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The synchronization between two coupled FitzHugh-Nagumo (FHN) neurons with or without external current is studied by using the Laplace transform and the Adomian decomposition method. Different from other researches, the synchronization error system is expressed as sets of Volterra integral equations based on the convolution theorem in the Laplace transform. Then, it is easy to analytically obtain the conditions that synchronization errors disappear based on the successive approximation method in integral equation theorem, the correctness of which is verified by numerical simulations. Furthermore, the synchronous dynamics of the two coupled FHN neurons also can be written in the form of Volterra integral equations, which is more convenient to analytically solve by using the Adomian decomposition method. It is found that the occurrence of synchronization between the two FHN neurons only depends on the coupling strength and is irrelevant to the external current. Only synchronous rest state in the two FHN neurons without external current can be achieved, while synchronous spikes appear if the external current is not zero. [ABSTRACT FROM AUTHOR]

In this paper, an inertial two-neuron system with multiple delays is analyzed to exhibit the effect of time delays on system dynamics. The parameter region with multiple equilibria is obtained employing the pitchfork bifurcation of trivial equilibrium. The stability analysis illustrates that two nontrivial equilibria are both stable for any delays. It implies that the neural system exhibits a stability coexistence of two resting states. Further, due to the existence of multiple delays, the neural system has a periodic activity around the trivial equilibrium via Hopf bifurcation. Finally, numerical simulations are employed to illustrate many richness coexistence for multitype activity patterns. Employing the period-adding route and fold bifurcation of periodic orbit, the neural system may have multistability coexistence of two resting states, two ASP-3s (anti-symmetric periodic activity with period three), one SSP-1 (self-symmetric periodic activity with period one), and one quasi-periodic spiking. Additionally, with increasing delay, quasi-periodic spiking evolves into chaos behavior. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a neural network system composed of two delay-coupled neural oscillators, where each of these can be regarded as the dynamical system describing the average activity of neural population. Analyzing the corresponding characteristic equation, the local stability of rest state is studied. The system exhibits the switch phenomenon between the rest state and periodic activity. Furthermore, the Hopf bifurcation is analyzed and the bifurcation curve is given in the parameters plane. The stability of the bifurcating periodic solutions and direction of the Hopf bifurcation are exhibited. Regarding time delay and coupled weight as the bifurcation parameters, the Fold–Hopf bifurcation is investigated in detail in terms of the central manifold reduction and normal form method. The neural system demonstrates the coexistence of the rest states and periodic activities in the different parameter regions. Employing the normal form of the original system, the coexistence regions are illustrated approximately near the Fold–Hopf singularity point. Finally, numerical simulations are performed to display more complex dynamics. The results illustrate that system may exhibit the rich coexistence of the different neuro-computational properties, such as the rest states, periodic activities, and quasi-periodic behavior. In particular, some periodic activities can evolve into the bursting-type behaviors with the varying time delay. It implies that the coexistence of the quasi-periodic activity and bursting-type behavior can be obtained if the suitable value of system parameter is chosen. [Copyright &y& Elsevier]

In this paper, some complex phenomena of dynamical bifurcations are shown for a two-neural network with delay coupling. The sigmoid activation function with slope ratio, a monotonically increasing function, is proposed to consider the relations of the sigmoid and Hardlim functions. The equilibrium points are studied analytically in detail in terms of the characteristic equation and static bifurcation. The central manifold reduction and normal form method are employed to determine Hopf bifurcation and its stability. The stable equilibrium points and periodic motions are observed in different parameter regions. Effects of slope ratio and delayed coupling on dynamic behaviors are investigated by the numerical simulation, such as Poincare map, phase portraits, and power spectrum. Various active transitions to chaos and the corresponding critical boundaries on the focused parameter regions are obtained to classify dynamical behaviors including stable equilibrium point, periodic solution, 2-torus, 3-torus, and then the chaotic motions. [ABSTRACT FROM AUTHOR]

Bursting behavior is one of the most important firing activities of neural system and plays an important role in signal encoding and transmission. In the present paper, a neural network with delay coupling is modeled to investigate the generation mechanism of bursting behavior. The Andronov-Hopf bifurcation is firstly studied and then the degenerated Andronov-Hopf bifurcation, namely Bautin bifurcation, is analyzed with the external input varying. Classifying dynamics in the neighborhood of the Bautin bifurcation, we obtain the bifurcation sets where the supercritical/subcritical Andronov-Hopf, or the fold limit cycle bifurcation may happen in the system under consideration. For a periodic disturbance occurring in the neighborhood of the Bautin bifurcation, it is seen that the Andronov-Hopf bifurcation and fold limit cycle bifurcation may lead to the transition from quiescent state to firing activities. Complex bursting phenomena, including Hopf/Hopf bursting, Hopf/Fold cycle bursting, SubHopf/Hopf bursting and SubHopf/Fold cycle bursting are found in the firing area. The results show that the dynamical properties of different burstings are related to the dynamical behaviors derived from the bifurcations of the system. Finally, it is seen that the bursting disappears but the periodic spiking appears in the delayed neural network for large values of delay. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies the existence and stability of the disease-free equilibrium and endemic equilibria for the SIRS epidemic model with the saturated incidence rate, considering the factor of population dynamics such as the disease-related, the natural mortality and the constant recruitment of population. Analytical techniques are used to show, for some parameter values, the periodic solutions can arise through the Hopf bifurcation, which is important to carry different strategies for the controlling disease. Then the codimension-two bifurcation, i.e. BT bifurcation, is investigated by using a global qualitative method and the curves of saddle-node bifurcation, Hopf bifurcation and homoclinic bifurcation are obtained at the degenerate equilibrium. Moreover, several numerical simulations are given to support the theoretical analysis. [Copyright &y& Elsevier]

In this paper, we consider a delayed Hopfield two-neural system with a monotonic activation function and find the periodic coexistence by bifurcation analysis. Firstly, we obtain the pitchfork bifurcation of the trivial equilibrium employing the central manifold and normal form methods. The neural system exhibits two pitchfork bifurcations near the trivial equilibrium. Then, analyzing the characteristic equation of the nontrivial equilibrium, we illustrate the saddle-node bifurcation of the nontrivial equilibria. The system exhibits the multi-coexistences of the stable and unstable equilibria. Further, we illustrate the plane regions of parameters having different numbers of equilibria. To obtain a time delay in neural system dynamics, we present the stability analysis and find the periodic orbit. The system exhibits stability switching by the Hopf bifurcation curves. Finally, the dynamic behaviors near the Hopf–Hopf bifurcation point are presented. The system exhibits coexistence of multiple periodic orbits with different frequencies. [ABSTRACT FROM AUTHOR]

In this paper, an analytical criterion is proposed to investigate the synchronization between two Hindmarsh-Rose neurons with linear and nonlinear coupling functions based on the Laplace transform method. Different from previous works, the synchronization error system is expressed in its integral form, which is more convenient to analyze. The synchronization problem of two HR coupled neurons is ultimately converted into the stability problem of roots to a nonlinear algebraic equation. Then, an analytical criterion for synchronization between the two HR neurons can be given by using the Routh-Hurwitz criterion. Numerical simulations show that the synchronization criterion derived in this paper is valid, regardless of the periodic spikes or burst-spike chaotic behavior of the two HR neurons. Furthermore, the analytical results have almost the same accuracy as the conditional Lyapunov method. In addition, the calculation quantities always are small no matter the linear and nonlinear coupling functions, which show that the approach presented in this paper is easy to be developed to study synchronization between a large number of HR neurons. [ABSTRACT FROM AUTHOR]

Complexity and dynamical analysis in neural systems play an important role in the application of optimization problem and associative memory. In this paper, we establish a delayed neural system with external stimulations. The complex dynamical behaviors induced by external simulations are investigated employing theoretical analysis and numerical simulation. Firstly, we illustrate number of equilibria by the saddle-node bifurcation of nontrivial equilibria. It implies that the neural system has one/three equilibria for the external stimulation. Then, analyzing characteristic equation to find Hopf bifurcation, we obtain the equilibrium's stability and illustrate periodic activity induced by the external stimulations and time delay. The neural system exhibits a periodic activity with the increased delay. Further, the external stimulations can induce and suppress the periodic activity. The system dynamics can be transformed from quiescent state (i.e., the stable equilibrium) to periodic activity and then quiescent state with stimulation increasing. Finally, inspired by ubiquitous rhythm in living organisms, we introduce periodic stimulations with low frequency as rhythm activity from sensory organs and other regions. The neural system subjected by the periodic stimulations exhibits some interesting activities, such as periodic spiking, subthreshold oscillation, and bursting-like activity. Moreover, the subthreshold oscillation can switch its position with delay increasing. The neural system may employ time delay to realize Winner-Take-All functionality. [ABSTRACT FROM AUTHOR]

Understanding internal vortex breakdowns (VBs) and their trajectories in sealed cylinders are important for the scientific and industrial applications with which they are linked. However, the fluids in sealed cylinders are often sheared, which makes it difficult to clearly and multidirectionally observe their internal flow patterns simultaneously with existing experimental tools; this results in some important features not being captured. In this work, we performed thousands of numerical simulations in a sealed cylinder utilizing the finite element approach. Abundant internal VB patterns were obtained for different aspect ratios as the Reynolds number (Re) increased. To further quantitatively study the morphological evolution of VBs, we focused on the axial lengths and trajectories of VBs with multiple aspect ratios. Surprisingly, the numbers of VBs in the rotating fluid were not fixed for the same aspect ratio, which also affected the complexity of the VB evolution. In particular, the stair-step changes of the locations of the VB and local extrema of the axial velocity, pressure, and vorticity of the key flows at the axis were revealed in detail. We used the theory of swirl decay to explain the VB formation and stair-step change from an energy perspective and clarified why the pressure minimum was under the center of the VB. The discovery of the stair-step change of the VB provided evidence of the existence of a new class of fluid behavior that may provide insight into vortex control. [ABSTRACT FROM AUTHOR]