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The mean field approach for populations of spiking neurons
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
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