On the Chaotic Pole of Attraction with Nonlocal and Nonsingular Operators in Neurobiology

  • Emile F. Doungmo GoufoEmail author
  • Abdon Atangana
  • Melusi Khumalo
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 194)


Until the neurologists J.L. Hindmarsh and R.M. Rose improved the Hodgkin–Huxley model to provide a better understanding on the diversity of neural response, features like pole of attraction unfolding complex bifurcation for the membrane potential was still a mystery. This work explores the possible existence of chaotic poles of attraction in the dynamics of Hindmarsh–Rose neurons with external current input. Combining with fractional differentiation, the model is generalized with introduction of an additional parameter, the non-integer order of the derivative \(\sigma \) and solved numerically thanks to the Haar Wavelets. Numerical simulations of the membrane potential dynamic show that in the standard case the control parameter is \(\sigma =1,\) the nerve cell’s behavior seems irregular with a pole of attraction generating a limit cycle. This irregularity accentuates as \(\sigma \) decreases (\(\sigma =0.8\) and \(\sigma =0.5\)) with the pole of attraction becoming chaotic.


Fractional calculus Atangana–Baleanu fractional derivative Hindmarsh Rose neuron 



The work of EF Doungmo Goufo was partially supported by the grant No: 105932 from the National Research Foundation (NRF) of South Africa.


  1. 1.
    Department of Biochemistry and Molecular Biophysics, Jessell, T., Siegelbaum, S., Hudspeth, A.J.: Principles of Neural Science. Kandel, E.R., Schwartz, J.H., Jessell, T.M. (eds.), vol. 4, pp. 1227–1246. McGraw-hill, New York (2000)Google Scholar
  2. 2.
    Logothetis, N.K., Pauls, J., Augath, M., Trinath, T., Oeltermann, A.: Neurophysiological investigation of the basis of the fMRI signal. Nature 412(6843), 150–157 (2001)Google Scholar
  3. 3.
    Ren, H.P., Bai, C., Baptista, M.S., Grebogi, C.: Weak connections form an infinite number of patterns in the brain. Sci. Rep. 7, 1–12 (2017)Google Scholar
  4. 4.
    Rizzolatti, G., Craighero, L.: The mirror-neuron system. Annu. Rev. Neurosci. 27, 169–192 (2004)Google Scholar
  5. 5.
    Thompson, R.F., Spencer, W.A.: Habituation: a model phenomenon for the study of neuronal substrates of behavior. Psychol. Rev. 73(1), 1–16 (1966)Google Scholar
  6. 6.
    Misiaszek, J.E.: The H-reflex as a tool in neurophysiology: its limitations and uses in understanding nervous system function. Muscle Nerve 28(2), 144–160 (2003)Google Scholar
  7. 7.
    Hodgkin, A., Huxley, A.: A quantitative description of membrane current and its application to conduction and excitation in nerve. Bull. Math. Biol. 52(1–2), 25–71 (1990)Google Scholar
  8. 8.
    Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117(4), 500–544 (1952)Google Scholar
  9. 9.
    Hindmarsh, J., Rose, R.: A model of the nerve impulse using two first-order differential equations. Nature 296(5853), 162–164 (1982)Google Scholar
  10. 10.
    Yamada, Y., Kashimori, Y.: Neural mechanism of dynamic responses of neurons in inferior temporal cortex in face perception. Cogn. Neurodynamics 7(1), 23–38 (2013)Google Scholar
  11. 11.
    Barrio, R., Angeles Martínez, M., Serrano, S., Shilnikov, A.: Macro-and micro-chaotic structures in the Hindmarsh-Rose model of bursting neurons. Chaos: Interdiscip. J. Nonlinear Sci. 24(2), 1–11 (2014)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hindmarsh, J.L., Rose, R.: A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond. B: Biol. Sci. 221(1222), 87–102 (1984)Google Scholar
  13. 13.
    Jun, D., Guang-jun, Z., Yong, X., Hong, Y., Jue, W.: Dynamic behavior analysis of fractional-order Hindmarsh-Rose neuronal model. Cogn. Neurodynamics 8(2), 167–175 (2014)Google Scholar
  14. 14.
    Che, Y.-Q., Wang, J., Tsang, K.-M., Chan, W.-L.: Unidirectional synchronization for Hindmarsh-Rose neurons via robust adaptive sliding mode control. Nonlinear Anal. R. World Appl. 11(2), 1096–1104 (2010)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Ostojic, S., Brunel, N., Hakim, V.: Synchronization properties of networks of electrically coupled neurons in the presence of noise and heterogeneities. J. Comput. Neurosci. 26(3), 1–24 (2009)MathSciNetGoogle Scholar
  16. 16.
    Storace, M., Linaro, D., de Lange, E.: The Hindmarsh–Rose neuron model: bifurcation analysis and piecewise-linear approximations. Chaos: Interdiscip. J. Nonlinear Sci. 18(3), 1–10 (2008)MathSciNetGoogle Scholar
  17. 17.
    Caputo, M.: Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. Int. 13(5), 529–539 (1967)Google Scholar
  18. 18.
    Doungmo Goufo, E.F., Atangana, A.: Analytical and numerical schemes for a derivative with filtering property and no singular kernel with applications to diffusion. Eur. Phys. J. Plus 131(8), 1–26 (2016)Google Scholar
  19. 19.
    Doungmo Goufo, E.F.: Chaotic processes using the two-parameter derivative with non-singular and nonlocal kernel: basic theory and applications. Chaos: Interdiscip. J. Nonlinear Sci. 26(8), 1–21 (2016)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Atangana, A., Baleanu, D.: New fractional derivatives with non-local and non-singular kernel. Therm. Sci. 20(2), 763–769 (2016)Google Scholar
  21. 21.
    Gómez-Aguilar, J.F.: Analytical and numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations. Phys. A: Stat. Mech. Its Appl. 494, 52–75 (2018)MathSciNetGoogle Scholar
  22. 22.
    Atangana, A., Gómez-Aguilar, J.F.: Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena. Eur. Phys. J. Plus 133, 1–22 (2018)Google Scholar
  23. 23.
    Atangana, A.: Non validity of index law in fractional calculus: a fractional differential operator with markovian and non-markovian properties. Phys. A: Stat. Mech. Its Appl. 505, 688–706 (2018)MathSciNetGoogle Scholar
  24. 24.
    Atangana, A., Nieto, J.: Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel. Adv. Mech. Eng. 7(10), 1–7 (2015)Google Scholar
  25. 25.
    Morales-Delgado, V.F., Taneco-Hernández, M.A., Gómez-Aguilar, J.F.: On the solutions of fractional order of evolution equations. Eur. Phys. J. Plus 132(1), 1–17 (2017)Google Scholar
  26. 26.
    Gómez-Aguilar, J.F., Escobar-Jiménez, R.F., López-López, M.G., Alvarado-Martínez, V.M.: Atangana-Baleanu fractional derivative applied to electromagnetic waves in dielectric media. J. Electromagn. Waves Appl. 30(15), 1937–1952 (2016)Google Scholar
  27. 27.
    Brockmann, D., Hufnagel, L.: Front propagation in reaction-superdiffusion dynamics: taming Lévy flights with fluctuations. Phys. Rev. Lett. 98(17), 178–301 (2007)Google Scholar
  28. 28.
    Doungmo Goufo, E.F.: Speeding up chaos and limit cycles in evolutionary language and learning processes. Math. Methods Appl. Sci. 40(8), 3055–3065 (2017)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Doungmo Goufo, E.F.: Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Bergers equation. Math. Model. Anal. 21(2), 188–198 (2016)MathSciNetGoogle Scholar
  30. 30.
    Das, S.: Convergence of Riemann-Liouville and caputo derivative definitions for practical solution of fractional order differential equation. Int. J. Appl. Math. Stat. 23(D11), 64–74 (2011)MathSciNetGoogle Scholar
  31. 31.
    Doungmo Goufo, E.F.: Solvability of chaotic fractional systems with 3D four-scroll attractors. Chaos Solitons Fractals 104, 443–451 (2017)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Hilfer, R.: Application of Fractional Calculus in Physics. World Scientific, Singapore (1999)Google Scholar
  33. 33.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science Limited (2006)Google Scholar
  34. 34.
    Coronel-Escamilla, A., Torres, F., Gómez-Aguilar, J.F., Escobar-Jiménez, R.F., Guerrero-Ramírez, G.V.: On the trajectory tracking control for an SCARA robot manipulator in a fractional model driven by induction motors with PSO tuning. Multibody Syst. Dyn. 43(3), 257–277 (2018)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Yépez-Martínez, H., Gómez-Aguilar, J.F.: A new modified definition of Caputo-Fabrizio fractional-order derivative and their applications to the multi step homotopy analysis method (MHAM). J. Comput. Appl. Math. 346, 247–260 (2018)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Gómez-Aguilar, J.F., Yépez-Martínez, H., Escobar-Jiménez, R.F., Astorga-Zaragoza, C.M., Reyes-Reyes, J.: Analytical and numerical solutions of electrical circuits described by fractional derivatives. Appl. Math. Model. 40(21–22), 9079–9094 (2016)MathSciNetGoogle Scholar
  37. 37.
    Rosales, J., Guía, M., Gómez, F., Aguilar, F., Martínez, J.: Two dimensional fractional projectile motion in a resisting medium. Open Phys. 12(7), 517–520 (2014)Google Scholar
  38. 38.
    Gómez-Aguilar, J.F., Yépez-Martínez, H., Escobar-Jiménez, R.F., Astorga-Zaragoza, C.M., Morales-Mendoza, L.J., González-Lee, M.: Universal character of the fractional space-time electromagnetic waves in dielectric media. J. Electromagn. Waves Appl. 29(6), 727–740 (2015)Google Scholar
  39. 39.
    Coronel-Escamilla, A., Gómez-Aguilar, J.F., Alvarado-Méndez, E., Guerrero-Ramírez, G.V., Escobar-Jiménez, R.F.: Fractional dynamics of charged particles in magnetic fields. Int. J. Mod. Phys. C 27(08), 1–16 (2016)MathSciNetGoogle Scholar
  40. 40.
    Babolian, E., Shahsavaran, A.: Numerical solution of nonlinear fredholm integral equations of the second kind using haar wavelets. J. Comput. Appl. Math. 225(1), 87–95 (2009)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Chen, Y., Yi, M., Yu, C.: Error analysis for numerical solution of fractional differential equation by Haar wavelets method. J. Comput. Sci. 3(5), 367–373 (2012)Google Scholar
  42. 42.
    Lepik, Ü., Hein, H.: Haar Wavelets: With Applications. Springer Science & Business Media (2014)Google Scholar
  43. 43.
    Tonelli, L.: Sullintegrazione per parti. Rend. Acc. Naz. Lincei 5(18), 246–253 (1909)zbMATHGoogle Scholar
  44. 44.
    Fubini, G.: Opere scelte II. Cremonese, Roma (1958)zbMATHGoogle Scholar
  45. 45.
    Morales-Delgado, V.F., Gómez-Aguilar, J.F., Kumar, S., Taneco-Hernández, M.A.: Analytical solutions of the Keller-Segel chemotaxis model involving fractional operators without singular kernel. Eur. Phys. J. Plus 133(5), 1–26 (2018)Google Scholar
  46. 46.
    Coronel-Escamilla, A., Gómez-Aguilar, J.F., Baleanu, D., Córdova-Fraga, T., Escobar-Jiménez, R.F., Olivares-Peregrino, V.H., Qurashi, MMAl: Bateman-Feshbach tikochinsky and Caldirola–Kanai oscillators with new fractional differentiation. Entropy 19(2), 1–21 (2017)Google Scholar
  47. 47.
    Cuahutenango-Barro, B., Taneco-Hernández, M.A., Gómez-Aguilar, J.F.: On the solutions of fractional-time wave equation with memory effect involving operators with regular kernel. Chaos Solitons Fractals 115, 283–299 (2018)MathSciNetGoogle Scholar
  48. 48.
    Doungmo Goufo, E.F., Nieto, J.J.: Attractors for fractional differential problems of transition to turbulent flows. J. Comput. Appl. Math. 339, 329–342 (2017)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Emile F. Doungmo Goufo
    • 1
    Email author
  • Abdon Atangana
    • 2
  • Melusi Khumalo
    • 1
  1. 1.Department of Mathematical Sciences University of South AfricaFloridaSouth Africa
  2. 2.Institute for Groundwater Studies, University of the Free StateBloemfonteinSouth Africa

Personalised recommendations