Chimera patterns in two-dimensional networks of coupled neurons

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.

Breathing chimera states in nonlocally coupled type-I excitable phase oscillators

We explore chimera states in a ring of nonlocally coupled type-I excitable phase oscillators, with each isolated oscillator being restricted to a homogeneous equilibrium state. Our study identifies the presence of breathing chimera states, characterized by their oscillatory dynamics and periodic fluctuations in the global order parameter. Beyond the breathing chimera states with a single coherent cluster, we find the 2n-cluster breathing chimera states, where 2n represents an even number of coherent clusters. These states exhibit the varying phase difference between adjacent clusters and a consistent phase among clusters separated by one intermediate cluster. The number of clusters is found to be modulated by the relative coupling radius. These dynamics for the finite number of oscillators are well confirmed by the Ott-Antonsen ansatz.

Floating-Point Approximation Enabling Cost-Effective and High-Precision Digital Implementation of FitzHugh-Nagumo Neural Networks

The study of neuron interactions and hardware implementations are crucial research directions in neuroscience, particularly in developing large-scale biological neural networks. The FitzHugh-Nagumo (FHN) model is a popular neuron model with highly biological plausibility, but its complexity makes it difficult to apply at scale. This paper presents a cost-saving and improved precision approximation algorithm for the digital implementation of the FHN model. By converting the computational data into floating-point numbers, the original multiplication calculations are replaced by adding the floating-point exponent part and fitting the mantissa part with piecewise linear. In the hardware implementation, shifters and adders are used, greatly reducing resource overhead. Implementing FHN neurons by this approximation calculations on FPGA reduces the normalized root mean square error (RMSE) to 3.5% of the state-of-the-art (SOTA) while maintaining a performance overhead ratio improvement of 1.09 times. Compared to implementations based on approximate multipliers, the proposed method achieves a 20% reduction in error at the cost of a 2.8% increase in overhead.This model gained additional biological properties compared to LIF while reducing the deployment scale by only 9%. Furthermore, the hardware implementation of nine coupled circular networks with eight nodes and directional diffusion was carried out to demonstrate the algorithm's effectiveness on neural networks. The error decreased to 60% compared to the single neuron of the SOTA. This hardware-friendly algorithm allows for the low-cost implementation of high-precision hardware simulation, providing a novel perspective for studying large-scale, biologically plausible neural networks.

Spiral wave chimeras in nonlocally coupled bicomponent oscillators

Chimera states in nonidentical oscillators have received extensive attention in recent years. Previous studies have demonstrated that chimera states can exist in a ring of nonlocally coupled bicomponent oscillators even in the presence of strong parameter heterogeneity. In this study, we investigate spiral wave chimeras in two-dimensional nonlocally coupled bicomponent oscillators where oscillators are randomly divided into two groups, with identical oscillators in the same group. Using phase oscillators and FitzHugh-Nagumo oscillators as examples, we numerically demonstrate that each group of oscillators supports its own spiral wave chimera and two spiral wave chimeras coexist with each other. We find that there exist three heterogeneity regimes: the synchronous regime at weak heterogeneity, the asynchronous regime at strong heterogeneity, and the transition regime in between. In the synchronous regime, spiral wave chimeras supported by different groups are synchronized with each other by sharing a same rotating frequency and a same incoherent core. In the asynchronous regime, the two spiral wave chimeras rotate at different frequencies and their incoherent cores are far away from each other. These phenomena are also observed in a nonrandom distribution of the two group oscillators and the continuum limit of infinitely many phase oscillators. The transition from synchronous to asynchronous spiral wave chimeras depends on the component oscillators. Specifically, it is a discontinuous transition for phase oscillators but a continuous one for FitzHugh-Nagumo oscillators. We also find that, in the asynchronous regime, increasing heterogeneity leads irregularly meandering spiral wave chimeras to rigidly rotating ones.

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.

Stability Analysis for a Fractional-Order Coupled FitzHugh–Nagumo-Type Neuronal Model

The aim of this work is to describe the dynamics of a fractional-order coupled FitzHugh–Nagumo neuronal model. The equilibrium states are analyzed in terms of their stability properties, both dependently and independently of the fractional orders of the Caputo derivatives, based on recently established theoretical results. Numerical simulations are shown to clarify and exemplify the theoretical results.

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.

Moving spiral wave chimeras

We consider a two-dimensional array of heterogeneous nonlocally coupled phase oscillators on a flat torus and study the bound states of two counter-rotating spiral chimeras, shortly two-core spiral chimeras, observed in this system. In contrast to other known spiral chimeras with motionless incoherent cores, the two-core spiral chimeras typically show a drift motion. Due to this drift, their incoherent cores become spatially modulated and develop specific fingerprint patterns of varying synchrony levels. In the continuum limit of infinitely many oscillators, the two-core spiral chimeras can be studied using the Ott-Antonsen equation. Numerical analysis of this equation allows us to reveal the stability region of different spiral chimeras, which we group into three main classes-symmetric, asymmetric, and meandering spiral chimeras.

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.

Structural anomalies in brain networks induce dynamical pacemaker effects

Dynamical effects on healthy brains and brains affected by tumor are investigated via numerical simulations. The brains are modeled as multilayer networks consisting of neuronal oscillators whose connectivities are extracted from Magnetic Resonance Imaging (MRI) data. The numerical results demonstrate that the healthy brain presents chimera-like states where regions with high white matter concentrations in the direction connecting the two hemispheres act as the coherent domain, while the rest of the brain presents incoherent oscillations. To the contrary, in brains with destructed structures, traveling waves are produced initiated at the region where the tumor is located. These areas act as the pacemaker of the waves sweeping across the brain. The numerical simulations are performed using two neuronal models: (a) the FitzHugh-Nagumo model and (b) the leaky integrate-and-fire model. Both models give consistent results regarding the chimera-like oscillations in healthy brains and the pacemaker effect in the tumorous brains. These results are considered a starting point for further investigation in the detection of tumors with small sizes before becoming discernible on MRI recordings as well as in tumor development and evolution.

Labyrinth chaos: Revisiting the elegant, chaotic, and hyperchaotic walks

Labyrinth chaos was discovered by Otto Rössler and René Thomas in their endeavor to identify the necessary mathematical conditions for the appearance of chaotic and hyperchaotic motion in continuous flows. Here, we celebrate their discovery by considering a single labyrinth walk system and an array of coupled labyrinth chaos systems that exhibit complex, chaotic behavior, reminiscent of chimera-like states, a peculiar synchronization phenomenon. We discuss the properties of the single labyrinth walk system and review the ability of coupled labyrinth chaos systems to exhibit chimera-like states due to the unique properties of their space-filling, chaotic trajectories, which amounts to elegant, hyperchaotic walks. Finally, we discuss further implications in relation to the labyrinth walk system by showing that even though it is volume-preserving, it is not force-conservative.

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.

Relay and complete synchronization in heterogeneous multiplex networks of chaotic maps

We study relay and complete synchronization in a heterogeneous triplex network of discrete-time chaotic oscillators. A relay layer and two outer layers, which are not directly coupled but interact via the relay layer, represent rings of nonlocally coupled two-dimensional maps. We consider for the first time the case when the spatiotemporal dynamics of the relay layer is completely different from that of the outer layers. Two different configurations of the triplex network are explored: when the relay layer consists of Lozi maps while the outer layers are given by Henon maps and vice versa. Phase and amplitude chimera states are observed in the uncoupled Henon map ring, while solitary state regimes are typical for the isolated Lozi map ring. We show for the first time relay synchronization of amplitude and phase chimeras, a solitary state chimera, and solitary state regimes in the outer layers. We reveal regimes of complete synchronization for the chimera structures and solitary state modes in all the three layers. We also analyze how the synchronization effects depend on the spatiotemporal dynamics of the relay layer and construct phase diagrams in the parameter plane of inter-layer vs intra-layer coupling strength of the relay layer.

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.

Transition from spiral wave chimeras to phase cluster states

Photochemically coupled Belousov-Zhabotinsky micro-oscillators are studied in experiments and simulations. Generally good agreement between the experimental and simulated dynamical behavior is found, with spiral wave chimeras exhibited at small values of the time delay in the coupling between the oscillators, spiral wave core splitting at higher values, and phase cluster states replacing the spiral wave dynamics at the highest values of the time delay. Spiral wave chimera dynamics is exhibited experimentally for much of the time delay range, while spiral wave phase cluster states are exhibited more in the model simulations. In addition to comparing the experimental and simulation behavior, we explore the novel spiral wave phase cluster states and develop a mechanism for this new and unusual dynamical behavior.

Two-dimensional optical chimera states in an array of coupled waveguide resonators

Two-dimensional arrays of coupled waveguides or coupled microcavities allow us to confine and manipulate light. Based on a paradigmatic envelope equation, we show that these devices, subject to a coherent optical injection, support coexistence between a coherent and incoherent emission. In this regime, we show that two-dimensional chimera states can be generated. Depending on initial conditions, the system exhibits a family of two-dimensional chimera states and interaction between them. We characterize these two-dimensional structures by computing their Lyapunov spectrum and Yorke-Kaplan dimension. Finally, we show that two-dimensional chimera states are of spatiotemporal chaotic nature.

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.

Spiral and target wave chimeras in a 2D lattice of map-based neuron models

We study the dynamics of a two-dimensional lattice of nonlocally coupled-map-based neuron models represented by Rulkov maps. It is firstly shown that this discrete-time neural network can exhibit spiral and target waves and corresponding chimera states when the control parameters (the coupling strength and the coupling radius) are varied. It is demonstrated that one-core, multicore, and ring-shaped core spiral chimeras can be realized in the network. We also reveal a novel type of chimera structure-a target wave chimera. We explore the transition from spiral wave chimeras to target wave structures when varying the coupling parameters. We report for the first time that the spiral wave regime can be suppressed by applying noise excitations, and the subsequent transition to the target wave mode occurs.

Solitary states and solitary state chimera in neural networks

We investigate solitary states and solitary state chimeras in a ring of nonlocally coupled systems represented by FitzHugh-Nagumo neurons in the oscillatory regime. We perform a systematic study of solitary states in this network. In particular, we explore the phase space structure, calculate basins of attraction, analyze the region of existence of solitary states in the system's parameter space, and investigate how the number of solitary nodes in the network depends on the coupling parameters. We report for the first time the occurrence of solitary state chimera in networks of coupled time-continuous neural systems. Our results disclose distinctive features characteristic of solitary states in the FitzHugh-Nagumo model, such as the flat mean phase velocity profile. On the other hand, we show that the mechanism of solitary states' formation in the FitzHugh-Nagumo model similar to chaotic maps and the Kuramoto model with inertia is related to the appearance of bistability in the system for certain values of coupling parameters. This indicates a general, probably a universal desynchronization scenario via solitary states in networks of very different nature.

Synchronization of spiral wave patterns in two-layer 2D lattices of nonlocally coupled discrete oscillators

The paper describes the effects of mutual and external synchronization of spiral wave structures in two coupled two-dimensional lattices of coupled discrete-time oscillators. Each lattice is given by a 2D N×N network of nonlocally coupled Nekorkin maps which model neuronal activity. We show numerically that spiral wave structures, including spiral wave chimeras, can be synchronized and establish the mechanism of the synchronization scenario. Our numerical studies indicate that when the coupling strength between the lattices is sufficiently weak, only a certain part of oscillators of the interacting networks is imperfectly synchronized, while the other part demonstrates a partially synchronous behavior. If the spatiotemporal patterns in the lattices do not include incoherent cores, imperfect synchronization is realized for most oscillators above a certain value of the coupling strength. In the regime of spiral wave chimeras, the imperfect synchronization of all oscillators cannot be achieved even for sufficiently large values of the coupling strength.

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.

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.

Complex dynamics of a neuron model with discontinuous magnetic induction and exposed to external radiation

The last two decades have seen many literatures on the mathematical and computational analysis of neuronal activities resulting in many mathematical models to describe neuron. Many of those models have described the membrane potential of a neuron in terms of the leakage current and the synaptic inputs. Only recently researchers have proposed a new neuron model based on the electromagnetic induction theorem, which considers inner magnetic fluctuation and external electromagnetic radiation as a significant missing part that can participate in neural activity. While the flux coupling of the membrane is considered equivalent to a memductance function of a memristor, standard memductance model of α + 3 β ϕ 2 has been used in the literatures, but in this paper we propose a new memductance function based on discontinuous flux coupling. Various dynamical properties of the neuron model with discontinuous flux coupling are studied and interestingly the proposed model shows hyperchaotic behavior which was not identified in the literatures. Furthermore, we consider a ring network of the proposed model and investigate whether the chimera state can emerge. The chimera state relates to the state with simultaneously coherence and incoherence in oscillatory networks and has received much attention in recent years.

Three-dimensional chimera patterns in networks of spiking neuron oscillators

We study the stable spatiotemporal patterns that arise in a three-dimensional (3D) network of neuron oscillators, whose dynamics is described by the leaky integrate-and-fire (LIF) model. More specifically, we investigate the form of the chimera states induced by a 3D coupling matrix with nonlocal topology. The observed patterns are in many cases direct generalizations of the corresponding two-dimensional (2D) patterns, e.g., spheres, layers, and cylinder grids. We also find cylindrical and "cross-layered" chimeras that do not have an equivalent in 2D systems. Quantitative measures are calculated, such as the ratio of synchronized and unsynchronized neurons as a function of the coupling range, the mean phase velocities, and the distribution of neurons in mean phase velocities. Based on these measures, the chimeras are categorized in two families. The first family of patterns is observed for weaker coupling and exhibits higher mean phase velocities for the unsynchronized areas of the network. The opposite holds for the second family, where the unsynchronized areas have lower mean phase velocities. The various measures demonstrate discontinuities, indicating criticality as the parameters cross from the first family of patterns to the second.

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.