Third-order nanocircuit elements for neuromorphic engineering


Current hardware approaches to biomimetic or neuromorphic artificial intelligence rely on elaborate transistor circuits to simulate biological functions. However, these can instead be more faithfully emulated by higher-order circuit elements that naturally express neuromorphic nonlinear dynamics1,2,3,4. Generating neuromorphic action potentials in a circuit element theoretically requires a minimum of third-order complexity (for example, three dynamical electrophysical processes)5, but there have been few examples of second-order neuromorphic elements, and no previous demonstration of any isolated third-order element6,7,8. Using both experiments and modelling, here we show how multiple electrophysical processes—including Mott transition dynamics—form a nanoscale third-order circuit element. We demonstrate simple transistorless networks of third-order elements that perform Boolean operations and find analogue solutions to a computationally hard graph-partitioning problem. This work paves a way towards very compact and densely functional neuromorphic computing primitives, and energy-efficient validation of neuroscientific models.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Element construction and static measurements.
Fig. 2: Experimental measurements and modelling of action potentials.
Fig. 3: Experimental demonstration of universal Boolean logic via nonmonotonic spiking behaviour.
Fig. 4: Experimental demonstration of neuromorphic analogue computing.

Data availability

Data presented in the main figures of the manuscript are available from the authors upon reasonable request.


  1. 1.

    Mead, C. Neuromorphic electronic systems. Proc. IEEE 78, 1629–1636 (1990).

    Article  Google Scholar 

  2. 2.

    Mainzer, K. & Chua, L. Local Activity Principle (Imperial College Press, 2013).

  3. 3.

    Kumar, S., Strachan, J. P. & Williams, R. S. Chaotic dynamics in nanoscale NbO2 Mott memristors for analogue computing. Nature 548, 318–321 (2017).

    ADS  CAS  Article  Google Scholar 

  4. 4.

    Chua, L. Memristor, Hodgkin–Huxley, and edge of chaos. Nanotechnology 24, 383001 (2013).

    ADS  Article  Google Scholar 

  5. 5.

    Izhikevich, E. M. Dynamical Systems in Neuroscience. (MIT Press, 2007).

  6. 6.

    Chua, L. Everything you wish to know about memristors but are afraid to ask. Radioengineering 24, 319–368 (2015).

    Article  Google Scholar 

  7. 7.

    Chua, L. Handbook of Memristor Networks (Springer Nature, 2019).

  8. 8.

    Bohaichuk, S. M. et al. Fast spiking of a Mott VO2–carbon nanotube composite device. Nano Lett. 19, 6751–6755 (2019).

    ADS  CAS  Article  Google Scholar 

  9. 9.

    Kendall, J. D. & Kumar, S. The building blocks of a brain-inspired computer. Appl. Phys. Rev. 7, 011305 (2020).

    ADS  CAS  Article  Google Scholar 

  10. 10.

    Zidan, M. A., Strachan, J. P. & Lu, W. D. The future of electronics based on memristive systems. Nature Electronics 1, 22 (2018).

    Article  Google Scholar 

  11. 11.

    Paugam-Moisy, H. & Bohte, S. Computing with spiking neuron networks. In Handbook of Natural Computing (eds Rozenberg, G. et al.) 335–376 (Springer, 2012).

  12. 12.

    Pickett, M. D., Borghetti, J., Yang, J. J., Medeiros-Ribeiro, G. & Williams, R. S. Coexistence of memristance and negative differential resistance in a nanoscale metal–oxide–metal system. Adv. Mater. 23, 1730–1733 (2011).

    CAS  Article  Google Scholar 

  13. 13.

    Pickett, M. D., Medeiros-Ribeiro, G. & Williams, R. S. A scalable neuristor built with Mott memristors. Nat. Mater. 12, 114–117 (2013).

    ADS  CAS  Article  Google Scholar 

  14. 14.

    Yi, W. et al. Biological plausibility and stochasticity in scalable VO2 active memristor neurons. Nat. Commun. 9, 4661 (2018).

    ADS  Article  Google Scholar 

  15. 15.

    Khanday, F. A., Kant, N. A., Dar, M. R., Zulkifli, T. Z. A. & Psychalinos, C. Low-voltage low-power integrable CMOS circuit implementation of integer- and fractional-order FitzHugh–Nagumo neuron model. IEEE Trans. Neural Netw. Learn. Syst. 30, 2108–2122 (2018).

    MathSciNet  Article  Google Scholar 

  16. 16.

    Markram, H. Seven challenges for neuroscience. Funct. Neurol. 28, 145–151 (2013).

    PubMed Central  Google Scholar 

  17. 17.

    Palmer, T. Modelling: build imprecise supercomputers. Nature 526, 32 (2015).

    ADS  CAS  Article  Google Scholar 

  18. 18.

    Gibson, G. A. et al. An accurate locally active memristor model for S-type negative differential resistance in NbOx. Appl. Phys. Lett. 108, 023505 (2016).

    ADS  Article  Google Scholar 

  19. 19.

    Slesazeck, S. et al. Physical model of threshold switching in NbO2-based memristors. RSC Adv. 5, 102318–102322 (2015).

    CAS  Article  Google Scholar 

  20. 20.

    Kumar, S. et al. Physical origins of current- and temperature-controlled negative differential resistances in NbO2. Nat. Commun. 8, 658 (2017).

    ADS  Article  Google Scholar 

  21. 21.

    Li, S., Liu, X., Nandi, S. K., Nath, S. K. & Elliman, R. G. Origin of current-controlled negative differential resistance modes and the emergence of composite characteristics with high complexity. Adv. Funct. Mater. 29, 1905060 (2019).

    CAS  Article  Google Scholar 

  22. 22.

    Goodwill, J. M. et al. Spontaneous current constriction in threshold switching devices. Nat. Commun. 10, 1628 (2019).

    ADS  Article  Google Scholar 

  23. 23.

    Zhang, J. et al. Thermally induced crystallization in NbO2 thin films. Sci. Rep. 6, 34294 (2016).

    ADS  CAS  Article  Google Scholar 

  24. 24.

    Seta, K. & Naito, K. Calorimetric study of the phase transition in NbO2. J. Chem. Thermodyn. 14, 921–935 (1982).

    CAS  Article  Google Scholar 

  25. 25.

    Kumar, S. et al. Spatially uniform resistance switching of low current, high endurance titanium–niobium–oxide memristors. Nanoscale 9, 1793 (2017).

    CAS  Article  Google Scholar 

  26. 26.

    Kumar, S. et al. The phase transition in VO2 probed using X-ray, visible and infrared radiations. Appl. Phys. Lett. 108, 073102 (2016).

    ADS  Article  Google Scholar 

  27. 27.

    Gibson, G. A. Designing negative differential resistance devices based on self-heating. Adv. Funct. Mater. 28, 1704175 (2018).

    Article  Google Scholar 

  28. 28.

    Pickett, M. D. & Williams, R. S. Phase transitions enable computational universality in neuristor-based cellular automata. Nanotechnology 24, 384002 (2013).

    ADS  Article  Google Scholar 

  29. 29.

    Kopell, N. & Somers, D. Anti-phase solutions in relaxation oscillators coupled through excitatory interactions. J. Math. Biol. 33, 261–280 (1995).

    MathSciNet  CAS  Article  Google Scholar 

  30. 30.

    Hoppensteadt, F. C. & Izhikevich, E. M. Thalamo-cortical interactions modeled by weakly connected oscillators: could the brain use FM radio principles? Biosystems 48, 85–94 (1998).

    CAS  Article  Google Scholar 

  31. 31.

    Bansal, K. et al. Cognitive chimera states in human brain networks. Sci. Adv. 5, eaau8535 (2019).

    ADS  Article  Google Scholar 

  32. 32.

    Steriade, M. Synchronized activities of coupled oscillators in the cerebral cortex and thalamus at different levels of vigilance. Cereb. Cortex 7, 583–604 (1997).

    CAS  Article  Google Scholar 

  33. 33.

    Csaba, G. & Porod, W. Coupled oscillators for computing: a review and perspective. Appl. Phys. Rev. 7, 011302 (2020).

    CAS  Article  Google Scholar 

  34. 34.

    Chou, J., Bramhavar, S., Ghosh, S. & Herzog, W. Analog coupled oscillator based weighted Ising machine. Sci. Rep. 9, 14786 (2019).

    ADS  Article  Google Scholar 

  35. 35.

    Parihar, A., Shukla, N., Jerry, M., Datta, S. & Raychowdhury, A. Vertex coloring of graphs via phase dynamics of coupled oscillatory networks. Sci. Rep. 7, 911 (2017); correction 8, 6120 (2018).

    ADS  Article  Google Scholar 

  36. 36.

    Maffezzoni, P., Bahr, B., Zhang, Z. & Daniel, L. Oscillator array models for associative memory and pattern recognition. IEEE Trans. Circuits Syst. I 62, 1591–1598 (2015).

    MathSciNet  Article  Google Scholar 

  37. 37.

    Romera, M. et al. Vowel recognition with four coupled spin-torque nano-oscillators. Nature 563, 230–234 (2018).

    ADS  CAS  Article  Google Scholar 

  38. 38.

    Mahmoodi, M., Prezioso, M. & Strukov, D. Versatile stochastic dot product circuits based on nonvolatile memories for high performance neurocomputing and neurooptimization. Nat. Commun. 10, 5113 (2019).

    ADS  CAS  Article  Google Scholar 

  39. 39.

    Cai, F. et al. Power-efficient combinatorial optimization using intrinsic noise in memristor Hopfield neural networks. Nat. Electron. 3, 409–418 (2020).

    Article  Google Scholar 

  40. 40.

    Huang, A., Kantor, R., DeLong, A., Schreier, L. & Istrail, S. QColors: an algorithm for conservative viral quasispecies reconstruction from short and non-contiguous next generation sequencing reads. In Silico Biol. 11, 193–201 (2011).

    PubMed  PubMed Central  Google Scholar 

  41. 41.

    Pang, J. et al. Potential rapid diagnostics, vaccine and therapeutics for 2019 novel coronavirus (2019-nCoV): a systematic review. J. Clin. Med. 9, 623 (2020).

    CAS  Article  Google Scholar 

  42. 42.

    Mangul, S. et al. Accurate viral population assembly from ultra-deep sequencing data. Bioinformatics 30, i329–i337 (2014).

    CAS  Article  Google Scholar 

  43. 43.

    Hamerly, R. et al. Experimental investigation of performance differences between coherent Ising machines and a quantum annealer. Sci. Adv. 5, eaau0823 (2019).

    ADS  Article  Google Scholar 

Download references


We thank L. O. Chua and H. S. P. Wong for comments on the manuscript. We also thank K. J. Cremata, A. L. D. Kilcoyne, T. Tyliszczak, D. Shapiro, G. Gibson, X. Sheng and J. Zhang for assistance in collecting experimental data or construction of the models. We acknowledge S. M. Bohaichuk, J. C. Nino, A. Conklin and J. L. Andrews for discussions on specific topics and suggestions for illustrations. Work was performed in part in the nano@Stanford labs, which are supported by the National Science Foundation under award ECCS-1542152. Synchrotron measurements were conducted at the Advanced Light Source, a US DOE Office of Science User Facility under contract no. DE-AC02-05CH11231, and at the Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, a US DOE Office of Science User Facility under contract no. DE-AC02-76SF00515. R.S.W. acknowledges the X-Grants Program of the President’s Excellence Fund at Texas A&M University.

Author information




S.K. and R.S.W. conceived the project and planned the various measurements. Z.W. performed the materials growth, compositional analysis and parts of the chip fabrication. S.K. and Z.W. performed the in operando spectroscopic measurements and collected the electrical data. S.K. and R.S.W. constructed the model and wrote the manuscript.

Corresponding author

Correspondence to Suhas Kumar.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature thanks Adnan Mehonic, Syed Ghazi Sarwat and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

9 sections of discussion and 31 figures. Document contains additional information on fabrication and construction, material and electrical measurements, data analysis, compact model and other general discussions supporting the main text.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kumar, S., Williams, R.S. & Wang, Z. Third-order nanocircuit elements for neuromorphic engineering. Nature 585, 518–523 (2020).

Download citation


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing