We perform a stability and bifurcation analysis of an oscillatory cortical network. An explicit stability constraint for the individual coupling strength between oscillatory units is given, which if satisfied, ensures the stability of the background fixed-point state. It is shown that, under increasing levels of input, departures from the fixed point occur because of a supercritical Hopf bifurcation, giving rise to limit cycle dynamics. Explicit conditions for the onset of the Hopf bifurcation are derived. The full range of dynamical behavior of the network, under typical scenarios of input variations, is illustrated. The result of this work is expected to be applicable to general oscillatory networks coupled through sigmoid type of nonlinear functions.