Network Synchronization of Piecewise-Linear Oscillators with Mean-Field Commutation Function

Synchronous behavior in networks of coupled systems : with applications to neuronal dynamics

This paper studies the stability and boundedness of solutions of certain nonlinear third-order delay differential equations. Sufficient conditions for the stability and boundedness of solutions for the equations considered are obtained by constructing a Lyapunov functional. Mathematical subject classification: 34K20.

The general purpose of this paper is to build up on our understanding of the basic mathematical principles that underlie the emergence of synchronous biological rhythms, in particular, the circadian clock. To do so, we study the role that the coupling strength, coupling type, and noise play in the synchronization of a system of coupled, nonlinear oscillators. First, we study a deterministic model based on Van der Pol coupled oscillators, modeling a population of diffusively coupled cells, to find regions in the parameter space for which synchronous oscillations emerge and to provide conditions under which diffusive coupling kills the synchronous oscillation. Second, we study how noise and coupling interact and lead to synchronous oscillations in linearly coupled oscillators, modeling the interaction between various pacemaker populations, each having an endogenous circadian clock. To do so, we use the Fokker-Planck equation associated to the system. We show how coupling can tune the frequency of the emergent synchronous oscillation, which provides a general mechanism to make fast (ultradian) pacemakers slow (circadian) and synchronous via coupling. The basic mechanisms behind the generation of oscillations and the emergence of synchrony that we describe here can be used to guide further studies of coupled oscillations in biophysical nonlinear models.

This paper is devoted to the mathematical analysis of the stability, boundedness and uniform boundedness of solutions of a class of nonlinear differential equations of second order with multiple constant deviating arguments. We use Lyapunov functionals to verify the stability and boundedness of the solutions and some examples are given to illustrate the theoretical analysis in this work.

We present two qualitative results concerning the solutions of the following equation: ; the first result covers the stochastic asymptotic stability of the zero solution for the above equation in case p ≡ 0, while the second one discusses the uniform stochastic boundedness of all solutions in case p≢0. Sufficient conditions for the stability and boundedness of solutions for the considered equation are obtained by constructing a Lyapunov functional. Two examples are also discussed to illustrate the efficiency of the obtained results.

Recently, vibration energy harvesting has been seen as a viable energy source to provide for our energy dependent society. Researchers have studied systems ranging from civil structures like bridges to biomechanical systems including human motion as potential sources of vibration energy. In this work, a bench-top system of a piecewise-linear (PWL) nonlinear vibration harvester is studied. A similar idealized model of the harvester was previously looked at numerically, and in this work the method is adjusted to handle physical systems to construct a realistic harvester design. With the physically realizable harvester design, the resonant frequency of the system is able to be tuned by changing the gap size between the oscillator and mechanical stopper, ensuring optimal performance over a broad frequency range. Current nonlinear harvester designs show decreased performance at certain excitation conditions, but this design overcomes these issues while also still maintaining the performance of a linear harvester at resonance. In this investigation, the system is tested at various excitation conditions and gap sizes. The computational response of the resonance behavior of the PWL system is validated against the experiments. Additionally, the electromechanical response is also validated with the experiments by comparing the output power generated from the experiments with the computational prediction.

Motivated by the recent interests in implementing both grid-forming (GFM) and grid-following (GFL) inverters in the grid, this work studies the stability of such inverter-dominated systems by analyzing the equivalent model using networked oscillators. The equivalent oscillator models for both GFM and GFL inverters are developed, respectively, which characterize the behavior of the electric network of inverters and loads. The synchronization of the networked oscillators is studied. It is proved that such a system has a set of unique and locally stable operating states when it is synchronized. Moreover, a necessary condition for the system to reach synchronization is derived, which provides a large-signal stability assessment for the system. At last, the proposed work is validated using simulation results.

Synchronized oscillations in networks of inhibitory and excitatory coupled bursting neurons are common in a variety of neural systems from central pattern generators to human brain circuits. One example of the latter is the subcortical network of the basal ganglia, formed by excitatory and inhibitory bursters of the subthalamic nucleus and globus pallidus, involved in motor control and affected in Parkinson’s disease. Recent experiments have demonstrated the intermittent nature of the phase-locking of neural activity in this network. Here, we explore one potential mechanism to explain the intermittent phase-locking in a network. We simplify the network to obtain a model of two inhibitory coupled elements and explore its dynamics. We used geometric analysis and singular perturbation methods for dynamical systems to reduce the full model to a simpler set of equations. Mathematical analysis was completed using three slow variables with two different time scales. Intermittently, synchronous oscillations are generated by overlapped spiking which crucially depends on the geometry of the slow phase plane and the interplay between slow variables as well as the strength of synapses. Two slow variables are responsible for the generation of activity patterns with overlapped spiking, and the other slower variable enhances the robustness of an irregular and intermittent activity pattern. While the analyzed network and the explored mechanism of intermittent synchrony appear to be quite generic, the results of this analysis can be used to trace particular values of biophysical parameters (synaptic strength and parameters of calcium dynamics), which are known to be impacted in Parkinson’s disease.

A scalar nonlinear integro-differential equation with time-variable and bounded delays and generalized Caputo proportional fractional derivative is considered. The main goal of this paper is to study the stability properties of the zero solution. Results are given concerning stability, exponential stability, asymptotic stability, and boundedness of solutions. The investigations are based on an application of a quadratic Lyapunov function, its generalized Caputo proportional derivative, and a modification of the Razumikhin approach. Some auxiliary properties of the generalized Caputo proportional derivative are proved. Five illustrative examples are included.

In the paper definitions of various kinds of stability and boundedness of solutions of linear controllable systems of partial differential equations are introduced and their interconnections are derived. By means of Ljapunov's functions theorems are proved which give necessary and sufficient conditions for particular kinds of stability and boundedness of the solutions.

In this paper, we consider a certain non-linear Volterra integro-differential equations with delay. We study stability and boundedness of solutions. The technique of proof involves defining suitable Lyapunov functionals. Our results improve and extend the results obtained in literature.

Dynamics of a tumor-immune model considering targeted chemotherapy

It is usually believed that strong diffusive coupling in one of the dynamical variables is well-suited for imposing synchronization of oscillators. But it was recently shown that weak diffusive coupling, counter-intuitively, can lead to dephasing of coupled neural oscillators. In this paper, we investigate how diffusively coupled oscillators become dephasing. For this we study a system of coupled neural oscillators on a limit cycle generated through a homoclinic bifurcation. We examine the asymptotic behavior of diffusive coupling as the control parameter approaches the critical value for which the homoclinic bifurcation occurs. In this study, we show that the gradient of phase velocity near the limit cycle is essential in generating dephasing through diffusive interaction.

Existence and exponential stability of periodic solution for BAM neural networks with periodic coefficients and delays

In this work, we first establish a new generalized Halanay inequality with impulses, which have two advantages: one is that impulses can be applied to stabilize the unstable continuous system, the other is that the requirements on the coefficients of inequality are more relaxed than those in previous works. Then, by using the inequality and Banach fixed point theorem, we will obtain some sufficient conditions ensuring the existence and exponential stability of periodic solution for impulsive delay differential equations. The sufficient conditions are easily checked in practice and have a wider adaptive range. Some examples are given to illustrate our results.