Pinning control of chimera states

Taming chimeras in coupled oscillators using soft actor-critic based reinforcement learning

We propose a universal method based on deep reinforcement learning (specifically, soft actor-critic) to control the chimera state in the coupled oscillators. The policy for control is learned by maximizing the expectation of the cumulative reward in the reinforcement learning framework. With the aid of the local order parameter, we design a class of reward functions for controlling the chimera state, specifically confining the spatial position of coherent and incoherent domains to any desired lateral position of oscillators. The proposed method is model-free, in contrast to the control schemes that require complete knowledge of the system equations. We test the method on the locally coupled Kuramoto oscillators and the nonlocally coupled FitzHugh-Nagumo model. Results show that the control is independent of initial conditions and coupling schemes. Not only the single-headed chimera, but also the multi-headed chimera and even the alternating chimera can be obtained by the method, and only the desired position needs to be changed. Beyond that, we discuss the influence of hyper-parameters, demonstrate the universality of the method to network sizes, and show that the proposed method can stabilize the drift of chimera and prevent its collapse in small 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.

Alternating chimera states in complex networks with modular structures

Chimera, the coexistence state of synchronization and non-synchronization, widely exists in complex networks. It has a great potentially explanatory power for the unihemispheric sleep of birds and some mammals, in which the synchronizations of the hemispheres of the cerebral cortex are evolving alternately. In this study, a coupled nonlinear oscillator system with a topology of the modular complex network was constructed to simulate the left and right hemispheres of the brain. The results showed that a stable chimera, an alternating chimera, and a breathing chimera were produced when the coupling strength and connection probability of the left and right hemispheres were changed. Further, we studied the effect of noise on rich synchronous patterns and found that the alternating chimera was robust to Gaussian white noise when the strength was not very large. Finally, our study was extended to a complex network with three sub-networks, and an alternating chimera could exist in two or three sub-networks. Our research provides a deeper insight into the mechanism of brain function like unihemispheric sleep.

Chimera Patterns of Synchrony in a Frustrated Array of Hebb Synapses

The union of the Kuramoto–Sakaguchi model and the Hebb dynamics reproduces the Lisman switch through a bistability in synchronized states. Here, we show that, within certain ranges of the frustration parameter, the chimera pattern can emerge, causing a different, time-evolving, distribution in the Hebbian synaptic strengths. We study the stability range of the chimera as a function of the frustration (phase-lag) parameter. Depending on the range of the frustration, two different types of chimeras can appear spontaneously, i.e., from randomized initial conditions. In the first type, the oscillators in the coherent region rotate, on average, slower than those in the incoherent region; while in the second type, the average rotational frequencies of the two regions are reversed, i.e., the coherent region runs, on average, faster than the incoherent region. We also show that non-stationary behavior at finite N can be controlled by adjusting the natural frequency of a single pacemaker oscillator. By slowly cycling the frequency of the pacemaker, we observe hysteresis in the system. Finally, we discuss how we can have a model for learning and memory.

Extended mean-field approach for chimera states in random complex networks

Identical oscillators in the chimera state exhibit a mixture of coherent and incoherent patterns simultaneously. Nonlocal interactions and phase lag are critical factors in forming a chimera state within the Kuramoto model in Euclidean space. Here, we investigate the contributions of nonlocal interactions and phase lag to the formation of the chimera state in random networks. By developing an extended mean-field approximation and using a numerical approach, we find that the emergence of a chimera state in the Erdös-Rényi network is due mainly to degree heterogeneity with nonzero phase lag. For a regularly random network, although all nodes have the same degree, we find that disordered connections may yield the chimera state in the presence of long-range interactions. Furthermore, we show a nontrivial dynamic state in which all the oscillators drift more slowly than a defined frequency due to connectivity disorder at large phase lags beyond the mean-field solutions.

Effects of Synaptic Pruning on Phase Synchronization in Chimera States of Neural Network

This research explores the effect of synaptic pruning on a ring-shaped neural network of non-locally coupled FitzHugh–Nagumo (FHN) oscillators. The neurons in the pruned region synchronize with each other, and they repel the coherent domain of the chimera states. Furthermore, the width of the pruned region decides the precision and efficiency of the control effect on the position of coherent domains. This phenomenon gives a systematic comprehension of the relation between pruning and synchronization in neural networks from a new aspect that has never been addressed. An explanation of this mechanism is also given.

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.

Remote pacemaker control of chimera states in multilayer networks of neurons

Networks of coupled nonlinear oscillators allow for the formation of nontrivial partially synchronized spatiotemporal patterns, such as chimera states, in which there are coexisting coherent (synchronized) and incoherent (desynchronized) domains. These complementary domains form spontaneously, and it is impossible to predict where the synchronized group will be positioned within the network. Therefore, possible ways to control the spatial position of the coherent and incoherent groups forming the chimera states are of high current interest. In this work we investigate how to control chimera patterns in multiplex networks of FitzHugh-Nagumo neurons, and in particular we want to prove that it is possible to remotely control chimera states exploiting the multiplex structure. We introduce a pacemaker oscillator within the network: this is an oscillator that does not receive input from the rest of the network but is sending out information to its neighbors. The pacemakers can be positioned in one or both layers. Their presence breaks the spatial symmetry of the layer in which they are introduced and allows us to control the position of the incoherent domain. We demonstrate how the remote control is possible for both uni- and bidirectional coupling between the layers. Furthermore we show which are the limitations of our control mechanisms when it is generalized from single-layer to multilayer networks.

Chimera dynamics in an array of coupled FitzHugh-Nagumo system with shift of close neighbors

In this paper, we consider an array of FitzHugh-Nagumo (FHN) systems with R close neighbors. Each element (j) connects to another (m) and its 2R neighbors. Shifting these neighbors produces particular phenomena such as chimera and multi-chimera. Step traveling chimera is observed for a time dependent shift. Results show that, basing oneself on both shift parameter m and close neighbors R, a full control on the chimera dynamics of the network can be ensured.

A two-layered brain network model and its chimera state

Based on the data of cerebral cortex, we present a two-layered brain network model of coupled neurons where the two layers represent the left and right hemispheres of cerebral cortex, respectively, and the links between the two layers represent the inter-couplings through the corpus callosum. By this model we show that abundant patterns of synchronization can be observed, especially the chimera state, depending on the parameters of system such as the coupling strengths and coupling phase. Further, we extend the model to a more general two-layered network to better understand the mechanism of the observed patterns, where each hemisphere of cerebral cortex is replaced by a highly clustered subnetwork. We find that the number of inter-couplings is another key parameter for the emergence of chimera states. Thus, the chimera states come from a matching between the structure parameters such as the number of inter-couplings and clustering coefficient etc and the dynamics parameters such as the intra-, inter-coupling strengths and coupling phase etc. A brief theoretical analysis is provided to explain the borderline of synchronization. These findings may provide helpful clues to understand the mechanism of brain functions.

Master-slave synchronization of hyperchaotic systems through a linear dynamic coupling

The development of synchronization strategies for dynamical systems is an important research activity that can be applied in several different fields from locomotion control of multilimbed structures to secure communication. In the presence of chaotic systems, synchronization is more difficult to accomplish and there are different techniques that can be adopted. In this paper we considered a master-slave topology where the coupling mechanism is realized through a second-order linear dynamical system. This control scheme, recently applied to chaotic systems, is here analyzed in the presence of hyperchaotic dynamics that represent a more challenging scenario. The possibility to reach a complete synchronization and the range of allowable coupling strength is investigated comparing the effects of the dynamical coupling with a standard configuration characterized by a static gain. This methodology is also applied to weighted networks to reach synchronization regimes otherwise not obtainable with a static coupling.

Controlling chimera states via minimal coupling modification

We propose a method to control chimera states in a ring-shaped network of nonlocally coupled phase oscillators. This method acts exclusively on the network's connectivity. Using the idea of a pacemaker oscillator, we investigate which is the minimal action needed to control chimeras. We implement the pacemaker choosing one oscillator and making its links unidirectional. Our results show that a pacemaker induces chimeras for parameters and initial conditions for which they do not form spontaneously. Furthermore, the pacemaker attracts the incoherent part of the chimera state, thus controlling its position. Beyond that, we find that these control effects can be achieved with modifications of the network's connectivity that are less invasive than a pacemaker, namely, the minimal action of just modifying the strength of one connection allows one to control chimeras.

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.

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.

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.

Generalized synchronization between chimera states

Networks of coupled oscillators in chimera states are characterized by an intriguing interplay of synchronous and asynchronous motion. While chimera states were initially discovered in mathematical model systems, there is growing experimental and conceptual evidence that they manifest themselves also in natural and man-made networks. In real-world systems, however, synchronization and desynchronization are not only important within individual networks but also across different interacting networks. It is therefore essential to investigate if chimera states can be synchronized across networks. To address this open problem, we use the classical setting of ring networks of non-locally coupled identical phase oscillators. We apply diffusive drive-response couplings between pairs of such networks that individually show chimera states when there is no coupling between them. The drive and response networks are either identical or they differ by a variable mismatch in their phase lag parameters. In both cases, already for weak couplings, the coherent domain of the response network aligns its position to the one of the driver networks. For identical networks, a sufficiently strong coupling leads to identical synchronization between the drive and response. For non-identical networks, we use the auxiliary system approach to demonstrate that generalized synchronization is established instead. In this case, the response network continues to show a chimera dynamics which however remains distinct from the one of the driver. Hence, segregated synchronized and desynchronized domains in individual networks congregate in generalized synchronization across networks.