Synchronization of neural activity plays an important role for processing of information in the brain. In this work, we study the dynamics of a neural network with plastic synaptic connections under the action of a spatially localized stimulus. It is established that synchronization of all the network neurons by a local action is possible in a certain range of periodic stimulation. The network forms the signals in the form of quasi-synchronous bursts of pulses and synchronization of the bursts with the applied stimuli occurs under an external periodic action. It is shown that the network synchronization takes place in a certain frequency range of external action. An extension of this range is observed with increasing geometric size of the stimulated region. The synchronization range decreases with increasing number of the neuron connections in the network. It is also established that the plastic (adaptive) connections among the elements of the network can improve its sensitivity to external action, i.e., extend the frequency range of stimulation which causes the network synchronization, and decrease the minimum size of the network segment exposed to external action.