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, to a low-dimensional description in terms of the Kuramoto order parameter, called Ott-Antonsen theory. We anticipate that 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.
- Received 30 December 2014
DOI:https://doi.org/10.1103/PhysRevX.5.021028
This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Popular Summary
A major challenge in neuroscience, statistical physics, and nonlinear dynamics during the last half century has been understanding the self-organizing principles governing the dynamics of large networks of spiking neurons. Researchers have proposed macroscopic descriptions of neural networks in terms of a relevant observable: the firing rate (i.e., the rate at which neurons emit action potentials, or spikes). Firing-rate descriptions are broadly used in theoretical neuroscience and have been shown to be extremely useful for understanding general computational principles underlying functions such as memory and visual processing. However, to date, all of these theoretical approaches have been either heuristic or approximate, and, furthermore, they were limited since firing-rate descriptions cannot describe neuronal states in which large fractions of neurons fire in concert, i.e., synchronous states. Here, we obtain an exact firing-rate description for a network of spiking neurons in terms of a few ordinary differential equations.
We derive exact macroscopic equations with the goal of relating the individual cell’s membrane potential (a microscopic observable) to the firing rate and population mean membrane potential (macroscopic observables). This macroscopic description is highly relevant given that medical measurement techniques provide data that are necessarily averaged over the activity of a large number of neurons (e.g., fMRI). We adopt a particular distribution of neuron membrane potentials that allows for a drastic dimensionality reduction and leads to the desired firing-rate equations. Our results show that the firing rate alone is not sufficient to describe the network dynamics. Rather, we find that the single-cell spike generation mechanism dynamically couples the firing rate to another macroscopic variable: the mean membrane potential. The resulting firing-rate equations fully describe the collective states of the network, including synchronization.
We anticipate our results will be important for investigating how large networks of spiking neurons self-organize in time to process and encode information in the brain.
