Noise enhanced hormonal signal transduction through intracellular calcium oscillations.

In a wide range of non-linear dynamical systems, noise may enhance the detection of weak deterministic input signals. Here, we demonstrate this phenomenon for transmembrane signaling in a hormonal model system of intracellular Ca(2+) oscillations. Adding Gaussian noise to a subthreshold extracellular pulsatile stimulus increased the sensitivity in the dose-response relation of the Ca(2+) oscillations compared to the same noise signal added as a constant mean level. These findings may have important physiological consequences for the operation of hormonal and other physiological signal transduction systems close to the threshold level.     

Analysis of neuronal networks in the visual system of the cat using statistical signals — Simple and complex cells

Superimposing additively a two-dimensional noise process to deterministic input signals (bars) the neurons of area 17 show a class-specific reaction for the task of signal extraction. Moving both parts of the signals simultaneously and varying the signal to noise ratio (S/N) the simple cells achieve the same performance as resulted from the psychophysical experiment. Type I complex cells extract moving deterministic signals (i.e. bars) from the stationary noise, whereas in the answers of Type II complex cells the statistical parts of the signals predominate. Considering the different cell types each as a series of a linear and a nonlinear system one obtains the cell specific space-time frequency and the amplitude characteristics.This work was supported by DFG Grant Ho 450/6 and Grant Se 251/9     

Information coding by ensembles of resonant neurons

In the present paper, we propose a novel neural procedure for signal processing and coding based on the subthreshold oscillations and resonance of the neural membrane potential that could be used by real neurons to perform frequency spectra analysis and information coding of incoming signals. Taking into account the biophysical properties of the neural membranes, we note that the subthreshold resonant behaviour they exhibit can be used to analyse incoming signals and represent them in the frequency domain. We study the reliability of the representation of signals depending on the biophysical parameters of the neurons, the fault-tolerance of this coding scheme and its robustness against noise and in the presence of spikes. The principal characteristics of our system are the use of the physical phenomenon of neural resonance (rarely considered in the literature for signal coding); it fits well with the biophysical parameters of most neurons that exhibit subthreshold oscillations; it is compatible with experimental data; and it can be easily integrated in a more general model of information processing and coding that includes communication between neurons based on spikes.     

Internal noise-driven circadian oscillator in Drosophila

An internal noise-driven oscillator was studied in a two-variable Drosophila model, where both positive feedback and negative feedback are crucial to the circadian oscillations. It is shown that internal noise could sustain reliable oscillations for the parameter which produces a stable steady state in the deterministic system. The noise-sustained oscillations are interpreted by using phase plane analysis. The period of such oscillations fluctuates slightly around the period of deterministic oscillations and the coherence of oscillations becomes the best at an optimal internal noise intensity, indicating the occurrence of intrinsic coherence resonance. In addition, in the oscillatory region, the coherence of noisy circadian oscillations is suppressed by the internal noise, but the period is hardly affected, demonstrating the robustness of the Drosophila model for circadian rhythms to the intrinsic noise.     

Internal noise-sustained circadian rhythms in a Drosophila model

Circadian rhythmic processes, mainly regulated by gene expression at the molecular level, have inherent stochasticity. Their robustness or resistance to internal noise has been extensively investigated by most of the previous studies. This work focuses on the constructive roles of internal noise in a reduced Drosophila model, which incorporates negative and positive feedback loops, each with a time delay. It is shown that internal noise sustains reliable oscillations with periods close to 24 h in a region of parameter space, where the deterministic kinetics would evolve to a stable steady state. The amplitudes of noise-sustained oscillations are significantly affected by the variation of internal noise level, and the best performance of the oscillations could be found at an optimal noise intensity, indicating the occurrence of intrinsic coherence resonance. In the oscillatory region of the deterministic model, the coherence of noisy circadian oscillations is suppressed by internal noise, while the period remains nearly constant over a large range of noise intensity, demonstrating robustness of the Drosophila model for circadian rhythms to intrinsic noise. In addition, the effects of time delay in the positive feedback on the oscillations are also investigated. It is found that the time delay could efficiently tune the performance of the noise-sustained oscillations. These results might aid understanding of the exploitation of intracellular noise in biochemical and genetic regulatory systems.     

A versatile data acquisition system for physiological modelling of laboratory fermentation processes

Summary A versatile data acquisition system for physiological modelling of laboratory fermentation processes is given. This system meets the stringent qualifications of modern modelling techniques in terms of signal to noise ratios and sampling rates using readily available equipment. The data are portable to any data processing system for modelling and analysing the culture characteristics. Two completely equipped fermentation systems are sampled with a sampling period of six seconds. The data acquisition system is used to acquire data of a set of continuous fermenters showing sustained oscillations. Oscillation data was used to show the performance of the data acquisition system. With the high sampling rate signal to noise ratios of 30 dB or higher are achieved. It was possible to determine periods of oscillatory behaviour and delay times. The system has shown to be versatile in alaboratory set up. Addition or subtraction of sensors or complete fermentation systems can be done without major alterations to the system.     

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.     

Nonlinearity in normal human EEG: cycles,temporal asymmetry,nonstationarity and randomness,not chaos

Two-hour vigilance and sleep electroencephalogram (EEG) recordings from five healthy volunteers were analyzed using a method for identifying nonlinearity and chaos which combines the redundancy–linear redundancy approach with the surrogate data technique. A nonlinear component in the EEG was detected, however, inconsistent with the hypothesis of low-dimensional chaos. A possibility that a temporally asymmetric process may underlie or influence the EEG dynamics was indicated. A process that merges nonstationary nonlinear deterministic oscillations with randomness is proposed for an explanation of observed properties of the analyzed EEG signals. Taking these results into consideration, the use of dimensional and related chaos-based algorithms in quantitative EEG analysis is critically discussed. Received: 25 September 1994 / Accepted in revised form: 10 July 1996     

Insights into the behaviour of systems biology models from dynamic sensitivity and identifiability analysis: a case study of an NF-kappaB signalling pathway

Mathematical modelling offers a variety of useful techniques to help in understanding the intrinsic behaviour of complex signal transduction networks. From the system engineering point of view, the dynamics of metabolic and signal transduction models can always be described by nonlinear ordinary differential equations (ODEs) following mass balance principles. Based on the state-space formulation, many methods from the area of automatic control can conveniently be applied to the modelling, analysis and design of cell networks. In the present study, dynamic sensitivity analysis is performed on a model of the IkappaB-NF-kappaB signal pathway system. Univariate analysis of the Euclidean-form overall sensitivities shows that only 8 out of the 64 parameters in the model have major influence on the nuclear NF-kappaB oscillations. The sensitivity matrix is then used to address correlation analysis, identifiability assessment and measurement set selection within the framework of least squares estimation and multivariate analysis. It is shown that certain pairs of parameters are exactly or highly correlated to each other in terms of their effects on the measured variables. The experimental design strategy provides guidance on which proteins should best be considered for measurement such that the unknown parameters can be estimated with the best statistical precision. The whole analysis scheme we describe provides efficient parameter estimation techniques for complex cell networks.     

Time-integrated fluorescence cumulant analysis in fluorescence fluctuation spectroscopy

We introduce a new analysis technique for fluorescence fluctuation data. Time-integrated fluorescence cumulant analysis (TIFCA) extracts information from the cumulants of the integrated fluorescence intensity. TIFCA builds on our earlier FCA theory, but in contrast to FCA or photon counting histogram (PCH) analysis is valid for arbitrary sampling times. The motivation for long sampling times lies in the improvement of the signal/noise ratio of the data. Because FCA and PCH theory are not valid in this regime, we first derive a theoretical model of cumulant functions for arbitrary sampling times. TIFCA is the first exact theory that describes the effects of sampling time on fluorescence fluctuation experiments. We calculate factorial cumulants of the photon counts for various sampling times by rebinning of the original data. Fits of the data to models determine the brightness, the occupation number, and the diffusion time of each species. To provide the tools for a rigorous error analysis of TIFCA, expressions for the variance of cumulants are developed and tested. We demonstrate that over a limited range rebinning reduces the relative error of higher order cumulants, and therefore improves the signal/noise ratio. The first four cumulant functions are explicitly calculated and are applied to simple dye systems to test the validity of TIFCA and demonstrate its ability to resolve species.     

Analysis of Time-Varying Biological Data Using Rainflow Cycle Counting

A wide range of biological investigations lead to time-history data. The characterization of such data can be difficult particularly in the presence of signal noise or superimposed signals. Several methods are described which can be brought to bear including FFT, thresholding, peak counting, and range counting. However, each of these approaches has significant disadvantages. In this paper we describe a novel method, known as rainflow cycle counting, for characterizing time varying biological time-history data in terms of spiking or oscillation amplitude and frequency. Rainflow counting is a straightforward algorithm for identifying complete cycles in the data and determining their amplitudes. The approach is simple, reliable, easily lends itself to automation, and robust even in the presence of superimposed signals or background noise. After describing the method, its use and behavior are demonstrated on three sample histories of intracellular calcium concentration in chondrocytes exposed to fluid shear stress. The method is also applied to a more challenging data set that has had an artificial random error included. The results demonstrate that the rainflow counting algorithm identifies signal oscillations and appropriately determines their amplitudes even when superimposed or distorted by background noise. These attractive properties make rainflow counting a powerful approach for quantifying and characterizing biological time histories.     

Stochastic signal processing and transduction in chemotactic response of eukaryotic cells

Single-molecule imaging analysis of chemotactic response in eukaryotic cells has revealed a stochastic nature in the input signals and the signal transduction processes. This leads to a fundamental question about the signaling processes: how does the signaling system operate under stochastic fluctuations or noise? Here, we report a stochastic model of chemotactic signaling in which noise and signal propagation along the transmembrane signaling pathway by chemoattractant receptors can be analyzed quantitatively. The results obtained from this analysis reveal that the second-messenger-production reactions by the receptors generate noisy signals that contain intrinsic noise inherently generated at this reaction and extrinsic noise propagated from the ligand-receptor binding. Such intrinsic and extrinsic noise limits the directional sensing ability of chemotactic cells, which may explain the dependence of chemotactic accuracy on chemical gradients that has been observed experimentally. Our analysis also reveals regulatory mechanisms for signal improvement in the stochastically operating signaling system by analyzing how the SNR of chemotactic signals can be improved on or deteriorated by the stochastic properties of receptors and second-messenger molecules. Theoretical consideration of noisy signal transduction by chemotactic signaling systems can further be applied to signaling systems in general.     

Modeling the euglycemic hyperinsulinemic clamp by stochastic differential equations

The Euglycemic Hyperinsulinemic Clamp (EHC) is the most widely used experimental procedure for the determination of insulin sensitivity. In the present study, 16 subjects with BMI between 18.5 and 63.6 kg/m² have been studied with a long-duration (5 hours) EHC. In order to explain the oscillations of glycemia occurring in response to the hyperinsulinization and to the continuous glucose infusion at varying speeds, we first hypothesized a system of ordinary differential equations (ODEs), with limited success. We then extended the model and represented the experiment using a system of stochastic differential equations (SDEs). The latter allow for distinction between (i) random variation imputable to observation error and (ii) system noise (intrinsic variability of the metabolic system), due to a variety of influences which change over time. The stochastic model of the EHC was fitted to data and the system noise was estimated by means of a (simulated) maximum likelihood procedure, for a series of different hypothetical measurement error values. We showed that, for the whole range of reasonable measurement error values: (i) the system noise estimates are non-negligible; and (ii) these estimates are robust to changes in the likely value of the measurement error. Explicit expression of system noise is physiologically relevant in this case, since glucose uptake rate is known to be affected by a host of additive influences, usually neglected when modeling metabolism. While in some of the studied subjects system noise appeared to only marginally affect the dynamics, in others the system appeared to be driven more by the erratic oscillations in tissue glucose transport rather than by the overall glucose-insulin control system. It is possible that the quantitative relevance of the unexpressed effects (system noise) should be considered in other physiological situations, represented so far only with deterministic models.The work was supported by grants from the Danish Medical Research Council and the Lundbeck Foundation to S. Ditlevsen.     

Visual information processing using bacteriorhodopsin-based complex LB films

Image extraction and visual information processing using bacteriorhodopsin (bR)-based bioelectronic devices is presented. Image extraction was achieved using a photoreceptor consisting of bR and spiropyran films. The undesired signals from the photoreceptor were automatically eliminated from the whole signal by spiropyran films acting as an optical noise filter that increases the target signal to an undesired signal ratio. For the information processing, the photoreceptor consisting of bR and lipid films deposited with different configurations was used and the target signals were processed to achieve the pattern recognition. The pattern recognition was based on not only the response variability of bacteriorhodopsin, induced by different film configurations, but also on the initial learning process. The input patterns were predicted by simple calculation with the known signals through the initial learning process.     

An integrated diabetic index using heart rate variability signal features for diagnosis of diabetes

Electrocardiogram (ECG) signals are difficult to interpret, and clinicians must undertake a long training process to learn to diagnose diabetes from subtle abnormalities in these signals. To facilitate these diagnoses, we have developed a technique based on the heart rate variability signal obtained from ECG signals. This technique uses digital signal processing methods and, therefore, automates the detection of diabetes from ECG signals. In this paper, we describe the signal processing techniques that extract features from heart rate (HR) signals and present an analysis procedure that uses these features to diagnose diabetes. Through statistical analysis, we have identified the correlation dimension, Poincaré geometry properties (SD2), and recurrence plot properties (REC, DET, L _mean) as useful features. These features differentiate the HR data of diabetic patients from those of patients who do not have the illness, and have been validated by using the AdaBoost classifier with the perceptron weak learner (yielding a classification accuracy of 86%). We then developed a novel diabetic integrated index (DII) that is a combination of these nonlinear features. The DII indicates whether a particular HR signal was taken from a person with diabetes. This index aids the automatic detection of diabetes, thereby allowing a more objective assessment and freeing medical professionals for other tasks.     

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.     

Adaptive processing techniques based on hidden Markov models for characterizing very small channel currents buried in noise and deterministic interferences.

Techniques for characterizing very small single-channel currents buried in background noise are described and tested on simulated data to give confidence when applied to real data. Single channel currents are represented as a discrete-time, finite-state, homogeneous, Markov process, and the noise that obscures the signal is assumed to be white and Gaussian. The various signal model parameters, such as the Markov state levels and transition probabilities, are unknown. In addition to white Gaussian noise, the signal can be corrupted by deterministic interferences of known form but unknown parameters, such as the sinusoidal disturbance stemming from AC interference and a drift of the base line owing to a slow development of liquid-junction potentials. To characterize the signal buried in such stochastic and deterministic interferences, the problem is first formulated in the framework of a Hidden Markov Model and then the Expectation Maximization algorithm is applied to obtain the maximum likelihood estimates of the model parameters (state levels, transition probabilities), signals, and the parameters of the deterministic disturbances. Using fictitious channel currents embedded in the idealized noise, we first show that the signal processing technique is capable of characterizing the signal characteristics quite accurately even when the amplitude of currents is as small as 5-10 fA. The statistics of the signal estimated from the processing technique include the amplitude, mean open and closed duration, open-time and closed-time histograms, probability of dwell-time and the transition probability matrix. With a periodic interference composed, for example, of 50 Hz and 100 Hz components, or a linear drift of the baseline added to the segment containing channel currents and white noise, the parameters of the deterministic interference, such as the amplitude and phase of the sinusoidal wave, or the rate of linear drift, as well as all the relevant statistics of the signal, are accurately estimated with the algorithm we propose. Also, if the frequencies of the periodic interference are unknown, they can be accurately estimated. Finally, we provide a technique by which channel currents originating from the sum of two or more independent single channels are decomposed so that each process can be separately characterized. This process is also formulated as a Hidden Markov Model problem and solved by applying the Expectation Maximization algorithm. The scheme relies on the fact that the transition matrix of the summed Markov process can be construed as a tensor product of the transition matrices of individual processes.     

Didactic discussion of stochastic resonance effects and weak signals

A simple, paradigmatic, model is used to illustrate some general properties of effects subsumed under the label “stochastic resonance.” In particular, analyses of the transparent model show that 1) a small amount of noise added to a much larger signal can greatly increase the response to the signal, but 2) a weak signal added to much larger noise will not generate a substantial added response. The conclusions drawn from the model illustrate the general result that stochastic resonance effects do not provide an avenue for signals that are much smaller than noise to affect biology. A further analysis demonstrates the effects of small signals in the shifting of biologically important chemical equilibria under conditions where stochastic resonance effects are significant. © 1996 Wiley-Liss, Inc.     

A theoretical approach to G-protein modulation of cellular responsiveness

 Structure and function of cells often depend critically on molecular signals arriving at their surface. There are universal mechanisms of signal transduction and signal processing across cell membranes. In this paper the mechanisms involving guanine-nucleotide regulatory proteins (“G-proteins”) and certain receptor-kinases are considered. On the basis of recent findings in molecular biology a mathematical model is developed taking into account all essential components in the biochemical network between first and second messenger. There are two coupled feedback loops inherent in this process. The model finally consists of three nonlinear equations, which are obtained from a system of originally ten equations by using conservation laws and quasi-steady state conditions. The second part of the paper contains a mathematical analysis of the model. Solutions describing the temporal development of the involved biochemical species are shown to be bounded, more specifically to remain, independent of the size of the input signal, in a bounded domain of the state space. For the situation of stationary input signals existence, uniqueness and asymptotic stability of steady states are derived. We also demonstrate biologically relevant stimulus-response properties like monotonicity and saturation effects. For temporally non-constant input signals we show numerically that the model is able to produce phenomena of hypersensitivity and desensitization which are important characteristics of cellular responsiveness. Received 18 March 1996; received in revised form 15 April 1996