The Mean Field Approach for Populations of Spiking Neurons

Abstract

Mean field theory is a device to analyze the collective behavior of a dynamical system comprising many interacting particles. The theory allows to reduce the behavior of the system to the properties of a handful of parameters. In neural circuits, these parameters are typically the firing rates of distinct, homogeneous subgroups of neurons. Knowledge of the firing rates under conditions of interest can reveal essential information on both the dynamics of neural circuits and the way they can subserve brain function. The goal of this chapter is to provide an elementary introduction to the mean field approach for populations of spiking neurons. We introduce the general idea in networks of binary neurons, starting from the most basic results and then generalizing to more relevant situations. This allows to derive the mean field equations in a simplified setting. We then derive the mean field equations for populations of integrate-and-fire neurons. An effort is made to derive the main equations of the theory using only elementary methods from calculus and probability theory. The chapter ends with a discussion of the assumptions of the theory and some of the consequences of violating those assumptions. This discussion includes an introduction to balanced and metastable networks and a brief catalogue of successful applications of the mean field approach to the study of neural circuits.

Co-existence of synaptic plasticity and metastable dynamics in a spiking model of cortical circuits

“…As customary for spiking networks, we performed the mean field analysis under the diffusion approximation, where the input current is characterized by the infinitesimal mean and variance of the synaptic inputs (Amit and Brunel, 1997; La Camera, 2022). The infinitesimal mean and variance to the neurons of the k -th excitatory cluster are given by where denotes the firing rate of the excitatory neurons in the k -th cluster and ν I is the firing rate of the inhibitory neurons.…”