Edge of Chaos Theory Unveils the First and Simplest Ever Reported Hodgkin–Huxley Neuristor

Abstract

The Hodgkin-Huxley model is an accurate yet convoluted mathematical description of the complex nonlinear dynamics of a biological neuronal axon. Employing four degrees of freedom, three of which embodied by the sodium and potassium memristive ion channels, it is capable to capture the cascade of three fundamental bifurcations, specifically a Hopf supercritical, a Hopf subcritical, and a saddle-node limit cycle bifurcation, marking the life cycle from birth to extinction via All-to-None effect of an electrical spike, also referred to as Action Potential in the literature, across biological axon membranes under monotonic change in the net synaptic current. This paper recurs to powerful concepts from the Local Activity and Edge of Chaos Principle and to methods from Circuit Theory and Nonlinear Dynamics to design the first and simplest ever-reported electrical circuit, which, leveraging the peculiar Negative Differential Resistance effects in a volatile NbOx threshold switch from NaMLab, and including additionally just one capacitor and one DC current source in its minimal topology, undergoes the three-bifurcation cascade, emerging across the fourth-order Hodgkin-Huxley neuron model under monotonic current sweep, while requiring half the number of degrees of freedom, which reveals the promising potential of Memristors on “Edge of Chaos” for energy-efficient bio-inspired electronics.