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A biologically inspired neural network approach is proposed for real-time collision-free path planning and tracking control of a nonholonomic mobile robot in a nonstationary environment. The real-time robot trajectory with obstacle avoidance is rated by a topologically organized neural network, where the dynamics of each neuron is characterized by a shunting equation. The varying environment is represented by the dynamic activity landscape of the neural network. Where the neural activity propagation is subject to the kinematic constraint of the nonholonomic mobile robots. The real-time tracking velocities are generated by a novel neural dynamics based controller, which is based on two shunting models and the backstepping technique. Unlike the backstepping controllers that produce non-smooth velocity commands with sharp jumps, the proposed tracking controller is capable of generating smooth, continuous commands not suffering from velocity jumps. The effectiveness and efficiency of the proposed approach are demonstrated through simulation and comparison studies.