Skip to main content
SpringerLink
Log in

Physical implementation of asynchronous cellular automata networks: mathematical models and preliminary experimental results

  • Original paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Physical implementation of asynchronous cellular automata networks has shown stably random oscillations under certain conditions. We present two simple mathematical models to describe transient and stationary regimes. The models are based on simple assumptions taking into account several aspects such as number of inputs of the cellular automata, rule balance, and technological frequency limitation. Numerical simulations reveal the possibility of chaotic dynamics of the average transition rate of the cellular automata in a stationary regime. With physical implementations on FPGA (field programmable gate array), preliminary experimental results show very good qualitative agreement with model’s prediction and numerical simulations. Several networks of interconnected 5-input asynchronous cellular automata have been successfully implemented in different FPGA devices, and we present some preliminary experimental results. This work aims at finding fundamental mechanisms of randomness such that the collective behavior of the cellular automata system does not depend on physical implementation details.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

Similar content being viewed by others

Availablility of data materials

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

References

  1. Schiff, J.L.: Cellular Automata: A Discrete View of the World, vol. 45. Wiley, New Jersey (2011)

    Google Scholar 

  2. Ilachinski, A.: Cellular Automata: A Discrete Universe. World Scientific Publishing Company, Singapore (2001)

    Book  Google Scholar 

  3. Chen, F.F., Chen, F.Y., Chen, G.R., Jin, W.F., Chen, L.: Symbolics dynamics of elementary cellular automata rule 88. Nonlinear Dyn. 58(1), 431 (2009)

    Article  MathSciNet  Google Scholar 

  4. Schönfisch, B., de Roos, A.: Synchronous and asynchronous updating in cellular automata. BioSystems 51(3), 123 (1999)

    Article  Google Scholar 

  5. Baetens, J.M., Van der Weeën, P., De Baets, B.: Effect of asynchronous updating on the stability of cellular automata. Chaos Solitons Fractals 45(4), 383 (2012)

    Article  Google Scholar 

  6. Bouré, O., Fates, N., Chevrier, V.: Probing robustness of cellular automata through variations of asynchronous updating. Nat. Comput. 11(4), 553 (2012)

    Article  MathSciNet  Google Scholar 

  7. Fates, N.: in International Workshop on Cellular Automata and Discrete Complex Systems, Springer, Berlin (2013), pp. 15–30

  8. Fatès, N.A., Morvan, M.: arXiv preprint nlin/0402016 (2004)

  9. Matsubara, T., Torikai, H.: Asynchronous cellular automaton-based neuron: theoretical analysis and on-FPGA learning. IEEE Transact. Neural Netw. Learn. sys. 24(5), 736 (2013)

    Article  Google Scholar 

  10. Takeda, K., Torikai, H.: A novel hardware-efficient cochlea model based on asynchronous cellular automaton dynamics: theoretical analysis and FPGA implementation. IEEE Trans. Circuits Syst. II Express Briefs 64(9), 1107 (2017)

    Article  Google Scholar 

  11. Wolfram, S.: Cellular automata as models of complexity. Nature 311(5985), 419 (1984)

    Article  Google Scholar 

  12. Wolfram, S.: Statistical mechanics of cellular automata. Rev. Mod. Phys. 55, 601 (1983)

    Article  MathSciNet  Google Scholar 

  13. Dee, D., Ghil, M.: Boolean difference equations, I: formulation and dynamic behavior. SIAM J. Appl. Math. 44(1), 111 (1984)

    Article  MathSciNet  Google Scholar 

  14. Ghil, M., Mullhaupt, A.: Boolean delay equations. II. Periodic and aperiodic solutions. J. Stat. Phys. 41(1–2), 125 (1985)

    Article  MathSciNet  Google Scholar 

  15. Ghil, M., Zaliapin, I., Coluzzi, B.: Boolean delay equations: a simple way of looking at complex systems. Phys. D 237(23), 2967 (2008)

    Article  MathSciNet  Google Scholar 

  16. Xilinx, Vivado Design Suite User Guide Design Analysis and Closure Techniques. Xilinx, Inc

  17. Kilts, S.: Advanced FPGA Design: Architecture, Implementation, and Optimization. Wiley - IEEE (Wiley, 2007). https://books.google.it/books?id=k9T-Q8RgEHcC

  18. Chen, D., Cong, J., Pan, P.: FPGA Design Automation: A Survey. Foundations and Trends(r) in E (Now Publishers, 2006). https://books.google.it/books?id=4k7rdxbymYsC

  19. Navid, R., Lee, T.H., Dutton, R.W.: An analytical formulation of phase noise of signals with Gaussian-distributed jitter. IEEE Trans. Circuits Syst. II Express Briefs 52(3), 149 (2005). https://doi.org/10.1109/TCSII.2004.842038

    Article  Google Scholar 

  20. Antonelli, M., De Micco, L., Larrondo, H.: Measuring the jitter of ring oscillators by means of information theory quantifiers. Commun. Nonlinear Sci. Numer. Simul. 43, 139 (2017)

    Article  MathSciNet  Google Scholar 

  21. Chen, S.C., Chang, Y.W.: in Proceedings of the 36th International Conference on Computer-Aided Design (IEEE Press, 2017), ICCAD ’17, p. 914–921

  22. McNeill, J.A.: Jitter in ring oscillators. IEEE J. Solid-State Circuits 32(6), 870 (1997). https://doi.org/10.1109/4.585289

    Article  Google Scholar 

  23. Maruyama, M., Suzuki, H., Hato, T., Yoshida, A., Tanabe, K.: Effect of thermal noise and trigger jitter on the operation of HTS sampler. IEEE Trans. Appl. Supercond. 16(4), 1959 (2006). https://doi.org/10.1109/TASC.2006.881818

    Article  Google Scholar 

  24. Gazzano, J., Crespo, M., Cicuttin, A., Calle, F.: Field-Programmable Gate Array (FPGA) Technologies for High Performance Instrumentation. Advances in cimputer and electrical engineering (IGI Global, 2016). https://books.google.it/books?id=ksC3DAEACAAJ

  25. Unger, S.H.: Hazards, critical races, and metastability. IEEE Trans. Comput. 44(6), 754 (1995). https://doi.org/10.1109/12.39118

    Article  MATH  Google Scholar 

  26. May, R.M.: Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976). https://doi.org/10.1038/261459a0

    Article  MATH  Google Scholar 

  27. Xilinx. Zynq 7000 Familly Overview. https://www.xilinx.com/support/documentation/data_sheets/ds190-Zynq-7000-Overview.pdf

  28. Wolfram, S.: A new kind of science (2002)

  29. digilent. ZedBoard. https://reference.digilentinc.com/reference/programmable-logic/zedboard/start

  30. Xilinx. Zynq-7000 SoC DC and AC Switching Characteristics. https://www.xilinx.com/support/documentation/data_sheets/ds191-XC7Z030-XC7Z045-data-sheet.pdf

Download references

Acknowledgements

This work was partially supported by the ICTP Associates and TRIL programmes, the ICTP Multidisciplinary Laboratory, the CONICET (National Council for Science and Technology), the National Agency of Scientific and Technological Promotion (ANPCyT, PICT2019-2019-03024), and the Department of Engineering and Architecture of the University of Trieste.

Author information

Authors and Affiliations

  1. Multidisciplinary Laboratory, International Centre for Theoretical Physics (ICTP), 34100, Trieste, Italy

    A. Cicuttin, M. L. Crespo, L. Garcia & W. Florian

  2. Department of Engineering and Architecture, Università degli Studi di Trieste, 34127, Trieste, Italy

    L. Garcia & W. Florian

  3. Instituto de Investigaciones Científicas y Tecnológicas en Electrónica (ICyTE), Facultad de Ingeniería, Universidad Nacional de Mar del Plata (UNMDP) - CONICET, Mar del Plata, Argentina

    L. De Micco & M. Antonelli

Authors
  1. A. Cicuttin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. De Micco
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. L. Crespo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. Antonelli
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. L. Garcia
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. W. Florian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. De Micco.

Ethics declarations

Conflict of interest

The authors have no conflicts of interest to declare that are relevant to the content of this article.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cicuttin, A., De Micco, L., Crespo, M.L. et al. Physical implementation of asynchronous cellular automata networks: mathematical models and preliminary experimental results. Nonlinear Dyn 105, 2431–2452 (2021). https://doi.org/10.1007/s11071-021-06754-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-06754-z

Keywords

Search

Navigation