Memristors on ‘edge of chaos’

Rather than echoing the vision and perspectives proffered by numerous previous publications, this Review focuses on the recent resolution of four unsolved classic problems — Galvani’s ‘irritability’, the Hodgkin–Huxley ‘all-or-none’ mystery, the Turing instability and the Smale paradox — the oldest dating back 243 years to Galvani in 1781. Unlike advances reported previously, which tend to be ephemeral, our resolution of these problems is timeless, because they are a manifestation of a new law of nature, called the ‘principle of local activity’, which, within a certain relatively small parameter space, could harbour a physical state dubbed the ‘edge of chaos’. In this Review, we provide an explicit formula for calculating, via matrix algebra, the precise parameter range where a nonlinear device, or system, is locally active or operating on the edge of chaos. Unlike numerous unsuccessful attempts by luminaries, such as Boltzmann’s assay for decreasing entropy, Schrödinger’s futile search for negentropy, Prigogine’s quest for the ‘instability of the homogeneous’ and Gell-Mann’s musing on ‘amplification of fluctuations’, the principle of local activity provides an explicit formula to identify the parameter space where the edge of chaos reigns supreme. This Review resolves the age-old problems of Galvani’s irritability, the Hodgkin–Huxley ‘all-or-none’ mystery, the Turing instability and the Smale paradox, by applying the findings in 2023 that memristors operating on the ‘edge of chaos’ can model the nonlinear dynamics of these problems, complementing the second law of thermodynamics.