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Chaotic dynamics in nanoscale NbO2 Mott memristors for analogue computing


At present, machine learning systems use simplified neuron models that lack the rich nonlinear phenomena observed in biological systems, which display spatio-temporal cooperative dynamics. There is evidence that neurons operate in a regime called the edge of chaos1 that may be central to complexity, learning efficiency, adaptability and analogue (non-Boolean) computation in brains2,3,4,5,6,7. Neural networks have exhibited enhanced computational complexity when operated at the edge of chaos2, and networks of chaotic elements have been proposed for solving combinatorial or global optimization problems8. Thus, a source of controllable chaotic behaviour that can be incorporated into a neural-inspired circuit may be an essential component of future computational systems. Such chaotic elements have been simulated using elaborate transistor circuits that simulate known equations of chaos9,10,11,12, but an experimental realization of chaotic dynamics from a single scalable electronic device has been lacking5,6,13. Here we describe niobium dioxide (NbO2) Mott memristors each less than 100 nanometres across that exhibit both a nonlinear-transport-driven current-controlled negative differential resistance and a Mott-transition-driven temperature-controlled negative differential resistance. Mott materials have a temperature-dependent metal–insulator transition that acts as an electronic switch, which introduces a history-dependent resistance into the device. We incorporate these memristors into a relaxation oscillator14 and observe a tunable range of periodic and chaotic self-oscillations15. We show that the nonlinear current transport coupled with thermal fluctuations at the nanoscale generates chaotic oscillations. Such memristors could be useful in certain types of neural-inspired computation by introducing a pseudo-random signal that prevents global synchronization and could also assist in finding a global minimum during a constrained search. We specifically demonstrate that incorporating such memristors into the hardware of a Hopfield computing network can greatly improve the efficiency and accuracy of converging to a solution for computationally difficult problems.

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Figure 1: Device structure and static behaviour.
Figure 2: Experimental dynamical behaviour.
Figure 3: Simulations of thermal fluctuations and relaxation oscillator dynamics.
Figure 4: Chaos-aided global minimization.


  1. Chua, L., Sbitnev, V. & Kim, H. Neurons are poised near the edge of chaos. Int. J. Bifurc. Chaos 22, 1250098 (2012)

    Article  Google Scholar 

  2. Bertschinger, N. & Natschläger, T. Real-time computation at the edge of chaos in recurrent neural networks. Neural Comput. 16, 1413–1436 (2004)

    Article  Google Scholar 

  3. Seifter, J. & Reggia, J. A. Lambda and the edge of chaos in recurrent neural networks. Artif. Life 21, 55–71 (2015)

    Article  Google Scholar 

  4. Kauffman, S. A. Requirements for evolvability in complex systems: orderly dynamics and frozen components. Physica D 42, 135–152 (1990)

    ADS  Article  Google Scholar 

  5. Suzuki, H., Imura, J.-i., Horio, Y. & Aihara, K. Chaotic Boltzmann machines. Sci. Rep. 3, 1610 (2013)

    CAS  Article  Google Scholar 

  6. Crutchfield, J. P. Between order and chaos. Nat. Phys. 8, 17–24 (2012)

    CAS  Article  Google Scholar 

  7. Whitfield, J. Complex systems: order out of chaos. Nature 436, 905–907 (2005)

    CAS  ADS  Article  Google Scholar 

  8. Chen, L. & Aihara, K. Chaotic simulated annealing by a neural network model with transient chaos. Neural Netw. 8, 915–930 (1995)

    Article  Google Scholar 

  9. Hu, X., Chen, G., Duan, S. & Feng, G. in Memristor Networks (eds Adamatzky, A. & Chua, L. ) 351–364 (Springer, 2013)

  10. Ditto, W. L., Murali, K. & Sinha, S. Chaos computing: ideas and implementations. Phil. Trans. R. Soc. Lond. A 366, 653–664 (2008)

    ADS  MathSciNet  Article  Google Scholar 

  11. Driscoll, T., Pershin, Y. V., Basov, D. N. & Di Ventra, M. Chaotic memristor. Appl. Phys. A 102, 885–889 (2011)

    CAS  ADS  Article  Google Scholar 

  12. Wang, G., Cui, M., Cai, B., Wang, X. & Hu, T. A chaotic oscillator based on HP memristor model. Math. Probl. Eng. 2015, 561901 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Muthuswamy, B. & Chua, L. O. Simplest chaotic circuit. Int. J. Bifurc. Chaos 20, 1567–1580 (2010)

    Article  Google Scholar 

  14. Pickett, M. D. & Williams, R. S. Sub-100 fJ and sub-nanosecond thermally driven threshold switching in niobium oxide crosspoint nanodevices. Nanotechnology 23, 215202 (2012)

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  16. 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 

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

  18. Ascoli, A., Slesazeck, S., Mahne, H., Tetzlaff, R. & Mikolajick, T. Nonlinear dynamics of a locally-active memristor. IEEE Trans. Circ. Syst. 62, 1165–1174 (2015)

    MathSciNet  MATH  Google Scholar 

  19. Guckenheimer, J . & Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields 117–165 (Springer, 1983)

  20. Mannan, Z. I., Choi, H. & Kim, H. Chua corsage memristor oscillator via Hopf bifurcation. Int. J. Bifurc. Chaos 26, 1630009 (2016)

    MathSciNet  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  22. Chua, L. O. Local activity is the origin of complexity. Int. J. Bifurc. Chaos 15, 3435–3456 (2005)

    MathSciNet  Article  Google Scholar 

  23. Ott, E. Chaos in Dynamical Systems (Cambridge Univ. Press, 2002)

  24. 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 

  25. Hopfield, J. J. & Tank, D. W. “Neural” computation of decisions in optimization problems. Biol. Cybern. 52, 141–152 (1985)

    CAS  PubMed  MATH  Google Scholar 

  26. Kruse, R ., Borgelt, C ., Braune, C ., Mostaghim, S . & Steinbrecher, M. Computational Intelligence: A Methodological Introduction Ch. 8, 131–157 (Springer, 2016)

  27. Hu, M . et al. Dot-product engine for neuromorphic computing: programming 1T1M crossbar to accelerate matrix-vector multiplication. In IEEE Conf. Design Automation (IEEE, 2016)

  28. Kumar, S. et al. Spatially uniform resistance switching of low current, high endurance titanium–niobium-oxide memristors. Nanoscale 9, 1793–1798 (2017)

    CAS  Article  Google Scholar 

  29. Shafiee, A. et al. ISAAC: a convolutional neural network accelerator with in-situ analog arithmetic in crossbars. In Proc. 43rd Int. Symp. Computer Architecture 14–26, (IEEE Press, 2016)

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The research is in part based on work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via contract number 2017-17013000002. We thank L. O. Chua for reviewing the manuscript, discussions and data analysis. We also thank G. Gibson for discussions and insights.

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Authors and Affiliations



All authors contributed to the conception of the idea, design of experiments, construction of the model, data analysis and writing of the manuscript. S.K. primarily set up experiments and collected experimental data. S.K. and J.P.S. together ran the simulations of the models. R.S.W. conceptualized the static model, the inclusion of thermal noise in the dynamical model, and determined the relevance of chaos in computational systems. J.P.S. had the specific idea of using chaos for accelerating solutions in Hopfield networks.

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Correspondence to Suhas Kumar or R. Stanley Williams.

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The authors declare no competing financial interests.

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Reviewer Information Nature thanks W. Ditto, Z. Toroczkai and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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

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