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 ...

We analyze synchronization between two interacting populations of different phase oscillators. For the important case of asymmetric coupling functions, we find a much richer dynamical behavior compared to that of symmetrically coupled populations of identical oscillators [1]. It includes three types of bistabilities, higher order entrainment and the existence of states with unusual stability properties. All possible routes to synchronization of the populations are presented and some stability boundaries are obtained analytically. The impact of these findings for neuroscience is discussed.

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.

Recurrently coupled networks of inhibitory neurons robustly generate oscillations in the gamma band. Nonetheless, the corresponding Wilson-Cowan type firing rate equation for such an inhibitory population does not generate such oscillations without an explicit time delay. We show that this discrepancy is due to a voltage-dependent spike-synchronization mechanism inherent in networks of spiking neurons which is not captured by standard firing rate equations. Here we investigate an exact low-dimensional description for a network of heterogeneous canonical Class 1 inhibitory neurons which includes the sub-threshold dynamics crucial for generating synchronous states. In the limit of slow synaptic kinetics the spike-synchrony mechanism is suppressed and the standard Wilson-Cowan equations are formally recovered as long as external inputs are also slow. However, even in this limit synchronous spiking can be elicited by inputs which fluctuate on a time-scale of the membrane time-constant of the neurons. Our meanfield equations therefore represent an extension of the standard Wilson-Cowan equations in which spike synchrony is also correctly described.

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.