  We analytically derive mean-field models for all-to-all coupled networks of
heterogeneous, adapting, two-dimensional integrate and fire neurons. The class
of models we consider includes the Izhikevich, adaptive exponential, and
quartic integrate and fire models. The heterogeneity in the parameters leads to
different moment closure assumptions that can be made in the derivation of the
mean-field model from the population density equation for the large network.
Three different moment closure assumptions lead to three different mean-field
systems. These systems can be used for distinct purposes such as bifurcation
analysis of the large networks, prediction of steady state firing rate
distributions, parameter estimation for actual neurons, and faster exploration
of the parameter space. We use the mean-field systems to analyze adaptation
induced bursting under realistic sources of heterogeneity in multiple
parameters. Our analysis demonstrates that the presence of heterogeneity causes
the Hopf bifurcation associated with the emergence of bursting to change from
sub-critical to super-critical. This is confirmed with numerical simulations of
the full network for biologically reasonable parameter values. This change
decreases the plausibility of adaptation being the cause of bursting in
hippocampal area CA3, an area with a sizable population of heavily coupled,
strongly adapting neurons.
