scholarly journals Stability Analysis of Regular and Chaotic Ca2+ Oscillations in Astrocytes

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Min Ye ◽  
Hongkun Zuo
Keyword(s):  
Equilibrium Points ◽  
Cell Types ◽  
Mechanism Analysis ◽  
Biological Experiment ◽  
Dynamical Mechanism ◽  
Nonlinear Dynamical ◽  
Dynamical Behaviors ◽  
The Stability

Ca2+ oscillations play an important role in various cell types. Thus, understanding the dynamical mechanisms underlying astrocytic Ca2+ oscillations is of great importance. The main purpose of this article was to investigate dynamical behaviors and bifurcation mechanisms associated with astrocytic Ca2+ oscillations, including stability of equilibrium and classification of different dynamical activities including regular and chaotic Ca2+ oscillations. Computation results show that part of the reason for the appearance and disappearance of spontaneous astrocytic Ca2+ oscillations is that they embody the subcritical Hopf and the supercritical Hopf bifurcation points. In more details, we theoretically analyze the stability of the equilibrium points and illustrate the regular and chaotic spontaneous calcium firing activities in the astrocytes model, which are qualitatively similar to actual biological experiment. Then, we investigate the effectiveness and the accuracy of our nonlinear dynamical mechanism analysis via computer simulations. These results suggest the important role of spontaneous Ca2+ oscillations in conjunction with the adjacent neuronal input that may help correlate the connection of both the glia and neuron.

scholarly journals Dynamical Analysis of SIR Epidemic Models with Distributed Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Wencai Zhao ◽  
Tongqian Zhang ◽  
Zhengbo Chang ◽  
Xinzhu Meng ◽  
Yulin Liu
Keyword(s):  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Distributed Delay ◽  
Epidemic Models ◽  
Dynamical Analysis ◽  
Sir Epidemic ◽  
Dynamical Behaviors ◽  
The Stability ◽  
Sir Epidemic Models

SIR epidemic models with distributed delay are proposed. Firstly, the dynamical behaviors of the model without vaccination are studied. Using the Jacobian matrix, the stability of the equilibrium points of the system without vaccination is analyzed. The basic reproduction numberRis got. In order to study the important role of vaccination to prevent diseases, the model with distributed delay under impulsive vaccination is formulated. And the sufficient conditions of globally asymptotic stability of “infection-free” periodic solution and the permanence of the model are obtained by using Floquet’s theorem, small-amplitude perturbation skills, and comparison theorem. Lastly, numerical simulation is presented to illustrate our main conclusions that vaccination has significant effects on the dynamical behaviors of the model. The results can provide effective tactic basis for the practical infectious disease prevention.

Bifurcation Control of Ro¨ssler System

2003 ◽  
Author(s):  
Z. Q. Wu ◽  
P. Yu
Keyword(s):  
Feedback Control ◽  
Nonlinear Dynamical Systems ◽  
Equilibrium Points ◽  
New Method ◽  
Control Law ◽  
Equilibrium Solutions ◽  
Nonlinear Dynamical ◽  
The Stability ◽  
Parameter Values ◽  
Feasible System

In this paper, a new method is proposed for controlling bifurcations of nonlinear dynamical systems. This approach employs the idea used in deriving the transition variety sets of bifurcations with constraints to find the stability region of equilibrium points in parameter space. With this method, one can design, via a feedback control, appropriate parameter values to delay either static, or dynamic or both bifurcations as one wishes. The approach is applied to consider controlling bifurcations of the Ro¨ssler system. The uncontrolled Ro¨ssler has two equilibrium solutions, one of which exhibits static bifurcation while the other has Hopf bifurcation. When a feedback control based on the new method is applied, one can delay the bifurcations and even change the type of bifurcations. An optimal control law is obtained to stabilize the Ro¨ssler system using all feasible system parameter values. Numerical simulations are used to verify the analytical results.

Mathematical Modeling and Stability Analysis of Japanese Encephalitis

2020 ◽  
Vol 12 (1) ◽  
pp. 120-127
Author(s):  
Vinod Baniya ◽  
Ram Keval
Keyword(s):  
Mathematical Modeling ◽  
Japanese Encephalitis ◽  
Disease Transmission ◽  
Equilibrium Points ◽  
Numerical Verification ◽  
Transmission Potential ◽  
Dynamical Behaviors ◽  
Proposed Model ◽  
The Stability ◽  
Stability Issues

Mathematical modeling of Japanese encephalitis (JE) disease in human population with pig and mosquito has been presented in this paper. The proposed model, which involves three compartments of human (Susceptible, Vaccinated, Infected), two compartments of mosquito (Susceptible, Infected) and three compartments of the pig (Susceptible, Vaccinated, Infected). In this work, it is assumed that JE spreads between susceptible class and infected mosquitoes only. Basic results like boundedness of the model, the existence of equilibrium and local stability issues are investigated. Here, to measure the disease transmission potential in the population the basic reproduction number (R0) from the system has been analyzed w.r.t. control parameters both numerically and theoretically. The dynamical behaviors of the system have been analyzed by using the stability theory of differential equations and numerical simulations at equilibrium points. A numerical verification of results is carried out of the model under consideration.

Some stability results for a model of Hepatitis C including alanine aminotransferase and immune system

2020 ◽  
pp. 2050080
Author(s):  
Salvo Danilo Lombardo ◽  
Sebastiano Lombardo
Keyword(s):  
Immune System ◽  
Hepatitis C ◽  
Alanine Aminotransferase ◽  
Equilibrium Points ◽  
Asymptotical Stability ◽  
Direct Acting Antivirals ◽  
The Stability ◽  
Direct Acting ◽  
Stability Results

In clinical practice, many cirrhosis scores based on alanine aminotransferase (ALT) levels exist. Although the most recent direct acting antivirals (DAAs) reduce fibrosis and ALT levels, the Hepatitis C virus (HCV) is not always removed. In this paper, we study a mathematical model of the HCV virus, which takes into account the role of the immune system, to investigate the ALT behavior during therapy. We find five equilibrium points and analyze their stability. A sufficient condition for global asymptotical stability of the infection-free equilibrium is obtained and local asymptotical stability conditions are given for the immune-free infection and cytotoxic T lymphocytes (CTL) response equilibria. The stability of the infection equilibrium with the full immune response is numerically performed.

scholarly journals Symbolic Methods for Analysing Bifurcations and Chaos of Two Five-Parameter Families of Planar Quadratic Maps

2020 ◽  
Vol 8 (2) ◽  
pp. 72-79
Author(s):  
Sarbast H. Mikaeel ◽  
Bewar H. Othman
Keyword(s):  
Stable Equilibrium ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Algorithmic Approach ◽  
Quadratic Maps ◽  
Dynamical Behaviors ◽  
Planar Maps ◽  
Symbolic Methods ◽  
The Stability

In this work, we analyze the dynamical behaviors of two five-parameter families of planar quadratic maps by utilizing strategies of symbolic computation. We are going to use computer algebra methods to clarify how to detect the stability of equilibrium points to analyze chaos and also the bifurcation of planar maps. Based on strategies for solving the systems in types of semi-algebraic and by utilizing an algorithmic approach, we obtain respectively for the two maps, sufficient conditions on the parameters to have a prescribed number of (stable) equilibrium points; necessary conditions on the parameters to undergo a certain type of bifurcation or to have chaotic behavior induced by snapback repeller.

MODELING AND NONLINEAR DYNAMICAL ANALYSIS OF THE ĆUK CONVERTER UNDER FREE-RUNNING CURRENT-PROGRAMMED CONTROL

2008 ◽  
Vol 18 (10) ◽  
pp. 3093-3099 ◽  
Author(s):  
YI-BO ZHAO ◽  
DU-QU WEI ◽  
XIAO-SHU LUO
Keyword(s):  
Sliding Mode ◽  
Variable Structure ◽  
Dynamical Analysis ◽  
Nonlinear Dynamical ◽  
Cuk Converter ◽  
Dynamical Behaviors ◽  
Free Running ◽  
Nonlinear Dynamical Analysis ◽  
Sliding Mode Variable Structure ◽  
The Stability

In this paper, a sliding-mode variable structure model is proposed for studying the nonlinear dynamical behaviors of the free-running current-programmed Ćuk converter. Conditions for the occurrence of various bifurcations are derived analytically, along with simulation results that illustrate the theoretical findings. This paper provides useful information for the stability design of Ćuk converters under free-running current-programmed control and illustrates the application of the nonlinear analysis method to a specific type of plant.

COMPLEX DYNAMICAL ANALYSIS OF A COUPLED NETWORK FROM INNATE IMMUNE RESPONSES

2013 ◽  
Vol 23 (11) ◽  
pp. 1350180 ◽  
Author(s):  
JINYING TAN ◽  
XIUFEN ZOU
Keyword(s):  
Immune Responses ◽  
Negative Feedback ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Innate Immune ◽  
Dynamical Analysis ◽  
Innate Immune Responses ◽  
Dynamical Behaviors ◽  
Positive And Negative Feedback ◽  
The Stability

In this paper, we investigate the complex dynamical behaviors of a biological network that is derived from innate immune responses and that couples positive and negative feedback loops. The stability conditions of the non-negative equilibrium points (EPs) of the system are obtained, using the theory of dynamical systems, and we deduce that no more than three stable EPs exist in this system. Through bifurcation analysis and numerical simulations, we find that the system presents rich dynamical behaviors, such as monostability, bistability and oscillations. These results reveal how positive and negative feedback cooperatively regulate the dynamical behavior of the system.

scholarly journals Stabilization and Analytic Approximate Solutions of an Optimal Control Problem

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 476-487 ◽  
Author(s):  
Camelia Pop ◽  
Camelia Petrişor ◽  
Remus-Daniel Ene
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Equilibrium Points ◽  
Approximate Solutions ◽  
Linear Control ◽  
Runge Kutta Method ◽  
The Stability ◽  
Stability Results

Abstract This paper analyses a dynamical system derived from a left-invariant, drift-free optimal control problem on the Lie group SO(3) × ℝ3 × ℝ3 in deep connection with the important role of the Lie groups in tackling the various problems occurring in physics, mathematics, engineering and economic areas [1, 2, 3, 4, 5]. The stability results for the initial dynamics were inconclusive for a lot of equilibrium points (see [6]), so a linear control has been considered in order to stabilize the dynamics. The analytic approximate solutions of the resulting nonlinear system are established and a comparison with the numerical results obtained via the fourth-order Runge-Kutta method is achieved.

scholarly journals Electrogenic properties of the Na+/K+ATPase control transitions between normal and pathological brain states

2015 ◽  
Vol 113 (9) ◽  
pp. 3356-3374 ◽  
Author(s):  
Giri P. Krishnan ◽  
Gregory Filatov ◽  
Andrey Shilnikov ◽  
Maxim Bazhenov
Keyword(s):  
Epileptic Seizures ◽  
Pump Current ◽  
Progressive Increase ◽  
Neuronal Dynamics ◽  
Dynamical Mechanism ◽  
Dynamical Systems Analysis ◽  
Brain States ◽  
The Stability

Ionic concentrations fluctuate significantly during epileptic seizures. In this study, using a combination of in vitro electrophysiology, computer modeling, and dynamical systems analysis, we demonstrate that changes in the potassium and sodium intra- and extracellular ion concentrations ([K+] and [Na+], respectively) during seizure affect the neuron dynamics by modulating the outward Na+/K+pump current. First, we show that an increase of the outward Na+/K+pump current mediates termination of seizures when there is a progressive increase in the intracellular [Na+]. Second, we show that the Na+/K+pump current is crucial in maintaining the stability of the physiological network state; a reduction of this current leads to the onset of seizures via a positive-feedback loop. We then present a novel dynamical mechanism for bursting in neurons with a reduced Na+/K+pump. Overall, our study demonstrates the profound role of the current mediated by Na+/K+ATPase on the stability of neuronal dynamics that was previously unknown.

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