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High-performance modeling of neurophysiological processes is an urgent task that requires new approaches to information processing. In this context, two- and three-junction superconducting quantum interferometers with Josephson weak links based on gold nanowires are fabricated and investigated experimentally. The studied cells are proposed for the implementation of bio-inspired neurons—high-performance, energy-efficient, and compact elements of neuromorphic processor. The operation modes of an advanced artificial neuron capable of generating the burst firing activation patterns are explored theoretically. A comparison with the Izhikevich mathematical model of biological neurons is carried out.