Cognitive Neurodynamics

, Volume 13, Issue 4, pp 393–407 | Cite as

Bifurcation analysis and diverse firing activities of a modified excitable neuron model

  • Argha Mondal
  • Ranjit Kumar UpadhyayEmail author
  • Jun Ma
  • Binesh Kumar Yadav
  • Sanjeev Kumar Sharma
  • Arnab Mondal
Research Article


Electrical activities of excitable cells produce diverse spiking-bursting patterns. The dynamics of the neuronal responses can be changed due to the variations of ionic concentrations between outside and inside the cell membrane. We investigate such type of spiking-bursting patterns under the effect of an electromagnetic induction on an excitable neuron model. The effect of electromagnetic induction across the membrane potential can be considered to analyze the collective behavior for signal processing. The paper addresses the issue of the electromagnetic flow on a modified Hindmarsh–Rose model (H–R) which preserves biophysical neurocomputational properties of a class of neuron models. The different types of firing activities such as square wave bursting, chattering, fast spiking, periodic spiking, mixed-mode oscillations etc. can be observed using different injected current stimulus. The improved version of the model includes more parameter sets and the multiple electrical activities are exhibited in different parameter regimes. We perform the bifurcation analysis analytically and numerically with respect to the key parameters which reveals the properties of the fast-slow system for neuronal responses. The firing activities can be suppressed/enhanced using the different external stimulus current and by allowing a noise induced current. To study the electrical activities of neural computation, the improved neuron model is suitable for further investigation.


Improved H–R model Electromagnetic induction effect Bifurcation Various neuronal responses Noise 



This work is supported by the Council of Scientific and Industrial Research (CSIR), Govt. of India under Grant No. 25(0277)/17/EMR-II) to the author (R. K. Upadhyay).


  1. Baltanas JP, Casado JM (2002) Noise-induced resonances in the Hindmarsh-Rose neuronal model. Phys Rev E 65(4):041915Google Scholar
  2. Bao B, Jiang T, Xu Q et al (2016) Coexisting infinitely many attractors in active band-pass filter-based memristive circuit. Nonlinear Dyn 86(3):1711–1723Google Scholar
  3. Bao BC, Liu Z, Xu JP (2010) Steady periodic memristor oscillator with transient chaotic behaviors. Electron Lett 46(3):237–238Google Scholar
  4. Barrio R, Shilnikov A (2011) Parameter-sweeping techniques for temporal dynamics of neuronal systems: case study of Hindmarsh-Rose model. J Math Neurosci 1(1):6Google Scholar
  5. Bekkers JM (2003) Synaptic transmission: functional autapses in the cortex. Curr Biol 13(11):R433–R435Google Scholar
  6. Bertram R, Rubin JE (2017) Multi-timescale systems and fast-slow analysis. Math Biosci 287:105–121Google Scholar
  7. Chik DTW, Wang Y, Wang ZD (2001) Stochastic resonance in a Hodgkin-Huxley neuron in the absence of external noise. Phys Rev E 64(2):021913Google Scholar
  8. Coombes S, Bressloff PC (2005) Bursting: the genesis of rhythm in the nervous system. World Scientific, SingaporeGoogle Scholar
  9. Ditlevsen S, Samson A (2013) Introduction to stochastic models in biology. In: Bachar M, Batzel J (eds) Stochastic biomathematical models with applications to neuronal modeling, vol 2058. Lecture notes in mathematics series (biosciences subseries). Springer, BerlinGoogle Scholar
  10. Faisal AA, Selen LPJ, Wolpert DM (2008) Noise in the nervous system. Nat Rev Neurosci 9(4):292Google Scholar
  11. Gu HG, Jia B, Chen GR (2013) Experimental evidence of a chaotic region in a neural pacemaker. Phys Lett A 377(9):718–720Google Scholar
  12. Gu H, Pan B (2015) A four-dimensional neuronal model to describe the complex nonlinear dynamics observed in the firing patterns of a sciatic nerve chronic constriction injury model. Nonlinear Dyn 81(4):2107–2126Google Scholar
  13. Gu H, Pan B, Chen G et al (2014) Biological experimental demonstration of bifurcations from bursting to spiking predicted by theoretical models. Nonlinear Dyn 78(1):391–407Google Scholar
  14. Herrmann CS, Klaus A (2004) Autapse turns neuron into oscillator. Int J Bifurc Chaos 14(2):623–633Google Scholar
  15. Herz AVM, Gollisch T, Machens CK et al (2006) Modeling single-neuron dynamics and computations: a balance of detail and abstraction. Science 314(5796):80–85Google Scholar
  16. Higham DJ (2001) An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev 43(3):525–546Google Scholar
  17. Hsü ID, Kazarinoff ND (1976) An applicable Hopf bifurcation formula and instability of small periodic solutions of the Field-Noyes model. J Math Anal Appl 55(1):61–89Google Scholar
  18. Hsü ID, Kazarinoff ND (1977) Existence and stability of periodic solutions of a third-order non-linear autonomous system simulating immune response in animals. Proc R Soc Edinburgh Sect A 77(1–2):163–175Google Scholar
  19. Izhikevich EM (2007) Dynamical systems in neuroscience. MIT Press, CambridgeGoogle Scholar
  20. Izhikevich EM (2004) Which model to use for cortical spiking neurons? IEEE Trans Neural Netw 15(5):1063–1070Google Scholar
  21. Izhikevich EM, Desai NS, Walcott EC et al (2003) Bursts as a unit of neural information: selective communication via resonance. Trends Neurosci 26(3):161–167Google Scholar
  22. Kuznetsov YA (1998) Elements of applied bifurcation theory, 2nd edn. Springer, New YorkGoogle Scholar
  23. Larter R, Speelman B, Worth RM (1999) A coupled ordinary differential lattice model for the simulation of epileptic seizures. Chaos 9(3):795–804Google Scholar
  24. Lee SG, Kim S (1999) Parameter dependence of stochastic resonance in the stochastic Hodgkin-Huxley neuron. Phys Rev E 60(1):826Google Scholar
  25. Li J, Tang J, Ma J et al (2016) Dynamic transition of neuronal firing induced by abnormal astrocytic glutamate oscillation. Sci Rep 6:32343Google Scholar
  26. Li Q, Zeng H, Li J (2015) Hyperchaos in a 4D memristive circuit with infinitely many stable equilibria. Nonlinear Dyn 79(4):2295–2308Google Scholar
  27. Lindner B, Garcia-Ojalvo J, Neiman A et al (2004) Effects of noise in excitable systems. Phys Rep 392(6):321–424Google Scholar
  28. Longtin A (2010) Stochastic dynamical systems. Scholarpedia 5(4):1619Google Scholar
  29. Lv M, Ma J (2016) Multiple modes of electrical activities in a new neuron model under electromagnetic radiation. Neurocomputing 205:375–381Google Scholar
  30. Lv M, Wang C, Ren G et al (2016) Model of electrical activity in a neuron under magnetic flow effect. Nonlinear Dyn 85(3):1479–1490Google Scholar
  31. Ma J, Tang J (2015) A review for dynamics of collective behaviours of network of neurons. Sci China Technol Sci 58(12):2038–2045Google Scholar
  32. Ma J, Wu F, Hayat T et al (2017a) Electromagnetic induction and radiation-induced abnormality of wave propagation in excitable media. Phys A 486:508–516Google Scholar
  33. Ma J, Wang Y, Wang C et al (2017b) Mode selection in electrical activities of myocardial cell exposed to electromagnetic radiation. Chaos Solitons Fractals 99:219–225Google Scholar
  34. Muthuswamy B (2010) Implementing memristor based chaotic circuits. Int J Bifurc Chaos 20(5):1335–1350Google Scholar
  35. Perc M (2006) Thoughts out of noise. Eur J Phys 27(2):451Google Scholar
  36. Perc M, Marhl M (2005) Amplification of information transfer in excitable systems that reside in a steady state near a bifurcation point to complex oscillatory behavior. Phys Rev E 71(2):026229Google Scholar
  37. Qin HX, Ma J, Jin WY et al (2014) Dynamics of electrical activities in neuron and neurons of network induced by autapses. Sci China Technol Sci 57(5):936–946Google Scholar
  38. Shilnikov A, Kolomiets M (2008) Methods of the qualitative theory for the Hindmarsh-Rose model: a case study-a tutorial. Int J Bifurc Chaos 18(8):2141–2168Google Scholar
  39. Song XL, Wang CN, Ma J et al (2015) Transition of electric activity of neurons induced by chemical and electric autapses. Sci China Technol Sci 58(6):1007–1014Google Scholar
  40. Storace M, Linaro D, de Lange E (2008) The Hindmarsh-Rose neuron model: bifurcation analysis and piecewise linear approximations. Chaos 18(3):033128Google Scholar
  41. Tang J, Liu TB, Ma J et al (2016) Effect of calcium channel noise in astrocytes on neuronal transmission. Commun Nonlinear Sci Numer Simul 32:262–272Google Scholar
  42. Tang J, Luo JM, Ma J (2013) Information transmission in a neuron-astrocyte coupled model. PLoS ONE 8(11):e80324Google Scholar
  43. Tsaneva-Atanasova K, Osinga HM, Rie T et al (2010) Full system bifurcation analysis of endocrine bursting models. J Theor Biol 264(4):1133–1146Google Scholar
  44. Upadhyay RK, Mondal A, Teka WW (2017) Mixed mode oscillations and synchronous activity in noise induced modified Morris-Lecar neural system. Int J Bifurc Chaos 27(5):1730019Google Scholar
  45. Wang C, Ma J (2018) A review and guidance for pattern selection in spatiotemporal system. Int J Mod Phys B 32(6):1830003Google Scholar
  46. Wang Y, Ma J, Xu Y et al (2017) The electrical activity of neurons subject to electromagnetic induction and Gaussian white noise. Int J Bifurc Chaos 27(2):1750030Google Scholar
  47. Wig GS, Schlaggar BL, Petersen SE (2011) Concepts and principles in the analysis of brain networks. Ann N Y Acad Sci 1224(1):126–146Google Scholar
  48. Wu F, Wang C, Xu Y et al (2016a) Model of electrical activity in cardiac tissue under electromagnetic induction. Sci Rep 6(1):28Google Scholar
  49. Wu H, Bao B, Liu Z et al (2016b) Chaotic and periodic bursting phenomena in a memristive Wien-bridge oscillator. Nonlinear Dyn 83(1–2):893–903Google Scholar
  50. Xiao-Bo W, Juan M, Ming-Hao Y (2008) Two different bifurcation scenarios in neural firing rhythms discovered in biological experiments by adjusting two parameters. Chin Phys Lett 25(8):2799Google Scholar
  51. Xu Y, Ying H, Jia Y et al (2017) Autaptic regulation of electrical activities in neuron under electromagnetic induction. Sci Rep 7:43452Google Scholar
  52. Zhan F, Liu S (2017) Response of electrical activity in an improved neuron model under electromagnetic radiation and noise. Front Comput Neurosci 11(107):1–12Google Scholar
  53. Zhou J, Wu Q, Xiang L (2012) Impulsive pinning complex dynamical networks and applications to firing neuronal synchronization. Nonlinear Dyn 69(3):1393–1403Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsIndian Institute of Technology (Indian School of Mines)DhanbadIndia
  2. 2.Computational Neuroscience CenterUniversity of WashingtonSeattleUSA
  3. 3.Department of PhysicsLanzhou University of TechnologyLanzhouPeople’s Republic of China

Personalised recommendations