Recurrent neurodynamics methodology, which finds solutions by regulating state vectors to zero, has been frequently investigated in recent years to solve a variety of complicated computational problems. In this paper, recurrent neurodynamics based methods are proposed to compute positions of nodes during wireless sensor network (WSN) node localization. The proposed neurodynamics models are capable of both finding minimal square solutions of original problems and solving them as time-variant quadratic programming (TVQP) problems. The aim of the proposed models is to reduce the transmission cost of WSN during localization procedures. The convergence and effectiveness of the proposed models are verified via rigorous theoretical proofs, and then further substantiated through computer simulations. It is demonstrated via simulation results that the proposed methods can utilize extra information as constraints to further improve the accuracy of localization.