Bistable Model Research Articles

On the precision of quasi steady state assumptions in stochastic dynamics

Many biochemical networks have complex multidimensional dynamics and there is a long history of methods that have been used for dimensionality reduction for such reaction networks. Usually a deterministic mass action approach is used; however, in small volumes, there are significant fluctuations from the mean which the mass action approach cannot capture. In such cases stochastic simulation methods should be used. In this paper, we evaluate the applicability of one such dimensionality reduction method, the quasi-steady state approximation (QSSA) [L. Menten and M. Michaelis, "Die kinetik der invertinwirkung," Biochem. Z 49, 333369 (1913)] for dimensionality reduction in case of stochastic dynamics. First, the applicability of QSSA approach is evaluated for a canonical system of enzyme reactions. Application of QSSA to such a reaction system in a deterministic setting leads to Michaelis-Menten reduced kinetics which can be used to derive the equilibrium concentrations of the reaction species. In the case of stochastic simulations, however, the steady state is characterized by fluctuations around the mean equilibrium concentration. Our analysis shows that a QSSA based approach for dimensionality reduction captures well the mean of the distribution as obtained from a full dimensional simulation but fails to accurately capture the distribution around that mean. Moreover, the QSSA approximation is not unique. We have then extended the analysis to a simple bistable biochemical network model proposed to account for the stability of synaptic efficacies; the substrate of learning and memory [J. E. Lisman, "A mechanism of memory storage insensitive to molecular turnover: A bistable autophosphorylating kinase," Proc. Natl. Acad. Sci. U.S.A. 82, 3055-3057 (1985)]. Our analysis shows that a QSSA based dimensionality reduction method results in errors as big as two orders of magnitude in predicting the residence times in the two stable states.

Robustness-based Model Validation of an Apoptosis Signalling Network Model

Models of intracellular biochemical reaction networks are difficult to parameterise due to the low number of quantitative time series experimental values. Therefore, model validation or invalidation plays an important role, as it allows to check qualitatively whether a model structure is suited or not to reproduce qualitatively the experimental findings.This paper analyses the robustness of an experimentally validated polynomial differential equation model of TNF-induced pro-and anti-apoptotic signalling. The bistability of the median model is shown robust to large single parameter variations. Only two parameters (XIAP and Procaspase-3 production rates) are shown to be fragile, in particular when changed simultaneously. Therefore, the model seems valid from the point of view of robustness analysis of the bistability.Many biological experiments quantify average concentrations or the percentage of viable cells, while other methods such as microscopy-based experiments observe single cells. The integration of single cell and cell population behaviour of TNF-induced pro- and anti-apoptotic signalling has been achieved via a cell ensemble model, whose robustness is also analysed here. We show that within the cell population there are cells with not only quantitative differences, but also qualitative ones. In particular, all cells are not bistable. The degree of robustness applicable for the median cell is expanded to combine mono- and bistable models. This measure, applied solely to the two-dimensional subspace of fragile parameters, is shown to correlate well with the time of death. While robustness of bistability can serve for model validation of the median cell model, it cannot for the model of the cell population.

Weak Signal Detection Based on Stochastic Resonance of Piecewise Model

Based on traditional theory of bistable stochastic resonance, this paper puts forward a piecewise stochastic resonance model which used to detect medium- low-frequency weak periodic signal and deducts analytical results of signal-to-noise ratio. Besides, we have done simulation experiments and contrast analysis of the two models also. The result shows that the piecewise model is better to detect medium- low-frequency weak periodic signal in background of strong noise with higher SNR and wider frequency range. It has raised adaptability to detect periodic signal which is common in mechanical failure. Combined with the piecewise bistable stochastic resonance model and the modulate method, it is more effectively to achieve judgment and detection of the mechanical fault signals.

Azimuthal dependence of switching field strength for nematic liquid crystal bistability on patterned alignment layers

We investigate the azimuthal dependence of the switching field strength for nematic liquid crystal bistability on patterned alignment layers to determine the validity of a switching theory proposed by Kim et al. [Appl. Phys. Lett. 78, 3055 (2001)] and to evaluate higher-order azimuthal anchoring coefficients. Director behavior during switching is described in detail for general azimuthal directions of an applied electric field and the experimental results are explained on the basis of the theory. We obtain azimuthal anchoring coefficients up to the second higher-order term in an expansion of the anchoring energy function. Detailed analysis of the director behavior reveals a close analogy between the proposed model of nematic liquid crystal bistability and a coherent rotation (Stoner-Wohlfarth) model of ferromagnetism. This suggests a reversal-asymmetric property of the substrate-nematic interface director, differing from the reversal symmetry of the bulk nematic director, as far as the nematic bistability switching is concerned.

Storage of Correlated Patterns in Standard and Bistable Purkinje Cell Models

The cerebellum has long been considered to undergo supervised learning, with climbing fibers acting as a ‘teaching’ or ‘error’ signal. Purkinje cells (PCs), the sole output of the cerebellar cortex, have been considered as analogs of perceptrons storing input/output associations. In support of this hypothesis, a recent study found that the distribution of synaptic weights of a perceptron at maximal capacity is in striking agreement with experimental data in adult rats. However, the calculation was performed using random uncorrelated inputs and outputs. This is a clearly unrealistic assumption since sensory inputs and motor outputs carry a substantial degree of temporal correlations. In this paper, we consider a binary output neuron with a large number of inputs, which is required to store associations between temporally correlated sequences of binary inputs and outputs, modelled as Markov chains. Storage capacity is found to increase with both input and output correlations, and diverges in the limit where both go to unity. We also investigate the capacity of a bistable output unit, since PCs have been shown to be bistable in some experimental conditions. Bistability is shown to enhance storage capacity whenever the output correlation is stronger than the input correlation. Distribution of synaptic weights at maximal capacity is shown to be independent on correlations, and is also unaffected by the presence of bistability.

Noise-induced contrast enhancement using stochastic resonance on singular values

In this paper, a dynamic stochastic resonance (DSR)-based technique in singular value domain for contrast enhancement of dark images has been presented. The internal noise due to the lack of illumination is utilized using a DSR iterative process to obtain enhancement in contrast, colorfulness as well as perceptual quality. DSR is a phenomenon that has been strategically induced and exploited and has been found to give remarkable response when applied on the singular values of a dark low-contrast image. When an image is represented as a summation of image layers comprising of eigen vectors and values, the singular values denote luminance information of each such image layer. By application of DSR on the singular values using the analogy of a bistable double-well potential model, each of the singular values is scaled to produce an image with enhanced contrast as well as visual quality. When compared with performance of some existing spatial domain enhancement techniques, the proposed DSR-SVD technique is found to give noteworthy better performance in terms of contrast enhancement factor, color enhancement factor and perceptual quality measure.

Global Analysis of Dynamical Decision-Making Models through Local Computation around the Hidden Saddle

Bistable dynamical switches are frequently encountered in mathematical modeling of biological systems because binary decisions are at the core of many cellular processes. Bistable switches present two stable steady-states, each of them corresponding to a distinct decision. In response to a transient signal, the system can flip back and forth between these two stable steady-states, switching between both decisions. Understanding which parameters and states affect this switch between stable states may shed light on the mechanisms underlying the decision-making process. Yet, answering such a question involves analyzing the global dynamical (i.e., transient) behavior of a nonlinear, possibly high dimensional model. In this paper, we show how a local analysis at a particular equilibrium point of bistable systems is highly relevant to understand the global properties of the switching system. The local analysis is performed at the saddle point, an often disregarded equilibrium point of bistable models but which is shown to be a key ruler of the decision-making process. Results are illustrated on three previously published models of biological switches: two models of apoptosis, the programmed cell death and one model of long-term potentiation, a phenomenon underlying synaptic plasticity.

A Bistable Model of Cell Polarity

Ultrasensitivity, as described by Goldbeter and Koshland, has been considered for a long time as a way to realize bistable switches in biological systems. It is not as well recognized that when ultrasensitivity and reinforcing feedback loops are present in a spatially distributed system such as the cell plasmamembrane, they may induce bistability and spatial separation of the system into distinct signaling phases. Here we suggest that bistability of ultrasensitive signaling pathways in a diffusive environment provides a basic mechanism to realize cell membrane polarity. Cell membrane polarization is a fundamental process implicated in several basic biological phenomena, such as differentiation, proliferation, migration and morphogenesis of unicellular and multicellular organisms. We describe a simple, solvable model of cell membrane polarization based on the coupling of membrane diffusion with bistable enzymatic dynamics. The model can reproduce a broad range of symmetry-breaking events, such as those observed in eukaryotic directional sensing, the apico-basal polarization of epithelium cells, the polarization of budding and mating yeast, and the formation of Ras nanoclusters in several cell types.

On the mechanism for edge localized mode mitigation by supersonic molecular beam injection

We construct a diffusive, bi-stable cellular automata model to elucidate the physical mechanisms underlying observed edge localized mode (ELM) mitigation by supersonic molecular beam injection (SMBI). The extended cellular automata model reproduces key qualitative features of ELM mitigation experiments, most significantly the increase in frequency of grain ejection events (ELMs), and the decrease in the number of grains ejected by these transport events. The basic mechanism of mitigation is the triggering of small scale pedestal avalanches by additional grain injection directly into the H-mode pedestal. The small scale avalanches prevent the gradient from building-up to marginality throughout the pedestal, thus avoiding large scale transport events which span the full extent of that region. We explore different grain injection parameters to find an optimal SMBI scenario. We show that shallow SMBI deposition is sufficient for ELM mitigation.

Deterministic Continuation of Stochastic Metastable Equilibria via Lyapunov Equations and Ellipsoids

Numerical continuation methods for deterministic dynamical systems have been one of the most successful tools in applied dynamical systems theory. Continuation techniques have been employed in all branches of the natural sciences as well as in engineering to analyze ordinary, partial and delay differential equations. Here we show that the deterministic continuation algorithm for equilibrium points can be extended to track information about metastable equilibrium points of stochastic differential equations (SDEs). We stress that we do not develop a new technical tool but that we combine results and methods from probability theory, dynamical systems, numerical analysis, optimization and control theory into an algorithm that augments classical equilibrium continuation methods. In particular, we use ellipsoids defining regions of high concentration of sample paths. It is shown that these ellipsoids and the distances between them can be efficiently calculated using iterative methods that take advantage of the numerical continuation framework. We apply our method to a bistable neural competition model and a classical predator-prey system. Furthermore, we show how global assumptions on the flow can be incorporated - if they are available - by relating numerical continuation, Kramers' formula and Rayleigh iteration.

A Bistable Field Model of Cancer Dynamics

AbstractCancer spread is a dynamical process occurring not only in time but also in space which, for solid tumors at least, can be modeled quantitatively by reaction and diffusion equations with a bistable behavior: tumor cell colonization happens in a portion of tissue and propagates, but in some cases the process is stopped. Such a cancer proliferation/extintion dynamics is obtained in many mathematical models as a limit of complicated interacting biological fields. In this article we present a very basic model of cancer proliferation adopting thebistableequation for a single tumor cell dynamics. The reaction-diffusion theory is numerically and analytically studied and then extended in order to take into account dispersal effects in cancer progression in analogy with ecological models based on the porous medium equation. Possible implications of this approach for explanation and prediction of tumor development on the lines of existing studies on brain cancer progression are discussed. The potential role of continuum models in connecting the two predominant interpretative theories about cancer, once formalized in appropriate mathematical terms, is discussed.

ANALYTICAL MODEL FOR OPTICAL BISTABILITY IN LASER WITH SATURABLE ABSORBER

In this paper, we develop a theoretical model that accurately describes the nonlinear phenomenon of optical bistability in Fabry-Perot laser containing a saturable absorber "LSA". We present an analytical model for optical bistability in a LSA. The physics of optical bistability in a classical optical resonant system such as Fabry-Perot etalon has been well-established, and can be described by simple analytical models. The photon intensities were determined as a function of the pumping rate of the active medium, and we analyzed the linear stability of the stationary solutions obtained.

Evoking transitions from states of motor readiness to motor action in a phase-coupled oscillator model of motor cortex

Background Rapid changes in behavior require rapid changes in brain states, yet the brain must also remain stable in the face of noise and uncertainty. The problem is exemplified by motor readiness in which execution of a planned motor behavior is voluntarily delayed until the onset of a ‘go’ stimulus. Achieving such rapid changes between otherwise stable brain states remains a fundamental problem in computational neuroscience. Single-state attractor networks provide stability but require global changes to the attractor landscape to alter network state and such transitions are invariably slow. Multi-state attractor networks promise rapid transitions between co-existing stable states by direct perturbation of the state variables however the nature of the perturbation is critical. Here we explore the efficacy of a global perturbation for invoking state transitions in a bistable model of motor cortex based on phase-coupled neural oscillators [1]. Methods Motor cortex was modeled as a two-dimensional sheet (128x128) of phase-coupled neural oscillators that were locally coupled using a center-surround connection topology. This model exhibits bistability when the strength of the inhibitory surround is within a specific range. In such cases the oscillators spontaneously selforganize into either spatially synchronous patterns or spatiotemporal traveling wave patterns depending upon initial conditions. These patterns are reminiscent of those observed in primate motor cortex during voluntary [2] and self-paced movements [3] where the timing of the evoked waves are dependent on both the onset of movement and the phase of the local-field potential [2,3]. Results We show that a global perturbation that is sinusoidally coupled to the phase of the mean-field can evoke a rapid transition from the synchronous state to the wave state. Our results demonstrate that an appropriately timed global perturbation can induce rapid transitions between distinct stable firing patterns and we posit that motor cortex may use a similar mechanism to transition rapidly from motor readiness to action.

Modeling and in situ identification of material parameters for layered structures based on carbon nanotube arrays

We test and model the mechanical response of a multilayer composite structure composed of alternating layers of aligned carbon nanotubes and copper foils under compression. We employ a bistable mass-spring model to capture the three-phase hysteretic response of the loading curve with excellent agreement with the experimental observations. An in situ identification procedure is proposed to quantify the material parameters corresponding to the mesoscopic scale of the structure. We refine the results using a genetic algorithm and compare the response of two different models based on three and four springs in series. The localization of deformation can be accurately captured by these simplified models, which hold promise for the design of novel materials with tailored deformation responses.

Near-oscillatory relaxation behavior of levitation force in infiltration and growth processed bulk YBCO/Ag superconducting composites

Time relaxation behavior of levitation force has been studied in IGP bulk YBCO/Ag superconductor using levitation force measurements as these measurements throw light on the magnetic relaxation in superconductors and the underlying vortex dynamics, pinning mechanisms and the nature of pinning forces. The measurements have revealed a hitherto unknown near-oscillatory relaxation of the levitation force with varying magnetic field. This kind of behavior is found to be more pronounced at smaller gap distances between the permanent magnet and the superconductor. A switch-type polarity bistable equilibrium model for the supercurrent structure has been proposed based on the understanding that even the permanent magnet gets magnetized in the presence of the superconductor, which has also been verified and reported here. This model satisfactorily explains the observed oscillatory behavior of relaxation rates.

Investigation of oscillation modes based on numerical solutions of nonlinear models

A single-frequency approximation according to the Prony method for numerical solution of a nonlinear model estimates the central fundamental frequency that determines the period of oscillations, i.e., the minimal time scale of this numerical solution. Then, subharmonic modes of this numerical solution can be identified in the time slots multiple to the oscillation period. Numerical solutions of a bistable oscillator model are considered as an example of spectral identification of a sub-harmonic oscillation mode according to the Prony method.

A bistable computational model of recurring epileptiform activity as observed in rodent slice preparations

We describe a computational model of epileptiform activity mimicking the activity exhibited by an animal model of epilepsy in vitro. The computational model permits generation of synthetic data to assist in the evaluation of new algorithms for epilepsy treatment via adaptive neurostimulation. The model implements both single-compartment pyramidal neurons and fast-spiking interneurons, arranged in a one-dimensional network using both excitatory and inhibitory synapses. The model tracks changes in extracellular ion concentrations, which determine the reversal potentials of membrane currents. Changes in simulated ion concentration provide positive feedback which drives the system towards the epileptiform state. One mechanism of positive feedback explored by this model is the conversion of pyramidal cells from regular spiking to intrinsic bursting as extracellular potassium concentration increases. One of the main contributions of this work is the development of a slow depression mechanism that enforces seizure termination. The network spontaneously leaves the seizure-like state as the slow depression variable decreases. This is one of the first detailed computational models of epileptiform activity, which exhibits realistic transitions between inter-seizure and seizure states, and back, with state durations similar to the in vitro model. We validate the computational model by comparing its state durations to those of the biological model. We also show that electrical stimulation of the computational model achieves seizure suppression comparable to that observed in the in vitro model.

Logic signals driven stochastic resonance in bistable dynamics subjected to 1/f noise floor

In the presence of 1/f β noise, we investigate the logical stochastic resonance (LSR) in an asymmetric bistable model driven by various cycling combinations of two logic inputs. The probability of correct logic outputs is calculated according to true table of logic relationships. Two major results are presented. Firstly, it is shown that the LSR effect can be obtained by changing noise strength. Over entire range of noise variance, white noise can be considered to be better than 1/f noise or 1/f 2 noise to obtain clean logic operation. At a smaller noise level, 1/f noise can realize higher output probability than white noise or 1/f 2 noise. In the sense, 1/f noise can be considered to be better than white noise or 1/f 2. On the other hand, the correct probability can evolves nonmonotonically as noise exponent β increases, and a kind of SR-like effect can be obtained as a result of β. At certain intermediate noise variance, the output probability is able to attain its minimum at β = 1. It is also shown that actually some finite β sometime can be better than β = 0 at small range of noise variance. The study might provide some potential complement to LSR effect in the presence of 1/f β noise.

Modelling of piezoelectrically actuated bistable composites

Bistable asymmetric composites with piezoelectric actuators that induce ‘snap-through’ from one stable state to another are candidate smart materials for shape-change (morphing) applications. This paper combines a homogenised piezoelectric model of a Macro Fibre Composite with a bistable asymmetric laminate model. Both predicted shape and snap-through voltage of a piezo-actuated [0/90] T laminate compare favourably with experimental results.

Neuronal Adaptation Effects in Decision Making

Recently, there has been an increased interest on the neural mechanisms underlying perceptual decision making. However, the effect of neuronal adaptation in this context has not yet been studied. We begin our study by investigating how adaptation can bias perceptual decisions. We considered behavioral data from an experiment on high-level adaptation-related aftereffects in a perceptual decision task with ambiguous stimuli on humans. To understand the driving force behind the perceptual decision process, a biologically inspired cortical network model was used. Two theoretical scenarios arose for explaining the perceptual switch from the category of the adaptor stimulus to the opposite, nonadapted one. One is noise-driven transition due to the probabilistic spike times of neurons and the other is adaptation-driven transition due to afterhyperpolarization currents. With increasing levels of neural adaptation, the system shifts from a noise-driven to an adaptation-driven modus. The behavioral results show that the underlying model is not just a bistable model, as usual in the decision-making modeling literature, but that neuronal adaptation is high and therefore the working point of the model is in the oscillatory regime. Using the same model parameters, we studied the effect of neural adaptation in a perceptual decision-making task where the same ambiguous stimulus was presented with and without a preceding adaptor stimulus. We find that for different levels of sensory evidence favoring one of the two interpretations of the ambiguous stimulus, higher levels of neural adaptation lead to quicker decisions contributing to a speed-accuracy trade off.