  The higher-order interactions of complex systems, such as the brain are
captured by their simplicial complex structure and have a significant effect on
dynamics. However, the existing dynamical models defined on simplicial
complexes make the strong assumption that the dynamics resides exclusively on
the nodes. Here we formulate the higher-order Kuramoto model which describes
the interactions between oscillators placed not only on nodes but also on
links, triangles, and so on. We show that higher-order Kuramoto dynamics can
lead to an explosive synchronization transition by using an adaptive coupling
dependent on the solenoidal and the irrotational component of the dynamics.
