The memristor is a novel kind of electronic device with dynamic variable resistance that is dependent on the past history of the input current or voltage. As the fourth fundamental circuit element, the memristor captures a number of unique properties that have been found to possess attractive potentials in some promising fields such as nonvolatile memory, nonlinear circuit and system, and neuromorphic system. Additionally, compared with a circuit of single memristor, series-parallel circuit of memristors possesses more abundant device characteristics which arouses increasingly extensive interest from numerous researchers. In this paper, the mathematical closed-form charge-governed and flux-governed HP memristor nonlinear models are presented with constructive procedures. In particular, these models are more realistic by taking into account the nonlinear dopant drift effect nearby the terminals and the boundary conditions, and by adding a simple and effective window function. Furthermore, based on the internal parameters and threshold of the memristor respectively, the theoretical derivation and numerical analysis of the memristor-based series-parallel connection circuits have been made comprehensively. For obtaining the characteristics of the memristor-based combinational circuits intuitively, a graphical user interface is designed based on Matlab software, which is beneficial to displaying the properties of the memristive system clearly. The results in the present paper may provide theoretical reference and reliable experimental basis for the further development of the memristor-based combinational circuits.