Advertisement

Revealing Bistability in Neurological Disorder Models By Solving Parametric Polynomial Systems Geometrically

  • Changbo Chen
  • Wenyuan WuEmail author
Conference paper
  • 388 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11110)

Abstract

Understanding the mechanisms of the brain is a common theme for both computational neuroscience and artificial intelligence. Machine learning technique, like artificial neural network, has been benefiting from a better understanding of the neuronal network in human brains. In the study of neurons, mathematical modeling plays a vital role. In this paper, we analyze the important phenomenon of bistability in neurological disorders modeled by ordinary differential equations in virtue of our recently developed method for solving bi-parametric polynomial systems. Unlike the algebraic symbolic approach, our numeric method solves parametric systems geometrically. With respect to the classical bifurcation analysis approach, our method naturally has good initial points thanks to the critical point technique in real algebraic geometry.

Special heuristic strategies are proposed for addressing the multiscale problem of parameters and variables occurring in biological models, as well as taking into account the fact that the variables representing concentrations are non-negative. Comparing with its symbolic algebraic counterparts, one merit of this geometrical method is that it may compute smaller boundaries.

Keywords

Neurological Disease Models Numerical Algebraic Geometry Ordinary Differential Equations Fold Bifurcation Hopf Bifurcation Boundary 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work is partially supported by the projects NSFC (11471307, 11671377, 61572024), and the Key Research Program of Frontier Sciences of CAS (QYZDB-SSW-SYS026).

References

  1. 1.
    Bard Ermentrout, G., Terman, D.H.: Mathematical Foundations of Neuroscience. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-0-387-87708-2CrossRefzbMATHGoogle Scholar
  2. 2.
    Bower, J.M. (ed.): 20 Years of Computational Neuroscience. Springer Series in Computational Neuroscience, vol. 9. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-1-4614-1424-7CrossRefGoogle Scholar
  3. 3.
    Bradford, R.J., et al.: A case study on the parametric occurrence of multiple steady states. In: ISSAC 2017, pp. 45–52 (2017)Google Scholar
  4. 4.
    Chen, C., Wu, W.: A numerical method for analyzing the stability of bi-parametric biological systems. In: SYNASC 2016, pp. 91–98 (2016)Google Scholar
  5. 5.
    Chen, C., Wu, W.: A numerical method for computing border curves of bi-parametric real polynomial systems and applications. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2016. LNCS, vol. 9890, pp. 156–171. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-45641-6_11CrossRefGoogle Scholar
  6. 6.
    Chen, C., Moreno Maza, M.: Quantifier elimination by cylindrical algebraic decomposition based on regular chains. J. Symb. Comput. 7(5), 74–93 (2016)CrossRefGoogle Scholar
  7. 7.
    Chen, C., Maza, M.M.: Semi-algebraic description of the equilibria of dynamical systems. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2011. LNCS, vol. 6885, pp. 101–125. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-23568-9_9CrossRefGoogle Scholar
  8. 8.
    Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decompostion. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975).  https://doi.org/10.1007/3-540-07407-4_17CrossRefGoogle Scholar
  9. 9.
    De Caluwé, J., Dupont, G.: The progression towards Alzheimer’s disease described as a bistable switch arising from the positive loop between amyloids and \({C}a^{2+}\). J. Theor. Biol. 331, 12–18 (2013)CrossRefGoogle Scholar
  10. 10.
    Garfinkel, A., Shevtsov, J., Guo, Y.: Modeling Life: The Mathematics of Biological Systems. Springer, Heidelberg (2017).  https://doi.org/10.1007/978-3-319-59731-7CrossRefzbMATHGoogle Scholar
  11. 11.
    Gerstner, W., Kistler, W.M., Naud, R., Paninski, L.: Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition. Cambridge University Press, Cambridge (2014)CrossRefGoogle Scholar
  12. 12.
    Govaerts, W.: Numerical Methods for Bifurcations of Dynamical Equilibria. Society for Industrial and Applied Mathematics, Philadelphia (2000)CrossRefGoogle Scholar
  13. 13.
    Hassabis, D., Kumaran, D., Summerfield, C., Botvinick, M.: Neuroscience-inspired artificial intelligence. Neuron 95(2), 245–258 (2017)CrossRefGoogle Scholar
  14. 14.
    Hauenstein, J.D.: Numerically computing real points on algebraic sets. Acta Applicandae Mathematicae 125(1), 105–119 (2012)CrossRefGoogle Scholar
  15. 15.
    Hong, H., Tang, X., Xia, B.: Special algorithm for stability analysis of multistable biological regulatory systems. J. Symb. Comput. 70, 112–135 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Springer, Heidelberg (1995).  https://doi.org/10.1007/978-1-4757-2421-9CrossRefzbMATHGoogle Scholar
  17. 17.
    Lazard, D., Rouillier, F.: Solving parametric polynomial systems. J. Symb. Comput. 42(6), 636–667 (2007)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Li, T.Y.: Numerical solution of multivariate polynomial systems by homotopy continuation methods. Acta Numerica 6, 399–436 (1997)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Niu, W., Wang, D.: Algebraic approaches to stability analysis of biological systems. Math. Comput. Sci. 1(3), 507–539 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ogasawara, H., Kawato, M.: The protein kinase M\(\zeta \) network as a bistable switch to store neuronal memory. BMC Syst. Biol. 4(1), 181 (2010)CrossRefGoogle Scholar
  21. 21.
    Érdi, P., Bhattacharya, B.S., Cochran, A.L. (eds.): Computational Neurology and Psychiatry. SSB, vol. 6. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-49959-8CrossRefzbMATHGoogle Scholar
  22. 22.
    Rouillier, F., Roy, M.F., Safey El Din, M.: Finding at least one point in each connected component of a real algebraic set defined by a single equation. J. Complex. 16(4), 716–750 (2000)CrossRefGoogle Scholar
  23. 23.
    Sacktor, T.C.: Memory maintenance by PKM\(\zeta \) – an evolutionary perspective. Mol. Brain 5(1), 31 (2012)CrossRefGoogle Scholar
  24. 24.
    Schwartz, R.: Biological Modeling and Simulation. The MIT Press, Cambridge (2008)Google Scholar
  25. 25.
    Sommese, A., Wampler, C.: The Numerical Solution of Systems of Polynomials Arising in Engineering and Science. World Scientific Press, Singapore (2005)CrossRefGoogle Scholar
  26. 26.
    Wang, D.M., Xia, B.: Stability analysis of biological systems with real solution classification. In: Kauers, M. (ed.) ISSAC 2005, pp. 354–361 (2005)Google Scholar
  27. 27.
    Wu, W., Reid, G.: Finding points on real solution components and applications to differential polynomial systems. ISSAC 2013, 339–346 (2013)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Wu, W., Reid, G., Feng, Y.: Computing real witness points of positive dimensional polynomial systems. Theor. Comput. Sci. 681, 217–231 (2017)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Yang, L., Xia, B.: Real solution classifications of a class of parametric semi-algebraic systems. In: A3L 2005, pp. 281–289 (2005)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Chongqing Key Laboratory of Automated Reasoning and CognitionChongqing Institute of Green and Intelligent Technology, Chinese Academy of SciencesChongqingChina
  2. 2.University of Chinese Academy of SciencesBeijingChina

Personalised recommendations