Bistable Dynamics Research Articles

Bistable dynamics and Hopf bifurcation in a refined model of early stage HIV infection

Recent clinical studies have shown that HIV disease pathogenesis can depend strongly on many factors at the time of transmission, including the strength of the initial viral load and the local availability of CD4+ T-cells. In this article, a new within-host model of HIV infection that incorporates the homeostatic proliferation of T-cells is formulated and analyzed. Due to the effects of this biological process, the influence of initial conditions on the proliferation of HIV infection is further elucidated. The identifiability of parameters within the model is investigated and a local stability analysis, which displays additional complexity in comparison to previous models, is conducted. The current study extends previous theoretical and computational work on the early stages of the disease and leads to interesting nonlinear dynamics, including a parameter region featuring bistability of infectious and viral clearance equilibria and the appearance of a Hopf bifurcation within biologically relevant parameter regimes.

Bifurcation analysis of bistable and oscillatory dynamics in biological networks using the root-locus method.

Most of the biological systems including gene regulatory networks can be described well by ordinary differential equation models with rational non-linearities. These models are derived either based on the reaction kinetics or by curve fitting to experimental data. This study demonstrates the applicability of the root-locus-based bifurcation analysis method for studying the complex dynamics of such models. The effectiveness of the bifurcation analysis in determining the exact parameter regions in each of which the system shows a certain dynamical behaviour, such as bistability, oscillation, and asymptotically equilibrium dynamics is shown by considering two mostly studied gene regulatory networks, namely Gardner's genetic toggle switch and p53 gene network possessing two-phase (mono-stable/oscillation) dynamics.

Experimental analysis of the effect of local base blowing on three-dimensional wake modes

Wake modes of a three-dimensional blunt-based body near a wall are investigated at a Reynolds number . The targeted modes are the static symmetry-breaking mode and two antisymmetric periodic modes. The static mode orientation is aligned with the horizontal major -axis of the base and randomly switches between a positive and a negative state leading to long-time bistable dynamics of the turbulent wake. The modifications of these modes are studied when continuous blowing is applied at different locations through four slits along the base edges (denoted L for left, R for right, T for top and B for bottom) in either four single asymmetric configurations or two double symmetric configurations (denoted LR and TB). Two regimes, referred to as mass and momentum, are clearly identifiable for all configurations. The mass regime, which is fairly insensitive to blowing momentum and location, is characterized by the growth of the recirculating bubble as the total injected flow rate is increased, and is associated with a base drag reduction and interpreted as resulting from the equilibrium between mass fluxes feeding and emptying the recirculating region. A simple budget model is shown to be in agreement with entrainment velocities measured for isolated turbulent mixing layers. The strength of the static mode is reduced up to 20 % when the bubble length is maximum, whereas no change in the periodic mode frequencies is found. On the other hand, the momentum regime is characterized by the deflating of the recirculating bubble, leading to base drag increase, and it is interpreted by the free shear layer forcing, which increases the entrainment velocity, thus emptying the recirculating bubble. In this regime the static mode orientation is imposed by the blowing symmetry. Lateral L and R (respectively top/bottom T and B) blowing configurations select or states in the horizontal (respectively vertical) direction, while bistable dynamics persists for the symmetric LR and TB configurations. The shape of periodic modes follows the changes in wake static orientation. The transition between the two regimes is governed by both the total injected flow rate and the location of the injection.

Perimetric blowing at the rear of a bluff body: consequences on the wake dynamics and drag reduction

AbstractIn a perspective of active flow control of the bistable wake of a square‐back Ahmed body, we show that the steady blowing through a perimetric slit at the rear of the body is not equivalent to a material rear perimetric cavity, the latter being able to suppress the bistable dynamics and symmetrize the wake flow, while the former cannot achieve any of the two targets, indicating that the steady perimetric blowing has no control authority on the symmetric unstable steady solution of the system.

The sign of traveling wave speed in bistable dynamics

We are concerned with the sign of traveling wave speed in bistable dynamics. This question is related to which species wins the competition in multiple species competition models. It is well-known that the wave speed is unique for traveling wave connecting two stable states. In this paper, we first review some known results on the sign of wave speed in bistable two species competition models. Then we derive rigorously the sign of bistable wave speed for a special three species competition model describing the competition in two different circumstances: (1) two species are weak competitors and one species is a strong competitor; (2) three species are very strong competitors. It is interesting to observe that, under certain conditions on the parameters, two weaker competitors can wipe out the strongest competitor.

Auditory streaming and bistability paradigm extended to a dynamic environment

We explore stream segregation with temporally modulated acoustic features using behavioral experiments and modelling. The auditory streaming paradigm in which alternating high- A and low-frequency tones B appear in a repeating ABA-pattern, has been shown to be perceptually bistable for extended presentations (order of minutes). For a fixed, repeating stimulus, perception spontaneously changes (switches) at random times, every 2–15 s, between an integrated interpretation with a galloping rhythm and segregated streams. Streaming in a natural auditory environment requires segregation of auditory objects with features that evolve over time. With the relatively idealized ABA-triplet paradigm, we explore perceptual switching in a non-static environment by considering slowly and periodically varying stimulus features. Our previously published model captures the dynamics of auditory bistability and predicts here how perceptual switches are entrained, tightly locked to the rising and falling phase of modulation. In psychoacoustic experiments we find that entrainment depends on both the period of modulation and the intrinsic switch characteristics of individual listeners. The extended auditory streaming paradigm with slowly modulated stimulus features presented here will be of significant interest for future imaging and neurophysiology experiments by reducing the need for subjective perceptual reports of ongoing perception.

Metastable Resting State Brain Dynamics.

Metastability refers to the fact that the state of a dynamical system spends a large amount of time in a restricted region of its available phase space before a transition takes place, bringing the system into another state from where it might recur into the previous one. beim Graben and Hutt (2013) suggested to use the recurrence plot (RP) technique introduced by Eckmann et al. (1987) for the segmentation of system's trajectories into metastable states using recurrence grammars. Here, we apply this recurrence structure analysis (RSA) for the first time to resting-state brain dynamics obtained from functional magnetic resonance imaging (fMRI). Brain regions are defined according to the brain hierarchical atlas (BHA) developed by Diez et al. (2015), and as a consequence, regions present high-connectivity in both structure (obtained from diffusion tensor imaging) and function (from the blood-level dependent-oxygenation—BOLD—signal). Remarkably, regions observed by Diez et al. were completely time-invariant. Here, in order to compare this static picture with the metastable systems dynamics obtained from the RSA segmentation, we determine the number of metastable states as a measure of complexity for all subjects and for region numbers varying from 3 to 100. We find RSA convergence toward an optimal segmentation of 40 metastable states for normalized BOLD signals, averaged over BHA modules. Next, we build a bistable dynamics at population level by pooling 30 subjects after Hausdorff clustering. In link with this finding, we reflect on the different modeling frameworks that can allow for such scenarios: heteroclinic dynamics, dynamics with riddled basins of attraction, multiple-timescale dynamics. Finally, we characterize the metastable states both functionally and structurally, using templates for resting state networks (RSNs) and the automated anatomical labeling (AAL) atlas, respectively.

Analysis of an in-host tuberculosis model for disease control

Tuberculosis is still a global threat to humans. In this letter, we analyzed a four-dimensional with-in host tuberculosis infection model and obtain thresholds for basic reproduction number, forward bifurcation, and backward bifurcation. Global sensitivity analysis provides parameters, which significantly influence the model dynamics. Bifurcation analysis and diagrams show parameter regions for disease elimination, latency, and active disease. Numerical simulations prove analytical results and demonstrate bistable model dynamics.

Oscillations and bistability in a model of ERK regulation.

This work concerns the question of how two important dynamical properties, oscillations and bistability, emerge in an important biological signaling network. Specifically, we consider a model for dual-site phosphorylation and dephosphorylation of extracellular signal-regulated kinase (ERK). We prove that oscillations persist even as the model is greatly simplified (reactions are made irreversible and intermediates are removed). Bistability, however, is much less robust-this property is lost when intermediates are removed or even when all reactions are made irreversible. Moreover, bistability is characterized by the presence of two reversible, catalytic reactions: as other reactions are made irreversible, bistability persists as long as one or both of the specified reactions is preserved. Finally, we investigate the maximum number of steady states, aided by a network's "mixed volume" (a concept from convex geometry). Taken together, our results shed light on the question of how oscillations and bistability emerge from a limiting network of the ERK network-namely, the fully processive dual-site network-which is known to be globally stable and therefore lack both oscillations and bistability. Our proofs are enabled by a Hopf bifurcation criterion due to Yang, analyses of Newton polytopes arising from Hurwitz determinants, and recent characterizations of multistationarity for networks having a steady-state parametrization.

Spatiotemporal dynamics of a reaction-diffusion model of pollen tube tip growth.

A reaction-diffusion model is proposed to describe the mechanisms underlying the spatial distributions of ROP1 and calcium on the pollen tube tip. The model assumes that the plasma membrane ROP1 activates itself through positive feedback loop, while the cytosolic calcium ions inhibit ROP1 via a negative feedback loop. Furthermore it is proposed that lateral movement of molecules on the plasma membrane are depicted by diffusion. It is shown that bistable or oscillatory dynamics could exist even in the non-spatial model, and stationary and oscillatory spatiotemporal patterns are found in the full spatial model which resemble the experimental data of pollen tube tip growth.

Classical and quantum dissipative dynamics in Josephson junctions: An Arnold problem, bifurcation, and capture into resonance.

We theoretically study the phase dynamics in Josephson junctions, which maps onto the oscillatory motion of a pointlike particle in the washboard potential. Under appropriate driving and damping conditions, the Josephson phase undergoes intriguing bistable dynamics near a saddle point in the quasienergy landscape. The bifurcation mechanism plays a critical role in superconducting quantum circuits with relevance to nondemolition measurements such as high-fidelity readout of qubit states. We address the question "what is the probability of capture into either basin of attraction?" and answer it concerning both classical and quantum dynamics. Consequently, we derive the Arnold probability and numerically analyze its implementation of the controlled dynamical switching between two steady states under the various nonequilibrium conditions.

A Network Activity Reconfiguration Underlies the Transition from Goal to Action

Neurons in prefrontal cortex (PF) represent mnemonic information about current goals until the action can be selected and executed. However, the neuronal dynamics underlying the transition from goal into specific actions are poorly understood. Here, we show that the goal-coding PF network is dynamically reconfigured from mnemonic to action selection states and that such reconfiguration is mediated by cell assemblies with heterogeneous excitability. We recorded neuronal activity from PF while monkeys selected their actions on the basis of memorized goals. Many PF neurons encoded the goal, but only a minority of them did so across both memory retention and action selection stages. Interestingly, about half of this minority of neurons switched their goal preference across the goal-action transition. Our computational model led us to propose a PF network composed of heterogeneous cell assemblies with single-state and bistable local dynamics able to produce both dynamical stability and input susceptibility simultaneously.

Modelling bistable tumour population dynamics to design effective treatment strategies

Despite recent advances in targeted drugs and immunotherapy, cancer remains “the emperor of all maladies” due to almost inevitable emergence of resistance. Drug resistance is thought to be driven by genetic alterations and/or dynamic plasticity that deregulate pathway activities and regulatory programs of a highly heterogeneous tumour. In this study, we propose a modelling framework to simulate population dynamics of heterogeneous tumour cells with reversible drug resistance. Drug sensitivity of a tumour cell is determined by its internal states, which are demarcated by coordinated activities of multiple interconnected oncogenic pathways. Transitions between cellular states depend on the effects of targeted drugs and regulatory relations between the pathways. Under this framework, we build a simple model to capture drug resistance characteristics of BRAF-mutant melanoma, where two cell states are determined by two mutually inhibitory – main and alternative – pathways. We assume that cells with an activated main pathway are proliferative yet sensitive to the BRAF inhibitor, and cells with an activated alternative pathway are quiescent but resistant to the drug. We describe a dynamical process of tumour growth under various drug regimens using the explicit solutions of mean-field equations. Based on these solutions, we compare efficacy of three treatment strategies from simulated data: static treatments with continuous and constant dosages, periodic treatments with regular intermittent active phases and drug holidays, and treatments derived from optimal control theory (OCT). Periodic treatments outperform static treatments with a considerable margin, while treatments based on OCT outperform the best periodic treatment. Our results provide insights regarding optimal cancer treatment modalities for heterogeneous tumours, and may guide the development of optimal therapeutic strategies to circumvent plastic drug resistance. They can also be used to evaluate the efficacy of suboptimal treatments that may account for side effects of the treatment and the cost of its application.

Transcutaneous Vagus Nerve Stimulation (tVNS) and the Dynamics of Visual Bistable Perception

Transcutaneous vagus nerve stimulation (tVNS) is widely used for clinical applications, but its mechanism of action is poorly understood. One candidate pathway that might mediate the effects of tVNS is an increase in GABAergic neurotransmission. In this study, we investigated the effect of tVNS on visual bistable perception, which is highly coupled to GABA. Participants were 34 healthy young subjects. We used a static (Necker cube) and a dynamic (structure from motion) bistable perception task. Each subject underwent tVNS as well as sham (placebo) stimulation for ∼45 min. We analyze effects of tVNS on percept durations by means of Bayesian multilevel regression. We find no evidence for a modulation of bistable perception dynamics through tVNS in either task, but the analyses do not ultimately confirm the null hypothesis either. We discuss different possible implications of our finding and propose that GABAergic effects of tVNS should be further investigated using more direct measures of GABA concentration, and, more generally, that a better understanding of the mechanisms of action of vagus nerve stimulation is needed. Finally, we discuss limitations of our study design, data analysis, and conclusions.

Stochastic nonlinear dynamics of confined cell migration in two-state systems

Migrating cells in physiological processes, including development, homeostasis and cancer, encounter structured environments and are forced to overcome physical obstacles. Yet, the dynamics of confined cell migration remains poorly understood, and thus there is a need to study the complex motility of cells in controlled confining microenvironments. Here, we develop two-state micropatterns, consisting of two adhesive sites connected by a thin constriction, in which migrating cells perform repeated stochastic transitions. This minimal system enables us to obtain a large ensemble of single-cell trajectories. From these trajectories, we infer an equation of cell motion, which decomposes the dynamics into deterministic and stochastic contributions in position–velocity phase space. Our results reveal that cells in two-state micropatterns exhibit intricate nonlinear migratory dynamics, with qualitatively similar features for a cancerous (MDA-MB-231) and a non-cancerous (MCF10A) cell line. In both cases, the cells drive themselves deterministically into the thin constriction; a process that is sped up by noise. Interestingly, however, these two cell lines have distinct deterministic dynamics: MDA-MB-231 cells exhibit a limit cycle, while MCF10A cells show excitable bistable dynamics. Our approach yields a conceptual framework that may be extended to understand cell migration in more complex confining environments. Two-state micropatterns offer a unique platform to study cell migration. An equation of motion is inferred from a large ensemble of trajectories, revealing key differences in the nonlinear dynamics of healthy and cancerous cells.

Sensitivity and Resolution Enhancement of Coupled-Core Fluxgate Magnetometer by Negative Feedback

Coupled-core fluxgate magnetometer (CCFM), which is based on the coupling-induced oscillations in bistable nonlinear dynamics, can achieve self-excitation and obtain superior performances compared with the single core fluxgate magnetometer under the premise of utilizing the same residence times difference readout method. One of the pivotal factors determining the performances of CCFM is the critical coupling factor. To get control of it and further enhance the sensor performances, the negative feedback is exploited and implemented by feeding the magnetization from induction coil to excitation coil of the same fluxgate element (self-feedback). Theoretical analyses, Simulink simulations with a quantitative model, and experimental results obtained in the magnetic shielded environment reveal the critical coupling factor adjustment effect of negative feedback on CCFM and confirm the substantial enhancement of sensor performances. Finally, the sensitivity of CCFM for the static magnetic field detection is improved from 0.0253 to $0.7895~\mu \text{s}$ /nT, and the magnetic resolution of CCFM is optimized from 0.395 to 0.049 nT. Therefore, the negative feedback is an effective and reliable method for performances enhancement of CCFM, and it has potential utilization on the oscillation systems with similar dynamic peculiarities for factor adjustment.

Optical bistability and nonlinear dynamics by saturation of cold Yb atoms in a cavity

We observed optical bistability as well as oscillations in the upper bistable branch when cold ytterbium atoms are dispersively coupled to a high finesse optical cavity. Comparable previous observations were explained by atomic density oscillations in case of a Bose-Einstein-Condensate or optical pumping in case of a cold cloud of cesium. Both explanations do not apply in our case of thermal atoms with only a single ground state. We propose a simple two-level model including saturation to describe our experimental results. This paper introduces the experimental setup, derives the mentioned model and compares it to our experimental data. Good quantitative agreement supports our model.

Bi-stable dynamics of a host-pathogen model

Crop host-pathogen interaction have been a main issue for decades, in particular for food security. In this paper, we focus on the modeling and long term behavior of soil-borne pathogens. We first develop a new compartmental temporal model, which exhibits bi-stable asymptotical dynamics. To investigate the long term behavior, we use LaSalle Invariance Principle to derive sufficient conditions for global asymptotic stability of the pathogen free equilibrium and monotone dynamical systems theory to provide sufficient conditions for permanence of the system. Finally, we develop a partially degenerate reaction diffusion system, providing a numerical exploration based on the results obtained for the temporal system. We show that a traveling wave solution may exist where the speed of the wave follows a power law.

A threshold delay model of HIV infection of newborn infants through breastfeeding.

The breast milk of HIV infected women contains HIV virus particles, therefore children can become infected through breastfeeding. We develop a mathematical epidemiological model of HIV infection in infants, infected children and infected women that represents infection of an infant/child as a series of exposures, by incorporating within-host virus dynamics in the individuals exposed to the virus through breastfeeding. We show that repeated exposures of the infant/child via breastfeeding can cause bi-stability dynamics and, subsequently, infection persistence even when the epidemiological basic reproduction number is less than unity. This feature of the model, due to a backward bifurcation, gives new insight into the control mechanisms of HIV disease through breastfeeding.

Bifurcation analysis of a wild and sterile mosquito model.

The bifurcation of an ordinary differential equation model describing interaction of the wild and the released sterile mosquitoes is analyzed. It is shown that the model undergoes a sequence of bifurcations including saddle-node bifurcation, supercritical Hopf bifurcation, subcritical Hopf bifurcation, homoclinic bifurcation and Bogdanov-Takens bifurcation. We also find that the model displays monostable, bistable or tristable dynamics. This analysis suggests that the densities of the initial wild mosquitoes and the released sterile ones determine the asymptotic states of both populations. This study may give an insight into the estimation number of the released sterile mosquitoes.