Remote pacemaker control of chimera states in multilayer networks of neurons

Influence of sinusoidal forcing on the master FitzHugh-Nagumo neuron model and global dynamics of a unidirectionally coupled two-neuron system

In this paper, we investigate a seven-parameter, five-dimensional dynamical system, specifically a unidirectional coupling of two FitzHugh-Nagumo neuron models, with one neuron being sinusoidally driven. This master-slave configuration features neuron N1 as the master, subjected to an external sinusoidal electrical current, and neuron N2 as the slave, interacting with N1 through an electrical force. We report numerical results for three distinct scenarios where N1 operates in (i) periodic, (ii) quasiperiodic, and (iii) chaotic regimes. The primary objective is to explore how the dynamics of the master neuron N1 influence the coupled system's behavior. To achieve this, we generated cross sections of the seven-dimensional parameter space, known as parameter planes. Our findings reveal that in the periodic regime of N1, the coupled system exhibits period-adding sequences of Arnold tongue-like structures in the parameter planes. Furthermore, regions of multistability can also be identified in these parameter planes of the coupled system. In the quasiperiodic regime, regions of periodic motion are absent, with only regions of quasiperiodic and chaotic dynamics present. In the chaotic regime of N1, the parameter planes display regions of chaos, hyperchaos, and transient hyperchaos.

Delay-induced amplitude death in multiplex oscillator network with frequency-mismatched layers

The present paper analytically investigates the stability of amplitude death in a multiplex Stuart-Landau oscillator network with a delayed interlayer connection. The network consists of two frequency-mismatched layers, and all oscillators in each layer have identical frequencies. We show that, if the matrices describing the network topologies of each layer commute, then the characteristic equation governing the stability can be reduced to a simple form. This form reveals that the stability of amplitude death in the multiplex network is equally or more conservative than that in a pair of frequency-mismatched oscillators coupled by a delayed connection. In addition, we provide a procedure for designing the delayed interlayer connection such that amplitude death is stable for any commuting matrices and for any intralayer coupling strength. These analytical results are verified through numerical examples. Moreover, we numerically discuss the results for the case in which the commutative property does not hold.

Chimera states in multiplex networks: Chameleon-like across-layer synchronization

Different across-layer synchronization types of chimera states in multilayer networks have been discovered recently. We investigate possible relations between them, for example, if the onset of some synchronization type implies the onset of some other type. For this purpose, we use a two-layer network with multiplex inter-layer coupling. Each layer consists of a ring of non-locally coupled phase oscillators. While oscillators in each layer are identical, the layers are made non-identical by introducing mismatches in the oscillators' mean frequencies and phase lag parameters of the intra-layer coupling. We use different metrics to quantify the degree of various across-layer synchronization types. These include phase-locking between individual interacting oscillators, amplitude and phase synchronization between the order parameters of each layer, generalized synchronization between the driver and response layer, and the alignment of the incoherent oscillator groups' position on the two rings. For positive phase lag parameter mismatches, we get a cascaded onset of synchronization upon a gradual increase of the inter-layer coupling strength. For example, the two order parameters show phase synchronization before any of the interacting oscillator pairs does. For negative mismatches, most synchronization types have their onset in a narrow range of the coupling strength. Weaker couplings can destabilize chimera states in the response layer toward an almost fully coherent or fully incoherent motion. Finally, in the absence of a phase lag mismatch, sufficient coupling turns the response dynamics into a replica of the driver dynamics with the phases of all oscillators shifted by a constant lag.

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.

Noise-induced stabilization of the FitzHugh-Nagumo neuron dynamics: Multistability and transient chaos

The nonlinear dynamics of a FitzHugh-Nagumo (FHN) neuron driven by an oscillating current and perturbed by a Gaussian noise signal with different intensities D is investigated. In the noiseless case, stable periodic structures [Arnold tongues (ATS), cuspidal and shrimp-shaped] are identified in the parameter space. The periods of the ATSs obey specific generating and recurrence rules and are organized according to linear Diophantine equations responsible for bifurcation cascades. While for small values of D, noise starts to destroy elongations ("antennas") of the cuspidals, for larger values of D, the periodic motion expands into chaotic regimes in the parameter space, stabilizing the chaotic motion, and a transient chaotic motion is observed at the periodic-chaotic borderline. Besides giving a detailed description of the neuronal dynamics, the intriguing novel effect observed for larger D values is the generation of a regular dynamics for the driven FHN neuron. This result has a fundamental importance if the complex local dynamics is considered to study the global behavior of the neural networks when parameters are simultaneously varied, and there is the necessity to deal the intrinsic stochastic signal merged into the time series obtained from real experiments. As the FHN model has crucial properties presented by usual neuron models, our results should be helpful in large-scale simulations using complex neuron networks and for applications.

What Models and Tools can Contribute to a Better Understanding of Brain Activity?

Despite impressive scientific advances in understanding the structure and function of the human brain, big challenges remain. A deep understanding of healthy and aberrant brain activity at a wide range of temporal and spatial scales is needed. Here we discuss, from an interdisciplinary network perspective, the advancements in physical and mathematical modeling as well as in data analysis techniques that, in our opinion, have potential to further advance our understanding of brain structure and function.

Synchronization in Multiplex Leaky Integrate-and-Fire Networks With Nonlocal Interactions

We study synchronization phenomena in a multiplex network composed of two rings with identical Leaky Integrate-and-Fire (LIF) oscillators located on the nodes of the rings. Within each ring the LIF oscillators interact nonlocally, while between rings there are one-to-one inter-ring interactions. This structure is motivated by the observed connectivity between the two hemispheres of the brain: within each hemisphere the various brain regions interact with neighboring regions, while across hemispheres each region interacts, primarily, with the functionally homologous region. We consider both positive (excitatory) and negative (inhibitory) linking. We identify numerically various parameter regimes where the multiplex network develops coexistence of active and subthreshold domains, chimera states, solitary states, full coherence or incoherence. In particular, for weak inter-ring coupling (weak multiplexing) different synchronization patterns on the two rings are supported. These are stable and are obtained when the intra-ring coupling values are near the critical points separating qualitatively distinct synchronization regimes, e.g., between the travelling fronts regime and the chimera state one.

Applications of optimal nonlinear control to a whole-brain network of FitzHugh-Nagumo oscillators

We apply the framework of optimal nonlinear control to steer the dynamics of a whole-brain network of FitzHugh-Nagumo oscillators. Its nodes correspond to the cortical areas of an atlas-based segmentation of the human cerebral cortex, and the internode coupling strengths are derived from diffusion tensor imaging data of the connectome of the human brain. Nodes are coupled using an additive scheme without delays and are driven by background inputs with fixed mean and additive Gaussian noise. Optimal control inputs to nodes are determined by minimizing a cost functional that penalizes the deviations from a desired network dynamic, the control energy, and spatially nonsparse control inputs. Using the strength of the background input and the overall coupling strength as order parameters, the network's state-space decomposes into regions of low- and high-activity fixed points separated by a high-amplitude limit cycle, all of which qualitatively correspond to the states of an isolated network node. Along the borders, however, additional limit cycles, asynchronous states, and multistability can be observed. Optimal control is applied to several state-switching and network synchronization tasks, and the results are compared to controllability measures from linear control theory for the same connectome. We find that intuitions from the latter about the roles of nodes in steering the network dynamics, which are solely based on connectome features, do not generally carry over to nonlinear systems, as had been previously implied. Instead, the role of nodes under optimal nonlinear control critically depends on the specified task and the system's location in state space. Our results shed new light on the controllability of brain network states and may serve as an inspiration for the design of new paradigms for noninvasive brain stimulation.

One-way dependent clusters and stability of cluster synchronization in directed networks

Cluster synchronization in networks of coupled oscillators is the subject of broad interest from the scientific community, with applications ranging from neural to social and animal networks and technological systems. Most of these networks are directed, with flows of information or energy that propagate unidirectionally from given nodes to other nodes. Nevertheless, most of the work on cluster synchronization has focused on undirected networks. Here we characterize cluster synchronization in general directed networks. Our first observation is that, in directed networks, a cluster A of nodes might be one-way dependent on another cluster B: in this case, A may remain synchronized provided that B is stable, but the opposite does not hold. The main contribution of this paper is a method to transform the cluster stability problem in an irreducible form. In this way, we decompose the original problem into subproblems of the lowest dimension, which allows us to immediately detect inter-dependencies among clusters. We apply our analysis to two examples of interest, a human network of violin players executing a musical piece for which directed interactions may be either activated or deactivated by the musicians, and a multilayer neural network with directed layer-to-layer connections.

Mechanisms of cluster formation in networks with directed links differ from those in undirected networks. Lodi et al. propose a method to compute interdependencies among clusters of nodes in directed networks. They show that clusters can be one-way dependent, as found in social and neural networks.

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.

Transient dynamics and multistability in two electrically interacting FitzHugh-Nagumo neurons

We analyze the existence of chaotic and regular dynamics, transient chaos phenomenon, and multistability in the parameter space of two electrically interacting FitzHugh-Nagumo (FHN) neurons. By using extensive numerical experiments to investigate the particular organization between periodic and chaotic domains in the parameter space, we obtained three important findings: (i) there are self-organized generic stable periodic structures along specific directions immersed in a chaotic portion of the parameter space; (ii) the existence of transient chaos phenomenon is responsible for long chaotic temporal evolution preceding the asymptotic (periodic) dynamics for particular parametric combinations in the parameter space; and (iii) the existence of various multistable domains in the parameter space with an arbitrary number of attractors. Additionally, we also prove through numerical simulations that chaos, transient chaos, and multistability prevail even for different coupling strengths between identical FHN neurons. It is possible to find multistable attractors in the phase and parameter spaces and to steer them apart by increasing the asymmetry in the coupling force between neurons. Such a strategy can be essential to experimental matters, as setting the right parameter ranges. As the FHN model shares the crucial properties presented by the more realistic Hodgkin-Huxley-like neurons, our results can be extended to high-dimensional coupled neuron models.

Chimeras confined by fractal boundaries in the complex plane

Complex-valued quadratic maps either converge to fixed points, enter into periodic cycles, show aperiodic behavior, or diverge to infinity. Which of these scenarios takes place depends on the map's complex-valued parameter c and the initial conditions. The Mandelbrot set is defined by the set of c values for which the map remains bounded when initiated at the origin of the complex plane. In this study, we analyze the dynamics of a coupled network of two pairs of two quadratic maps in dependence on the parameter c. Across the four maps, c is kept the same whereby the maps are identical. In analogy to the behavior of individual maps, the network iterates either diverge to infinity or remain bounded. The bounded solutions settle into different stable states, including full synchronization and desynchronization of all maps. Furthermore, symmetric partially synchronized states of within-pair synchronization and across-pair synchronization as well as a symmetry broken chimera state are found. The boundaries between bounded and divergent solutions in the domain of c are fractals showing a rich variety of intriguingly esthetic patterns. Moreover, the set of bounded solutions is divided into countless subsets throughout all length scales in the complex plane. Each individual subset contains only one state of synchronization and is enclosed within fractal boundaries by c values leading to divergence.

Smallest Chimeras Under Repulsive Interactions

We present an exemplary system of three identical oscillators in a ring interacting repulsively to show up chimera patterns. The dynamics of individual oscillators is governed by the superconducting Josephson junction. Surprisingly, the repulsive interactions can only establish a symmetry of complete synchrony in the ring, which is broken with increasing repulsive interactions when the junctions pass through serials of asynchronous states (periodic and chaotic) but finally emerge into chimera states. The chimera pattern first appears in chaotic rotational motion of the three junctions when two junctions evolve coherently, while the third junction is incoherent. For larger repulsive coupling, the junctions evolve into another chimera pattern in a periodic state when two junctions remain coherent in rotational motion and one junction transits to incoherent librational motion. This chimera pattern is sensitive to initial conditions in the sense that the chimera state flips to another pattern when two junctions switch to coherent librational motion and the third junction remains in rotational motion, but incoherent. The chimera patterns are detected by using partial and global error functions of the junctions, while the librational and rotational motions are identified by a libration index. All the collective states, complete synchrony, desynchronization, and two chimera patterns are delineated in a parameter plane of the ring of junctions, where the boundaries of complete synchrony are demarcated by using the master stability function.