  Synchronization and desynchronization are the two ends on the spectrum of
emergent phenomena that somehow often coexist in biological, neuronal, and
physical networks. However, previous studies essentially regard their
coexistence as a partition of the network units: those that are in relative
synchrony and those that are not. In real-world systems, desynchrony bears
subtler divisions because the interacting units are high-dimensional, like fish
schooling with varying speeds in a circle, synchrony and desynchrony may occur
to the different dimensions of the population. In this work, we show with an
ensemble of multidimensional Kuramoto oscillators, that this property is
generalizable to arbitrary dimensions: for a $d$ dimensional population, there
exist $2^d$ system modes where each dimension is either synchronized or
desynchronized, represented by a set of almost binary order parameters. Such
phenomena are induced by a matrix coupling mechanism that goes beyond the
conventional scalar-valued coupling by capturing the inter-dimensional
dependence amongst multidimensional individuals, which arises naturally from
physical, sociological and engineering systems. As verified by our theory, the
property of the coupling matrix thoroughly affects the emergent system modes
and the phase transitions toward them. By numerically demonstrating that these
system modes are interchangeable through matrix manipulation, we also observe
explosive synchronization/desynchronization that is induced without the
conditions that are previously deemed essential. Our discovery provides
theoretical analogy to the cerebral activity where the resting state and the
activated state coexist unihemispherically, it also evokes a new possibility of
information storage in oscillatory neural networks.
