  Adaptation plays a pivotal role in the evolution of natural and artificial
complex systems, and in the determination of their functionality. Here, we
investigate the impact of adaptive inter-layer processes on intra-layer
synchronization in multiplex networks. The considered adaptation mechanism is
governed by a Hebbian learning rule, i.e., the link weight between a pair of
interconnected nodes is enhanced if the two nodes are in phase. Such adaptive
coupling induces an irreversible first-order transition route to
synchronization accompanied with a hysteresis. We provide rigorous analytic
predictions of the critical coupling strengths for the onset of synchronization
and de-synchronization, and verify all our theoretical predictions by means of
extensive numerical simulations.
