  A system of two enzymes mechanically coupled to each other in a viscous
medium was recently studied, and conditions for obtaining synchronization and
an enhanced average rate of the thermally-activated catalytic reactions of the
enzymes were identified. The transition to synchronization occurred as the
result of a global bifurcation in the underlying dynamical system. Here, we
extend and generalize this idea to an arbitrary number of noise-activated
cyclic processes, or oscillators, that are all coupled to each other via a
dissipative coupling. The $N$ coupled oscillators are described by $N$ phase
coordinates driven in a tilted washboard potential. At low $N$ and strong
coupling, we find synchronization as well as an enhancement in the average
speed of the oscillators. In the large $N$ regime, we show that the collective
dynamics can be described through a mean-field theory, which predicts a great
enhancement in the average speed. In fact, beyond a critical value of the
coupling strength, noise activation becomes irrelevant and the dynamics switch
to an effectively deterministic "running" mode. Finally, we study the
stochastic thermodynamics of the coupled oscillators, in particular their
performance with regards to the thermodynamic uncertainty relation.
