Energy constraints are a fundamental limitation of the brain, which is physically embedded in a restricted space. The collective dynamics of neurons through connections enable the brain to achieve rich functionality, but building connections and maintaining activity come at a high cost. The effects of reducing these costs can be found in the characteristic structures of the brain network. Nevertheless, the mechanism by which energy constraints affect the organization and formation of the neuronal network in the brain is unclear. Here, it is shown that a simple model based on cost minimization can reproduce structures characteristic of the brain network. With reference to the behavior of neurons in real brains, the cost function was introduced in an activity-dependent form correlating the activity cost and the wiring cost as a simple ratio. Cost reduction of this ratio resulted in strengthening connections, especially at highly activated nodes, and induced the formation of large clusters. Regarding these network features, statistical similarity was confirmed by comparison to connectome datasets from various real brains. The findings indicate that these networks share an efficient structure maintained with low costs, both for activity and for wiring. These results imply the crucial role of energy constraints in regulating the network activity and structure of the brain.