived parameter estimates. Evaluation of reproducibility is then based on simulations capable of generating realistic variation patterns. To this end, we start from a relatively simple statistical model for the evolution of efficiency in a single PCR reaction and introduce additional error components, one at a time, to arrive at stochastic data generation capable of simulating the variation patterns witnessed in repeated reactions (technical repeats). Most of the variation in values was adequately captured by the statistical model in terms of foreseen components. To recreate the dispersion of the repeats'' plateau levels while keeping the other aspects of the PCR curves within realistic bounds, additional sources of reagent consumption (side reactions) enter into the model. Once an adequate data generating model is available, simulations can serve to evaluate various aspects of PCR under the assumptions of the model and beyond.     

Constructive effects of environmental noise in an excitable prey–predator plankton system with infected prey

An excitable model of fast phytoplankton and slow zooplankton dynamics is considered for the case of lysogenic viral infection of the phytoplankton population. The phytoplankton population is split into a susceptible (S) and an infected (I) part. Both parts grow logistically, limited by a common carrying capacity. Zooplankton (Z) is grazing on susceptibles and infected, following a Holling-type III functional response. The local analysis of the S–I–Z differential equations yields a number of stationary and/or oscillatory regimes and their combinations. Correspondingly interesting is the behaviour under multiplicative noise, modelled by stochastic differential equations. The external noise can enhance the survival of susceptibles and infected, respectively, that would go extinct in a deterministic environment. In the parameter range of excitability, noise can induce prey–predator oscillations and coherence resonance (CR). In the spatially extended case, synchronized global oscillations can be observed for medium noise intensities. Higher values of noise give rise to the formation of stationary spatial patterns.     

Nutrients and toxin producing phytoplankton control algal blooms — a spatio-temporal study in a noisy environment

A phytoplankton-zooplankton prey-predator model has been investigated for temporal, spatial and spatio-temporal dissipative pattern formation in a deterministic and noisy environment, respectively. The overall carrying capacity for the phytoplankton population depends on the nutrient level. The role of nutrient concentrations and toxin producing phytoplankton for controlling the algal blooms has been discussed. The local analysis yields a number of stationary and/or oscillatory regimes and their combinations. Correspondingly interesting is the spatio-temporal behaviour, modelled by stochastic reaction-diffusion equations. The present study also reveals the fact that the rate of toxin production by toxin producing phytoplankton (TPP) plays an important role for controlling oscillations in the plankton system. We also observe that different mortality functions of zooplankton due to TPP have significant influence in controlling oscillations, coexistence, survival or extinction of the zoo-plankton population. External noise can enhance the survival and spread of zooplankton that would go extinct in the deterministic system due to a high rate of toxin production.     

Stochastic models for inferring genetic regulation from microarray gene expression data

Microarray expression profiles are inherently noisy and many different sources of variation exist in microarray experiments. It is still a significant challenge to develop stochastic models to realize noise in microarray expression profiles, which has profound influence on the reverse engineering of genetic regulation. Using the target genes of the tumour suppressor gene p53 as the test problem, we developed stochastic differential equation models and established the relationship between the noise strength of stochastic models and parameters of an error model for describing the distribution of the microarray measurements. Numerical results indicate that the simulated variance from stochastic models with a stochastic degradation process can be represented by a monomial in terms of the hybridization intensity and the order of the monomial depends on the type of stochastic process. The developed stochastic models with multiple stochastic processes generated simulations whose variance is consistent with the prediction of the error model. This work also established a general method to develop stochastic models from experimental information.     

A model of stochastic variations of postsynaptic activity

Stochastic oscillations imitating postsynaptic activity in the excitatory neurons are produced by a nonlinear difference equation which does not contain any sources of noise. The given back inhibition via inhibitory interneurons presents a negative feedback loop due to which oscillations in the model system are realized. By means of variation of parameters of the system the patterns of stochastic oscillations can be changed in wide range of physiologically meaningful patterns of the neuronal activity.     

Deterministic Versus Stochastic Models for Circadian Rhythms

Circadian rhythms which occur with a period close to 24 h in nearly all living organisms originate from the negative autoregulation of gene expression.Deterministic models based on genetic regulatory processes account for theoccurrence of circadian rhythms in constant environmental conditions (e.g.constant darkness), for entrainment of these rhythms by light-dark cycles, and for their phase-shifting by light pulses. At low numbers of protein and mRNA molecules, it becomes necessary to resort to stochastic simulations to assess the influence of molecular noise on circadian oscillations. We address the effect of molecular noise by considering two stochastic versions of a core model for circadian rhythms. The deterministic version of this core modelwas previously proposed for circadian oscillations of the PER protein in Drosophila and of the FRQ protein in Neurospora. In the first, non-developed version of the stochastic model, we introduce molecular noise without decomposing the deterministic mechanism into detailed reaction steps while in the second, developed version we carry out such a detailed decomposition. Numerical simulations of the two stochastic versions of the model are performed by means of the Gillespie method. We compare the predictions of the deterministic approach with those of the two stochastic models, with respect both to sustained oscillations of the limit cycle type and to the influence of the proximity of a bifurcation point beyond which the system evolves to a stable steady state. The results indicate that robust circadian oscillations can occur even when the numbers of mRNA and nuclear protein involved in the oscillatory mechanism are reduced to a few tens orhundreds, respectively. The non-developed and developed versions of the stochastic model yield largely similar results and provide good agreement with the predictions of the deterministic model for circadian rhythms.     

Stochastic Dynamic Model for Estimation of Rate Constants and Their Variances from Noisy and Heterogeneous PET Measurements

Tissue heterogeneity, radioactive decay and measurement noise are the main error sources in compartmental modeling used to estimate the physiologic rate constants of various radiopharmaceuticals from a dynamic PET study. We introduce a new approach to this problem by modeling the tissue heterogeneity with random rate constants in compartment models. In addition, the Poisson nature of the radioactive decay is included as a Poisson random variable in the measurement equations. The estimation problem will be carried out using the maximum likelihood estimation. With this approach, we do not only get accurate mean estimates for the rate constants, but also estimates for tissue heterogeneity within the region of interest and other possibly unknown model parameters, e.g. instrument noise variance, as well. We also avoid the problem of the optimal weighting of the data related to the conventionally used weighted least-squares method. The new approach was tested with simulated time–activity curves from the conventional three compartment – three rate constants model with normally distributed rate constants and with a noise mixture of Poisson and normally distributed random variables. Our simulation results showed that this new model gave accurate estimates for the mean of the rate constants, the measurement noise parameter and also for the tissue heterogeneity, i.e. for the variance of the rate constants within the region of interest.     

Stochastic vs. deterministic uptake of dodecanedioic acid by isolated rat livers

Deterministic and stochastic differential equations models of the uptake of dodecanedioic acid (C12) are fitted to experimental data obtained on nine isolated, perfused rat livers. 11500 μg of C12 were injected as a bolus into the perfusing liver solution. The concentrations of C12 in perfusate samples taken over 2 h from the beginning of the experiments were analyzed by High Performance Liquid Chromatography (HPLC). A two-compartment deterministic model is studied. To include spontaneous erratic variations in the metabolic processes the parameter for the uptake rate is randomized to obtain a stochastic differential equations model. Parameters are estimated in a two-step procedure: first, parameters in the drift part are estimated by least squares; then, the diffusion parameter is estimated using Monte-Carlo simulations to approximate the unknown likelihood function. Parameter estimation is carried out over a wide range of reasonable measurement error variances to check robustness of estimates. It is concluded that the kinetics of dodecanedioic acid, in the experimental conditions discussed, is well approximated by a model including spontaneous erratic variations in the liver uptake rate.     

Thermodynamic and kinetic studies of a stoichiometric model of energetic metabolism under starvation conditions

Abstract A stoichiometric model of anaerobic glycolysis is presented and the influence on its dynamics by the ATP-consuming membrane transport processes and substrate input rate are studied. The model is represented by a system of four ODE (ordinary differential equations), mass conservation equations and functions of state variables, such as thermodynamic efficiency. A low substrate input rate provokes damped oscillations while a high enrgy load determines sustained oscillations in all the metabolites and in thermodynamic efficiency. Due to the lack of linearity between fluxes and forces in the oscillatory region it may be stated that oscillations appear when the system is kinetically controlled.     

Signal transduction and amplification in a circadian oscillator: Interaction between two colored noises

The signal transduction and amplification in a Neurospora circadian clock system is studied by using the mechanism of internal signal stochastic resonance (ISSR). Two cases have been investigated: the case of no correlations between multiplicative and additive colored noises and the case of correlations between two noises. The results show that, in both cases, the noise-induced circadian oscillations can be transduced with the phenomenon of internal signal stochastic resonance (ISSR). However, the correlation time and intensity of an additive colored noise play different roles for the ISSR, driven by multiplicative colored noise, while the correlation time and intensity of multiplicative colored noise hardly influence the ISSR driven by additive colored noise. In addition, the ISSR can be amplified or suppressed at an appropriate range of the correlation intensity between two colored noises. The fundamental frequency of noise-induced circadian oscillations is hardly shifted with the increment of the intensity and correlation time of colored noises, which implies that the Neurospora system could be resistant to colored noises, exhibit strong vitality and sustain intrinsic circadian rhythms.     

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.     

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 withsuperimposed 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.     

Fisher information as a metric of locally optimal processing and stochastic resonance

The origins of Fisher information are in its use as a performance measure for parametric estimation. We augment this and show that the Fisher information can characterize the performance in several other significant signal processing operations. For processing of a weak signal in additive white noise, we demonstrate that the Fisher information determines (i) the maximum output signal-to-noise ratio for a periodic signal; (ii) the optimum asymptotic efficacy for signal detection; (iii) the best cross-correlation coefficient for signal transmission; and (iv) the minimum mean square error of an unbiased estimator. This unifying picture, via inequalities on the Fisher information, is used to establish conditions where improvement by noise through stochastic resonance is feasible or not.     

Adjusting phenotypes by noise control

Genetically identical cells can show phenotypic variability. This is often caused by stochastic events that originate from randomness in biochemical processes involving in gene expression and other extrinsic cellular processes. From an engineering perspective, there have been efforts focused on theory and experiments to control noise levels by perturbing and replacing gene network components. However, systematic methods for noise control are lacking mainly due to the intractable mathematical structure of noise propagation through reaction networks. Here, we provide a numerical analysis method by quantifying the parametric sensitivity of noise characteristics at the level of the linear noise approximation. Our analysis is readily applicable to various types of noise control and to different types of system; for example, we can orthogonally control the mean and noise levels and can control system dynamics such as noisy oscillations. As an illustration we applied our method to HIV and yeast gene expression systems and metabolic networks. The oscillatory signal control was applied to p53 oscillations from DNA damage. Furthermore, we showed that the efficiency of orthogonal control can be enhanced by applying extrinsic noise and feedback. Our noise control analysis can be applied to any stochastic model belonging to continuous time Markovian systems such as biological and chemical reaction systems, and even computer and social networks. We anticipate the proposed analysis to be a useful tool for designing and controlling synthetic gene networks.     

Process and measurement errors of population size: their mutual effects on precision and bias of estimates for demographic parameters

Knowing the parameters of population growth and regulation is fundamental for answering many ecological questions and the successful implementation of conservation strategies. Moreover, detecting a population trend is often a legal obligation. Yet, inherent process and measurement errors aggravate the ability to estimate these parameters from population time-series. We use numerical simulations to explore how the lengths of the time-series, process and measurement error influence estimates of demographic parameters. We first generate time-series of population sizes with given demographic parameters for density-dependent stochastic population growth, but assume that these population sizes are estimated with measurement errors. We then fit parameters for population growth, habitat capacity, total error and long-term trends to the ‘measured’ time-series data using non-linear regression. The length of the time-series and measurement error introduce a substantial bias in the estimates for population growth rate and to a lesser degree on estimates for habitat capacity, while process error has little effect on parameter bias. The total error term of the statistical model is dominated by process error as long as the latter is larger than the measurement error. A decline in population size is difficult to document as soon as either error becomes moderate, trends are not very pronounced, and time-series are short (<10–15 seasons). Detecting an annual decline of 1% within 6-year reporting periods, as required for the European Union for the species of Community Interest, appears unachievable.