A new approach to partial synchronization in globally coupled rotators

How synaptic function controls critical transitions in spiking neuron networks: insight from a Kuramoto model reduction

The dynamics of synaptic interactions within spiking neuron networks play a fundamental role in shaping emergent collective behavior. This paper studies a finite-size network of quadratic integrate-and-fire neurons interconnected via a general synaptic function that accounts for synaptic dynamics and time delays. Through asymptotic analysis, we transform this integrate-and-fire network into the Kuramoto-Sakaguchi model, whose parameters are explicitly expressed via synaptic function characteristics. This reduction yields analytical conditions on synaptic activation rates and time delays determining whether the synaptic coupling is attractive or repulsive. Our analysis reveals alternating stability regions for synchronous and partially synchronous firing, dependent on slow synaptic activation and time delay. We also demonstrate that the reduced microscopic model predicts the emergence of synchronization, weakly stable cyclops states, and non-stationary regimes remarkably well in the original integrate-and-fire network and its theta neuron counterpart. Our reduction approach promises to open the door to rigorous analysis of rhythmogenesis in networks with synaptic adaptation and plasticity.

Spatiotemporal spread of perturbations in a driven dissipative Duffing chain: An out-of-time-ordered correlator approach

Out-of-time-ordered correlators (OTOCs) have been extensively used as a major tool for exploring quantum chaos, and recently there has been a classical analog. Studies have been limited to closed systems. In this work, we probe an open classical many-body system, more specifically, a spatially extended driven dissipative chain of coupled Duffing oscillators using the classical OTOC to investigate the spread and growth (decay) of an initially localized perturbation in the chain. Correspondingly, we find three distinct types of dynamical behavior: the sustained chaos, transient chaos, and nonchaotic region, as clearly exhibited by different geometrical shapes in the OTOC heat map. To quantify such differences, we look at instantaneous speed (IS), finite-time Lyapunov exponents (FTLEs), and velocity-dependent Lyapunov exponents (VDLEs) extracted from OTOCs. Introduction of these quantities turns out to be instrumental in diagnosing and demarcating different regimes of dynamical behavior. To gain control over open nonlinear systems, it is important to look at the variation of these quantities with respect to parameters. As we tune drive, dissipation, and coupling, FTLEs and IS exhibit transition between sustained chaos and nonchaotic regimes with intermediate transient chaos regimes and highly intermittent sustained chaos points. In the limit of zero nonlinearity, we present exact analytical results for the driven dissipative harmonic system, and we find that our analytical results can very well describe the nonchaotic regime as well as the late-time behavior in the transient regime of the Duffing chain. We believe that this analysis is an important step forward towards understanding nonlinear dynamics, chaos, and spatiotemporal spread of perturbations in many-particle open systems.

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.

Reconstructing networks of pulse-coupled oscillators from spike trains

We present an approach for reconstructing networks of pulse-coupled neuronlike oscillators from passive observation of pulse trains of all nodes. It is assumed that units are described by their phase response curves and that their phases are instantaneously reset by incoming pulses. Using an iterative procedure, we recover the properties of all nodes, namely their phase response curves and natural frequencies, as well as strengths of all directed connections.

Chimeras and complex cluster states in arrays of spin-torque oscillators

We consider synchronization properties of arrays of spin-torque nano-oscillators coupled via an RC load. We show that while the fully synchronized state of identical oscillators may be locally stable in some parameter range, this synchrony is not globally attracting. Instead, regimes of different levels of compositional complexity are observed. These include chimera states (a part of the array forms a cluster while other units are desynchronized), clustered chimeras (several clusters plus desynchronized oscillators), cluster state (all oscillators form several clusters), and partial synchronization (no clusters but a nonvanishing mean field). Dynamically, these states are also complex, demonstrating irregular and close to quasiperiodic modulation. Remarkably, when heterogeneity of spin-torque oscillators is taken into account, dynamical complexity even increases: close to the onset of a macroscopic mean field, the dynamics of this field is rather irregular.

Synchronization patterns: from network motifs to hierarchical networks

We investigate complex synchronization patterns such as cluster synchronization and partial amplitude death in networks of coupled Stuart-Landau oscillators with fractal connectivities. The study of fractal or self-similar topology is motivated by the network of neurons in the brain. This fractal property is well represented in hierarchical networks, for which we present three different models. In addition, we introduce an analytical eigensolution method and provide a comprehensive picture of the interplay of network topology and the corresponding network dynamics, thus allowing us to predict the dynamics of arbitrarily large hierarchical networks simply by analysing small network motifs. We also show that oscillation death can be induced in these networks, even if the coupling is symmetric, contrary to previous understanding of oscillation death. Our results show that there is a direct correlation between topology and dynamics: hierarchical networks exhibit the corresponding hierarchical dynamics. This helps bridge the gap between mesoscale motifs and macroscopic networks.This article is part of the themed issue 'Horizons of cybernetical physics'.

Cluster synchronization in networks of identical oscillators with α-function pulse coupling

We study a network of N identical leaky integrate-and-fire model neurons coupled by α-function pulses, weighted by a coupling parameter K. Studies of the dynamics of this system have mostly focused on the stability of the fully synchronized and the fully asynchronous splay states, which naturally depends on the sign of K, i.e., excitation vs inhibition. We find that there is also a rich set of attractors consisting of clusters of fully synchronized oscillators, such as fixed (N-1,1) states, which have synchronized clusters of sizes N-1 and 1, as well as splay states of clusters with equal sizes greater than 1. Additionally, we find limit cycles that clarify the stability of previously observed quasiperiodic behavior. Our framework exploits the neutrality of the dynamics for K=0 which allows us to implement a dimensional reduction strategy that simplifies the dynamics to a continuous flow on a codimension 3 subspace with the sign of K determining the flow direction. This reduction framework naturally incorporates a hierarchy of partially synchronized subspaces in which the new attracting states lie. Using high-precision numerical simulations, we describe completely the sequence of bifurcations and the stability of all fixed points and limit cycles for N=2-4. The set of possible attracting states can be used to distinguish different classes of neuron models. For instance from our previous work [Chaos 24, 013114 (2014)CHAOEH1054-150010.1063/1.4858458] we know that of the types of partially synchronized states discussed here, only the (N-1,1) states can be stable in systems of identical coupled sinusoidal (i.e., Kuramoto type) oscillators, such as θ-neuron models. Upon introducing a small variation in individual neuron parameters, the attracting fixed points we discuss here generalize to equivalent fixed points in which neurons need not fire coincidently.

From Quasiperiodic Partial Synchronization to Collective Chaos in Populations of Inhibitory Neurons with Delay

Collective chaos is shown to emerge, via a period-doubling cascade, from quasiperiodic partial synchronization in a population of identical inhibitory neurons with delayed global coupling. This system is thoroughly investigated by means of an exact model of the macroscopic dynamics, valid in the thermodynamic limit. The collective chaotic state is reproduced numerically with a finite population, and persists in the presence of weak heterogeneities. Finally, the relationship of the model's dynamics with fast neuronal oscillations is discussed.

Marginal chimera state at cross-frequency locking of pulse-coupled neural networks

We consider two coupled populations of leaky integrate-and-fire neurons. Depending on the coupling strength, mean fields generated by these populations can have incommensurate frequencies or become frequency locked. In the observed 2:1 locking state of the mean fields, individual neurons in one population are asynchronous with the mean fields, while in another population they have the same frequency as the mean field. These synchronous neurons form a chimera state, where part of them build a fully synchronized cluster, while other remain scattered. We explain this chimera as a marginal one, caused by a self-organized neutral dynamics of the effective circle map.

Dynamics of globally coupled oscillators: Progress and perspectives

In this paper, we discuss recent progress in research of ensembles of mean field coupled oscillators. Without an ambition to present a comprehensive review, we outline most interesting from our viewpoint results and surprises, as well as interrelations between different approaches.

Two types of quasiperiodic partial synchrony in oscillator ensembles

We analyze quasiperiodic partially synchronous states in an ensemble of Stuart-Landau oscillators with global nonlinear coupling. We reveal two types of such dynamics: in the first case the time-averaged frequencies of oscillators and of the mean field differ, while in the second case they are equal, but the motion of oscillators is additionally modulated. We describe transitions from the synchronous state to both types of quasiperiodic dynamics, and a transition between two different quasiperiodic states. We present an example of a bifurcation diagram, where we show the borderlines for all these transitions, as well as domain of bistability.

Equivalence of phase-oscillator and integrate-and-fire models

A quantitative comparison of various classes of oscillators (integrate-and-fire, Winfree, and Kuramoto-Daido type) is performed in the weak-coupling limit for a fully connected network of identical units. An almost perfect agreement is found, with only tiny differences among the models. We also show that the regime of self-consistent partial synchronization is rather general and can be observed for arbitrarily small coupling strength in any model class. As a byproduct of our study, we are able to show that an integrate-and-fire model with a generic pulse shape can be always transformed into a similar model with δ pulses and a suitable phase response curve.

Microscopic mechanism for self-organized quasiperiodicity in random networks of nonlinear oscillators

Self-organized quasiperiodicity is one of the most puzzling dynamical phases observed in systems of nonlinear coupled oscillators. The single dynamical units are not locked to the periodic mean field they produce, but they still feature a coherent behavior, through an unexplained complex form of correlation. We consider a class of leaky integrate-and-fire oscillators on random sparse and massive networks with dynamical synapses, featuring self-organized quasiperiodicity, and we show how complex collective oscillations arise from constructive interference of microscopic dynamics. In particular, we find a simple quantitative relationship between two relevant microscopic dynamical time scales and the macroscopic time scale of the global signal. We show that the proposed relation is a general property of collective oscillations, common to all the partially synchronous dynamical phases analyzed. We argue that an analogous mechanism could be at the origin of similar network dynamics.

Synchronization of degrade-and-fire oscillations via a common activator

The development of synthetic gene oscillators has not only demonstrated our ability to forward engineer reliable circuits in living cells, but it has also proven to be an excellent testing ground for the statistical behavior of coupled noisy oscillators. Previous experimental studies demonstrated that a shared positive feedback can reliably synchronize such oscillators, though the theoretical mechanism was not studied in detail. In the present work, we examine an experimentally motivated stochastic model for coupled degrade-and-fire gene oscillators, where a core delayed negative feedback establishes oscillations within each cell, and a shared delayed positive feedback couples all cells. We use analytic and numerical techniques to investigate conditions for one cluster and multicluster synchrony. A nonzero delay in the shared positive feedback, as expected for the experimental systems, is found to be important for synchrony to occur.

Sisyphus effect in pulse-coupled excitatory neural networks with spike-timing-dependent plasticity

The collective dynamics of excitatory pulse-coupled neural networks with spike-timing-dependent plasticity (STDP) is studied. Depending on the model parameters stationary states characterized by high or low synchronization can be observed. In particular, at the transition between these two regimes, persistent irregular low frequency oscillations between strongly and weakly synchronized states are observable, which can be identified as infraslow oscillations with frequencies ≃0.02-0.03 Hz. Their emergence can be explained in terms of the Sisyphus effect, a mechanism caused by a continuous feedback between the evolution of the coherent population activity and of the average synaptic weight. Due to this effect, the synaptic weights have oscillating equilibrium values, which prevents the neuronal population from relaxing into a stationary macroscopic state.

Chimeralike states in an ensemble of globally coupled oscillators

We demonstrate the emergence of a complex state in a homogeneous ensemble of globally coupled identical oscillators, reminiscent of chimera states in nonlocally coupled oscillator lattices. In this regime some part of the ensemble forms a regularly evolving cluster, while all other units irregularly oscillate and remain asynchronous. We argue that the chimera emerges because of effective bistability, which dynamically appears in the originally monostable system due to internal delayed feedback in individual units. Additionally, we present two examples of chimeras in bistable systems with frequency-dependent phase shift in the global coupling.

Average synaptic activity and neural networks topology: a global inverse problem

The dynamics of neural networks is often characterized by collective behavior and quasi-synchronous events, where a large fraction of neurons fire in short time intervals, separated by uncorrelated firing activity. These global temporal signals are crucial for brain functioning. They strongly depend on the topology of the network and on the fluctuations of the connectivity. We propose a heterogeneous mean–field approach to neural dynamics on random networks, that explicitly preserves the disorder in the topology at growing network sizes, and leads to a set of self-consistent equations. Within this approach, we provide an effective description of microscopic and large scale temporal signals in a leaky integrate-and-fire model with short term plasticity, where quasi-synchronous events arise. Our equations provide a clear analytical picture of the dynamics, evidencing the contributions of both periodic (locked) and aperiodic (unlocked) neurons to the measurable average signal. In particular, we formulate and solve a global inverse problem of reconstructing the in-degree distribution from the knowledge of the average activity field. Our method is very general and applies to a large class of dynamical models on dense random networks.

Robust synchronization of spin-torque oscillators with an LCR load

We study dynamics of a serial array of spin-torque oscillators with a parallel inductor-capacitor-resistor (LCR) load. In a large range of parameters the fully synchronous regime, where all the oscillators have the same state and the output field is maximal, is shown to be stable. However, not always such a robust complete synchronization develops from a random initial state; in many cases nontrivial clustering is observed, with a partial synchronization resulting in a quasiperiodic or chaotic mean-field dynamics.

Autonomous and forced dynamics of oscillator ensembles with global nonlinear coupling: an experimental study

We perform experiments with 72 electronic limit-cycle oscillators, globally coupled via a linear or nonlinear feedback loop. While in the linear case we observe a standard Kuramoto-like synchronization transition, in the nonlinear case, with increase of the coupling strength, we first observe a transition to full synchrony and then a desynchronization transition to a quasiperiodic state. However, in this state the ensemble remains coherent so that the amplitude of the mean field is nonzero, but the frequency of the mean field is larger than frequencies of all oscillators. Next, we analyze effects of common periodic forcing of the linearly or nonlinearly coupled ensemble and demonstrate regimes when the mean field is entrained by the force whereas the oscillators are not.

Stability of the splay state in networks of pulse-coupled neurons

We analytically investigate the stability of splay states in the networks of N globally pulse-coupled phase-like models of neurons. We develop a perturbative technique which allows determining the Floquet exponents for a generic velocity field and implement the method for a given pulse shape. We find that in the case of discontinuous velocity fields, the Floquet spectrum scales as 1/N2 and the stability is determined by the sign of the jump at the discontinuity. Altogether, the form of the spectrum depends on the pulse shape, but it is independent of the velocity field.PACS: 05.45.Xt, 84.35.+i, 87.19.lj.

Coherent periodic activity in excitatory Erdös-Renyi neural networks: the role of network connectivity

In this article, we investigate the role of connectivity in promoting coherent activity in excitatory neural networks. In particular, we would like to understand if the onset of collective oscillations can be related to a minimal average connectivity and how this critical connectivity depends on the number of neurons in the networks. For these purposes, we consider an excitatory random network of leaky integrate-and-fire pulse coupled neurons. The neurons are connected as in a directed Erdös-Renyi graph with average connectivity <k> scaling as a power law with the number of neurons in the network. The scaling is controlled by a parameter γ, which allows to pass from massively connected to sparse networks and therefore to modify the topology of the system. At a macroscopic level, we observe two distinct dynamical phases: an asynchronous state corresponding to a desynchronized dynamics of the neurons and a regime of partial synchronization (PS) associated with a coherent periodic activity of the network. At low connectivity, the system is in an asynchronous state, while PS emerges above a certain critical average connectivity <k>(c). For sufficiently large networks, <k>(c) saturates to a constant value suggesting that a minimal average connectivity is sufficient to observe coherent activity in systems of any size irrespectively of the kind of considered network: sparse or massively connected. However, this value depends on the nature of the synapses: reliable or unreliable. For unreliable synapses, the critical value required to observe the onset of macroscopic behaviors is noticeably smaller than for reliable synaptic transmission. Due to the disorder present in the system, for finite number of neurons we have inhomogeneities in the neuronal behaviors, inducing a weak form of chaos, which vanishes in the thermodynamic limit. In such a limit, the disordered systems exhibit regular (non chaotic) dynamics and their properties correspond to that of a homogeneous fully connected network for any γ-value. Apart for the peculiar exception of sparse networks, which remain intrinsically inhomogeneous at any system size.

Experiments on oscillator ensembles with global nonlinear coupling

We experimentally analyze collective dynamics of a population of 20 electronic Wien-bridge limit-cycle oscillators with a nonlinear phase-shifting unit in the global feedback loop. With an increase in the coupling strength we first observe formation and then destruction of a synchronous cluster, so that the dependence of the order parameter on the coupling strength is not monotonic. After destruction of the cluster the ensemble remains nevertheless coherent, i.e., it exhibits an oscillatory collective mode (mean field). We show that the system is now in a self-organized quasiperiodic state, predicted in Rosenblum and Pikovsky [Phys. Rev. Lett. 98, 064101 (2007)]. In this state, frequencies of all oscillators are smaller than the frequency of the mean field, so that the oscillators are not locked to the mean field they create and their dynamics is quasiperiodic. Without a nonlinear phase-shifting unit, the system exhibits a standard Kuramoto-like transition to a fully synchronous state. We demonstrate a good correspondence between the experiment and previously developed theory. We also propose a simple measure which characterizes the macroscopic incoherence-coherence transition in a finite-size ensemble.

Macroscopic entrainment of periodically forced oscillatory ensembles

Large-amplitude oscillations of macroscopic neuronal signals, such as local field potentials and electroencephalography or magnetoencephalography signals, are commonly considered as being generated by a population of mutually synchronized neurons. In a computational study in generic networks of phase oscillators and bursting neurons, however, we show that this common belief may be wrong if the neuronal population receives an external rhythmic input. The latter may stem from another neuronal population or an external, e.g., sensory or electrical, source. In that case the population field potential may be entrained by the rhythmic input, whereas the individual neurons are phase desynchronized both mutually and with their field potential. Intriguingly, the corresponding large-amplitude oscillations of the population mean field are generated by pairwise desynchronized neurons oscillating at frequencies shifted far away from the frequency of the macroscopic field potential.

Collective oscillations in disordered neural networks

We investigate the onset of collective oscillations in a excitatory pulse-coupled network of leaky integrate-and-fire neurons in the presence of quenched and annealed disorder. We find that the disorder induces a weak form of chaos that is analogous to that arising in the Kuramoto model for a finite number N of oscillators [O. V. Popovych, Phys. Rev. E 71 065201(R) (2005)]. In fact, the maximum Lyapunov exponent turns out to scale to zero for N-->infinity , with an exponent that is different for the two types of disorder. In the thermodynamic limit, the random-network dynamics reduces to that of a fully homogeneous system with a suitably scaled coupling strength. Moreover, we show that the Lyapunov spectrum of the periodically collective state scales to zero as 1/N{2}, analogously to the scaling found for the "splay state."

Collective atomic recoil laser as a synchronization transition

We consider here a model previously introduced to describe the collective behavior of an ensemble of cold atoms interacting with a coherent electromagnetic field. The atomic motion along the self-generated spatially periodic force field can be interpreted as the rotation of a phase oscillator. This suggests a relationship with synchronization transitions occurring in globally coupled rotators. In fact, we show that whenever the field dynamics can be adiabatically eliminated, the model reduces to a self-consistent equation for the probability distribution of the atomic "phases." In this limit, there exists a formal equivalence with the Kuramoto model, though with important differences in the self-consistency conditions. Depending on the field-cavity detuning, we show that the onset of synchronized behavior may occur through either a first- or second-order phase transition. Furthermore, we find a secondary threshold, above which a periodic self-pulsing regime sets in, that is immediately followed by the unlocking of the forward-field frequency. At yet higher, but still experimentally meaningful, input intensities, irregular, chaotic oscillations may eventually appear. Finally, we derive a simpler model, involving only five scalar variables, which is able to reproduce the entire phenomenology exhibited by the original model.

Stability of the splay state in pulse-coupled networks

The stability of the dynamical states characterized by a uniform firing rate (splay states) is analyzed in a network of globally coupled leaky integrate-and-fire neurons. This is done by reducing the set of differential equations to a map that is investigated in the limit of large network size. We show that the stability of the splay state depends crucially on the ratio between the pulse width and the interspike interval. More precisely, the spectrum of Floquet exponents turns out to consist of three components: (i) one that coincides with the predictions of the mean-field analysis [Abbott and van Vreesvijk, Phys. Rev. E 48, 1483 (1993)], (ii) a component measuring the instability of "finite-frequency" modes, (iii) a number of "isolated" eigenvalues that are connected to the characteristics of the single pulse and may give rise to strong instabilities (the Floquet exponent being proportional to the network size). Finally, as a side result, we find that the splay state can be stable even for inhibitory coupling.

Self-organized quasiperiodicity in oscillator ensembles with global nonlinear coupling

We describe a transition from fully synchronous periodic oscillations to partially synchronous quasiperiodic dynamics in ensembles of identical oscillators with all-to-all coupling that nonlinearly depends on the generalized order parameters. We present an analytically solvable model that predicts a regime where the mean field does not entrain individual oscillators, but has a frequency incommensurate to theirs. The self-organized onset of quasiperiodicity is illustrated with Landau-Stuart oscillators and a Josephson junction array with a nonlinear coupling.

Desynchronization in diluted neural networks

The dynamical behavior of a weakly diluted fully inhibitory network of pulse-coupled spiking neurons is investigated. Upon increasing the coupling strength, a transition from regular to stochasticlike regime is observed. In the weak-coupling phase, a periodic dynamics is rapidly approached, with all neurons firing with the same rate and mutually phase locked. The strong-coupling phase is characterized by an irregular pattern, even though the maximum Lyapunov exponent is negative. The paradox is solved by drawing an analogy with the phenomenon of "stable chaos," i.e., by observing that the stochasticlike behavior is "limited" to an exponentially long (with the system size) transient. Remarkably, the transient dynamics turns out to be stationary.