Glassy synchronization in a population of coupled oscillators

A phase model for a population of oscillators with random excitatory and inhibitory mean-field coupling and subject to external white noise random forces is proposed and studied. In the thermodynamic limit different stable phases for the oscillator population may be found: (i) an incoherent state where all possible values of an oscillator phase are equally probable, (ii) a synchronized state where the population has a nonzero collective phase; (iii) a glassy phase where the global synchronization is zero but the oscillators are in phase with the random disorder; and (iv) a mixed phase where the oscillators are partially synchronized and partially in phase with the disorder. These predictions are based upon bifurcation analysis of the reduced equation valid at the thermodynamic limit and confirmed by Brownian simulation.