Amplitude death in nonlinear oscillators with mixed time-delayed coupling

Using critical curves to compute master stability islands for amplitude death in networks of delay-coupled oscillators

In this paper, we present a method to compute master stability islands (MSIs) for amplitude death in networks of delay-coupled oscillators using critical curves. We first demonstrate how critical curves can be used to compute boundaries and contours of MSIs in delay-coupling parameter space and then provide a general study on the effects of the oscillator dynamics and network topology on the number, size, and contour types of all MSIs. We find that the oscillator dynamics can be used to determine the number and size of MSIs and that there are six possible contour types that depend on the choice of oscillator dynamics and the network topology. We introduce contour sequences and use these sequences to study the contours of all MSIs. Finally, we provide example MSIs for several classical nonlinear systems including the van der Pol system, the Rucklidge system, and the Rössler system.

Fast regular firings induced by intra- and inter-time delays in two clustered neuronal networks

In this paper, we consider two clustered neuronal networks with dense intra-synaptic links within each cluster and sparse inter-synaptic links between them. We focus on the effects of intra- and inter-time delays on the spiking regularity and timing in both clusters. With the aid of simulation results, we show that intermediate intra- and inter-time delays are able to separately induce fast regular firing - spiking activity with a high firing rate as well as a high spiking regularity. Moreover, when both intra- and inter-time delays are present, we find that fast regular firings are induced much more frequently than if only a single type of delay is present in the system. Our results indicate that appropriately adjusted intra- and inter-time delays can significantly facilitate fast regular firing in neuronal networks. Based on a detailed analysis, we conjecture that this is most likely when the largest value of common divisors of the intra- and inter-time delays falls into a range where fast regular firings are induced by suitable intra- or inter-time delays alone.

Mathematical Modeling for Description of Oscillation Suppression Induced by Deep Brain Stimulation

A mathematical modeling for description of oscillation suppression by deep brain stimulation (DBS) is explored in this paper. High-frequency DBS introduced to the basal ganglia network can suppress pathological neural oscillations that occur in the Parkinsonian state. However, selecting appropriate stimulation parameters remains a challenging issue due to the limited understanding of the underlying mechanisms of the Parkinsonian state and its control. In this paper, we use a describing function analysis to provide an intuitive way to select the optimal stimulation parameters based on a biologically plausible computational model of the Parkinsonian neural network. By the stability analysis using the describing function method, effective DBS parameter regions for inhibiting the pathological oscillations can be predicted. Additionally, it is also found that a novel sinusoidal-shaped DBS may become an alternative stimulation pattern and expends less energy, but with a different mechanism. This paper provides new insight into the possible mechanisms underlying DBS and a prediction of optimal DBS parameter settings, and even suggests how to select novel DBS wave patterns for the treatment of movement disorders, such as Parkinson's disease.

Design of coupling parameters for inducing amplitude death in Cartesian product networks of delayed coupled oscillators

The present study investigates amplitude death in Cartesian product networks of two subnetworks, where each subnetwork has a different coupling delay. The property of the Cartesian product helps us to analyze the stability of amplitude death. Our analysis reveals that amplitude death can occur for long coupling delays if there is a suitable difference in the coupling delays in the two subnetworks. Furthermore, based on the edge theorem in robust control theory, we propose two design procedures of coupling parameters for inducing amplitude death in the Cartesian product networks. Our procedures do not require any information of topologies of the subnetworks. The validity of these procedures is numerically confirmed.

The impact of propagation and processing delays on amplitude and oscillation deaths in the presence of symmetry-breaking coupling

We numerically investigate the impacts of both propagation and processing delays on the emergences of amplitude death (AD) and oscillation death (OD) in one system of two Stuart-Landau oscillators with symmetry-breaking coupling. In either the absence of or the presence of propagation delay, the processing delay destabilizes both AD and OD by revoking the stability of the stable homogenous and inhomogenous steady states. In the AD to OD transition, the processing delay destabilizes first OD from large values of coupling strength until its stable regime completely disappears and then AD from both the upper and lower bounds of the stable coupling interval. Our numerical study sheds new insight lights on the understanding of nontrivial effects of time delays on dynamic activity of coupled nonlinear systems.

Aging transition in systems of oscillators with global distributed-delay coupling

We consider a globally coupled network of active (oscillatory) and inactive (nonoscillatory) oscillators with distributed-delay coupling. Conditions for aging transition, associated with suppression of oscillations, are derived for uniform and gamma delay distributions in terms of coupling parameters and the proportion of inactive oscillators. The results suggest that for the uniform distribution increasing the width of distribution for the same mean delay allows aging transition to happen for a smaller coupling strength and a smaller proportion of inactive elements. For gamma distribution with sufficiently large mean time delay, it may be possible to achieve aging transition for an arbitrary proportion of inactive oscillators, as long as the coupling strength lies in a certain range.

Master stability islands for amplitude death in networks of delay-coupled oscillators

This paper presents a master stability function (MSF) approach for analyzing the stability of amplitude death (AD) in networks of delay-coupled oscillators. Unlike the familiar MSFs for instantaneously coupled networks, which typically have a single input encoding for the effects of the eigenvalues of the network Laplacian matrix, for delay-coupled networks we show that such MSFs generally require two additional inputs: the time delay and the coupling strength. To utilize the MSF for determining the stability of AD of general networks for a chosen nonlinear system (node dynamics) and coupling function, we introduce the concept of master stability islands (MSIs), which are two-dimensional stability islands of the delay-coupling parameter space together with a third dimension ("altitude") encoding for eigenvalues that result in stable AD. We numerically compute the MSFs and visualize the corresponding MSIs for several common chaotic systems including the Rössler, the Lorenz, and Chen's system and find that it is generally possible to achieve AD and that a nonzero time delay is necessary for the stabilization of the AD states.

Feedback as a mechanism for the resurrection of oscillations from death states

The quenching of oscillations in interacting systems leads to several unwanted situations, which necessitate a suitable remedy to overcome the quenching. In this connection, this work addresses a mechanism that can resurrect oscillations in a typical situation. Through both numerical and analytical studies, we show that the candidate which is capable of resurrecting oscillations is nothing but the feedback, the one which is profoundly used in dynamical control and in biotherapies. Even in the case of a rather general system, we demonstrate analytically the applicability of the technique over one of the oscillation quenched states called amplitude death states. We also discuss some of the features of this mechanism such as adaptability of the technique with the feedback of only a few of the oscillators.

Exceptional points in coupled dissipative dynamical systems

We study the transient behavior in coupled dissipative dynamical systems based on the linear analysis around the steady state. We find that the transient time is minimized at a specific set of system parameters and show that at this parameter set, two eigenvalues and two eigenvectors of the Jacobian matrix coalesce at the same time; this degenerate point is called the exceptional point. For the case of coupled limit-cycle oscillators, we investigate the transient behavior into the amplitude death state, and clarify that the exceptional point is associated with a critical point of frequency locking, as well as the transition of the envelope oscillation.

Design of time-delayed connection parameters for inducing amplitude death in high-dimensional oscillator networks

The present paper studies time-delayed-connection induced amplitude death in high-dimensional oscillator networks. We provide two procedures for design of a coupling strength and a transmission delay: these procedures do not depend on the topology of oscillator networks (i.e., network structure and number of oscillators). A graphical procedure based on the Nyquist criterion is proposed and then is numerically confirmed for the case of five-dimensional oscillators, called generalized Rössler oscillators, which have two pairs of complex conjugate unstable roots. In addition, for the case of high-dimensional oscillators having two unstable roots, the procedure can be systematically carried out using only a simple algebraic calculation. This systematic procedure is numerically confirmed for the case of three-dimensional oscillators, called Moore-Spiegel oscillators, which have two positive real unstable roots.

Time-delay effects on the aging transition in a population of coupled oscillators

We investigate the influence of time-delayed coupling on the nature of the aging transition in a system of coupled oscillators that have a mix of active and inactive oscillators, where the aging transition is defined as the gradual loss of collective synchrony as the proportion of inactive oscillators is increased. We start from a simple model of two time-delay coupled Stuart-Landau oscillators that have identical frequencies but are located at different distances from the Hopf bifurcation point. A systematic numerical and analytic study delineates the dependence of the critical coupling strength (at which the system experiences total loss of synchrony) on time delay and the average distance of the system from the Hopf bifurcation point. We find that time delay can act to facilitate the aging transition by lowering the threshold coupling strength for amplitude death in the system. We then extend our study to larger systems of globally coupled active and inactive oscillators including an infinite system in the thermodynamic limit. Our model system and results can provide a useful paradigm for understanding the functional robustness of diverse physical and biological systems that are prone to aging transitions.