Macroscopic equations governing noisy spiking neuronal populations

TL;DR

This paper derives a rigorous, low-dimensional firing-rate model from complex spiking neuron networks, accurately capturing their dynamics and linking microscopic noise-driven activity to macroscopic behavior.

Contribution

It provides a principled reduction method connecting detailed spiking neuron models to simplified firing-rate equations, with parameters derived from underlying cellular properties.

Findings

Reduced models accurately reproduce large network dynamics

Parameters and functions are available for various neuron models

The approach links microscopic noise to macroscopic activity

Abstract

At functional scales, cortical behavior results from the complex interplay of a large number of excitable cells operating in noisy environments. Such systems resist to mathematical analysis, and computational neurosciences have largely relied on heuristic partial (and partially justified) macroscopic models, which successfully reproduced a number of relevant phenomena. The relationship between these macroscopic models and the spiking noisy dynamics of the underlying cells has since then been a great endeavor. Based on recent mean-field reductions for such spiking neurons, we present here {a principled reduction of large biologically plausible neuronal networks to firing-rate models, providing a rigorous} relationship between the macroscopic activity of populations of spiking neurons and popular macroscopic models, under a few assumptions (mainly linearity of the synapses). {The reduced model we derive consists of simple, low-dimensional ordinary differential equations with} parameters and {nonlinearities derived from} the underlying properties of the cells, and in particular the noise level. {These simple reduced models are shown to reproduce accurately the dynamics of large networks in numerical simulations}. Appropriate parameters and functions are made available {online} for different models of neurons: McKean, Fitzhugh-Nagumo and Hodgkin-Huxley models.