Skip to main content
SpringerLink
Log in

Mathematical Consistency and Long-Term Behaviour of a Dynamical System with a Self-Organising Vector Field

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

A dynamical system with a plastic self-organising velocity vector field was introduced in Janson and Marsden (Sci Rep 7:17007, 2017) as a mathematical prototype of new explainable intelligent systems. Although inspired by the brain plasticity, it does not model or explain any specific brain mechanisms or processes, but instead expresses a hypothesised principle possibly implemented by the brain. The hypothesis states that, by means of its plastic architecture, the brain creates a plastic self-organising velocity vector field, which embodies self-organising rules governing neural activity and through that the behaviour of the whole body. The model is represented by a two-tier dynamical system, in which the observable behaviour obeys a velocity field, which is itself controlled by another dynamical system. Contrary to standard brain models, in the new model the sensory input affects the velocity field directly, rather than indirectly via neural activity. However, this model was postulated without sufficient explication or theoretical proof of its mathematical consistency. Here we provide a more rigorous mathematical formulation of this problem, make several simplifying assumptions about the form of the model and of the applied stimulus, and perform its mathematical analysis. Namely, we explore the existence, uniqueness, continuity and smoothness of both the plastic velocity vector field controlling the observable behaviour of the system, and the of the behaviour itself. We also analyse the existence of pullback attractors and of forward limit sets in such a non-autonomous system of a special form. Our results verify the consistency of the problem and pave the way to constructing more models with specific pre-defined cognitive functions.

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.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. He required the system to be defined in the whole past and the convergence to be uniform in \(t_0 \in {\mathbb {R}}\).

References

  1. Abbot, L.F., Nelson, S.B.: Synaptic plasticity taming the beast. Nat. Neurosci. 3, 1178–1183 (2000)

    Article  Google Scholar 

  2. Ardiansyah, S., Majid, M.A., Zain, J.M.: Knowledge of extraction from trained neural network by using decision tree. In: 2nd International Conference on Science in Information Technology (ICSITech), pp. 220–225 (2016)

  3. Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998)

    Book  Google Scholar 

  4. Barron, A.B., Hebets, E.A., Cleland, T.A., Fitzpatrick, C.L., Hauber, M.E., Stevens, J.R.: Embracing multiple definitions of learning. Trends Neurosci. 38(7), 405–407 (2015)

    Article  Google Scholar 

  5. Bi, G., Poo, M.: Synaptic modification of correlated activity: Hebb’s postulate revisited. Ann. Rev. Neurosci. 24, 139–166 (2001)

    Article  Google Scholar 

  6. Bleicher, A.: Demystifying the black box that is AI. Sci. Am. 9, 8 (2017)

    Google Scholar 

  7. Boz, O.: Extracting decision trees from trained neural networks. In: Proceedings of the Eighth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, ACM, New York, pp. 456–461 (2002)

  8. Brinker, T.J., Hekler, A., Enk, A.H., Klode, J., Hauschild, A., Berking, C., Schilling, B., Haferkamp, S., Schadendorf, D., Holland-Letz, T., Utikal, J.S., von Kalle, C., et al.: Deep learning outperformed 136 of 157 dermatologists in a head-to-head dermoscopic melanoma image classification task. Eur. J. Cancer 113, 47–54 (2019)

    Article  Google Scholar 

  9. Crauel, H., Kloeden, P.E.: Nonautonomous and random attractors. Jahresbericht der Deutschen Mathematiker-Vereinigung 117, 173–206 (2015)

    Article  MathSciNet  Google Scholar 

  10. Crutchfield, J.P.: Dynamical embodiments of computation in cognitive processes. Behav. Brain Sci. 21, 635 (1998)

    Article  Google Scholar 

  11. Cui, H., Langa, J.A.: Uniform attractors for non-autonommous random dynamical systems. J. Differ. Equ. 263, 1225–1268 (2017)

    Article  Google Scholar 

  12. Cui, H., Kloeden, P.E.: Invariant forward random attractors of non-autonomous random dynamical systems. J. Differ. Eqn. 65, 6166–6186 (2018)

    Article  MathSciNet  Google Scholar 

  13. Djurfeldt, M., Johansson, C., Ekeberg, Ö., Rehn, M., Lundqvist, M., Lansner, A.: Massively parallel simulation of brain-scale neuronal network models. Computational biology and neurocomputing, School of Computer Science and Communication. Royal Institute of Technology, Stockholm. TRITA-NA-P0513 (2005)

  14. Dong, D.W., Hopfield, J.J.: Dynamic properties of neural networks with adapting synapses. Netw. Comput. Neural Syst. 3, 267–283 (1992)

    Article  MathSciNet  Google Scholar 

  15. Esteva, A., Kuprel, B., Novoa, R.A., Ko, J., Swetter, S.M., Blau, H.M., Thrun, S.: Dermatologist-level classification of skin cancer with deep neural networks. Nature 542, 115 (2017)

    Article  Google Scholar 

  16. van Gelder, T.: The dynamical hypothesis in cognitive science. Behav. Brain Sci. 21, 615–665 (1998)

    Article  Google Scholar 

  17. Hammarlund, P., Ekeberg, Ö.: Large neural network simulations on multiple hardware platforms. J. Comput. Neurosci. 5, 443–459 (1998)

    Article  Google Scholar 

  18. Han, X., Kloeden, P.E.: Random Ordinary Differential Equations and their Numerical Solution. Springer, Singapore (2017)

    Book  Google Scholar 

  19. Janson, N.B., Marsden, C.J.: Dynamical system with plastic self-organized velocity field as an alternative conceptual model of a cognitive system. Sci. Rep. 7, 17007 (2017)

    Article  Google Scholar 

  20. Janson, N.B., Marsden, C.J.: Supplementary Note to: Dynamical system with plastic self-organized velocity field as an alternative conceptual model of a cognitive system. Sci. Rep. 7, 17007 (2017)

    Article  Google Scholar 

  21. Kloeden, P.E.: Pullback attractors of nonautonomous semidynamical systems. Stoch. Dyn. 3, 101–112 (2003)

    Article  MathSciNet  Google Scholar 

  22. Kloeden, P.E., Rasmussen, M.: Nonautonomous Dynamical Systems. American Mathematical Society, Providence (2011)

    Book  Google Scholar 

  23. Kloeden, P.E.: Asymptotic invariance and the discretisation of nonautonomous forward attracting sets. J. Comput. Dyn. 3, 179–189 (2016)

    Article  MathSciNet  Google Scholar 

  24. Marr, B.: 5 Important Artificial Intelligence predictions (for 2019) everyone should read. Forbes, 3 December (2018). https://www.forbes.com/sites/bernardmarr/2018/12/03/5-important-artificial-intelligence-predictions-for-2019-everyone-should-read/#6b4e590c319f

  25. Moravčík, M., Schmid, M., Burch, N., Lisý, V., Morrill, D., Bard, N., Davis, T., Waugh, K., Johanson, M., Bowling, M.: DeepStack: expert-level artificial intelligence in heads-up no-limit poker. Science 356, 508–513 (2017)

    Article  MathSciNet  Google Scholar 

  26. McGough, M.: How bad is Sacramento’s air, exactly? Google results appear at odds with reality, some say. Sacramento Bee, 7 August (2018). https://www.sacbee.com/news/california/fires/article216227775.html

  27. Olcese, U., Oude Lohius, M.N., Pennartz, C.M.A.: Sensory processing across conscious and nonconscious brain states: from single neurons to distributed networks for inferential representation. Front. Syst. Neurosci. 12, 49 (2018)

    Article  Google Scholar 

  28. Peng, T.: AI hasn’t found its Isaac Newton: Gary Marcus on deep learning defects and “Frenemy” Yann LeCun. Synced AI Technology and Industry Review, 15 February (2019). https://syncedreview.com/2019/02/15/ai-hasnt-found-its-isaac-newton-gary-marcus-on-deep-learning-defects-frenemy-yann-lecun/

  29. Romeiras, F., Grebogi, C., Ott, E.: Multifractal properties of snap-shot attractors of random maps. Phys. Rev. A 41, 784–799 (1990)

    Article  MathSciNet  Google Scholar 

  30. Rudin, C.: Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead. Nat. Mach. Intell. 1, 206–215 (2019)

    Article  Google Scholar 

  31. Silver, D., Schrittwieser, J., Simonyan, K., Antonoglou, I., Huang, A., Guez, A., Hubert, T., Baker, L.R., Lai, M., Bolton, A., Chen, Y., Lillicrap, T.P., Hui, F.F., Sifre, L., Driessche, G.V., Graepel, T., Hassabis, D.: Mastering the game of Go without human knowledge. Nature 550, 354–359 (2017)

    Article  Google Scholar 

  32. Stickgold, R.: Sleep-dependent memory consolidation. Nature 437, 1272–1278 (2005)

    Article  Google Scholar 

  33. Taigman, Y., Yang, M., Ranzato, M., Wolf, L.: DeepFace: closing the gap to human-level performance in face verification. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 1701–1708 (2014)

  34. Varshney, K.R., Alemzadeh, H.: On the safety of machine learning: cyber-physical systems, decision sciences, and data products. Big Data 5, 246–255 (2017)

    Article  Google Scholar 

  35. Velluti, R.: Interactions between sleep and sensory physiology, ain states: from single neurons to distributed networks for inferential representation. J. Sleep Res. 6, 61–77 (1997)

    Article  Google Scholar 

  36. Vincent, J.: The state of AI in 2019. The Verge, 28 January (2019)

  37. Vincent, J.: AI systems should be accountable, explainable, and unbiased, says EU. The Verge, 8 April (2019)

  38. Vishik, M.I.: Asymptotic Behaviour of Solutions of Evolutionary Equations. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

  39. Wexler, R.: When a computer program keeps you in jail: How computers are harming criminal justice. New York Times, 13 June (2017)

  40. Walter, W.: Ordinary Differential Equations. Springer, New York (1998)

    Book  Google Scholar 

  41. Zhu, J., Liapis, A., Risi, S., Bidarra, R., Youngblood, G.M.: Explainable AI for designers: a human-centered perspective on mixed-initiative co-creation. In: IEEE Conference on Computational Intelligence and Games (CIG), pp 1–8 (2018)

Download references

Acknowledgements

The visit of PEK to Loughborough University was supported by London Mathematical Society.

Author information

Authors and Affiliations

  1. Department of Mathematics, Loughborough University, Loughborough, LE11 3TU, UK

    N. B. Janson

  2. Mathematics Department, University of Tübingen, Tübingen, 72074, Germany

    P. E. Kloeden

Authors
  1. N. B. Janson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. E. Kloeden
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. E. Kloeden.

Additional information

Dedicated to the memory of Russell Johnson.

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

Janson, N.B., Kloeden, P.E. Mathematical Consistency and Long-Term Behaviour of a Dynamical System with a Self-Organising Vector Field. J Dyn Diff Equat 34, 63–78 (2022). https://doi.org/10.1007/s10884-020-09834-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-020-09834-7

Keywords

Mathematics Subject Classification

Search

Navigation