Bifurcation and Frequency Properties of S-Type Neuronic Oscillators

Oscillators are nonlinear elements or systems that generate periodic signals in an electronic circuit. In neuromorphic circuits, oscillators are used to replicate essential neural processes, such as synchronization, spiking, and rhythmic activity. To obtain these functions, a broad range of systems is investigated for artificial neurons, such as electrochemical autocatalytic systems, organic electrochemical transistors, and binary oxides memristors with an insulator-metal transition. The general features of oscillators controlled by a single internal physical variable, which produces an S-type current-voltage curve with a negative differential resistance, with matched external R and C elements, are discussed. The paper provides a classification of dynamical behaviors that will be found in the practical investigation and applications. A Hopf bifurcation ensures the existence of a limit cycle where the oscillations have small amplitude and nearly sinusoidal form. Slower relaxation oscillations occur at large values of the capacitor time constant since the internal variable becomes very fast in comparison to the variation of voltage. The values of the oscillation frequencies across the whole variation of bifurcation parameters are described, which facilitates the application of the oscillators in coupled configurations such as oscillating neural networks.