Dynamics, synchronization and analog circuit implementation of a discrete neuron-like map with pulsating spiral dynamics

Owing to their computational efficiency and generative potential, discrete neural models have been gaining interest, both theoretically and towards engineering applications. In this paper, we propose a discrete map endowed with neuron-like dynamics and two distinguishing properties. Namely, the attractor has a spiral topology, and it has a cavity. The size of this cavity shows a non-monotonic relationship to a control parameter, delineating a pulsating behavior. The map shows a multitude of qualitatively-different periodic, chaotic, and hyperchaotic behaviors as a function of the control parameter settings, and its jump points restrict the internal and external size of the attractor. The oscillation frequency is also controllable and also shows a non-monotonic effect. In preparation for future work realizing neural-like networks, we demonstrate the synchronizability by coupling two nodes through a quadratic discrete memristor as the synapse. Finally, the analog circuit realizing this map is implemented and its dynamics are studied experimentally via a comprehensive set of measurements. As the control parameter is varied, the dynamic contraction and expansion of the attractor are observed alongside its well-developed spiral structure.