  The dynamics of a system formed by a finite number $N$ of globally coupled
bistable oscillators and driven by external forces is studied focusing on a
global variable defined as the arithmetic mean of each oscillator variable.
Several models based on truncation schemes of a hierarchy of stochastic
equations for a set of fluctuating cumulant variables are presented. This
hierarchy is derived using It\^o stochastic calculus, and the noise terms in it
are treated using an asymptotic approximation valid for large $N$. In addition,
a simplified one-variable model based on an effective potential is also
considered. These models are tested in the framework of the phenomenon of
stochastic resonance. In turn, they are used to explain in simple terms the
very large gains recently observed in these finite systems.
