  We analyze the emergent regimes and the stimulus-response relationship of a
population of noisy map neurons by means of a mean-field model, derived within
the framework of cumulant approach complemented by the Gaussian closure
hypothesis. It is demonstrated that the mean-field model can qualitatively
account for stability and bifurcations of the exact system, capturing all the
generic forms of collective behavior, including macroscopic excitability,
subthreshold oscillations, periodic or chaotic spiking and chaotic bursting
dynamics. Apart from qualitative analogies, we find a substantial quantitative
agreement between the exact and the approximate system, as reflected in
matching of the parameter domains admitting the different dynamical regimes, as
well as the characteristic properties of the associated time series. The
effective model is further shown to reproduce with sufficient accuracy the
phase response curves of the exact system and the assembly's response to
external stimulation of finite amplitude and duration.
