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In this paper, a novel biologically inspired neural network approach is proposed for dynamic collision-free path planning and tracking control of a nonholonomic mobile robot in a nonstationary environment. The real-time collision-free trajectory of the mobile robot with obstacle avoidance is generated by a topologically organized neural network, where the dynamics of each neuron is characterized by a shunting equation derived from Hodgkin and Huxley's biological membrane equation. The configuration space of the mobile robot constitutes the state space of the neural network. 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 robot. Thus no local collision checking procedures are needed. The tracking velocities are generated by a novel neural dynamics based controller, which is based on two shunting models and the conventional backstepping technique. Unlike the backstepping controllers that produce velocity commands with sharp jumps, the proposed tracking controller can generate smooth, continuous commands, not suffering from the velocity jump problem. The effectiveness and efficiency of the proposed approach are demonstrated through simulation and comparison studies.