Three-dimensional chimera patterns in networks of spiking neuron oscillators

Mechanisms for bump state localization in two-dimensional networks of leaky integrate-and-fire neurons

Networks of nonlocally coupled leaky integrate-and-fire neurons exhibit a variety of complex collective behaviors, such as partial synchronization, frequency or amplitude chimeras, solitary states, and bump states. In particular, the bump states consist of one or many regions of asynchronous elements within a sea of subthreshold (quiescent) elements. The asynchronous domains travel in the network in a direction predetermined by the initial conditions. In the present study, we investigate the occurrence of bump states in networks of leaky integrate-and-fire neurons in two-dimensions using nonlocal toroidal connectivity, and we explore possible mechanisms for stabilizing the moving asynchronous domains. Our findings indicate that (I) incorporating a refractory period can effectively anchor the position of these domains in the network, and (II) the switching off of some randomly preselected nodes (i.e., making them permanently idle/inactive) can likewise contribute to stabilizing the positions of the asynchronous domains. In particular, in case II for large values of the coupling strength and a large percentage of idle elements, all nodes acquire different fixed (frozen) values in the quiescent region and oscillations cease throughout the network due to self-organization. For the special case of stationary bump states, we propose an analytical approach to predict their properties. This approach is based on the self-consistency argument and is valid for infinitely large networks. Case I is of particular biomedical interest in view of the importance of refractoriness for biological neurons, while case II can be biomedically relevant when designing therapeutic methods for stabilizing moving signals in the brain.

Multistable Synaptic Plasticity Induces Memory Effects and Cohabitation of Chimera and Bump States in Leaky Integrate-and-Fire Networks

Chimera states and bump states are collective synchronization phenomena observed independently (in different parameter regions) in networks of coupled nonlinear oscillators. And while chimera states are characterized by coexistence of coherent and incoherent domains, bump states consist of alternating active and inactive domains. The idea of multistable plasticity in the network connections originates from brain dynamics where the strength of the synapses (axons) connecting the network nodes (neurons) may change dynamically in time; when reaching the steady state the network connections may be found in one of many possible values depending on various factors, such as local connectivity, influence of neighboring cells etc. The sign of the link weights is also a significant factor in the network dynamics: positive weights are characterized as excitatory connections and negative ones as inhibitory. In the present study we consider the simplest case of bistable plasticity, where the link dynamics has only two fixed points. During the system/network integration, the link weights change and as a consequence the network organizes in excitatory or inhibitory domains characterized by different synaptic strengths. We specifically explore the influence of bistable plasticity on collective synchronization states and we numerically demonstrate that the dynamics of the linking may, under special conditions, give rise to co-existence of bump-like and chimera-like states simultaneously in the network. In the case of bump and chimera co-existence, confinement effects appear: the different domains stay localized and do not travel around the network. Memory effects are also reported in the sense that the final spatial arrangement of the coupling strengths reflects some of the local properties of the initial link distribution. For the quantification of the system's spatial and temporal features, the global and local entropy functions are employed as measures of the network organization, while the average firing rates account for the network evolution and dynamics. In particular, the spatial minima of the local entropy designate the transition points between domains of different synaptic weights in the hybrid states, while the number of minima corresponds to the number of different domains. In addition, the entropy deviations signify the presence of chimera-like or bump-like states in the network.

From Turing patterns to chimera states in the 2D Brusselator model

The Brusselator has been used as a prototype model for autocatalytic reactions and, in particular, for the Belousov-Zhabotinsky reaction. When coupled at the diffusive limit, the Brusselator undergoes a Turing bifurcation resulting in the formation of classical Turing patterns, such as spots, stripes, and spirals in two spatial dimensions. In the present study, we use generic nonlocally coupled Brusselators and show that in the limit of the coupling range R→1 (diffusive limit), the classical Turing patterns are recovered, while for intermediate coupling ranges and appropriate parameter values, chimera states are produced. This study demonstrates how the parameters of a typical nonlinear oscillator can be tuned so that the coupled system passes from spatially stable Turing structures to dynamical spatiotemporal chimera states.

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.

Shooting solitaries due to small-world connectivity in leaky integrate-and-fire networks

We study the synchronization properties in a network of leaky integrate-and-fire oscillators with nonlocal connectivity under probabilistic small-world rewiring. We demonstrate that the random links lead to the emergence of chimera-like states where the coherent regions are interrupted by scattered, short-lived solitaries; these are termed "shooting solitaries." Moreover, we provide evidence that random links enhance the appearance of chimera-like states for values of the parameter space that otherwise support synchronization. This last effect is counter-intuitive because by adding random links to the synchronous state, the system locally organizes into coherent and incoherent domains.

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.

Identification of chimera using machine learning

Chimera state refers to the coexistence of coherent and non-coherent phases in identically coupled dynamical units found in various complex dynamical systems. Identification of chimera, on one hand, is essential due to its applicability in various areas including neuroscience and, on the other hand, is challenging due to its widely varied appearance in different systems and the peculiar nature of its profile. Therefore, a simple yet universal method for its identification remains an open problem. Here, we present a very distinctive approach using machine learning techniques to characterize different dynamical phases and identify the chimera state from given spatial profiles generated using various different models. The experimental results show that the performance of the classification algorithms varies for different dynamical models. The machine learning algorithms, namely, random forest, oblique random forest based on Tikhonov, axis-parallel split, and null space regularization achieved more than 96% accuracy for the Kuramoto model. For the logistic maps, random forest and Tikhonov regularization based oblique random forest showed more than 90% accuracy, and for the Hénon map model, random forest, null space, and axis-parallel split regularization based oblique random forest achieved more than 80% accuracy. The oblique random forest with null space regularization achieved consistent performance (more than 83% accuracy) across different dynamical models while the auto-encoder based random vector functional link neural network showed relatively lower performance. This work provides a direction for employing machine learning techniques to identify dynamical patterns arising in coupled non-linear units on large-scale and for characterizing complex spatiotemporal patterns in real-world systems for various applications.

Chimeras and solitary states in 3D oscillator networks with inertia

We report the diversity of scroll wave chimeras in the three-dimensional (3D) Kuramoto model with inertia for N³ identical phase oscillators placed in a unit 3D cube with periodic boundary conditions. In the considered model with inertia, we have found patterns that do not exist in a pure system without inertia. In particular, a scroll ring chimera is obtained from random initial conditions. In contrast to this system without inertia, where all chimera states have incoherent inner parts, these states can have partially coherent or fully coherent inner parts as exemplified by a scroll ring chimera. Solitary states exist in the considered model as separate states or can coexist with scroll wave chimeras in the oscillatory space. We also propose a method of construction of 3D images using solitary states as solutions of the 3D Kuramoto model with inertia.

Finite Size Effects in Networks of Coupled Neurons

We use extensive computer simulations to study synchronization phenomena in networks of biological neurons. Each individual neuron is modeled using the leaky integrate-and-fire (LIF) scheme, while many neurons are coupled nonlocally in a network. In this system chimera states develop, which are complex states consisting of coexisting synchronous and asynchronous network areas. We study the influence of the network size on the properties and the form of chimera states. We show that for constant coupling strength, the number of the synchronous/asynchronous domains depends quantitatively on the coupling ratio. This dependence allows to extract synchronization properties in large ensembles of neurons after extrapolating from simulations of small networks. Since computer simulations of even small neuron networks are highly demanding in memory and CPU time, this property is particularly important in view of the large number of neurons involved in any cognitive function. In total, the number of neurons in the human brain is of the order of 10¹⁰, and each of them is connected with an average of 10³ other neurons.

Pattern formation and chimera states in 2D SQUID metamaterials

The Superconducting QUantum Interference Device (SQUID) is a highly nonlinear oscillator with rich dynamical behavior, including chaos. When driven by a time-periodic magnetic flux, the SQUID exhibits extreme multistability at frequencies around the geometric resonance, which is manifested by a "snakelike" form of the resonance curve. Repeating motifs of SQUIDs form metamaterials, i.e., artificially structured media of weakly coupled discrete elements that exhibit extraordinary properties, e.g., negative diamagnetic permeability. We report on the emergent collective dynamics in two-dimensional lattices of coupled SQUID oscillators, which involves a rich menagerie of spatiotemporal dynamics, including Turing-like patterns and chimera states. Using Fourier analysis, we characterize these patterns and identify characteristic spatial and temporal periods. In the low coupling limit, the Turing-like patterns occur near the synchronization-desynchronization transition, which can be related to the bifurcation scenarios of the single SQUID. Chimeras emerge due to the multistability near the geometric resonance, and by varying the dc component of the external force, we can make them appear and reappear and, also, control their location. A detailed analysis of the parameter space reveals the coexistence of Turing-like patterns and chimera states in our model, as well as the ability to transform between these states by varying the system parameters.

Chimera patterns and subthreshold oscillations in two-dimensional networks of fractally coupled leaky integrate-and-fire neurons

We discuss the effects that fractal coupling induces on chimera states in a network of leaky integrate-and-fire (LIF) oscillators arranged in a two-dimensional toroidal geometry. We provide evidence that the introduction of a hierarchical coupling topology in the form of a Sierpinski carpet gives rise to complex spatial structures such as multiple spots, stripe-and-grid chimeras, as well as traveling waves and subthreshold oscillations. Unlike in the case of typical nonlocal connectivity, when tuning the coupling strength to small positive values a spot chimera is formed with internal structure reminiscent of the fractal connectivity scheme. This is in line with previous results for one-dimensional networks, where hierarchical connectivity also induces chimeras with stratified spatial arrangements. In the case of negative coupling, cooperative effects produce subthreshold oscillating regions with traveling active islands crossing through them. Subthreshold oscillations and traveling waves are frequently reported in biological neural network experiments.

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.

Chimeras in digital phase-locked loops

Digital phase-locked loops (DPLLs) are nonlinear feedback-controlled systems that are widely used in electronic communication and signal processing applications. In most of the applications, they work in coupled mode; however, a vast amount of the studies on DPLLs concentrate on the dynamics of a single isolated unit. In this paper, we consider both one- and two-dimensional networks of DPLLs connected through a practically realistic nonlocal coupling and explore their collective dynamics. For the one-dimensional network, we analytically derive the parametric zone of a stable phase-locked state in which DPLLs essentially work in their normal mode of operation. We demonstrate that apart from the stable phase-locked state, a variety of spatiotemporal structures including chimeras arise in a broad parameter zone. For the two-dimensional network under nonlocal coupling, we identify several variants of chimera patterns, such as strip and spot chimeras. We identify and characterize the chimera patterns through suitable measures like local curvature and correlation function. Our study reveals the existence of chimeras in a widely used engineering system; therefore, we believe that these chimera patterns can be observed in experiments as well.

Networks of coupled oscillators: From phase to amplitude chimeras

We show that amplitude-mediated phase chimeras and amplitude chimeras can occur in the same network of nonlocally coupled identical oscillators. These are two different partial synchronization patterns, where spatially coherent domains coexist with incoherent domains and coherence/incoherence referring to both amplitude and phase or only the amplitude of the oscillators, respectively. By changing the coupling strength, the two types of chimera patterns can be induced. We find numerically that the amplitude chimeras are not short-living transients but can have a long lifetime. Also, we observe variants of the amplitude chimeras with quasiperiodic temporal oscillations. We provide a qualitative explanation of the observed phenomena in the light of symmetry breaking bifurcation scenarios. We believe that this study will shed light on the connection between two disparate chimera states having different symmetry-breaking properties.