We consider a 4D system obtained via center manifold reduction from equations for Boussinesq convection. For an open set of parameter values the system possesses a heteroclinic network, which includes a connection from an equilibrium to a conventional heteroclinic cycle. We investigate dynamics near this heteroclinic network. In particular, we show that the subcycle involved into the depth two connection is essentially asymptotically stable. Without an added noise typical trajectories are attracted by this subcycle, while in the presence of noise trajectories make excursions along the depth two connection. The effects of addition of noise are compared with numerical simulations of equations of Boussinesq convection with stochastic forcing.