Spiking neural networks (SNNs) are considered a crucial research direction to address the “storage wall” and “power wall” challenges faced by traditional artificial intelligence computing. However, developing SNN chips based on CMOS (complementary metal oxide semiconductor) circuits remains a challenge. Although memristor process technology is the best alternative to synapses, it is still undergoing refinement. In this study, a novel approach is proposed that employs tools to automatically generate HDL (hardware description language) code for constructing neuron and memristor circuits after using Python to describe the neuron and memristor models. Based on this approach, HR (Hindmash–Rose), LIF (leaky integrate-and-fire), and IZ (Izhikevich) neuron circuits, as well as HP, EG (enhanced generalized), and TB (the behavioral threshold bipolar) memristor circuits are designed to construct the most basic connection of a SNN: the neuron–memristor–neuron circuit that satisfies the STDP (spike-timing-dependent-plasticity) learning rule. Through simulation experiments and FPGA (field programmable gate array) prototype verification, it is confirmed that the IZ and LIF circuits are suitable as neurons in SNNs, while the X variables of the EG memristor model serve as characteristic synaptic weights. The EG memristor circuits best satisfy the STDP learning rule and are suitable as synapses in SNNs. In comparison to previous works on hardware spiking neurons, the proposed method needed fewer area resources for creating spiking neurons models on FPGA. The proposed SNN basic components design method, and the resulting circuits, are beneficial for architectural exploration and hardware–software co-design of SNN chips.