Clustering in globally coupled phase oscillators

Higher-order interactions induce anomalous transitions to synchrony

We analyze the simplest model of identical coupled phase oscillators subject to two-body and three-body interactions with permutation symmetry and phase lags. This model is derived from an ensemble of weakly coupled nonlinear oscillators by phase reduction, where the first and second harmonic interactions with phase lags naturally appear. Our study indicates that the higher-order interactions induce anomalous transitions to synchrony. Unlike the conventional Kuramoto model, higher-order interactions lead to anomalous phenomena such as multistability of full synchronization, incoherent, and two-cluster states, and transitions to synchrony through slow switching and clustering. Phase diagrams of the dynamical regimes are constructed theoretically and verified by direct numerical simulations. We also show that similar transition scenarios are observed even if a small heterogeneity in the oscillators' frequency is included.

Extreme transient dynamics

We study the extreme transient dynamics of four self-excited pendula coupled via the movable beam. A slight difference in the pendula lengths induces the appearance of traveling phase behavior, within which the oscillators synchronize, but the phases between the nodes change in time. We discuss various scenarios of traveling states (involving different pendula) and their properties, comparing them with classical synchronization patterns of phase-locking. The research investigates the problem of transient dynamics preceding the stabilization of the network on a final synchronous attractor, showing that the width of transient windows can become extremely long. The relation between the behavior of the system within the transient regime and its initial conditions is examined and described. Our results include both identical and non-identical pendula masses, showing that the distribution of the latter ones is related to the transients. The research performed in this paper underlines possible transient problems occurring during the analysis of the systems when the slow evolution of the dynamics can be misinterpreted as the final behavior.

Patterns of synchronization in 2D networks of inhibitory neurons

Neural firing in many inhibitory networks displays synchronous assembly or clustered firing, in which subsets of neurons fire synchronously, and these subsets may vary with different inputs to, or states of, the network. Most prior analytical and computational modeling of such networks has focused on 1D networks or 2D networks with symmetry (often circular symmetry). Here, we consider a 2D discrete network model on a general torus, where neurons are coupled to two or more nearest neighbors in three directions (horizontal, vertical, and diagonal), and allow different coupling strengths in all directions. Using phase model analysis, we establish conditions for the stability of different patterns of clustered firing behavior in the network. We then apply our results to study how variation of network connectivity and the presence of heterogeneous coupling strengths influence which patterns are stable. We confirm and supplement our results with numerical simulations of biophysical inhibitory neural network models. Our work shows that 2D networks may exhibit clustered firing behavior that cannot be predicted as a simple generalization of a 1D network, and that heterogeneity of coupling can be an important factor in determining which patterns are stable.

Connecting minimal chimeras and fully asymmetric chaotic attractors through equivariant pitchfork bifurcations

Highly symmetric networks can exhibit partly symmetry-broken states, including clusters and chimera states, i.e., states of coexisting synchronized and unsynchronized elements. We address the S_{4} permutation symmetry of four globally coupled Stuart-Landau oscillators and uncover an interconnected web of solutions with different symmetries. Among these are chaotic 2-1-1 minimal chimeras that arise from 2-1-1 periodic solutions in a period-doubling cascade, as well as fully asymmetric chaotic states arising similarly from periodic 1-1-1-1 solutions. A backbone of equivariant pitchfork bifurcations mediate between the two cascades, culminating in equivariant pitchforks of chaotic attractors.

Spatially localized cluster solutions in inhibitory neural networks

Neurons in the inhibitory network of the striatum display cell assembly firing patterns which recent results suggest may consist of spatially compact neural clusters. Previous computational modeling of striatal neural networks has indicated that non-monotonic, distance-dependent coupling may promote spatially localized cluster firing. Here, we identify conditions for the existence and stability of cluster firing solutions in which clusters consist of spatially adjacent neurons in inhibitory neural networks. We consider simple non-monotonic, distance-dependent connectivity schemes in weakly coupled 1-D networks where cells make stronger connections with their kth nearest neighbors on each side and weaker connections with closer neighbors. Using the phase model reduction of the network system, we prove the existence of cluster solutions where neurons that are spatially close together are also synchronized in the same cluster, and find stability conditions for these solutions. Our analysis predicts the long-term behavior for networks of neurons, and we confirm our results by numerical simulations of biophysical neuron network models. Our results demonstrate that an inhibitory network with non-monotonic, distance-dependent connectivity can exhibit cluster solutions where adjacent cells fire together.

Diversity of dynamical behaviors due to initial conditions: Extension of the Ott-Antonsen ansatz for identical Kuramoto-Sakaguchi phase oscillators

The Ott-Antonsen ansatz is a powerful tool to extract the behaviors of coupled phase oscillators, but it imposes a strong restriction on the initial condition. Herein, an extension of the Ott-Antonsen ansatz is proposed to relax the restriction, enabling the systematic approximation of the behavior of a globally coupled phase oscillator system with an arbitrary initial condition. The proposed method is validated on the Kuramoto-Sakaguchi model of identical phase oscillators. The method yields cluster and chimera-like solutions that are not obtained by the conventional ansatz.

Synchronization in starlike networks of phase oscillators

We fully describe the mechanisms underlying synchronization in starlike networks of phase oscillators. In particular, the routes to synchronization and the critical points for the associated phase transitions are determined analytically. In contrast to the classical Kuramoto theory, we unveil that relaxation rates to each equilibrium state indeed exist and remain invariant under three levels of descriptions corresponding to different geometric implications. The special symmetry in the coupling determines a quasi-Hamiltonian property, which is further unveiled on the basis of singular perturbation theory. Since starlike coupling configurations constitute the building blocks of technological and biological real world networks, our paper paves the way towards the understanding of the functioning of such real world systems in many practical situations.

Phase reduction beyond the first order: The case of the mean-field complex Ginzburg-Landau equation

Phase reduction is a powerful technique that makes possible to describe the dynamics of a weakly perturbed limit-cycle oscillator in terms of its phase. For ensembles of oscillators, a classical example of phase reduction is the derivation of the Kuramoto model from the mean-field complex Ginzburg-Landau equation (MF-CGLE). Still, the Kuramoto model is a first-order phase approximation that displays either full synchronization or incoherence, but none of the nontrivial dynamics of the MF-CGLE. This fact calls for an expansion beyond the first order in the coupling constant. We develop an isochron-based scheme to obtain the second-order phase approximation, which reproduces the weak-coupling dynamics of the MF-CGLE. The practicality of our method is evidenced by extending the calculation up to third order. Each new term of the power-series expansion contributes with additional higher-order multibody (i.e., nonpairwise) interactions. This points to intricate multibody phase interactions as the source of pure collective chaos in the MF-CGLE at moderate coupling.

A coupled oscillator model for the origin of bimodality and multimodality

Perhaps because of the elegance of the central limit theorem, it is often assumed that distributions in nature will approach singly-peaked, unimodal shapes reminiscent of the Gaussian normal distribution. However, many systems behave differently, with variables following apparently bimodal or multimodal distributions. Here, we argue that multimodality may emerge naturally as a result of repulsive or inhibitory coupling dynamics, and we show rigorously how it emerges for a broad class of coupling functions in variants of the paradigmatic Kuramoto model.

Between phase and amplitude oscillators

We analyze an intermediate collective regime where amplitude oscillators distribute themselves along a closed, smooth, time-dependent curve C, thereby maintaining the typical ordering of (identical) phase oscillators. This is achieved by developing a general formalism based on two partial differential equations, which describe the evolution of the probability density along C and of the shape of C itself. The formalism is specifically developed for Stuart-Landau oscillators, but it is general enough to apply to other classes of amplitude oscillators. The main achievements consist in (i) identification and characterization of a transition to self-consistent partial synchrony (SCPS), which confirms the crucial role played by higher Fourier harmonics in the coupling function; (ii) an analytical treatment of SCPS, including a detailed stability analysis; and (iii) the discovery of a different form of collective chaos, which can be seen as a generalization of SCPS and characterized by a multifractal probability density. Finally, we are able to describe given dynamical regimes at both the macroscopic and the microscopic level, thereby shedding additional light on the relationship between the two different levels of description.

Repulsively coupled Kuramoto-Sakaguchi phase oscillators ensemble subject to common noise

We consider the Kuramoto-Sakaguchi model of identical coupled phase oscillators with a common noisy forcing. While common noise always tends to synchronize the oscillators, a strong repulsive coupling prevents the fully synchronous state and leads to a nontrivial distribution of oscillator phases. In previous numerical simulations, the formation of stable multicluster states has been observed in this regime. However, we argue here that because identical phase oscillators in the Kuramoto-Sakaguchi model form a partially integrable system according to the Watanabe-Strogatz theory, the formation of clusters is impossible. Integrating with various time steps reveals that clustering is a numerical artifact, explained by the existence of higher order Fourier terms in the errors of the employed numerical integration schemes. By monitoring the induced change in certain integrals of motion, we quantify these errors. We support these observations by showing, on the basis of the analysis of the corresponding Fokker-Planck equation, that two-cluster states are non-attractive. On the other hand, in ensembles of general limit cycle oscillators, such as Van der Pol oscillators, due to an anharmonic phase response function as well as additional amplitude dynamics, multiclusters can occur naturally.

Phase response theory explains cluster formation in sparsely but strongly connected inhibitory neural networks and effects of jitter due to sparse connectivity

We show how to predict whether a neural network will exhibit global synchrony (a one-cluster state) or a two-cluster state based on the assumption of pulsatile coupling and critically dependent upon the phase response curve (PRC) generated by the appropriate perturbation from a partner cluster. Our results hold for a monotonically increasing (meaning longer delays as the phase increases) PRC, which likely characterizes inhibitory fast-spiking basket and cortical low-threshold-spiking interneurons in response to strong inhibition. Conduction delays stabilize synchrony for this PRC shape, whereas they destroy two-cluster states, the former by avoiding a destabilizing discontinuity and the latter by approaching it. With conduction delays, stronger coupling strength can promote a one-cluster state, so the weak coupling limit is not applicable here. We show how jitter can destabilize global synchrony but not a two-cluster state. Local stability of global synchrony in an all-to-all network does not guarantee that global synchrony can be observed in an appropriately scaled sparsely connected network; the basin of attraction can be inferred from the PRC and must be sufficiently large. Two-cluster synchrony is not obviously different from one-cluster synchrony in the presence of noise and may be the actual substrate for oscillations observed in the local field potential (LFP) and the electroencephalogram (EEG) in situations where global synchrony is not possible. Transitions between cluster states may change the frequency of the rhythms observed in the LFP or EEG. Transitions between cluster states within an inhibitory subnetwork may allow more effective recruitment of pyramidal neurons into the network rhythm. NEW & NOTEWORTHY We show that jitter induced by sparse connectivity can destabilize global synchrony but not a two-cluster state with two smaller clusters firing alternately. On the other hand, conduction delays stabilize synchrony and destroy two-cluster states. These results hold if each cluster exhibits a phase response curve similar to one that characterizes fast-spiking basket and cortical low-threshold-spiking cells for strong inhibition. Either a two-cluster or a one-cluster state might provide the oscillatory substrate for neural computations.

Low-dimensional dynamics in non-Abelian Kuramoto model on the 3-sphere

This paper deals with the low-dimensional dynamics in the general non-Abelian Kuramoto model of mutually interacting generalized oscillators on the 3-sphere. If all oscillators have identical intrinsic generalized frequencies and the coupling is global, the dynamics is fully determined by several global variables. We state that these generalized oscillators evolve by the action of the group GH of (quaternionic) Möbius transformations that preserve S3 . The global variables satisfy a certain system of quaternion-valued ordinary differential equations, that is an extension of the Watanabe-Strogatz system. If the initial distribution of oscillators is uniform on S3 , additional symmetries arise and the dynamics can be restricted further to invariant submanifolds of (real) dimension four.

Ubiquity of collective irregular dynamics in balanced networks of spiking neurons

We revisit the dynamics of a prototypical model of balanced activity in networks of spiking neurons. A detailed investigation of the thermodynamic limit for fixed density of connections (massive coupling) shows that, when inhibition prevails, the asymptotic regime is not asynchronous but rather characterized by a self-sustained irregular, macroscopic (collective) dynamics. So long as the connectivity is massive, this regime is found in many different setups: leaky as well as quadratic integrate-and-fire neurons; large and small coupling strength; and weak and strong external currents.

Phase-locking and bistability in neuronal networks with synaptic depression

We consider a recurrent network of two oscillatory neurons that are coupled with inhibitory synapses. We use the phase response curves of the neurons and the properties of short-term synaptic depression to define Poincaré maps for the activity of the network. The fixed points of these maps correspond to phase-locked modes of the network. Using these maps, we analyze the conditions that allow short-term synaptic depression to lead to the existence of bistable phase-locked, periodic solutions. We show that bistability arises when either the phase response curve of the neuron or the short-term depression profile changes steeply enough. The results apply to any Type I oscillator and we illustrate our findings using the Quadratic Integrate-and-Fire and Morris-Lecar neuron models.

Emergence of localized patterns in globally coupled networks of relaxation oscillators with heterogeneous connectivity

Oscillations in far-from-equilibrium systems (e.g., chemical, biochemical, biological) are generated by the nonlinear interplay of positive and negative feedback effects operating at different time scales. Relaxation oscillations emerge when the time scales between the activators and the inhibitors are well separated. In addition to the large-amplitude oscillations (LAOs) or relaxation type, these systems exhibit small-amplitude oscillations (SAOs) as well as abrupt transitions between them (canard phenomenon). Localized cluster patterns in networks of relaxation oscillators consist of one cluster oscillating in the LAO regime or exhibiting mixed-mode oscillations (LAOs interspersed with SAOs), while the other oscillates in the SAO regime. Because the individual oscillators are monostable, localized patterns are a network phenomenon that involves the interplay of the connectivity and the intrinsic dynamic properties of the individual nodes. Motivated by experimental and theoretical results on the Belousov-Zhabotinsky reaction, we investigate the mechanisms underlying the generation of localized patterns in globally coupled networks of piecewise-linear relaxation oscillators where the global feedback term affects the rate of change of the activator (fast variable) and depends on the weighted sum of the inhibitor (slow variable) at any given time. We also investigate whether these patterns are affected by the presence of a diffusive type of coupling whose synchronizing effects compete with the symmetry-breaking global feedback effects.

Flocking ferromagnetic colloids

Assemblages of microscopic colloidal particles exhibit fascinating collective motion when energized by electric or magnetic fields. The behaviors range from coherent vortical motion to phase separation and dynamic self-assembly. Although colloidal systems are relatively simple, understanding their collective response, especially under out-of-equilibrium conditions, remains elusive. We report on the emergence of flocking and global rotation in the system of rolling ferromagnetic microparticles energized by a vertical alternating magnetic field. By combing experiments and discrete particle simulations, we have identified primary physical mechanisms, leading to the emergence of large-scale collective motion: spontaneous symmetry breaking of the clockwise/counterclockwise particle rotation, collisional alignment of particle velocities, and random particle reorientations due to shape imperfections. We have also shown that hydrodynamic interactions between the particles do not have a qualitative effect on the collective dynamics. Our findings shed light on the onset of spatial and temporal coherence in a large class of active systems, both synthetic (colloids, swarms of robots, and biopolymers) and living (suspensions of bacteria, cell colonies, and bird flocks).

Drifting States and Synchronization Induced Chaos in Autonomous Networks of Excitable Neurons

The study of balanced networks of excitatory and inhibitory neurons has led to several open questions. On the one hand it is yet unclear whether the asynchronous state observed in the brain is autonomously generated, or if it results from the interplay between external drivings and internal dynamics. It is also not known, which kind of network variabilities will lead to irregular spiking and which to synchronous firing states. Here we show how isolated networks of purely excitatory neurons generically show asynchronous firing whenever a minimal level of structural variability is present together with a refractory period. Our autonomous networks are composed of excitable units, in the form of leaky integrators spiking only in response to driving currents, remaining otherwise quiet. For a non-uniform network, composed exclusively of excitatory neurons, we find a rich repertoire of self-induced dynamical states. We show in particular that asynchronous drifting states may be stabilized in purely excitatory networks whenever a refractory period is present. Other states found are either fully synchronized or mixed, containing both drifting and synchronized components. The individual neurons considered are excitable and hence do not dispose of intrinsic natural firing frequencies. An effective network-wide distribution of natural frequencies is however generated autonomously through self-consistent feedback loops. The asynchronous drifting state is, additionally, amenable to an analytic solution. We find two types of asynchronous activity, with the individual neurons spiking regularly in the pure drifting state, albeit with a continuous distribution of firing frequencies. The activity of the drifting component, however, becomes irregular in the mixed state, due to the periodic driving of the synchronized component. We propose a new tool for the study of chaos in spiking neural networks, which consists of an analysis of the time series of pairs of consecutive interspike intervals. In this space, we show that a strange attractor with a fractal dimension of about 1.8 is formed in the mentioned mixed state.

Collective dynamics of identical phase oscillators with high-order coupling

In this paper, we propose a framework to investigate the collective dynamics in ensembles of globally coupled phase oscillators when higher-order modes dominate the coupling. The spatiotemporal properties of the attractors in various regions of parameter space are analyzed. Furthermore, a detailed linear stability analysis proves that the stationary symmetric distribution is only neutrally stable in the marginal regime which stems from the generalized time-reversal symmetry. Moreover, the critical parameters of the transition among various regimes are determined analytically by both the Ott-Antonsen method and linear stability analysis, the transient dynamics are further revealed in terms of the characteristic curves method. Finally, for the more general initial condition the symmetric dynamics could be reduced to a rigorous three-dimensional manifold which shows that the neutrally stable chaos could also occur in this model for particular parameters. Our theoretical analysis and numerical results are consistent with each other, which can help us understand the dynamical properties in general systems with higher-order harmonics couplings.

Star-type oscillatory networks with generic Kuramoto-type coupling: A model for "Japanese drums synchrony"

We analyze star-type networks of phase oscillators by virtue of two methods. For identical oscillators we adopt the Watanabe-Strogatz approach, which gives full analytical description of states, rotating with constant frequency. For nonidentical oscillators, such states can be obtained by virtue of the self-consistent approach in a parametric form. In this case stability analysis cannot be performed, however with the help of direct numerical simulations we show which solutions are stable and which not. We consider this system as a model for a drum orchestra, where we assume that the drummers follow the signal of the leader without listening to each other and the coupling parameters are determined by a geometrical organization of the orchestra.

Coarse-Grained Clustering Dynamics of Heterogeneously Coupled Neurons

The formation of oscillating phase clusters in a network of identical Hodgkin-Huxley neurons is studied, along with their dynamic behavior. The neurons are synaptically coupled in an all-to-all manner, yet the synaptic coupling characteristic time is heterogeneous across the connections. In a network of N neurons where this heterogeneity is characterized by a prescribed random variable, the oscillatory single-cluster state can transition-through [Formula: see text] (possibly perturbed) period-doubling and subsequent bifurcations-to a variety of multiple-cluster states. The clustering dynamic behavior is computationally studied both at the detailed and the coarse-grained levels, and a numerical approach that can enable studying the coarse-grained dynamics in a network of arbitrarily large size is suggested. Among a number of cluster states formed, double clusters, composed of nearly equal sub-network sizes are seen to be stable; interestingly, the heterogeneity parameter in each of the double-cluster components tends to be consistent with the random variable over the entire network: Given a double-cluster state, permuting the dynamical variables of the neurons can lead to a combinatorially large number of different, yet similar "fine" states that appear practically identical at the coarse-grained level. For weak heterogeneity we find that correlations rapidly develop, within each cluster, between the neuron's "identity" (its own value of the heterogeneity parameter) and its dynamical state. For single- and double-cluster states we demonstrate an effective coarse-graining approach that uses the Polynomial Chaos expansion to succinctly describe the dynamics by these quickly established "identity-state" correlations. This coarse-graining approach is utilized, within the equation-free framework, to perform efficient computations of the neuron ensemble dynamics.

Insights into collective cell behaviour from populations of coupled chemical oscillators

Biological systems such as yeast show coordinated activity driven by chemical communication between cells. Experiments with coupled chemical oscillators can provide insights into the collective behaviour.

Dynamics and Reliability of Bistable Neurons Driven with Time-Dependent Stimuli

The reliability of a spiking neuron depends on the frequency content of the driving input signal. Previous studies have shown that well above threshold, regularly firing neurons generate reliable responses when the input signal resonates with the firing frequency of the cell. Instead, well below threshold, reliable responses are obtained when the input frequency resonates with the subthreshold oscillations of the neuron. Previous theories, however, provide no clear prediction for the input frequency giving rise to maximally reliable spiking at threshold, which is probably the most relevant firing regime in mammalian cortex under physiological conditions. In particular, when the firing onset is governed by a subcritical Hopf bifurcation, the frequency of subthreshold oscillations often differs from the firing rate at threshold. The predictions of previous studies, hence, cannot be smoothly merged at threshold. Here we explore the behavior of reliability in bistable neurons near threshold using three types of driving stimuli: constant, periodic, and stochastic. We find that the two natural frequencies of the system, associated with the two coexisting attractors, provide a rich variety of possible locking modes with the external signal. Reliability is determined by the sensitivity to noise of each locking mode and by the transition probabilities between modes. Noise increases the amount of spike time jitter, and minimal jitter is obtained for input frequencies coinciding with the suprathreshold firing rate of the cell. In addition, noise may either enhance or inhibit transitions between the two attractors, depending on the input frequency. The dual role played by noise in bistable systems implies that reliability is determined by a delicate balance between spike time jitter and the rate of transitions between attractors.

Experimental synchronization of chaos in a large ring of mutually coupled single-transistor oscillators: phase, amplitude, and clustering effects

In this paper, experimental evidence of multiple synchronization phenomena in a large (n = 30) ring of chaotic oscillators is presented. Each node consists of an elementary circuit, generating spikes of irregular amplitude and comprising one bipolar junction transistor, one capacitor, two inductors, and one biasing resistor. The nodes are mutually coupled to their neighbours via additional variable resistors. As coupling resistance is decreased, phase synchronization followed by complete synchronization is observed, and onset of synchronization is associated with partial synchronization, i.e., emergence of communities (clusters). While component tolerances affect community structure, the general synchronization properties are maintained across three prototypes and in numerical simulations. The clusters are destroyed by adding long distance connections with distant notes, but are otherwise relatively stable with respect to structural connectivity changes. The study provides evidence that several fundamental synchronization phenomena can be reliably observed in a network of elementary single-transistor oscillators, demonstrating their generative potential and opening way to potential applications of this undemanding setup in experimental modelling of the relationship between network structure, synchronization, and dynamical properties.

Mutually synchronized bottom-up multi-nanocontact spin-torque oscillators

Spin-torque oscillators offer a unique combination of nanosize, ultrafast modulation rates and ultrawide band signal generation from 100 MHz to close to 100 GHz. However, their low output power and large phase noise still limit their applicability to fundamental studies of spin-transfer torque and magnetodynamic phenomena. A possible solution to both problems is the spin-wave-mediated mutual synchronization of multiple spin-torque oscillators through a shared excited ferromagnetic layer. To date, synchronization of high-frequency spin-torque oscillators has only been achieved for two nanocontacts. As fabrication using expensive top-down lithography processes is not readily available to many groups, attempts to synchronize a large number of nanocontacts have been all but abandoned. Here we present an alternative, simple and cost-effective bottom-up method to realize large ensembles of synchronized nanocontact spin-torque oscillators. We demonstrate mutual synchronization of three high-frequency nanocontact spin-torque oscillators and pairwise synchronization in devices with four and five nanocontacts.

Classification of attractors for systems of identical coupled Kuramoto oscillators

We present a complete classification of attractors for networks of coupled identical Kuramoto oscillators. In such networks, each oscillator is driven by the same first-order trigonometric function, with coefficients given by symmetric functions of the entire oscillator ensemble. For [Formula: see text] oscillators, there are four possible types of attractors: completely synchronized fixed points or limit cycles, and fixed points or limit cycles where all but one of the oscillators are synchronized. The case N = 3 is exceptional; systems of three identical Kuramoto oscillators can also posses attracting fixed points or limit cycles with all three oscillators out of sync, as well as chaotic attractors. Our results rely heavily on the invariance of the flow for such systems under the action of the three-dimensional group of Möbius transformations, which preserve the unit disc, and the analysis of the possible limiting configurations for this group action.

The emergence of two anti-phase oscillatory neural populations in a computational model of the Parkinsonian globus pallidus

Experiments in rodent models of Parkinson's disease have demonstrated a prominent increase of oscillatory firing patterns in neurons within the Parkinsonian globus pallidus (GP) which may underlie some of the motor symptoms of the disease. There are two main pathways from the cortex to GP: via the striatum and via the subthalamic nucleus (STN), but it is not known how these inputs sculpt the pathological pallidal firing patterns. To study this we developed a novel neural network model of conductance-based spiking pallidal neurons with cortex-modulated input from STN neurons. Our results support the hypothesis that entrainment occurs primarily via the subthalamic pathway. We find that as a result of the interplay between excitatory input from the STN and mutual inhibitory coupling between GP neurons, a homogeneous population of GP neurons demonstrates a self-organizing dynamical behavior where two groups of neurons emerge: one spiking in-phase with the cortical rhythm and the other in anti-phase. This finding mirrors what is seen in recordings from the GP of rodents that have had Parkinsonism induced via brain lesions. Our model also includes downregulation of Hyperpolarization-activated Cyclic Nucleotide-gated (HCN) channels in response to burst firing of GP neurons, since this has been suggested as a possible mechanism for the emergence of Parkinsonian activity. We found that the downregulation of HCN channels provides even better correspondence with experimental data but that it is not essential in order for the two groups of oscillatory neurons to appear. We discuss how the influence of inhibitory striatal input will strengthen our results.

Chimera states in mechanical oscillator networks

The synchronization of coupled oscillators is a fascinating manifestation of self-organization that nature uses to orchestrate essential processes of life, such as the beating of the heart. Although it was long thought that synchrony and disorder were mutually exclusive steady states for a network of identical oscillators, numerous theoretical studies in recent years have revealed the intriguing possibility of "chimera states," in which the symmetry of the oscillator population is broken into a synchronous part and an asynchronous part. However, a striking lack of empirical evidence raises the question of whether chimeras are indeed characteristic of natural systems. This calls for a palpable realization of chimera states without any fine-tuning, from which physical mechanisms underlying their emergence can be uncovered. Here, we devise a simple experiment with mechanical oscillators coupled in a hierarchical network to show that chimeras emerge naturally from a competition between two antagonistic synchronization patterns. We identify a wide spectrum of complex states, encompassing and extending the set of previously described chimeras. Our mathematical model shows that the self-organization observed in our experiments is controlled by elementary dynamical equations from mechanics that are ubiquitous in many natural and technological systems. The symmetry-breaking mechanism revealed by our experiments may thus be prevalent in systems exhibiting collective behavior, such as power grids, optomechanical crystals, or cells communicating via quorum sensing in microbial populations.

Typical trajectories of coupled degrade-and-fire oscillators: from dispersed populations to massive clustering

We consider the dynamics of a piecewise affine system of degrade-and-fire oscillators with global repressive interaction, inspired by experiments on synchronization in colonies of bacteria-embedded genetic circuits. Due to global coupling, if any two oscillators happen to be in the same state at some time, they remain in sync at all subsequent times; thus clusters of synchronized oscillators cannot shrink as a result of the dynamics. Assuming that the system is initiated from random initial configurations of fully dispersed populations (no clusters), we estimate asymptotic cluster sizes as a function of the coupling strength. A sharp transition is proved to exist that separates a weak coupling regime of unclustered populations from a strong coupling phase where clusters of extensive size are formed. Each phenomena occurs with full probability in the thermodynamics limit. Moreover, the maximum number of asymptotic clusters is known to diverge linearly in this limit. In contrast, we show that with positive probability, the number of asymptotic clusters remains bounded, provided that the coupling strength is sufficiently large.

Swing, release, and escape mechanisms contribute to the generation of phase-locked cluster patterns in a globally coupled FitzHugh-Nagumo model

We investigate the mechanism of generation of phase-locked cluster patterns in a globally coupled FitzhHugh-Nagumo model where the fast variable (activator) receives global feedback from the slow variable (inhibitor). We identify three qualitatively different mechanisms (swing-and-release, hold-and-release, and escape-and-release) that contribute to the generation of these patterns. We describe these mechanisms and use this framework to explain under what circumstances two initially out-of-phase oscillatory clusters reach steady phase-locked and in-phase synchronized solutions, and how the phase difference between these steady state cluster patterns depends on the clusters relative size, the global coupling intensity, and other model parameters.

Dynamic mechanisms of generation of oscillatory cluster patterns in a globally coupled chemical system

We use simulations and dynamical systems tools to investigate the mechanisms of generation of phase-locked and localized oscillatory cluster patterns in a globally coupled Oregonator model where the activator receives global feedback from the inhibitor, mimicking experimental results observed in the photosensitive Belousov-Zhabotinsky reaction. A homogeneous two-cluster system (two clusters with equal cluster size) displays antiphase patterns. Heterogenous two-cluster systems (two clusters with different sizes) display both phase-locked and localized patterns depending on the parameter values. In a localized pattern the oscillation amplitude of the largest cluster is roughly an order of magnitude smaller than the oscillation amplitude of the smaller cluster, reflecting the effect of self-inhibition exerted by the global feedback term. The transition from phase-locked to localized cluster patterns occurs as the intensity of global feedback increases. Three qualitatively different basic mechanisms, described previously for a globally coupled FitzHugh-Nagumo model, are involved in the generation of the observed patterns. The swing-and-release mechanism is related to the canard phenomenon (canard explosion of limit cycles) in relaxation oscillators. The hold-and-release and hold-and-escape mechanisms are related to the release and escape mechanisms in synaptically connected neural models. The methods we use can be extended to the investigation of oscillatory chemical reactions with other types of non-local coupling.

Adjoint method provides phase response functions for delay-induced oscillations

Limit-cycle oscillations induced by time delay are widely observed in various systems, but a systematic phase-reduction theory for them has yet to be developed. Here we present a practical theoretical framework to calculate the phase response function Z(θ), a fundamental quantity for the theory, of delay-induced limit cycles with infinite-dimensional phase space. We show that Z(θ) can be obtained as a zero eigenfunction of the adjoint equation associated with an appropriate bilinear form for the delay differential equations. We confirm the validity of the proposed framework for two biological oscillators and demonstrate that the derived phase equation predicts intriguing multimodal locking behavior.

Persistent fluctuations in synchronization rate in globally coupled oscillators with periodic external forcing

A system of phase oscillators with repulsive global coupling and periodic external forcing undergoing asynchronous rotation is considered. The synchronization rate of the system can exhibit persistent fluctuations depending on parameters and initial phase distributions, and the amplitude of the fluctuations scales with the system size for uniformly random initial phase distributions. Using the Watanabe-Strogatz transformation that reduces the original system to low-dimensional macroscopic equations, we show that the fluctuations are collective dynamics of the system corresponding to low-dimensional trajectories of the reduced equations. It is argued that the amplitude of the fluctuations is determined by the inhomogeneity of the initial phase distribution, resulting in system-size scaling for the random case.

Interactions between network synchrony and the dynamics of neuronal threshold

Synchronous activity impacts on a range of functional brain capacities in health and disease. To address the interrelations between cellular level activity and network-wide synchronous events, we implemented in vitro a recently introduced technique, the response clamp, which enables online monitoring of single neuron threshold dynamics while ongoing network synchronous activity continues uninterrupted. We show that the occurrence of a synchronous network event causes a significant biphasic change in the single neuron threshold. These threshold dynamics are correlated across the neurons constituting the network and are entailed by the input to the neurons rather than by their own spiking (i.e., output) activity. The magnitude of network activity during a synchronous event is correlated with the threshold state of individual neurons at the event's onset. Recovery from the impact of a given synchronous event on the neuronal threshold lasts several seconds and seems to be a key determinant of the time to the next spontaneously occurring synchronous event. Moreover, the neuronal threshold is shown to be correlated with the excitability dynamics of the entire network. We conclude that the relations between the two levels (network activity and the single neuron threshold) should be thought of in terms that emphasize their interactive nature.

Self-organized network of phase oscillators coupled by activity-dependent interactions

We investigate a network of coupled phase oscillators whose interactions evolve dynamically depending on the relative phases between the oscillators. We found that this coevolving dynamical system robustly yields three basic states of collective behavior with their self-organized interactions. The first is the two-cluster state, in which the oscillators are organized into two synchronized groups. The second is the coherent state, in which the oscillators are arranged sequentially in time. The third is the chaotic state, in which the relative phases between oscillators and their coupling weights are chaotically shuffled. Furthermore, we demonstrate that self-assembled multiclusters can be designed by controlling the weight dynamics. Note that the phase patterns of the oscillators and the weighted network of interactions between them are simultaneously organized through this coevolving dynamics. We expect that these results will provide new insight into self-assembly mechanisms by which the collective behavior of a rhythmic system emerges as a result of the dynamics of adaptive interactions.

Sparse gamma rhythms arising through clustering in adapting neuronal networks

Gamma rhythms (30-100 Hz) are an extensively studied synchronous brain state responsible for a number of sensory, memory, and motor processes. Experimental evidence suggests that fast-spiking interneurons are responsible for carrying the high frequency components of the rhythm, while regular-spiking pyramidal neurons fire sparsely. We propose that a combination of spike frequency adaptation and global inhibition may be responsible for this behavior. Excitatory neurons form several clusters that fire every few cycles of the fast oscillation. This is first shown in a detailed biophysical network model and then analyzed thoroughly in an idealized model. We exploit the fact that the timescale of adaptation is much slower than that of the other variables. Singular perturbation theory is used to derive an approximate periodic solution for a single spiking unit. This is then used to predict the relationship between the number of clusters arising spontaneously in the network as it relates to the adaptation time constant. We compare this to a complementary analysis that employs a weak coupling assumption to predict the first Fourier mode to destabilize from the incoherent state of an associated phase model as the external noise is reduced. Both approaches predict the same scaling of cluster number with respect to the adaptation time constant, which is corroborated in numerical simulations of the full system. Thus, we develop several testable predictions regarding the formation and characteristics of gamma rhythms with sparsely firing excitatory neurons.

Cluster synchrony in systems of coupled phase oscillators with higher-order coupling

We study the phenomenon of cluster synchrony that occurs in ensembles of coupled phase oscillators when higher-order modes dominate the coupling between oscillators. For the first time, we develop a complete analytic description of the dynamics in the limit of a large number of oscillators and use it to quantify the degree of cluster synchrony, cluster asymmetry, and switching. We use a variation of the recent dimensionality-reduction technique of Ott and Antonsen [Chaos 18, 037113 (2008)] and find an analytic description of the degree of cluster synchrony valid on a globally attracting manifold. Shaped by this manifold, there is an infinite family of steady-state distributions of oscillators, resulting in a high degree of multistability in the cluster asymmetry. We also show how through external forcing the degree of asymmetry can be controlled, and suggest that systems displaying cluster synchrony can be used to encode and store data.

Generating macroscopic chaos in a network of globally coupled phase oscillators

We consider an infinite network of globally coupled phase oscillators in which the natural frequencies of the oscillators are drawn from a symmetric bimodal distribution. We demonstrate that macroscopic chaos can occur in this system when the coupling strength varies periodically in time. We identify period-doubling cascades to chaos, attractor crises, and horseshoe dynamics for the macroscopic mean field. Based on recent work that clarified the bifurcation structure of the static bimodal Kuramoto system, we qualitatively describe the mechanism for the generation of such complicated behavior in the time varying case.

Resonance tongues in a system of globally coupled FitzHugh-Nagumo oscillators with time-periodic coupling strength

We investigate the dynamics of a population of globally coupled FitzHugh-Nagumo oscillators with a time-periodic coupling strength. While for synchronizing global coupling, the in-phase state is always stable, the oscillators split into several cluster states for desynchronizing global coupling, most commonly in two, irrespective of the coupling strength. This confines the ability of the system to form n:m locked states considerably. The prevalence of two and four cluster states leads to large 2:1 and 4:1 subharmonic resonance regions, while at low coupling strength for a harmonic 1:1 or a superharmonic 1:m time-periodic coupling coefficient, any resonances are absent and the system exhibits nonresonant phase drifting cluster states. Furthermore, in the unforced, globally coupled system the frequency of the oscillators in a cluster state is in general lower than that of the uncoupled oscillator and strongly depends on the coupling strength. Periodic variation of the coupling strength at twice the natural frequency causes each oscillator to keep oscillating with its autonomous oscillation period.

Dynamics of one- and two-dimensional fronts in a bistable equation with time-delayed global feedback: Propagation failure and control mechanisms

We study the evolution of fronts in a bistable equation with time-delayed global feedback in the fast reaction and slow diffusion regime. This equation generalizes the Hodgkin-Grafstein and Allen-Cahn equations. We derive a nonlinear equation governing the motion of fronts, which includes a term with delay. In the one-dimensional case this equation is linear. We study the motion of one- and two-dimensional fronts, finding a much richer dynamics than for the previously studied cases (without time-delayed global feedback). We explain the mechanism by which localized fronts created by inhibitory global coupling loose stability in a Hopf bifurcation as the delay time increases. We show that for certain delay times, the prevailing phase is different from that corresponding to the system in the absence of global coupling. Numerical simulations of the partial differential equation are in agreement with the analytical predictions.

Transient process of cortical activity during Necker cube perception: from local clusters to global synchrony

BACKGROUND

It has been discussed that neural phase-synchrony across distant cortical areas (or global phase-synchrony) was correlated with various aspects of consciousness. The generating process of the synchrony, however, remains largely unknown. As a first step, we investigate transient process of global phase-synchrony, focusing on phase-synchronized clusters. We hypothesize that the phase-synchronized clusters are dynamically organized before global synchrony and clustering patterns depend on perceptual conditions.

METHODS

In an EEG study, Kitajo reported that phase-synchrony across distant cortical areas was selectively enhanced by top-down attention around 4 Hz in Necker cube perception. Here, we further analyzed the phase-synchronized clusters using hierarchical clustering which sequentially binds up the nearest electrodes based on similarity of phase locking between the cortical signals. First, we classified dominant components of the phase-synchronized clusters over time. We then investigated how the phase-synchronized clusters change with time, focusing on their size and spatial structure.

RESULTS

Phase-locked clusters organized a stable spatial pattern common to the perceptual conditions. In addition, the phase-locked clusters were modulated transiently depending on the perceptual conditions and the time from the perceptual switch. When top-down attention succeeded in switching perception as subjects intended, independent clusters at frontal and occipital areas grew to connect with each other around the time of the perceptual switch. However, the clusters in the occipital and left parietal areas remained divided when top-down attention failed in switching perception. When no primary biases exist, the cluster in the occipital area grew to its maximum at the time of the perceptual switch within the occipital area.

CONCLUSIONS

Our study confirmed the existence of stable phase-synchronized clusters. Furthermore, these clusters were transiently connected with each other. The connecting pattern depended on subjects' internal states. These results suggest that subjects' attentional states are associated with distinct spatio-temporal patterns of the phase-locked clusters.

Identical phase oscillators with global sinusoidal coupling evolve by Mobius group action

Systems of N identical phase oscillators with global sinusoidal coupling are known to display low-dimensional dynamics. Although this phenomenon was first observed about 20 years ago, its underlying cause has remained a puzzle. Here we expose the structure working behind the scenes of these systems by proving that the governing equations are generated by the action of the Mobius group, a three-parameter subgroup of fractional linear transformations that map the unit disk to itself. When there are no auxiliary state variables, the group action partitions the N-dimensional state space into three-dimensional invariant manifolds (the group orbits). The N-3 constants of motion associated with this foliation are the N-3 functionally independent cross ratios of the oscillator phases. No further reduction is possible, in general; numerical experiments on models of Josephson junction arrays suggest that the invariant manifolds often contain three-dimensional regions of neutrally stable chaos.

Clustering predicted by an electrophysiological model of the suprachiasmatic nucleus

Despite the wealth of experimental data on the electrophysiology of individual neurons in the suprachiasmatic nuclei (SCN), the neural code of the SCN remains largely unknown. To predict the electrical activity of the SCN, the authors simulated networks of 10,000 GABAergic SCN neurons using a detailed model of the ionic currents within SCN neurons. Their goal was to understand how neuronal firing, which occurs on a time scale faster than a second, can encode a set phase of the circadian (24-h) cycle. The authors studied the effects of key network properties including: 1) the synaptic density within the SCN, 2) the magnitude of postsynaptic currents, 3) the heterogeneity of circadian phase in the neuronal population, 4) the degree of synaptic noise, and 5) the balance between excitation and inhibition. Their main result was that under a wide variety of conditions, the SCN network spontaneously organized into (typically 3) groups of synchronously firing neurons. They showed that this type of clustering can lead to the silencing of neurons whose intracellular clocks are out of circadian phase with the rest of the population. Their results provide clues to how the SCN may generate a coherent electrical output signal at the tissue level to time rhythms throughout the body.

Stability of splay states in globally coupled rotators

The stability of dynamical states characterized by a uniform firing rate (splay states) is analyzed in a network of N globally pulse-coupled rotators (neurons) subject to a generic velocity field. In particular, we analyze short-wavelength modes that were known to be marginally stable in the infinite N limit and show that the corresponding Floquet exponent scale as 1/N2. Moreover, we find that the sign, and thereby the stability, of this spectral component is determined by the sign of the average derivative of the velocity field. For leaky-integrate-and-fire neurons, an analytic expression for the whole spectrum is obtained. In the intermediate case of continuous velocity fields, the Floquet exponents scale faster than 1/N2 (namely, as 1/N4) and we even find strictly neutral directions in a wider class than the sinusoidal velocity fields considered by Watanabe and Strogatz [Physica D 74, 197 (1994)].

Synchronization and clustering of synthetic genetic networks: a role for cis-regulatory modules

The effect of signal integration through cis-regulatory modules (CRMs) on synchronization and clustering of populations of two-component genetic oscillators coupled with quorum sensing is investigated in detail. We find that the CRMs play an important role in achieving synchronization and clustering. For this, we investigate six possible cis-regulatory input functions with AND, OR, ANDN, ORN, XOR, and EQU types of responses in two possible kinds of cell-to-cell communications: activator-regulated communication (i.e., the autoinducer regulates the activator) and repressor-regulated communication (i.e., the autoinducer regulates the repressor). Both theoretical analysis and numerical simulation show that different CRMs drive fundamentally different cellular patterns, such as complete synchronization, various cluster-balanced states and several cluster-nonbalanced states.

Experimentally observed route to spatiotemporal chaos in an extended one-dimensional array of convective oscillators

We report experimental evidence of the route to spatiotemporal chaos in a large one-dimensional array of hotspots in a thermoconvective system. As the driving force is increased, a stationary cellular pattern becomes unstable toward a mixed pattern of irregular clusters which consist of time-dependent localized patterns of variable spatiotemporal coherence. These irregular clusters coexist with the basic cellular pattern. The Fourier spectra corresponding to this synchronization transition reveal the weak coupling of a resonant triad. This pattern saturates with the formation of a unique domain of high spatiotemporal coherence. As we further increase the driving force, a supercritical bifurcation to a spatiotemporal beating regime takes place. The new pattern is characterized by the presence of two stationary clusters with a characteristic zig-zag geometry. The Fourier analysis reveals a stronger coupling than the previous mixed pattern and enables us to find out that this beating phenomenon is produced by the splitting of the fundamental spatiotemporal frequencies in a narrow band. Both secondary instabilities are phaselike synchronization transitions with global and absolute character. Far beyond this threshold, a new instability takes place when the system is not able to sustain the spatial frequency splitting, although the temporal beating remains inside these domains. These experimental results may support the understanding of other systems in nature undergoing similar clustering processes.

Invariant submanifold for series arrays of Josephson junctions

We study the nonlinear dynamics of series arrays of Josephson junctions in the large-N limit, where N is the number of junctions in the array. The junctions are assumed to be identical, overdamped, driven by a constant bias current, and globally coupled through a common load. Previous simulations of such arrays revealed that their dynamics are remarkably simple, hinting at the presence of some hidden symmetry or other structure. These observations were later explained by the discovery of N-3 constants of motion, the choice of which confines the resulting flow in phase space to a low-dimensional invariant manifold. Here we show that the dimensionality can be reduced further by restricting attention to a special family of states recently identified by Ott and Antonsen. In geometric terms, the Ott-Antonsen ansatz corresponds to an invariant submanifold of dimension one less than that found earlier. We derive and analyze the flow on this submanifold for two special cases: an array with purely resistive loading and another with resistive-inductive-capacitive loading. Our results recover (and in some instances improve) earlier findings based on linearization arguments.