A Verilog-A model of a spiking neural network is developed, and its dopamine-like learning is carried out in solving the problem of recognizing the simplest images. The necessity of using unipolar pulses from postsynaptic neurons to implement dynamic plasticity of the “bell-shaped” and “anti-bell-shaped” types, as well as the positive effect of the inhibitory layer of neurons on the operation of the system, is shown. A hardware and software complex implementing this neural network and the dynamics of changes in the conductivity window of a memristor synaptic connection obtained with its help when emulating different “dopamine levels” in a neuromorphic system are presented.