Basin stability for chimera states

Dynamics, synchronization and traveling wave patterns of flux coupled network of Chay neurons

Studies in the literature have demonstrated the significance of the synchronization of neuronal electrical activity for signal transmission and information encoding. In light of this importance, we investigate the synchronization of the Chay neuron model using both theoretical analysis and numerical simulations. The Chay model is chosen for its comprehensive understanding of neuronal behavior and computational efficiency. Additionally, we explore the impact of electromagnetic induction, leading to the magnetic flux Chay neuron model. The single neuron model exhibits rich and complex dynamics for various parameter choices. We explore the bifurcation structure of the model through bifurcation diagrams and Lyapunov exponents. Subsequently, we extend our study to two coupled magnetic flux Chay neurons, identifying mode locking and structures reminiscent of Arnold's tongue. We evaluate the stability of the synchronized manifold using Lyapunov theory and confirm our findings through simulations. Expanding our study to networks of diffusively coupled flux Chay neurons, we observe coherent, incoherent, and imperfect chimera patterns. Our investigation of three network types highlights the impact of network topology on the emergent dynamics of the Chay neuron network. Regular networks exhibit diverse patterns, small-world networks demonstrate a critical transition to coherence, and random networks showcase synchronization at specific coupling strengths. These findings significantly contribute to our understanding of the synchronization patterns exhibited by the magnetic flux Chay neuron. To assess the synchronization stability of the Chay neuron network, we employ master stability function analysis.

Chimera states in a large laterally coupled laser array with four different waveguide structures

Chimera states are rich and fascinating phenomena existing in many networks, where the identical oscillators self-organize into spatially separated coexisting domains of coherent and incoherent oscillations. Here, we report these states in the large laterally coupled laser array with four different waveguiding structures, with which a variety of chimera patterns can be revealed. We present the bifurcation diagrams giving birth to them and find that the chimeras exist in the boundary of the steady state and multi-period oscillation solutions, which applies to all the prevalent waveguiding structures considered. We also find that the waveguiding structures play an important role in the chimera states, e.g., the array composed of the index antiguiding with gain-guiding has a wider chimera region compared to other waveguides considered. Additionally, the effects of the crucial parameters including the laser separation ratio, pump rate, frequency detuning, and linewidth enhancement factor on the observed phenomena are discussed. Our analysis shows that the frequency detuning between lasers and the linewidth enhancement factor affects the lifetime and pattern of chimeras. The results could guide the design of laser arrays or introduce more insight into a new understanding of the dynamical behaviors of networks.

Short-lived chimera states

In the classic Kuramoto system of coupled two-dimensional rotators, chimera states characterized by the coexistence of synchronous and asynchronous groups of oscillators are long-lived because the average lifetime of these states increases exponentially with the system size. Recently, it was discovered that, when the rotators in the Kuramoto model are three-dimensional, the chimera states become short-lived in the sense that their lifetime scales with only the logarithm of the dimension-augmenting perturbation. We introduce transverse-stability analysis to understand the short-lived chimera states. In particular, on the unit sphere representing three-dimensional (3D) rotations, the long-lived chimera states in the classic Kuramoto system occur on the equator, to which latitudinal perturbations that make the rotations 3D are transverse. We demonstrate that the largest transverse Lyapunov exponent calculated with respect to these long-lived chimera states is typically positive, making them short-lived. The transverse-stability analysis turns the previous numerical scaling law of the transient lifetime into an exact formula: the "free" proportional constant in the original scaling law can now be precisely determined in terms of the largest transverse Lyapunov exponent. Our analysis reinforces the speculation that in physical systems, chimera states can be short-lived as they are vulnerable to any perturbations that have a component transverse to the invariant subspace in which they live.

Higher-order interactions promote chimera states

Since the discovery of chimera states, the presence of a nonzero phase lag parameter turns out to be an essential attribute for the emergence of chimeras in a nonlocally coupled identical Kuramoto phase oscillators' network with pairwise interactions. In this Letter, we report the emergence of chimeras without phase lag in a nonlocally coupled identical Kuramoto network owing to the introduction of nonpairwise interactions. The influence of added nonlinearity in the coupled system dynamics in the form of simplicial complexes mitigates the requisite of a nonzero phase lag parameter for the emergence of chimera states. Chimera states stimulated by the reciprocity of the pairwise and nonpairwise interaction strengths and their multistable nature are characterized with appropriate measures and are demonstrated in the parameter spaces.

Applying interval stability concept to empirical model of middle Pleistocene transition

Interval stability is a novel method for the study of complex dynamical systems, allowing for the estimation of their stability to strong perturbations. This method describes how large perturbation should be to disrupt the stable dynamical regime of the system (attractor). In our work, interval stability is used for the first time to study the properties of a real natural system: to analyze the stability of the earth's climate system during the last 2.6×10⁶ years. The main abrupt shift in global climate during this period is the middle Pleistocene transition (MPT), which occurred about 1×10⁶ years ago as a change of the periodicity of glacial cycles from 41 to 100 kyr. On the basis of the empirical nonlinear stochastic model proposed in our recent work, we demonstrate that the global climate stability to any perturbations decreases throughout the Pleistocene period (including the MPT), enhancing its response to fast (with a millennial scale or less) internal disturbances.

Basins with Tentacles

To explore basin geometry in high-dimensional dynamical systems, we consider a ring of identical Kuramoto oscillators. Many attractors coexist in this system; each is a twisted periodic orbit characterized by a winding number q, with basin size proportional to e^{-kq^{2}}. We uncover the geometry behind this size distribution and find the basins are octopuslike, with nearly all their volume in the tentacles, not the head of the octopus (the ball-like region close to the attractor). We present a simple geometrical reason why basins with tentacles should be common in high-dimensional systems.

Spiral wave chimera-like transient dynamics in three-dimensional grid of diffusive ecological systems

In the present article, we demonstrate the emergence and existence of the spiral wave chimera-like transient pattern in coupled ecological systems, composed of prey-predator patches, where the patches are connected in a three-dimensional medium through local diffusion. We explore the transition scenarios among several collective dynamical behaviors together with transient spiral wave chimera-like states and investigate the long time behavior of these states. The transition from the transient spiral chimera-like pattern to the long time synchronized or desynchronized pattern appears through the deformation of the incoherent region of the spiral core. We discuss the transient dynamics under the influence of the species diffusion at different time instants. By calculating the instantaneous strength of incoherence of the populations, we estimate the duration of the transient dynamics characterized by the persistence of the chimera-like spatial coexistence of coherent and incoherent patterns over the spatial domain. We generalize our observations on the transient dynamics in a three-dimensional grid of diffusive ecological systems by considering two different prey-predator systems.

Controlling chimera states in chaotic oscillator ensembles through linear augmentation

In this work, we show how "chimera states," namely, the dynamical situation when synchronized and desynchronized domains coexist in an oscillator ensemble, can be controlled through a linear augmentation (LA) technique. Specifically, in the networks of coupled chaotic oscillators, we obtain chimera states through induced multistability and demonstrate how LA can be used to control the size and spatial location of the incoherent and coherent populations in the ensemble. We examine basins of attraction of the system to analyze the effects of LA on its multistable behavior and thus on chimera states. Stability of the synchronized dynamics is analyzed through a master stability function. We find that these results are independent of a system's initial conditions and the strategy is applicable to the networks of globally, locally as well as nonlocally coupled oscillators. Our results suggest that LA control can be an effective method to control chimera states and to realize a desired collective dynamics in such ensembles.

Local connectivity effects in learning and coordination dynamics in a two-layer network

Anticoordination and chimera states occur in a two-layer model of learning and coordination dynamics in fully connected networks. Learning occurs in the intra-layer networks, while a coordination game is played in the inter-layer network. In this paper, we study the robustness of these states against local effects introduced by the local connectivity of random networks. We identify threshold values for the mean degree of the networks such that below these values, local effects prevent the existence of anticoordination and chimera states found in the fully connected setting. Local effects in the intra-layer learning network are more important than in the inter-layer network in preventing the existence of anticoordination states. The local connectivity of the intra- and inter-layer networks is important to avoid the occurrence of chimera states, but the local effects are stronger in the inter-layer coordination network than in the intra-layer learning network. We also study the effect of local connectivity on the problem of equilibrium selection in the asymmetric coordination game, showing that local effects favor the selection of the Pareto-dominant equilibrium in situations in which the risk-dominant equilibrium is selected in the fully connected network. In this case, again, the important local effects are those associated with the coordination dynamics inter-layer network. Indeed, lower average degree of the network connectivity between layers reduces the probability of achieving the risk-dominant strategy, favoring the Pareto-dominant equilibrium.

Basin of attraction for chimera states in a network of Rössler oscillators

Chimera states are spatiotemporal patterns in which coherent and incoherent dynamics coexist simultaneously. These patterns were observed in both locally and nonlocally coupled oscillators. We study the existence of chimera states in networks of coupled Rössler oscillators. The Rössler oscillator can exhibit periodic or chaotic behavior depending on the control parameters. In this work, we show that the existence of coherent, incoherent, and chimera states depends not only on the coupling strength, but also on the initial state of the network. The initial states can belong to complex basins of attraction that are not homogeneously distributed. Due to this fact, we characterize the basins by means of the uncertainty exponent and basin stability. In our simulations, we find basin boundaries with smooth, fractal, and riddled structures.

Minimal fatal shocks in multistable complex networks

Multistability is a common phenomenon which naturally occurs in complex networks. Often one of the coexisting stable states can be identified as being the desired one for a particular application. We present here a global approach to identify the minimal perturbation which will instantaneously kick the system out of the basin of attraction of its desired state and hence induce a critical or fatal transition we call shock-tipping. The corresponding Minimal Fatal Shock is a vector whose length can be used as a global stability measure and whose direction in state space allows us to draw conclusions on weaknesses of the network corresponding to critical network motifs. We demonstrate this approach in plant-pollinator networks and the power grid of Great Britain. In both system classes, tree-like substructures appear to be the most vulnerable with respect to the minimal shock perturbation.

Multistability and basin stability in coupled pendulum clocks

In this paper, we investigate the phenomenon of multistability and the concept of basin stability in two coupled pendula with escapement mechanisms, suspended on horizontally oscillating beam. The dynamics of a single pendulum clock is studied and described, showing possible responses of the unit. The basin stability maps are discussed in two-parameters plane, where we vary both the system's stiffness as well as the damping. The possible attractors for the investigated clocks are discussed, showing that different patterns of synchronization and desynchronization can occur. The oscillators may completely synchronize in one of the three possible combinations (including inphase and antiphase ones), practically synchronize with some fluctuations or stay in the irregular pattern, which includes chaotic motion. The transitions between solutions are studied, uncovering that the road from one type of dynamics into another may become very complex. Moreover, we examine the multistability property of our model using the bifurcation diagrams and the basins of attraction maps, discussing possible scenarios in which the states co-exist. The analysis of attractors' basins uncovers complicated structure of the latter ones, exhibiting that the final behavior of investigated model may be hard to determine and trace. Our results are discussed for the cases of identical and nonidentical pendula, as well as light and heavy beam, showing that depending on considered scenario, various patterns of behaviors and transitions may be observed. The research described in this paper proves that the mechanical properties of the system's suspension may play a crucial role in the possibility of the appearance of different types of attractors and that the basin stabilities of states strictly depend on the values of considered parameters.

Chimera state in complex networks of bistable Hodgkin-Huxley neurons

In this paper we study a chimera state in complex networks of bistable Hodgkin-Huxley neurons with excitatory coupling, which manifests as a termination of spiking activity of a part of interacting neurons. We provide a detailed investigation of this phenomenon in scale-free, small-world, and random networks and show that the chimera state is robust to the network topology. Nevertheless, network topological properties determine the stability of spatiotemporal states and therefore affect the excitability of the chimera state in the whole network. In particular, the scale-free network whose higher degree nodes are more stable to small perturbations is least exposed to chimera formation and exhibits an abrupt transition from a spiking to a silent regime. On the other hand, small-world and random networks are more likely to provide transitions to the chimera state.

Transitions from chimeras to coherence: An analytical approach by means of the coherent stability function

Chimera states have been a vibrant subject of research in the recent past, but the analytical treatment of transitions from chimeras to coherent states remains a challenge. Here we analytically derive the necessary conditions for this transition by means of the coherent stability function approach, which is akin to the master stability function approach that is traditionally used to study the stability of synchronization in coupled oscillators. In chimera states, there is typically at least one group of oscillators that evolves in a drifting, random manner, while other groups of oscillators follow a smoother, more coherent profile. In the coherent state, there thus exists a smooth functional relationship between the oscillators. This lays the foundation for the coherent stability function approach, where we determine the stability of the coherent state. We subsequently test the analytical prediction numerically by calculating the strength of incoherence during the transition point. We use leech neurons, which exhibit a coexistence of chaotic and periodic tonic spiking depending on initial conditions, coupled via nonlocal electrical synapses, to demonstrate our approach. We systematically explore various dynamical states with the focus on the transitions between chimeras and coherence, fully confirming the validity of the coherent stability function. We also observe complete synchronization for higher values of the coupling strength, which we verify by the master stability function approach.

Spike chimera states and firing regularities in neuronal hypernetworks

A complex spatiotemporal pattern with coexisting coherent and incoherent domains in a network of identically coupled oscillators is known as a chimera state. Here, we report the emergence and existence of a novel type of nonstationary chimera pattern in a network of identically coupled Hindmarsh-Rose neuronal oscillators in the presence of synaptic couplings. The development of brain function is mainly dependent on the interneuronal communications via bidirectional electrical gap junctions and unidirectional chemical synapses. In our study, we first consider a network of nonlocally coupled neurons where the interactions occur through chemical synapses. We uncover a new type of spatiotemporal pattern, which we call "spike chimera" induced by the desynchronized spikes of the coupled neurons with the coherent quiescent state. Thereafter, imperfect traveling chimera states emerge in a neuronal hypernetwork (which is characterized by the simultaneous presence of electrical and chemical synapses). Using suitable characterizations, such as local order parameter, strength of incoherence, and velocity profile, the existence of several dynamical states together with chimera states is identified in a wide range of parameter space. We also investigate the robustness of these nonstationary chimera states together with incoherent, coherent, and resting states with respect to initial conditions by using the basin stability measurement. Finally, we extend our study for the effect of firing regularity in the observed states. Interestingly, we find that the coherent motion of the neuronal network promotes the entire system to regular firing.

Emergence of synchronization in multiplex networks of mobile Rössler oscillators

Different aspects of synchronization emerging in networks of coupled oscillators have been examined prominently in the last decades. Nevertheless, little attention has been paid on the emergence of this imperative collective phenomenon in networks displaying temporal changes in the connectivity patterns. However, there are numerous practical examples where interactions are present only at certain points of time owing to physical proximity. In this work, we concentrate on exploring the emergence of interlayer and intralayer synchronization states in a multiplex dynamical network comprising of layers having mobile nodes performing two-dimensional lattice random walk. We thoroughly illustrate the impacts of the network parameters, in particular, the vision range ϕ and the step size u together with the inter- and intralayer coupling strengths ε and k on these synchronous states arising in coupled Rössler systems. The presented numerical results are very well validated by analytically derived necessary conditions for the emergence and stability of the synchronous states. Furthermore, the robustness of the states of synchrony is studied under both structural and dynamical perturbations. We find interesting results on interlayer synchronization for a continuous removal of the interlayer links as well as for progressively created static nodes. We demonstrate that the mobility parameters responsible for intralayer movement of the nodes can retrieve interlayer synchrony under such structural perturbations. For further analysis of survivability of interlayer synchrony against dynamical perturbations, we proceed through the investigation of single-node basin stability, where again the intralayer mobility properties have noticeable impacts. We also discuss the scenarios related mainly to effects of the mobility parameters in cases of varying lattice size and percolation of the whole network.

Chimera patterns in three-dimensional locally coupled systems

The coexistence of coherent and incoherent domains, namely the appearance of chimera states, has been studied extensively in many contexts of science and technology since the past decade, though the previous studies are mostly built on the framework of one-dimensional and two-dimensional interaction topologies. Recently, the emergence of such fascinating phenomena has been studied in a three-dimensional (3D) grid formation while considering only the nonlocal interaction. Here we study the emergence and existence of chimera patterns in a three-dimensional network of coupled Stuart-Landau limit-cycle oscillators and Hindmarsh-Rose neuronal oscillators with local (nearest-neighbor) interaction topology. The emergence of different types of spatiotemporal chimera patterns is investigated by taking two distinct nonlinear interaction functions. We provide appropriate analytical explanations in the 3D grid of the network formation and the corresponding numerical justifications are given. We extend our analysis on the basis of the Ott-Antonsen reduction approach in the case of Stuart-Landau oscillators containing infinite numbers of oscillators. Particularly, in the Hindmarsh-Rose neuronal network the existence of nonstationary chimera states is characterized by an instantaneous strength of incoherence and an instantaneous local order parameter. Besides, the condition for achieving exact neuronal synchrony is obtained analytically through a linear stability analysis. The different types of collective dynamics together with chimera states are mapped over a wide range of various parameter spaces.

Self-adaptation of chimera states

Chimera states in spatiotemporal dynamical systems have been investigated in physical, chemical, and biological systems, and have been shown to be robust against random perturbations. How do chimera states achieve their robustness? We uncover a self-adaptation behavior by which, upon a spatially localized perturbation, the coherent component of the chimera state spontaneously drifts to an optimal location as far away from the perturbation as possible, exposing only its incoherent component to the perturbation to minimize the disturbance. A systematic numerical analysis of the evolution of the spatiotemporal pattern of the chimera state towards the optimal stable state reveals an exponential relaxation process independent of the spatial location of the perturbation, implying that its effects can be modeled as restoring and damping forces in a mechanical system and enabling the articulation of a phenomenological model. Not only is the model able to reproduce the numerical results, it can also predict the trajectory of drifting. Our finding is striking as it reveals that, inherently, chimera states possess a kind of "intelligence" in achieving robustness through self-adaptation. The behavior can be exploited for the controlled generation of chimera states with their coherent component placed in any desired spatial region of the system.

Solitary states in multiplex networks owing to competing interactions

Recent researches in network science demonstrate the coexistence of different types of interactions among the individuals within the same system. A wide range of situations appear in ecological and neuronal systems that incorporate positive and negative interactions. Also, there are numerous examples of systems that are best represented by the multiplex configuration. The present article investigates a possible scenario for the emergence of a newly observed remarkable phenomenon named as solitary state in coupled dynamical units in which one or a few units split off and behave differently from the other units. For this, we consider dynamical systems connected through a multiplex architecture in the presence of both positive and negative couplings. We explore our findings through analysis of the paradigmatic FitzHugh-Nagumo system in both equilibrium and periodic regimes on the top of a multiplex network having positive inter-layer and negative intra-layer interactions. We further substantiate our proposition using a periodic Lorenz system with the same scheme and show that an opposite scheme of competitive interactions may also work for the Lorenz system in the chaotic regime.

Chimera states in neuronal networks: A review

Neuronal networks, similar to many other complex systems, self-organize into fascinating emergent states that are not only visually compelling, but also vital for the proper functioning of the brain. Synchronous spatiotemporal patterns, for example, play an important role in neuronal communication and plasticity, and in various cognitive processes. Recent research has shown that the coexistence of coherent and incoherent states, known as chimera states or simply chimeras, is particularly important and characteristic for neuronal systems. Chimeras have also been linked to the Parkinson's disease, epileptic seizures, and even to schizophrenia. The emergence of this unique collective behavior is due to diverse factors that characterize neuronal dynamics and the functioning of the brain in general, including neural bumps and unihemispheric slow-wave sleep in some aquatic mammals. Since their discovery, chimera states have attracted ample attention of researchers that work at the interface of physics and life sciences. We here review contemporary research dedicated to chimeras in neuronal networks, focusing on the relevance of different synaptic connections, and on the effects of different network structures and coupling setups. We also cover the emergence of different types of chimera states, we highlight their relevance in other related physical and biological systems, and we outline promising research directions for the future, including possibilities for experimental verification.

Riddling: Chimera's dilemma

We investigate the basin of attraction properties and its boundaries for chimera states in a circulant network of Hénon maps. It is known that coexisting basins of attraction lead to a hysteretic behaviour in the diagrams of the density of states as a function of a varying parameter. Chimera states, for which coherent and incoherent domains occur simultaneously, emerge as a consequence of the coexistence of basin of attractions for each state. Consequently, the distribution of chimera states can remain invariant by a parameter change, and it can also suffer subtle changes when one of the basins ceases to exist. A similar phenomenon is observed when perturbations are applied in the initial conditions. By means of the uncertainty exponent, we characterise the basin boundaries between the coherent and chimera states, and between the incoherent and chimera states. This way, we show that the density of chimera states can be not only moderately sensitive but also highly sensitive to initial conditions. This chimera's dilemma is a consequence of the fractal and riddled nature of the basin boundaries.

Alternating chimeras in networks of ephaptically coupled bursting neurons

The distinctive phenomenon of the chimera state has been explored in neuronal systems under a variety of different network topologies during the last decade. Nevertheless, in all the works, the neurons are presumed to interact with each other directly with the help of synapses only. But, the influence of ephaptic coupling, particularly magnetic flux across the membrane, is mostly unexplored and should essentially be dealt with during the emergence of collective electrical activities and propagation of signals among the neurons in a network. Through this article, we report the development of an emerging dynamical state, namely, the alternating chimera, in a network of identical neuronal systems induced by an external electromagnetic field. Owing to this interaction scenario, the nonlinear neuronal oscillators are coupled indirectly via electromagnetic induction with magnetic flux, through which neurons communicate in spite of the absence of physical connections among them. The evolution of each neuron, here, is described by the three-dimensional Hindmarsh-Rose dynamics. We demonstrate that the presence of such non-locally and globally interacting external environments induces a stationary alternating chimera pattern in the ensemble of neurons, whereas in the local coupling limit, the network exhibits a transient chimera state whenever the local dynamics of the neurons is of the chaotic square-wave bursting type. For periodic square-wave bursting of the neurons, a similar qualitative phenomenon has been witnessed with the exception of the disappearance of cluster states for non-local and global interactions. Besides these observations, we advance our work while providing confirmation of the findings for neuronal ensembles exhibiting plateau bursting dynamics and also put forward the fact that the plateau pattern actually favors the alternating chimera more than others. These results may deliver better interpretations for different aspects of synchronization appearing in a network of neurons through field coupling that also relaxes the prerequisite of synaptic connectivity for realizing the chimera state in neuronal networks.

Emergence of synchronization and regularity in firing patterns in time-varying neural hypernetworks

We study synchronization of dynamical systems coupled in time-varying network architectures, composed of two or more network topologies, corresponding to different interaction schemes. As a representative example of this class of time-varying hypernetworks, we consider coupled Hindmarsh-Rose neurons, involving two distinct types of networks, mimicking interactions that occur through the electrical gap junctions and the chemical synapses. Specifically, we consider the connections corresponding to the electrical gap junctions to form a small-world network, while the chemical synaptic interactions form a unidirectional random network. Further, all the connections in the hypernetwork are allowed to change in time, modeling a more realistic neurobiological scenario. We model this time variation by rewiring the links stochastically with a characteristic rewiring frequency f. We find that the coupling strength necessary to achieve complete neuronal synchrony is lower when the links are switched rapidly. Further, the average time required to reach the synchronized state decreases as synaptic coupling strength and/or rewiring frequency increases. To quantify the local stability of complete synchronous state we use the Master Stability Function approach, and for global stability we employ the concept of basin stability. The analytically derived necessary condition for synchrony is in excellent agreement with numerical results. Further we investigate the resilience of the synchronous states with respect to increasing network size, and we find that synchrony can be maintained up to larger network sizes by increasing either synaptic strength or rewiring frequency. Last, we find that time-varying links not only promote complete synchronization, but also have the capacity to change the local dynamics of each single neuron. Specifically, in a window of rewiring frequency and synaptic coupling strength, we observe that the spiking behavior becomes more regular.

Chimera states in two-dimensional networks of locally coupled oscillators

Chimera state is defined as a mixed type of collective state in which synchronized and desynchronized subpopulations of a network of coupled oscillators coexist and the appearance of such anomalous behavior has strong connection to diverse neuronal developments. Most of the previous studies on chimera states are not extensively done in two-dimensional ensembles of coupled oscillators by taking neuronal systems with nonlinear coupling function into account while such ensembles of oscillators are more realistic from a neurobiological point of view. In this paper, we report the emergence and existence of chimera states by considering locally coupled two-dimensional networks of identical oscillators where each node is interacting through nonlinear coupling function. This is in contrast with the existence of chimera states in two-dimensional nonlocally coupled oscillators with rectangular kernel in the coupling function. We find that the presence of nonlinearity in the coupling function plays a key role to produce chimera states in two-dimensional locally coupled oscillators. We analytically verify explicitly in the case of a network of coupled Stuart-Landau oscillators in two dimensions that the obtained results using Ott-Antonsen approach and our analytical finding very well matches with the numerical results. Next, we consider another type of important nonlinear coupling function which exists in neuronal systems, namely chemical synaptic function, through which the nearest-neighbor (locally coupled) neurons interact with each other. It is shown that such synaptic interacting function promotes the emergence of chimera states in two-dimensional lattices of locally coupled neuronal oscillators. In numerical simulations, we consider two paradigmatic neuronal oscillators, namely Hindmarsh-Rose neuron model and Rulkov map for each node which exhibit bursting dynamics. By associating various spatiotemporal behaviors and snapshots at particular times, we study the chimera states in detail over a large range of coupling parameter. The existence of chimera states is confirmed by instantaneous angular frequency, order parameter and strength of incoherence.

Analysis of chimera states as drive-response systems

Chimera states are spatiotemporal segregations – stably coexisting coherent and incoherent groups – that can occur in systems of identical phase oscillators. Here we demonstrate that this remarkable phenomenon can also be understood in terms of Pecora and Carroll’s drive-response theory. By calculating the conditional Lyapunov exponents, we show that the incoherent group acts to synchronize the coherent group; the latter playing the role of a response. We also compare the distributions of finite-time conditional Lyapunov exponents to the characteristic distribution that was reported previously for chimera states. The present analysis provides a unifying explanation of the inherently frustrated dynamics that gives rise to chimera states.

Time-varying multiplex network: Intralayer and interlayer synchronization

A large class of engineered and natural systems, ranging from transportation networks to neuronal networks, are best represented by multiplex network architectures, namely a network composed of two or more different layers where the mutual interaction in each layer may differ from other layers. Here we consider a multiplex network where the intralayer coupling interactions are switched stochastically with a characteristic frequency. We explore the intralayer and interlayer synchronization of such a time-varying multiplex network. We find that the analytically derived necessary condition for intralayer and interlayer synchronization, obtained by the master stability function approach, is in excellent agreement with our numerical results. Interestingly, we clearly find that the higher frequency of switching links in the layers enhances both intralayer and interlayer synchrony, yielding larger windows of synchronization. Further, we quantify the resilience of synchronous states against random perturbations, using a global stability measure based on the concept of basin stability, and this reveals that intralayer coupling strength is most crucial for determining both intralayer and interlayer synchrony. Lastly, we investigate the robustness of interlayer synchronization against a progressive demultiplexing of the multiplex structure, and we find that for rapid switching of intralayer links, the interlayer synchronization persists even when a large number of interlayer nodes are disconnected.

Coexisting synchronous and asynchronous states in locally coupled array of oscillators by partial self-feedback control

We report the emergence of coexisting synchronous and asynchronous subpopulations of oscillators in one dimensional arrays of identical oscillators by applying a self-feedback control. When a self-feedback is applied to a subpopulation of the array, similar to chimera states, it splits into two/more sub-subpopulations coexisting in coherent and incoherent states for a range of self-feedback strength. By tuning the coupling between the nearest neighbors and the amount of self-feedback in the perturbed subpopulation, the size of the coherent and the incoherent sub-subpopulations in the array can be controlled, although the exact size of them is unpredictable. We present numerical evidence using the Landau-Stuart system and the Kuramoto-Sakaguchi phase model.