During the past few decades, major structural damages due to natural disasters like earthquakes has led bridge engineers to develop structural control systems to mitigate damage and improve vibration reduction in real-time. Among different kinds of vibration control systems, base isolation is one of the most commonly used passive control strategies for civil structures. However, base isolators have their own limitations due to the lack of real-time adaptability and lower energy dissipation. In order to overcome this limitation, semi-active damping devices are installed between the deck and piers. In the present study, a semi-active control system comprised of magneto-rheological (MR) dampers is proposed for vibration mitigation of isolated bridge structures. Recently, inspired by evolutionary game theory, a replicator dynamic control algorithm was developed to allocate the input voltage of MR dampers. In this paper, a load balancing strategy is studied to reallocate additional resources and improve the power distribution over semi-active MR dampers. In order to achieve a high-performance design of the replicator controller, a modified patented Neural Dynamic (ND) model of Adeli and Park is used to optimize the load-balanced replicator control parameters. The ND model incorporates a penalty function, the Lyapunov stability theorem, and the Karush-Kuhn-Tucker conditions to guarantee the global convergence of the solution. The objective function is then defined to minimize the dynamic response of the bridge. The proposed methodology is evaluated using a benchmark control problem that is based on Interstate 5 overcrossing California State Route 91 bridge in Southern California subjected to near-field earthquake accelerograms. The performance of the proposed controller is evaluated and compared with conventional Lyapunov and fuzzy control algorithms in terms of 16 different performance criteria describing the reductions in dynamic response of the bridge structure.