  The synchronous dynamics of an array of excitable oscillators, coupled via a
generic graph, is studied. Non homogeneous perturbations can grow and destroy
synchrony, via a self-consistent instability which is solely instigated by the
intrinsic network dynamics. By acting on the characteristic time-scale of the
network modulation, one can make the examined system to behave as its
(partially) averaged analog. This result if formally obtained by proving an
extended version of the averaging theorem, which allows for partial averages to
be carried out. As a byproduct of the analysis, oscillation death are reported
to follow the onset of the network driven instability.
