Macroscopic Equations Governing Noisy Spiking Neuronal Populations with Linear Synapses

Mathieu N Galtier,Jonathan Touboul

doi:10.1371/journal.pone.0078917

Abstract

Deriving tractable reduced equations of biological neural networks capturing the macroscopic dynamics of sub-populations of neurons has been a longstanding problem in computational neuroscience. In this paper, we propose a reduction of large-scale multi-population stochastic networks based on the mean-field theory. We derive, for a wide class of spiking neuron models, a system of differential equations of the type of the usual Wilson-Cowan systems describing the macroscopic activity of populations, under the assumption that synaptic integration is linear with random coefficients. Our reduction involves one unknown function, the effective non-linearity of the network of populations, which can be analytically determined in simple cases, and numerically computed in general. This function depends on the underlying properties of the cells, and in particular the noise level. Appropriate parameters and functions involved in the reduction are given for different models of neurons: McKean, Fitzhugh-Nagumo and Hodgkin-Huxley models. Simulations of the reduced model show a precise agreement with the macroscopic dynamics of the networks for the first two models.

Highlights

The activity of the brain is characterized by large-scale macroscopic states resulting from the structured interaction of a very large number of neurons
Several relevant brain states and functions rely on the coordinated behaviors of large neural assemblies, and resulting collective phenomena recently raised the interest of physiologists and computational neuroscientists, among which we shall cite the rapid complex answers to specific stimuli [1], decorrelated activity citeecker-berens-etal:10,renart-de-la-rocha-etal:10, large scale oscillations [2], synchronization [3], and spatio-temporal pattern formation [4,5]. This motivates the development of models of the collective dynamics of neuronal populations, that are simple enough to be mathematically analyzed or efficiently simulated
Each neuron i in population a is described by the membrane potential vit and additional variables gathered in a ddimensional variable Zti, representing for instance ionic concentrations in the Hodgkin-Huxley model, or a recovery variable in the Fitzhugh-Nagumo or McKean models

Summary

Introduction

The activity of the brain is characterized by large-scale macroscopic states resulting from the structured interaction of a very large number of neurons. Several relevant brain states and functions rely on the coordinated behaviors of large neural assemblies, and resulting collective phenomena recently raised the interest of physiologists and computational neuroscientists, among which we shall cite the rapid complex answers to specific stimuli [1], decorrelated activity citeecker-berens-etal:10,renart-de-la-rocha-etal:, large scale oscillations [2], synchronization [3], and spatio-temporal pattern formation [4,5]. This motivates the development of models of the collective dynamics of neuronal populations, that are simple enough to be mathematically analyzed or efficiently simulated. The tenet of the present manuscript is precisely that theoretical approaches may allow rigorously deriving macroscopic models that can be efficiently implemented and which reproduce accurately the dynamics of large networks

Methods

Results

Conclusion