Global Dynamics of a Piecewise Smooth System for Brain Lactate Metabolism

  • J.-P. FrançoiseEmail author
  • Hongjun Ji
  • Dongmei Xiao
  • Jiang Yu


In this article, we study a piecewise smooth dynamical system inspired by a previous reduced system modeling compartimentalized brain metabolism. The piecewise system allows the introduction of an autoregulation induced by a feedback of the extracellular or capillary Lactate concentrations on the Capillary Blood Flow. New dynamical phenomena are uncovered and we discuss existence and nature of two equilibrium points, attractive segment, boundary equilibrium and periodic orbits depending of the Capillary Blood Flow.


Piecewise smooth system Qualitative analysis Modeling 

Mathematics Subject Classification

34D20 37N25 



This work was supported by the National Natural Science Foundations of China (N\(^\circ \) 11431008 and 11771282) and the NSF of Shanghai (N\(^\circ \) 15ZR1423700).


  1. 1.
    Aubert, A., Costalat, R.: Interaction between astrocytes and neurons studied using a mathematical model of compartmentalized energy metabolism. J. Cereb. Blood Flow Metab. 25, 1476–1490 (2005)CrossRefGoogle Scholar
  2. 2.
    Aubert, A., Costalat, R., Magistretti, P.J., Pellerin, L.: Brain Lactate kinetics: modeling evidence for neuronal Lactate uptake upon activation. Proc. Natl. Acad. Sci. USA 102, 16448–16453 (2005)CrossRefGoogle Scholar
  3. 3.
    di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications. Springer, Berlin (2007)zbMATHGoogle Scholar
  4. 4.
    Costalat, R., Françoise, J.-P., Menuel, C., Lahutte, M., Vallée, J.-N., de Marco, G., Chiras, J., Guillevin, R.: Mathematical modeling of metabolism and hemodynamics. Acta Biotheor. 60, 99–107 (2012)CrossRefGoogle Scholar
  5. 5.
    Françoise, J.-P., Ji, H.: The Stability Analysis of Brain Lactate Kinetics (to appear)Google Scholar
  6. 6.
    Guillevin, C., Guillevin, R., Miranville, A., Perrillat-Mercerot, A.: Analysis of a mathematical model for brain lactate kinetics. Math. Biosci. Eng. (to appear)Google Scholar
  7. 7.
    Guillevin, R., Miranville, A., Perrillat-Mercerot, A.: On a reaction-diffusion system associated with brain lactate kinetics. Electron. J. Differ. Equ. 23, 1–16 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hu, Y.: Wilson GS A temporary local energy pool coupled to neuronal activity: fluctuations of extracellular lactate levels in rat brain monitored with rapid-response enzyme-based sensor. J. Neurochem. 69, 1484–1490 (1997)CrossRefGoogle Scholar
  9. 9.
    Keener, J., Sneyd, J.: Mathematical Physiology. Interdisciplinary Applied Mathematics, vol. 8, 2nd edn. Springer, New York (2009)CrossRefGoogle Scholar
  10. 10.
    Lahutte-Auboin, M.: Modélisation biomathématique du métabolisme énergétique cérébral : réduction de modèle et approche multi-échelle, application à l’aide à la décision pour la pathologie des gliomes, PhD thesis, Université Pierre et Marie Curie (2015)Google Scholar
  11. 11.
    Lahutte-Auboin, M., Costalat, R., Françoise, J.-P., Guillevin, R.: Dip and buffering in a fast-slow system associated to brain lactate, kinetics. arXiv:1308.0486v1
  12. 12.
    Lahutte-Auboin, M., Guillevin, R., Françoise, J.-P., Vallée, J.-N., Costalat, R.: On a minimal model for hemodynamics and metabolism of lactate: application to low grade glioma and therapeutic strategies. Acta Biotheor. 61, 79–89 (2013)CrossRefGoogle Scholar
  13. 13.
    Miranville, A.: A singular reaction–diffusion equation associated with brain lactate kinetics. Math. Methods Appl. Sci. 40(7), 2452465 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Miranville, A.: Mathematical analysis of a parabolic-elliptic model for brain lactate kinetics, Springer INdAM Series 22, Special volume dedicated to G. Gilardi. Springer (2017)Google Scholar
  15. 15.
    Smith, H.L.: On the asymptotic behavior of a class of deterministic models of cooperating species. SIAM J. Appl. Math. 46, 368375 (1986)MathSciNetGoogle Scholar
  16. 16.
    Smith, H.L.: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Mathematical Surveys and Monographs. American Mathematical Society, Providence (1995)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • J.-P. Françoise
    • 1
    • 2
    Email author
  • Hongjun Ji
    • 1
  • Dongmei Xiao
    • 2
  • Jiang Yu
    • 2
  1. 1.Laboratoire Jacques–Louis Lions, UMR 7598 CNRSUniversité P.-M. Curie, Paris 6ParisFrance
  2. 2.School of Mathematical SciencesShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China

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