  Our main focus is on a general class of active rotators with mean field
interactions, that is globally coupled large families of dynamical systems on
the unit circle with non-trivial stochastic dynamics. Each isolated system is a
diffusion process on a circle, with drift -delta V', where V' is a periodic
function and delta is an intensity parameter. It is well known that the
interacting dynamics is accurately described, in the limit of infinitely many
interacting components, by a Fokker-Planck PDE and the model reduces for
delta=0 to a particular case of the Kuramoto synchronization model, for which
one can show the existence of a stable normally hyperbolic manifold of
stationary solutions for the corresponding Fokker-Planck equation (we are
interested in the case in which this manifold is non-trivial, that happens when
the interaction is sufficiently strong, that is in the synchronized regime of
the Kuramoto model). We use the robustness of normally hyperbolic structures to
infer qualitative and quantitative results on the |delta|< delta0 cases, with
delta0 a suitable threshold: as a matter of fact, we obtain an accurate
description of the dynamics on the invariant manifold for delta=0 and we link
it explicitly to the potential V . This approach allows to have a complete
description of the phase diagram of the active rotators model, at least for
|delta|< delta0, thus identifying for which values of the parameters (notably,
noise intensity and/or coupling strength) the system exhibits periodic pulse
waves or stabilizes at a quiescent resting state. Moreover, some of our results
are very explicit and this brings a new insight into the combined effect of
active rotator dynamics, noise and interaction. The links with the literature
on specific systems, notably neuronal models, are discussed in detail.
