A major goal of neuroscience, statistical physics and nonlinear dynamics is to understand how brain function arises from the collective dynamics of networks of spiking neurons. This challenge has been chiefly addressed through large-scale numerical simulations. Alternatively, researchers have formulated mean-field theories to gain insight into macroscopic states of large neuronal networks in terms of the collective firing activity of the neurons, or the firing rate. However, these theories have not succeeded in establishing an exact correspondence between the firing rate of the network and the underlying microscopic state of the spiking neurons. This has largely constrained the range of applicability of such macroscopic descriptions, particularly when trying to describe neuronal synchronization. Here we provide the derivation of a set of exact macroscopic equations for a network of spiking neurons. Our results reveal that the spike generation mechanism of individual neurons introduces an effective coupling between two biophysically relevant macroscopic quantities, the firing rate and the mean membrane potential, which together govern the evolution of the neuronal network. The resulting equations exactly describe all possible macroscopic dynamical states of the network, including states of synchronous spiking activity. Finally we show that the firing rate description is related, via a conformal map, with a lowdimensional description in terms of the Kuramoto order parameter, called Ott-Antonsen theory. We anticipate our results will be an important tool in investigating how large networks of spiking neurons self-organize in time to process and encode information in the brain.Processing and coding of information in the brain necessarily imply the coordinated activity of large ensembles of neurons. Within sensory regions of the cortex, many cells show similar responses to a given stimulus, indicating a high degree of neuronal redundancy at the local level. This suggests that information is encoded in the population response and hence can be captured via macroscopic measures of the network activity [1]. Moreover, the collective behavior of large neuronal networks is particularly relevant given that current brain measurement techniques, such as electroencephalography (EEG) or functional magnetic resonance imaging (fMRI), provide data which is necessarily averaged over the activity of a large number of neurons.The macroscopic dynamics of neuronal ensembles has been extensively studied through computational models of large networks of recurrently coupled spiking neurons, including Hodgkin-Huxley-type conductance-based neurons [2] as well as simplified neuron models, see e.g. [3][4][5]. In parallel, researchers have sought to develop statistical descriptions of neuronal networks, mainly in terms of a macroscopic observable that measures the mean rate at which neurons emit spikes, the firing rate [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. These descriptions, called firing-rate equations (FREs), have ...

In the Kuramoto model, a uniform distribution of the natural frequencies leads to a first-order (i.e., discontinuous) phase transition from incoherence to synchronization, at the critical coupling parameter Kc. We obtain the asymptotic dependence of the order parameter above criticality: r − rc ∝ (K − Kc) 2/3 . For a finite population, we demonstrate that the population size N may be included into a self-consistency equation relating r and K in the synchronized state. We analyze the convergence to the thermodynamic limit of two alternative schemes to set the natural frequencies. Other frequency distributions different from the uniform one are also considered.

Large communities of biological oscillators show a prevalent tendency to self-organize in time. This cooperative phenomenon inspired Winfree to formulate a mathematical model that originated the theory of macroscopic synchronization. Despite its fundamental importance, a complete mathematical analysis of the model proposed by Winfree -consisting of a large population of all-to-all pulse-coupled oscillators-is still missing. Here we show that the dynamics of the Winfree model evolves into the so-called Ott-Antonsen manifold. This important property allows for an exact description of this high-dimensional system in terms of a few macroscopic variables, and the full investigation of its dynamics. We find that brief pulses are capable of synchronizing heterogeneous ensembles which fail to synchronize with broad pulses, specially for certain phase response curves. Finally, to further illustrate the potential of our results, we investigate the possibility of 'chimera' states in populations of identical pulse-coupled oscillators. Chimeras are self-organized states in which the symmetry of a population is broken into a synchronous and an asynchronous part. Here we derive three ordinary differential equations describing two coupled populations, and uncover a variety of chimera states, including a new class with chaotic dynamics.

We study Lyapunov vectors (LVs) corresponding to the largest Lyapunov exponents in systems with spatiotemporal chaos. We focus on characteristic LVs and compare the results with backward LVs obtained via successive Gram-Schmidt orthonormalizations. Systems of a very different nature such as coupled-map lattices and the (continuous-time) Lorenz '96 model exhibit the same features in quantitative and qualitative terms. Additionally, we propose a minimal stochastic model that reproduces the results for chaotic systems. Our work supports the claims about universality of our earlier results [I. G. Szendro, Phys. Rev. E 76, 025202(R) (2007)] for a specific coupled-map lattice.

We investigate the transition to synchronization in the Kuramoto model with bimodal distributions of the natural frequencies. Previous studies have concluded that the model exhibits a hysteretic phase transition if the bimodal distribution is close to a unimodal one, due to the shallowness the central dip. Here we show that proximity to the unimodal-bimodal border does not necessarily imply hysteresis when the width, but not the depth, of the central dip tends to zero. We draw this conclusion from a detailed study of the Kuramoto model with a suitable family of bimodal distributions.

Large ensembles of heterogeneous oscillators often exhibit collective synchronization as a result of mutual interactions. If the oscillators have distributed natural frequencies and common shear (or nonisochronicity), the transition from incoherence to collective synchronization is known to occur at large enough values of the coupling strength. However, here we demonstrate that shear diversity cannot be counterbalanced by diffusive coupling leading to synchronization. We present the first analytical results for the Kuramoto model with distributed shear and show that the onset of collective synchronization is impossible if the width of the shear distribution exceeds a precise threshold.

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 multi-body (i.e. non-pairwise) interactions. This points to intricate multi-body phase interactions as the source of pure collective chaos in the MF-CGLE at moderate coupling. II. MEAN-FIELD COMPLEX GINZBURG-LANDAU EQUATIONThe MF-CGLE consists of N diffusively coupled Stuart-Landau oscillators governed by N coupled (complex-valued)

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. Electrical measurements of brain activity display a broad spectrum of oscillations, reflecting the complex coordination of spike discharges across large neuronal populations [1]. A particularly fruitful theoretical framework for investigating neuronal rhythms is to model networks of neurons as populations of heterogeneous oscillators [2][3][4]. These models exhibit a prevalent transition from incoherence to partial coherence, when a fraction of the oscillators becomes entrained to a common frequency. As a result a macroscopic oscillatory mode appears with the same frequency as that of the synchronized cluster [2,5].Yet, even populations of globally coupled identical oscillators are capable of exhibiting a much wider diversity of complex oscillatory states, see [6] for a recent survey. In general, this is due to the complexity of the coupling functions and of the individual oscillators. A relevant example is the so-called quasiperiodic partial synchronization (QPS), which has been extensively investigated in networks of excitatory leaky integrate-and-fire (LIF) neurons [7][8][9][10][11], as well as in populations of limit-cycle oscillators and phase oscillators [12][13][14][15][16][17][18]. In QPS, the network sets into a nontrivial dynamical regime in which oscillators display quasiperiodic dynamics while the collective observables oscillate periodically. Remarkably, the period of these oscillations differs from the mean period of the individual oscillators. As pointed out recently [17], this interesting property of QPS is shared by the collective chaos observed in populations of globally coupled limit-cycle oscillators [19][20][21][22][23][24][25]. Here, the collective chaotic mode is typically accompanied by microscopic chaotic dynamics at the level of the individual oscillators. However, as noticed in [20], populations of limit-cycle oscillators may also display pure collective chaos without trace of orbital instability at the microscopic level. In this state the coordinates of the oscillators fall on a smooth closed curve and no mixing occurs, what points to the existence of collective chaos in populations of oscillators governed by a single phase-like variable.In this Letter we uncover the spontaneous emergence of pure collective chaos from QPS, via a cascade of perioddoubling bifurcations. Notably, this is found in a simple population of identical integrate-and-fire oscillators with timedelayed pulse coupling, which is thoroughly analyzed within the framework of the s...