  We study the emergence of phase-locking for Winfree oscillators under the
effect of inertia. It is known that in a large coupling regime, oscillators
governed by the deterministic second-order Winfree model with inertia converge
to a unique equilibrium. In contrast, in this paper we show the asymptotic
emergence of non-trivial synchronization in a suitably small coupling regime.
Moreover, we study the effect of a new stochastically perturbed Winfree system
with multiplicative noise and obtain lower estimates in probability for the
pathwise emergence of such a synchronizing pattern, provided the noise is
sufficiently small. We also provide numerical simulations which hint at the
possibility of more general and stronger analytical results.
