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Networks of spiking neurons with adaption have been shown to be able toreproduce a wide range of neural activities, including the emergent populationbursting and spike synchrony that underpin brain disorders and normal function.Exact mean-field models derived from spiking neural networks are extremelyvaluable, as such models can be used to determine how individual neuron andnetwork parameters interact to produce macroscopic network behaviour. In thepaper, we derive and analyze a set of exact mean-field equations for the neuralnetwork with spike frequency adaptation. Specifically, our model is a networkof Izhikevich neurons, where each neuron is modeled by a two dimensional systemconsisting of a quadratic integrate and fire equation plus an equation whichimplements spike frequency adaptation. Previous work deriving a mean-fieldmodel for this type of network, relied on the assumption of sufficiently slowdynamics of the adaptation variable. However, this approximation did notsucceeded in establishing an exact correspondence between the macroscopicdescription and the realistic neural network, especially when the adaptationtime constant was not large. The challenge lies in how to achieve a closed setof mean-field equations with the inclusion of the mean-field expression of theadaptation variable. We address this challenge by using a Lorentzian ansatzcombined with the moment closure approach to arrive at the mean-field system inthe thermodynamic limit. The resulting macroscopic description is capable ofqualitatively and quantitatively describing the collective dynamics of theneural network, including transition between tonic firing and bursting.

Exact mean-field models for spiking neural networks with adaptation