  We consider the dynamics of a periodic chain of N coupled overdamped
particles under the influence of noise. Each particle is subjected to a
bistable local potential, to a linear coupling with its nearest neighbours, and
to an independent source of white noise. We show that as the coupling strength
increases, the number of equilibrium points of the system changes from 3^N to
3. While for weak coupling, the system behaves like an Ising model with
spin-flip dynamics, for strong coupling (of the order N^2), it synchronises, in
the sense that all oscillators assume almost the same position in their
respective local potential most of the time. We derive the exponential
asymptotics for the transition times, and describe the most probable transition
paths between synchronised states, in particular for coupling intensities below
the synchronisation threshold. Our techniques involve a centre-manifold
analysis of the desynchronisation bifurcation, with a precise control of the
stability of bifurcating solutions, allowing us to give a detailed description
of the system's potential landscape, in which the metastable behaviour is
encoded.
