The growth of connections in a quasi two-dimensional neuronal network in vitro is probed by the synchronized bursting frequency of the system. A model of an early stage of active search growth followed by a later stage of diffusive search mode is proposed. Our model can explain quantitatively the variation of the critical age(for synchronized firing) of the culture as a function of cell density. The later stage of growth implies a probability of connection that is exponentially decay in the mean connectivity. The biological significance of the model is discussed in the light of homeostatic feedback and plasticity.