  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 low-dimensional 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.
