In recent decades, with the rapid development of artificial intelligence technologies and bionic engineering, the spiking neural network (SNN), inspired by biological neural systems, has become one of the most promising research topics, enjoying numerous applications in various fields. Due to its complex structure, the simplification of SNN circuits requires serious consideration, along with their power consumption and space occupation. In this regard, the use of SSN circuits based on single-electron transistors (SETs) and modified memristor synapses is proposed herein. A prominent feature of SETs is Coulomb oscillation, which has characteristics similar to the pulses produced by spiking neurons. Here, a novel window function is used in the memristor model to improve the linearity of the memristor and solve the boundary and terminal lock problems. In addition, we modify the memristor synapse to achieve better weight control. Finally, to test the SNN constructed with SETs and memristor synapses, an associative memory learning process, including memory construction, loss, reconstruction, and change, is implemented in the circuit using the PSPICE simulator.