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The CPG neural system is an important local circuit to control rhythmic movement of vertebrates and invertebrates. The half-central oscillator (HCO), i.e. the building unit of the CPG neural circuit, should present the adjustable in-phase and anti-phase synchronous patterns. In this study, we introduce time delay into a pair of Hindmash-Rose (HR) neuronal model to construct a delayed bursting-HCO (DB-HCO) system. Employing numerical methods including phase-lag and its probability, we determine how the pattern of rhythmic activity of the DB-HCO evolves with time delay. The DB-HCO translates rhythmic activity from anti-phase to in-phase synchrony with adjusting time delay. Further, in the synchrony transition between in-phase and anti-phase states, there exist multiple types of stability coexistence including bi-stable and tri-stable stages. The DB-HCO model switches synchrony states among in-phase, anti-phase, and out-of-phase of the spiking and bursting activities. At last, based on the regulatory mechanism of coupling delay, we propose two models as application examples to replicate rhythmic patterns of CPG systems. The rhythm signals of neuronal cells having the fixed phase difference are reproduced in the pyloric and gastric mill CPGs. The DB-HCO model is regarded as a user-friendly building unit to obtain the expected rhythmic patterns of the whole CPG neural circuit. • Delayed bursting-HCO (DB-HCO) system consisting of a pair of Hindmash-Rose (HR) neuronal model is presented. • The phase-lag and its probability are used to determine the pattern evolution of rhythmic activity with time delay. • The DB-HCO system switches synchrony among in-phase, anti-phase, and out-of-phase of the spiking and bursting activities. • Rhythmic patterns of the pyloric and gastric mill CPG systems are replicated as the application examples. • The DB-HCO model is a user-friendly building unit to construct the expected rhythmic patterns of the CPG neural circuit. [ABSTRACT FROM AUTHOR]

In this paper, we consider a delay-coupled neural oscillator system with inhibitory-to- inhibitory connection. The effects of coupled weights and multiple delays are analyzed. It follows that the increasing coupled weights bring out the multiple coexistences of stable and unstable equilibria by a cascade of pitchfork bifurcations. Moreover, multiple delays control the unstable equilibria to stable periodic solutions. Thus, neural oscillator system exhibits the multiperiodicity coexistences. To this end, we firstly study the equilibria properties and give some equilibria patterns. Analyzing the multiple pitchfork bifurcations of trivial and nontrivial equilibria respectively, we find that there are one, three, five and nine equilibria for the increasing coupled weight. The neural system has a single trivial equilibrium and pairs of synchronous or anti-synchronous nontrivial equilibria, which exhibits the multiple equilibria coexistences. Secondly, employing the corresponding characteristic equation, the effects of coupled delays on stable and unstable equilibria are studied. We find that the stable equilibria keep their stability for the increasing delay, which is called the delay-independent stability. However, for the unstable equilibria, coupled delay suppresses system trajectories near these equilibria into some stable periodic solutions. Therefore, the neural oscillator system exhibits the stability coexistences with multiple equilibria and multiperiodicities. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an inertial two-neural coupling system with multiple delays. We analyze the number of equilibrium points and demonstrate the corresponding pitchfork bifurcation. Results show that the system has a unique equilibrium as well as three equilibria for different values of coupling weights. The local asymptotic stability of the equilibrium point is studied using the corresponding characteristic equation. We find that multiple delays can induce the system to exhibit stable switching between the resting state and periodic motion. Stability regions with delay-dependence are exhibited in the parameter plane of the time delays employing the Hopf bifurcation curves. To obtain the global perspective of the system dynamics, stability and periodic activity involving multiple equilibria are investigated by analyzing the intersection points of the pitchfork and Hopf bifurcation curves, called the Bogdanov-Takens (BT) bifurcation. The homoclinic bifurcation and the fold bifurcation of limit cycle are obtained using the BT theoretical results of the third-order normal form. Finally, numerical simulations are provided to support the theoretical analyses.

In this paper, a time-delayed coupled FitzHugh–Rinzel neuron system (two neurons) with electrical and chemical couplings is discussed. First, when choosing different strengths of electrical (or chemical) coupling, the number and position of equilibria of the neuron system without time delays affected by the variation of chemical (or electrical) coupling are investigated in detail. Next, based on the discussion about the equilibria of the coupled neuron system without time delays, the codimension one bifurcations of equilibria or limit cycles including static bifurcation, Hopf bifurcation and fold bifurcation of limit cycles are deduced when choosing the strength of electrical coupling or chemical coupling as the bifurcation parameter, respectively. Furthermore, the codimension two bifurcations of equilibria are discussed which exhibit more fascinating and complicated dynamical behaviors. Synchronization problems influenced by the strength of electrical coupling are also considered which contain the complete synchronous and asynchronous state. After the discussion of the neuron system without time delays, the impact of time delays on the stability of symmetric equilibria of coupled FitzHugh–Rinzel neuron system is explored. The phenomenon of stability switching by a Hopf bifurcation is well detected. Delay-dependent Hopf curves are found and the dynamical behaviors near the intersection point of Hopf bifurcation (Hopf–Hopf bifurcation point) in the parameter plane of two time delays are investigated where there exists the coexistence of two limit cycles with different frequencies. The numerical simulations are exhibited after the discussion of each part. • The dynamics of a time-delayed coupled FitzHugh-Rinzel neuron system with electrical and chemical couplings is discussed. • The co-existence of equilibria, codimension one and two bifurcations, synchronization problems are considered. • Delay-dependent Hopf curves and dynamical behaviors near the Hopf-Hopf bifurcation point are investigated. [ABSTRACT FROM AUTHOR]

In this paper transition curves and periodic solutions of a parametric pendulum system are calculated analytically by employing the energy method. In previous studies this problem usually was dealt with by using the asymptotic method which is limited by small parameter. In our research, the hypothesis of small number in the pendulum system is not necessary, some different conclusions are obtained on the impacts of nonlinearity in the pendulum system on the transition curves in the parametric plane. The results based on the asymptotic method suggested that nonlinearity in the pendulum system only significantly causes decrease of the area of the stable regions in the parametric plane when the angular displacement of the pendulum is not very small. However, our analysis according to the energy method shows that nonlinearity does not significantly change the area of the stable regions in the parametric plane, but notably alter positions of the stable regions. Furthermore, position of the stable regions to a large extent is related to the amplitude of periodic vibrations of the pendulum especially when the angular displacement of the pendulum is large enough. Our results are very different from that reported in previous studies, which have been verified by numerical simulations. [ABSTRACT FROM AUTHOR]