  Inter-layer synchronization is a distinctive process of multiplex networks
whereby each node in a given layer undergoes a synchronous evolution with all
its replicas in other layers, irrespective of whether or not it is synchronized
with the other units of the same layer. We analytically derive the necessary
conditions for the existence and stability of inter-layer synchronization, and
verify numerically the analytical predictions in several cases where such a
state emerges. We inspect the impact of the layer topology on the robustness of
such a state against a progressive de-multiplexing of the network. Finally, we
provide experimental evidence by means of multiplexes of nonlinear electronic
circuits, showing the stability of the synchronized manifold despite the
intrinsic noise and parameter mismatch in the experiment.
