Abstract
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.
Highlights
The processing and coding of information in the brain necessarily imply the coordinated activity of large ensembles of neurons
We propose a method to derive the firing-rate equations (FREs) for networks of heterogeneous, all-to-all coupled quadratic integrate-and-fire (QIF) neurons, which is exact in the thermodynamic limit, i.e., for large numbers of neurons
We present a method for deriving firing-rate equations for a network of heterogeneous QIF neurons, which is exact in the thermodynamic limit
Summary
The processing and coding of information in the brain necessarily imply the coordinated activity of large ensembles of neurons. 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] These descriptions, called firing-rate equations (FREs), have been proven to be extremely useful in understanding general computational principles underlying functions such as memory [21,22], visual processing [23,24,25], motor control [26], or decision making [27]. We show how the LA transforms, via a conformal mapping, into the so-called Ott-Antonsen ansatz (OA) that is used extensively to investigate the low-dimensional dynamics of large populations of phase oscillators in terms of the Kuramoto order parameter [35]
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