Bistable Model Research Articles

Lévy noise induced transition and enhanced stability in a gene regulatory network.

We investigate a quantitative bistable two-dimensional model (MeKS network) of gene expression dynamics describing the competence development in the Bacillus subtilis under the influence of Lévy as well as Brownian motions. To analyze the transitions between the vegetative and the competence regions therein, two dimensionless deterministic quantities, the mean first exit time (MFET) and the first escape probability, are determined from a microscopic perspective, as well as their averaged versions from a macroscopic perspective. The relative contribution factor λ, the ratio of non-Gaussian and Gaussian noise strengths, is adopted to identify an optimum choice in these transitions. Additionally, we use a recent geometric concept, the stochastic basin of attraction (SBA), to exhibit a pictorial comprehension about the influence of the Lévy motion on the basin stability of the competence state. Our main results indicate that (i) the transitions between the vegetative and the competence regions can be induced by the noise intensities, the relative contribution factor λ and the Lévy motion index α; (ii) a higher noise intensity and a larger α with smaller jump magnitude make the MFET shorter, and the MFET as a function of λ exhibits one maximum value, which is a signature of the noise-enhanced stability phenomenon for the vegetative state; (iii) a larger α makes the transition from the vegetative to the adjacent competence region to occur at the highest probability. The Lévy motion index α0≈0.5 (a larger jump magnitude with a lower frequency) is an ideal choice to implement the transition to the non-adjacent competence region; (iv) there is an expansion in SBA when α decreases.

Linking trophic cascades to changes in desert dune geomorphology using high-resolution drone data.

Vegetation cover is fundamental in the formation and maintenance of geomorphological features in dune systems. In arid Australia, increased woody shrub cover has been linked to removal of the apex predator (Dingoes, Canis dingo) via subsequent trophic cascades. We ask whether this increase in shrubs can be linked to altered physical characteristics of the dunes. We used drone-based remote sensing to measure shrub density and construct three-dimensional models of dune morphology. Dunes had significantly different physical characteristics either side of the 'dingo-proof fence', inside which dingoes are systematically eradicated and shrub density is higher over vast spatial extents. Generalized additive models revealed that dunes with increased shrub density were higher, differently shaped and more variable in height profile. We propose that low shrub density induces aeolian and sedimentary processes that result in greater surface erosion and sediment transport, whereas high shrub density promotes dune stability. We speculate that increased vegetation cover acts to push dunes towards an alternate stable state, where climatic variation no longer has a significant effect on their morphodynamic state within the bi-stable state model. Our study provides evidence that anthropogenically induced trophic cascades can indirectly lead to large-scale changes in landscape geomorphology.

Mean field model for collective motion bistability

<p style='text-indent:20px;'>We consider the Czirók model for collective motion of locusts along a one-dimensional torus. In the model, each agent's velocity locally interacts with other agents' velocities in the system, and there is also exogenous randomness to each agent's velocity. The interaction tends to create the alignment of collectivemotion. By analyzing the associated nonlinear Fokker-Planck equation, we obtain the condition for the existence of stationary order states and the conditions for their linear stability. These conditions depend on the noise level, which should be strong enough, and on the interaction between the agent's velocities, which should be neither too small, nor too strong. We carry out the fluctuation analysis of the interacting system and describe the large deviation principle to calculate the transition probability from one order state to the other. Numerical simulations confirm our analytical findings.

Stochastic resonance in an asymmetric bistable system driven by multiplicative and additive Gaussian noise and its application in bearing fault detection

The phenomenon of stochastic resonance (SR) in a new asymmetric bistable model is investigated. Firstly, a new asymmetric bistable model with an asymmetric term is proposed based on traditional bistable model and the influence of system parameters on the asymmetric bistable potential function is studied. Secondly, the signal-to-noise ratio (SNR) as the index of evaluating the model are researched. Thirdly, Applying the two-state theory and the adiabatic approximation theory, the analytical expressions of SNR is derived for the asymmetric bistable system driven by a periodic signal, unrelated multiplicative and additive Gaussian noise. Finally, the asymmetric bistable stochastic resonance (ABSR) is applied to the bearing fault detection and compared with classical bistable stochastic resonance (CBSR) and classical tri-stable stochastic resonance (CTSR). The numerical computations results show that:(1) the curve of SNR as a function of the additive Gaussian noise and multiplicative Gaussian noise first increased and then decreased with the different influence of the parameters a, b, r and A; This demonstrates that the phenomenon of SR can be induced by system parameters; (2) by parameter compensation method, the ABSR performs better in bearing fault detection than the CBSR and CTSR with merits of higher output SNR, better anti-noise and frequency response capability.

The tug-of-war behavior of a Brownian particle in an asymmetric double optical trap with stochastic fluctuations

A Brownian particle optically trapped in an asymmetric double potential surrounded by a thermal bath was simulated. Under the cooperative action of the resultant deterministic optical force and the stochastic fluctuations of the thermal bath, the confined particle undergoes Kramers transition, and oscillates between the two traps with a probability of trap occupancy that is asymmetrically distributed about the midpoint. The simulation results obtained at different temperatures indicate that the oscillation behavior of the particle can be treated as the result of a tug-of-war game played between the resultant deterministic force and the random force. We also employ a bistable model to explain the observed phenomena.

Mitotic waves in the early embryogenesis of Drosophila: Bistability traded for speed

Early embryogenesis of most metazoans is characterized by rapid and synchronous cleavage divisions. Chemical waves of Cdk1 activity were previously shown to spread across Drosophila embryos, and the underlying molecular processes were dissected. Here, we present the theory of the physical mechanisms that control Cdk1 waves in Drosophila The in vivo dynamics of Cdk1 are captured by a transiently bistable reaction-diffusion model, where time-dependent reaction terms account for the growing level of cyclins and Cdk1 activation across the cell cycle. We identify two distinct regimes. The first one is observed in mutants of the mitotic switch. There, waves are triggered by the classical mechanism of a stable state invading a metastable one. Conversely, waves in wild type reflect a transient phase that preserves the Cdk1 spatial gradients while the overall level of Cdk1 activity is swept upward by the time-dependent reaction terms. This unique mechanism generates a wave-like spreading that differs from bistable waves for its dependence on dynamic parameters and its faster speed. Namely, the speed of "sweep" waves strikingly decreases as the strength of the reaction terms increases and scales as the powers 3/4, -1/2, and 7/12 of Cdk1 molecular diffusivity, noise amplitude, and rate of increase of Cdk1 activity in the cell-cycle S phase, respectively. Theoretical predictions are supported by numerical simulations and experiments that couple quantitative measurements of Cdk1 activity and genetic perturbations of the accumulation rate of cyclins. Finally, our analysis bears upon the inhibition required to suppress Cdk1 waves at the cell-cycle pause for the maternal-to-zygotic transition.

Nonlinear model analysis of all-optical flip-flop and inverter operations of microring laser

We explore a theoretical model of bistability at two adjacent lasing wavelengths from an InGaAs/InGaAsP multiple quantum well (MQW) microring laser. We show that nonlinear effects on the phase and amplitude play significant roles in the lasing operations of the microring laser. Numerical simulations indicate that all-optical flip-flop operations and inverter operations can be observed within the same device by controlling the injection current. The validity of our analysis is confirmed by a comparison of the results for numerical simulations with experimental results of the lasing spectrum. We believe that the analysis presented in this paper will be useful for the future design of all-optical signal processing devices.

Perturbing the Hypothalamic-Pituitary-Adrenal Axis: A Mathematical Model for Interpreting PTSD Assessment Tests.

We use a dynamical systems model of the hypothalamic–pituitary–adrenal (HPA) axis to understand the mechanisms underlying clinical protocols used to probe patient stress response. Specifically, we address dexamethasone (DEX) and ACTH challenge tests, which probe pituitary and adrenal gland responses, respectively. We show that some previously observed features and experimental responses can arise from a bistable mathematical model containing two steady-states, rather than relying on specific and permanent parameter changes due to physiological disruption. Moreover, we show that the timing of a perturbation relative to the intrinsic oscillation of the HPA axis can affect challenge test responses. Conventional mechanistic hypotheses supported and refuted by the challenge tests are reexamined by varying parameters in our mathematical model associated with these hypotheses. We show that (a) adrenal hyposensitivity can give rise to the responses seen in ACTH challenge tests and (b) enhanced cortisol-mediated suppression of the pituitary in subjects with PTSD is not necessary to explain the responses observed in DEX stress tests. We propose a new two-stage DEX/external stressor protocol to more clearly distinguish between the conventional hypothesis of enhanced suppression of the pituitary and bistable dynamics hypothesized in our model.

Parameter estimation of stochastic models based on limited data

Progress in experimental techniques enables a more accurate quantification of genes, mRNA, and proteins at the single cell level. Provided with limited time series data from single-cell measurements, this note proposes a new quasi-Newton optimization algorithm (QNSTOP) for parameter estimation of stochastic models. To capture the stochasticity inside models and data, the random objective function is constructed based on the maximum log-likelihood of transition probabilities rather than summary statistics, which relies heavily on stochastic simulations. Simple to use and efficient, QNSTOP can find the "best" parameter vector from far away starting points in just a few iterations. Results on a bistable model match well the bistable dynamics that can only be obtained from stochastic models.

Fault Diagnosis of Rotating Machinery Based on Stochastic Resonance with a Bistable Confining Potential

Since the weak fault characteristics of mechanical equipment are often difficult to extract in strong background noise, stochastic resonance (SR) is widely used to extract the weak fault characteristics, which is able to utilize the noise to amplify weak fault characteristics. Although classical bistable stochastic resonance (CBSR) can enhance the weak characteristics by adjusting the parameters of potential model, when potential barrier height is adjusted potential well width is also changed and vice versa. The simultaneous change of both potential well width and barrier height is difficult to obtain a suitable potential model for better weak fault characteristic extraction and further fault diagnosis of machinery. For this reason, the output signal‐to‐noise ratio (SNR) of CBSR is greatly reduced, and the corresponding enhancement ability of weak fault characteristics is limited. In order to avoid the shortcomings, a new SR method is proposed to extract weak fault characteristics and further diagnose the faults of rotating machinery, where the classical bistable potential is replaced with a bistable confining potential to get the optimal SR. The bistable confining potential model not only has the characteristics of the classical bistable potential model but also has the ability to adjust the potential width, barrier height, and wall steepness independently. Simulated data are used to demonstrate the proposed new SR method. The results indicate that the weak fault characteristics can be effectively extracted from simulated signals with heavy noise. Experiments on the bearings and planetary gearboxes demonstrate that the proposed SR method can correctly diagnose the faults of rotating machinery and moreover has higher spectrum peak and better recognition degree compared with the CBSR method.

Stochastic Resonance in Second-Order Underdamped System With Exponential Bistable Potential for Bearing Fault Diagnosis

Stochastic resonance (SR) as effective approach for weak signal detection has been widely used in bearing fault signal diagnosis. In this paper, a new method was proposed to detect fault signals, which consists of an exponential bistable potential model generalized by using a harmonic Potential (HP) model, a Gaussian potential (GP) model and second-order underdamped system. As the classical bistable SR (CBSR) has the disadvantage of output saturation, which will suppress the optimal signal-to-noise ratio (SNR) of the system, therefore, underdamped SR with exponential potential (UESR) and underdamped SR with classical bistable potential (UCSR) are applied to detect fault signal, respectively. Under the adiabatic condition, the analytical expression of the SNR is calculated for the UESR system driven by Gaussian white noise and periodic signal. Then, the effects of system parameters and the damping factor on analytical expression of SNR as a function of noise intensity for different parameters are studied, respectively. Finally, the proposed method is applied to detect simulated fault signals and actual bearing fault signals. The experimental results show that the proposed UESR method is superior to the UCSR method in fault signal diagnosis, such as larger output SNR and higher spectrum peaks at fault characteristic frequencies.

A study on New Bi-stable Vibration Model and Its Vibration Characteristics

A study on New Bi-stable Vibration Model and Its Vibration Characteristics

A Novel Second Order Bistable Adaptive Random Resonance Noise Reduction Method

This paper aimed at the key problems which the monitoring and fault diagnosis of process system generated in the high performance composite components manufacturing, established a new type of second-order bistable model, proposed a kind of adaptive stochastic resonance noise reduction method based on the model, and experimented to verifiy, it has very good detection effect and higher operation effect.

Colored correlated multiplicative and additive Gaussian colored noises-induced transition of a piecewise nonlinear bistable model

In this paper, we investigate the steady-state properties of a piecewise nonlinear bistable model driven by multiplicative and additive Gaussian colored noises with colored cross-correlation. Using the unified colored noise approximation, we derive the analytical expression of the steady-state probability density (SPD) function. Then the effects of colored correlated Gaussian colored noises on SPD are presented. According to the research results, it is found that there appear some new nonlinear phenomena in this system. The multiplicative colored noise intensity, the additive colored noise intensity and the cross-correlation strength between noises can induce the transition. However, the transition cannot be induced by the auto-correlation time of multiplicative and additive Gaussian colored noises as well as the cross-correlation time between noises.

An alternate protocol to achieve stochastic and deterministic resonances.

Periodic and Aperiodic Stochastic Resonance (SR) and Deterministic Resonance (DR) are studied in this paper. To check for the ubiquitousness of the phenomena, two unrelated systems, namely, FitzHugh-Nagumo and a particle in a bistable potential well, are studied. Instead of the conventional scenario of noise amplitude (in the case of SR) or chaotic signal amplitude (in the case of DR) variation, a tunable system parameter ("a" in the case of FitzHugh-Nagumo model and the damping coefficient "j" in the bistable model) is regulated. The operating values of these parameters are defined as the "setpoint" of the system throughout the present work. Our results indicate that there exists an optimal value of the setpoint for which maximum information transfer between the input and the output signals takes place. This information transfer from the input sub-threshold signal to the output dynamics is quantified by the normalised cross-correlation coefficient ( |CCC|). |CCC| as a function of the setpoint exhibits a unimodal variation which is characteristic of SR (or DR). Furthermore, |CCC| is computed for a grid of noise (or chaotic signal) amplitude and setpoint values. The heat map of |CCC| over this grid yields the presence of a resonance region in the noise-setpoint plane for which the maximum enhancement of the input sub-threshold signal is observed. This resonance region could be possibly used to explain how organisms maintain their signal detection efficacy with fluctuating amounts of noise present in their environment. Interestingly, the method of regulating the setpoint without changing the noise amplitude was not able to induce Coherence Resonance (CR). A possible, qualitative reasoning for this is provided.

Strong dipole magnetic fields in fast rotating fully convective stars

M dwarfs are the most numerous stars in our Galaxy with masses between approximately 0.5 and 0.1 solar mass. Many of them show surface activity qualitatively similar to our Sun and generate flares, high X-ray fluxes, and large-scale magnetic fields. Such activity is driven by a dynamo powered by the convective motions in their interiors. Understanding properties of stellar magnetic fields in these stars finds a broad application in astrophysics, including, e.g., theory of stellar dynamos and environment conditions around planets that may be orbiting these stars. Most stars with convective envelopes follow a rotation-activity relationship where various activity indicators saturate in stars with rotation periods shorter than a few days. The activity gradually declines with rotation rate in stars rotating more slowly. It is thought that due to a tight empirical correlation between X-ray and magnetic flux, the stellar magnetic fields will also saturate, to values around ~4kG. Here we report the detection of magnetic fields above the presumed saturation limit in four fully convective M-dwarfs. By combining results from spectroscopic and polarimetric studies we explain our findings in terms of bistable dynamo models: stars with the strongest magnetic fields are those in a dipole dynamo state, while stars in a multipole state cannot generate fields stronger than about four kilogauss. Our study provides observational evidence that dynamo in fully convective M dwarfs generates magnetic fields that can differ not only in the geometry of their large scale component, but also in the total magnetic energy.

Mathematical analysis and modeling of DNA segregation mechanisms.

The precise regulation of cell life division is indispensable to the reliable inheritance of genetic material, i.e. DNA, in successive generations of cells. This is governed by dedicated biochemical networks which ensure that all requirements are met before transition from one phase to the next. The Spindle Assembly Checkpoint (SAC) is an evolutionarily mechanism that delays mitotic progression until all chromosomes are properly linked to the mitotic spindle. During some asymmetric cell divisions, such as those observed in budding yeast, an additional mechanism, the Spindle Position Checkpoint (SPOC), is required to delay exit from mitosis until the mitotic spindle is correctly aligned. These checkpoints are complex and their elaborate spatiotemporal dynamics are challenging to understand intuitively. In this study, bistable mathematical models for both activation and silencing of mitotic checkpoints were constructed and analyzed. A one-parameter bifurcation was computed to show the realistic biochemical switches considering all signals. Numerical simulations involving systems of ODEs and PDEs were performed over various parameters, to investigate the effect of the diffusion coefficient. The results provide systems-level insights into mitotic transition and demonstrate that mathematical analysis constitutes a powerful tool for investigation of the dynamic properties of complex biomedical systems.

Rigorous derivation of active plate models for thin sheets of nematic elastomers

In the context of finite elasticity, we propose plate models describing the spontaneous bending of nematic elastomer thin films due to variations along the thickness of the nematic order parameters. Reduced energy functionals are deduced from a three-dimensional description of the system using rigorous dimension reduction techniques, based on the theory of Γ-convergence. The two-dimensional models are non-linear plate theories, in which deviations from a characteristic target curvature tensor cost elastic energy. Moreover, the stored energy functional cannot be minimised to zero, thus revealing the presence of residual stresses, as observed in numerical simulations. Three nematic textures are considered: splay-bend and twisted orientations of the nematic director, and a uniform director perpendicular to the mid-plane of the film, with variable degree of nematic order along the thickness. These three textures realise three very different structural models: one with only one stable spontaneously bent configuration, a bistable model with two oppositely curved configurations of minimal energy, and a shell with zero stiffness to twisting.

Improving the bearing fault diagnosis efficiency by the adaptive stochastic resonance in a new nonlinear system

It is a challenging task to detect the weak character signal in the noisy background. The stochastic resonance (SR) method has been wildly adopted recently because it can not only reduce the noise, but also enhance the weak feature information simultaneously. However, the traditional bistable model for SR is not perfect. So, this paper presents a new model with periodic potential to induce the adaptive SR. In the new model, based on the adaptive SR theory, the system parameters are simultaneously optimized by the improved artificial fish swarm algorithm. Meanwhile, the improved signal-to-noise ratio (ISNR) is set as the evaluation index. When the ISNR reaches a maximum, the output is optimal. In order to eliminate interference to obtain more useful information, the signals are preprocessed by Hilbert transform and High-pass filter before being input to the adaptive SR system. To verify the effectiveness of the proposed method, both numerical simulation and the vibration signal of the rolling element bearing from the lab experimental are adopted. Both of the results indicate that the adaptive SR model proposed shows better performance in weak character signals detection than the traditional adaptive SR in the bistable model. Meanwhile, the experimental signals with different working conditions are also processed by the new method. The results show that the method proposed could be more widely applied.

Multi-frequency weak signal detection based on multi-segment cascaded stochastic resonance for rolling bearings

For rotating machinery, vibration signals excited by its faulty components provide rich condition information for its fault diagnosis and condition-based maintenance. However, strong noise severely influences the accurate detection of incipient faults. Thanks to the ability of enhancing weak input and suppressing the noise, the stochastic resonance (SR) has been applied to weak signal detection in some fields, and the improvement on its performance are still being concerned, especially in the mechanical engineering. For multi-frequency weak signals, this paper proposes an improved mechanism for the SR, called multi-segment cascaded stochastic resonance (MS-CSR). In this method, the input signal obtains segment enhancement by using some bistable SR models, and series connection of such a unit compose an improved cascaded SR (CSR) system, which can not only gradually enhance the weak signals of interest, but also pay more attention on the signal with relatively small amplitude at the initial. A modified measurement index, named alliance signal-to-noise ratio (ASNR) is defined to evaluate the detection performance of the proposed SR method, as well as the parameter selection for the MS-CSR system. In this index, a weight factor is introduced to influence the assignment of noise energy in the SR, so that the relatively weak signal in the multi-frequency input signal can obtain a high priority to make the resonance phenomenon happen and avoid the misdiagnosis. A simulated signal and an experimental vibration signal collected from a faulty bearing are used to verify the effectiveness of the proposed MS-CSR method. The results demonstrate that the MS-CSR is a useful tool for detecting weak signals with multiple characteristic frequencies.