  Synchronization is observed in many natural systems, with examples ranging
from neuronal activation to walking pedestrians. The models proposed by Winfree
and Kuramoto stand as the classic frameworks for investigating these phenomena.
The Kuramoto model, in particular, has been extended in different ways since
its original formulation to account for more general scenarios. One such
extension replaces the coupling parameter with a coupling matrix, describing a
form of generalized frustration with broken rotational symmetry. A key feature
of this model is the existence of {\it phase tuned states}, characterized by
having the phase of the order parameter pointing in the direction of the
dominant eigenvector of the coupling matrix. Here we investigate the matrix
coupled Kuramoto model on networks with two modules, such that one module is in
the phase tuned state and the other in a state where the order parameter
rotates. We identified different regimes in which one or the other module
dominates the dynamics. We found, in particular, that the phase tuned module
can create a bottleneck for the oscillation of the rotating module, leading to
a behavior similar to the charge and fire regimes of excitable systems. We also
found an extended region in the parameter space where motion is globally
disordered, even though one of the modules presented high levels of
synchronization when uncoupled.
