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  • Review Article
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Memristors on ‘edge of chaos’

Nature Reviews Electrical Engineering volume 1, pages 614–627 (2024)Cite this article

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A Publisher Correction to this article was published on 05 September 2024

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Abstract

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.

Key points

  • The Hodgkin–Huxley circuit model for neurons is poised near the edge of chaos.

  • The time-varying sodium and potassium conductances in the Hodgkin–Huxley circuit model are time-invariant sodium and potassium memristors, respectively.

  • Galvani discovered in 1781 that a frog’s leg contracts on application of an almost completely discharged Leyden jar. He searched in vain for an elucidating physical principle, which was identified by Chua in 2005 as the local activity principle.

  • When a biological neuron enters the edge-of-chaos operating regime, it is endowed with the capability to generate an ‘all-or-none’ spike, known as the action potential, which enables synaptic adaptations essential for the development of intelligence in the brain.

  • The edge of chaos is essential for generating the Turing instability observed in reaction–diffusion systems, which puzzled Alan Turing, the father of artificial intelligence.

  • The local activity principle is essential for resolving the celebrated Smale paradox, in which two identical mathematically ‘dead’ biological cells (resting in their equilibrium state) became alive by oscillating across the homogeneous cellular medium when allowed to interact via a diffusion process.

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Fig. 1: Exploiting dynamic route maps.
Fig. 2: Galvani’s lab inspires Hodgkin–Huxley and Mackinnon.
Fig. 3: Neurons function near the edge of chaos.
Fig. 4: Turing instability emerges from the edge of chaos.
Fig. 5: Smale paradox springs from the edge of chaos.
Fig. 6: Chua riddle as example par excellence of the edge of chaos.
Fig. 7: A variety of neuromorphic phenomena from the Chua corsage memristor.

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Acknowledgements

I thank S. Williams for sharing his fascinating recent exploitations of the edge of chaos to discover highly promising new memristive materials that could not have been found by trial and error, which had unique potential applications for advanced neurocomputing, AI and other exotic high-tech applications. I also thank R. Picos, A. James, G. Wang, P. Jin, A. Ascoli and Q. Xia for sharing their current, as yet unpublished research results. Last but not least, I thank M. Umraiz for his profound devotion and superb professionalism in all aspect of the preparation of this Review paper.

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  1. University of California Berkeley, Berkeley, CA, USA

    Leon O. Chua

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  1. Leon O. Chua
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Correspondence to Leon O. Chua.

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Nature Reviews Electrical Engineering thanks Yimin Wu; Wei Yi; and Jianhua Yang, who co-reviewed with Ruoyu Zhao, for their contribution to the peer review of this work.

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Chua, L.O. Memristors on ‘edge of chaos’. Nat Rev Electr Eng 1, 614–627 (2024). https://doi.org/10.1038/s44287-024-00082-1

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