  The onset of synchronization in networks of networks is investigated.
Specifically, we consider networks of interacting phase oscillators in which
the set of oscillators is composed of several distinct populations. The
oscillators in a given population are heterogeneous in that their natural
frequencies are drawn from a given distribution, and each population has its
own such distribution. The coupling among the oscillators is global, however,
we permit the coupling strengths between the members of different populations
to be separately specified. We determine the critical condition for the onset
of coherent collective behavior, and develop the illustrative case in which the
oscillator frequencies are drawn from a set of (possibly different)
Cauchy-Lorentz distributions. One motivation is drawn from neurobiology, in
which the collective dynamics of several interacting populations of oscillators
(such as excitatory and inhibitory neurons and glia) are of interest.
