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On the Darboux integrability of the Hindmarsh–Rose burster

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Abstract

We study the Hindmarsh–Rose burster which can be described by the differential system

$$\dot x = y - {x^3} + b{x^2} + I - z,\dot y = 1 - 5{x^2} - y,\dot z = \mu \left( {s\left( {x - {x_0}} \right) - z} \right),$$

where b, I, μ, s, x0 are parameters. We characterize all its invariant algebraic surfaces and all its exponential factors for all values of the parameters. We also characterize its Darboux integrability in function of the parameters. These characterizations allow to study the global dynamics of the system when such invariant algebraic surfaces exist.

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References

  1. Belykh, V. N., Belykh, I. V., Colding-Jorgensen, M., et al.: Homoclinic bifurcations leading to the emergence of bursting oscillations in cell models, The European Physical Journal E: Soft Matter and Biological Physics, 3, 205–219 (2000)

    Google Scholar 

  2. Christopher, C., Llibre, J., Pereira, J. V.: Multiplicity of invariant algebraic curves in polynomial vector fields. Pacific J. Math., 229, 63–117 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Darboux, G.: Mémoire sur les équations différentielles du premier ordre et du premier degreé (Mélanges). Bull. Sci. Math. 2éme série, 2, 60–96, 123–144, 151–200 (1878)

    MATH  Google Scholar 

  4. Darboux, G.: De l’emploi des solutions particuli`eres algébriques dans l’intégration des syst`emes d’équations différentielles algébriques. C. R. Math. Acad. Sci. Paris, 86, 1012–1014 (1878)

    MATH  Google Scholar 

  5. Dumortier, F., Llibre, J., Artés, J. C.: Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, Berlin, 2006

    MATH  Google Scholar 

  6. González-Miranda, J. M.: Observation of a continuous interior crisis in the Hindmarsh–Rose neuron model. Chaos, 13, 845–852 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. González-Miranda, J. M.: Complex bifurcation structures in the Hindmarsh–Rose neuron model. Int. J. Bifurcation and Chaos, 17, 3071–3083 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hindmarsh, J. L., Rose, R. M.: A model of neuronal bursting using three coupled first order differential equations, Proceedings of the Royal Society of London. Series B, Biological Sciences 221, 87–102 (1984) 246, 541–551 (2009)

    Article  Google Scholar 

  9. Hodgkin, A. L., Huxley, A. F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol.-London, 117, 500–544 (1952)

    Article  Google Scholar 

  10. Innocenti, G., Genesio, R.: On the dynamics of chaotic spiking-bursting transition in the Hindmarsh–Rose neuron. Chaos, 19, 023124–8 (2009)

    Article  MathSciNet  Google Scholar 

  11. Innocenti, G., Morelli, A., Genesio, R., et al.: Dynamical phases of the Hindmarsh–Rose neuronal model: Studies of the transition from bursting to spiking chaos. Chaos, 17, 043128–11 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Linaro, D., Champneys, A., Desroches, M., et al.: Codimension-two homoclinic bifurcations underlying spike adding in the Hindmarsh–Rose burster. SIAM J. Appl. Dyn. Syst., 11, 939–962 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Llibre, J., Valls, C.: On the integrability of the Bianchi IX system. J. Math. Phys., 46, 072901–13 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Llibre, J., Valls, C.: Liouvillian and analytic integrability of the quadratic vector fields having an invariant ellipse. Acta Math. Sinica, Engl. Ser., 30, 453–466 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Llibre, J., Zhang, X.: Darboux theory of integrability in Cn taking into account the multiplicity. J. Differential Equations, 246, 541–551 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Llibre, J., Zhang, X.: Darboux theory of integrability for polynomial vector fields in Rn taking into account the multiplicity at infinity. Bull. Sci. Math., 133, 765–778 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nguyen, L. H., Hong, K. S.: Adaptive synchronization of two coupled chaotic Hindmarsh–Rose neurons by controlling the membrane potential of a slave neuron. Appl. Math. Model., 37, 2460–2468 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Prelle, M. J., Singer, M. F.: Elementary first integrals of differential equations. Trans. Amer. Math. Soc., 279, 215–229 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  19. Silnikov, A., Kolomiets, M.: Methods of the qualitative theory for the Hindmarsh–Rose model: a case study. a tutorial. Int. J. Bifurcation and Chaos, 18, 2141–2168 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Singer, M. F.: Liouvillian first integrals of differential equations. Trans. Amer. Math. Soc., 333, 673–688 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. Storace, M., Linaro, D., de Lange, E.: The Hindmarsh–Rose neuron model: bifurcation analysis and piecewise-linear approximations. Chaos, 18, 033128–10 (2008)

    Article  MathSciNet  Google Scholar 

  22. Terman, D.: Chaotic spikes arising from a model of bursting in excitable membranes. SIAM J. Appl. Math., 51, 1418–1450 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  23. Terman, D.: The transition from bursting to continuous spiking in excitable membrane models. J. Nonlinear Science, 2, 135–182 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang, X. J.: Genesis of bursting oscillations in the Hindmarsh–Rose model and homoclinicity to a chaotic saddle. Physica D, 62, 263–274 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wu, Q., Zhou, J., Xiang, L., et al.: Impulsive control and synchronization of chaotic Hindmarsh–Rose models for neuronal activity. Chaos Solitons Fractals, 41, 2706–2715 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wu, Y., Xu, J., He, D., et al.: Generalized synchronization induced by noise and parameter mismatching in Hindmarsh–Rose neurons. Chaos Solitons Fractals, 23, 1605–1611 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yu, H., Peng, J.: Chaotic synchronization and control in nonlinear-coupled Hindmarsh–Rose neural systems. Chaos Solitons Fractals, 29, 342–348 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhang, X.: Liouvillian integrability of polynomial differential systems. Trans. Amer. Math. Soc., 368, 607–620 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Catalonia, Spain

    Jaume Llibre

  2. Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    Clàudia Valls

Authors
  1. Jaume Llibre
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  2. Clàudia Valls
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Correspondence to Jaume Llibre.

Additional information

The first author is partially supported by a MINECO-FEDER (Grant No. MTM2016-77278-P) a MINECO (Grant No. MTM2013-40998-P) and an AGAUR (Grant No. 2014SGR-568); the second author is partially supported by FCT/Portugal through UID/MAT/04459/2013

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Llibre, J., Valls, C. On the Darboux integrability of the Hindmarsh–Rose burster. Acta. Math. Sin.-English Ser. 34, 947–958 (2018). https://doi.org/10.1007/s10114-017-5661-1

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  • DOI: https://doi.org/10.1007/s10114-017-5661-1

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