Negative inductor effects in nonlinear two-dimensional systems: Oscillatory neurons and memristors

Many chemical and physical systems show self-sustained oscillations that can be described by a set of nonlinear differential equations. The system enters oscillatory behavior by an intrinsic instability that leads to bifurcation. We analyze conducting systems that present oscillating response under application of external voltage or current. Phenomena like electrochemical corrosion and the spiking response of a biological neuron are well-known examples. These systems have applications in artificial neurons and synapses for neuromorphic computation. Their dynamical properties can be characterized by normal mode analysis of small expansion of the constituent nonlinear equations. The linearized model leads to the technique of ac frequency response impedance spectroscopy that can be obtained experimentally. We show a general description of two-variable systems formed by a combination of a fast variable (the voltage) and a slowing down internal variable, which produce a chemical inductor. A classification of bifurcations and stability is obtained in terms of the parameters of the intrinsic equivalent circuit including the case of a negative inductor. Thereafter, we describe a number of physical examples and establish the characterization of their properties: The electrocatalytic reaction with adsorbed intermediate species, an oscillating metal oxide memristor, and finally we discuss the signs of the equivalent circuit elements in the central model of neuroscience, the Hodgkin–Huxley model for an oscillating neuron.