  Networks of spiking neurons with adaption have been shown to be able to
reproduce a wide range of neural activities, including the emergent population
bursting and spike synchrony that underpin brain disorders and normal function.
Exact mean-field models derived from spiking neural networks are extremely
valuable, as such models can be used to determine how individual neuron and
network parameters interact to produce macroscopic network behaviour. In the
paper, we derive and analyze a set of exact mean-field equations for the neural
network with spike frequency adaptation. Specifically, our model is a network
of Izhikevich neurons, where each neuron is modeled by a two dimensional system
consisting of a quadratic integrate and fire equation plus an equation which
implements spike frequency adaptation. Previous work deriving a mean-field
model for this type of network, relied on the assumption of sufficiently slow
dynamics of the adaptation variable. However, this approximation did not
succeeded in establishing an exact correspondence between the macroscopic
description and the realistic neural network, especially when the adaptation
time constant was not large. The challenge lies in how to achieve a closed set
of mean-field equations with the inclusion of the mean-field expression of the
adaptation variable. We address this challenge by using a Lorentzian ansatz
combined with the moment closure approach to arrive at the mean-field system in
the thermodynamic limit. The resulting macroscopic description is capable of
qualitatively and quantitatively describing the collective dynamics of the
neural network, including transition between tonic firing and bursting.
