A memristor is a nonlinear polarity-dependent fundamental circuit element. Due to these intrinsic properties of the device, analyzing a circuit that contains multiple memristors becomes complex. In this paper, we study the characteristics of multiple-memristor series (anti-series) and parallel (anti-parallel) connections, including their transient and stable state composite properties. Also, the existing phenomenological and physics-based memristor mathematical modeling techniques have been discussed for use in SPICE simulation environment. For making a standardized comparison between memristor stochastic and deterministic models, all models presented in this paper have been implemented in a single SPICE program. In addition to the well-known previously reported Joglekar and Biolek window functions, a modified Biolek window function and a novel generic scalable window function have been used to model the intrinsic nonlinearity of memristors effectively. Furthermore, electronic synapse circuits based on memristive devices in series and parallel connections and synaptic circuits based on CMOS transistor–memristor architecture have been presented and analyzed. Based on the obtained results, artificial synaptic circuit design limitation using a single memristor has been demonstrated.