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Symmetric bursting behaviors in the generalized FitzHugh–Nagumo model

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Abstract

In the current paper, we have investigated the generalized FitzHugh–Nagumo model. We have shown that symmetric bursting behaviors of different types could be observed in this model with an appropriate recovery term. A modified version of this system is used to construct bursting activities. Furthermore, we have shown some numerical examples of delayed Hopf bifurcation and canard phenomenon in the symmetric bursting of super-Hopf/homoclinic type near its super-Hopf and homoclinic bifurcations, respectively.

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References

  • Baer SM, Erneux T, Rinzel J (1989) The slow passage through a Hopf bifurcation: delay, memory effects, and resonance. SIAM J Appl Math 49:55–71

    Article  Google Scholar 

  • Benes GN, Barry AM, Kaper TJ, Kramer MA, Burke JM (2011) An elementary model of torus canards. Chaos 21:023131

    Article  PubMed  Google Scholar 

  • Booth V, Carr TW, Erneux T (1997) Near threshold bursting is delayed by a slow passage near a limit point. SIAM J Appl Math 57:1406–1420

    Article  Google Scholar 

  • Chay TR, Keizer J (1983) Minimal model for membrane oscillations in the pancreatic beta-cell. Biophys J 42:181–190

    Article  PubMed  CAS  Google Scholar 

  • Curtu R, Rubin J (2011) Interaction of canard and singular Hopf mechanisms in a neural model. SIAM J Appl Dyn Syst 10(4):1443–1479

    Article  Google Scholar 

  • Desroches M, Krauskopf B, Osinga HM (2008) Mixed-mode oscillations and slow manifolds in the self-coupled FitzHugh Nagumo system. Chaos 18(1):015107

    Article  PubMed  Google Scholar 

  • DeVries G (1998) Multiple bifurcations in a polynomial model of bursting oscillations. J Nonlinear Sci 8:281–316

    Article  Google Scholar 

  • Dolnik M, Epstein IR (1993) A coupled chemical burster: the chlorine dioxide-iodide reaction in two flow reactors. J Chem Phys 98:1149–1155

    Article  CAS  Google Scholar 

  • Ermentrout B (2002) Simulating, analyzing, and animating dynamical systems (A guide to XPPAUT for researchers and students). Siam Society for Industrial and Applied Mathematics, Philadelphia

    Book  Google Scholar 

  • Ermentrout GB, Terman D (2010) Mathematical foundations of neuroscience. Springer, New York

    Book  Google Scholar 

  • FitzHugh R (1969) Mathematical models for excitation and propagation in nerve. In: Schawn HP (ed) Biological engineering. McGraw-Hill, New York

    Google Scholar 

  • Georgiev NV (2003) Identifying generalized FitzHugh–Nagumo equation from a numerical solution of Hodgkin–Huxley model. J Appl Math 8:397–407

    Article  Google Scholar 

  • Gong PL, Xu JX (2001) Global dynamics and stochastic resonance of the forced FitzHugh–Nagumo neuron model. Phys Rev E 63:031906

    Article  CAS  Google Scholar 

  • Govaerts W, Dhooge A (2002) Bifurcation, bursting and spike generation in a neural model. Int J Bifurcat Chaos Appl Sci Eng 12(8):1731–1741

    Article  Google Scholar 

  • Guckenheimer J, Kuehn C (2009) Computing slow manifolds of saddle type. SIAM J Appl Dyn Syst 8(3):854–879

    Article  Google Scholar 

  • Han X, Jiang B, Bi Q (2010) 3-torus, quasi-periodic bursting, symmetric subHopf/fold-cycle bursting, subHopf/fold-cycle bursting and their relation. Nonlinear Dyn 61:667–676

    Article  Google Scholar 

  • Hindmarsh JL, Rose RM (1984) A model of neuronal bursting using three coupled first order differential equations. Proc R Soc Lond Biol 221:87–102

    Article  PubMed  CAS  Google Scholar 

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

    CAS  Google Scholar 

  • Izhikevich EM (2000) Neural excitability, spiking, and bursting. Int J Bifurcat Chaos 10:1171–1266

    Article  Google Scholar 

  • Izhikevich EM (2004) Which model to use for cortical spiking neurons? IEEE Trans Neural Netw 15:1063–1070

    Article  PubMed  Google Scholar 

  • Izhikevich EM (2007) Dynamical systems in neuroscience: the geometry of excitability and bursting. MIT Press, Cambridge

    Google Scholar 

  • Kinney WM (2000) An application of Conley index techniques to a model of bursting in excitable membranes. J Diff Equ 162:451–472

    Google Scholar 

  • Kinney WM (2008) Applying the Conley index to fast-slow systems with one slow variable and an attractor. J Math 38(4):1177

    Google Scholar 

  • Kramer M, Traub R, Kopell N (2008) New dynamics in cerebellar purkinje cells: torus canards. Phys Rev Lett 101:68103

    Article  Google Scholar 

  • Krupa M, Szmolyan P (2001) Extending geometric singular perturbation theory to nonhyperbolic points–fold and canard points in two dimensions. SIAM J Math Anal 33:286–314

    Article  Google Scholar 

  • Lee E, Terman D (1999) Uniqueness and stability of periodic bursting solutions. J Diff Equ 158:48–78

    Article  Google Scholar 

  • Linaro D, Champneys A, Desroches M, Storace M (2012) Codimension-two homoclinic bifurcations underlying spike adding in the Hindmarsh–Rose burster. SIAM J Appl Dyn Syst 11(3):939– 962

    Article  Google Scholar 

  • Meucci R, DiGarbo A, Allaria E, Arecchi FT (2002) Autonomous bursting in a homoclinic system. Phys Rev Lett 88:144101

    Article  PubMed  CAS  Google Scholar 

  • Morris C, Lecar H (1981) Voltage oscillations in the barnacle giant muscle fiber. Biophys J 35:193–213

    Article  PubMed  CAS  Google Scholar 

  • Neishtadt AI (1995) On calculation of stability loss delay time for dynamical bifurcations. In: Jacobnitzer D (ed) XIth international congress of mathematical physics. International Press, Boston

  • Neishtadt AI (1987) Persistence of stability loss for dynamical bifurcations I. Diff Equ 23:1385–1391

    Google Scholar 

  • Neishtadt AI (1988) Persistence of stability loss for dynamical bifurcations II. Diff Equ 24:171–176

    Google Scholar 

  • Nowacki J, Osinga HM, Tsaneva-Atanasova K (2012) Dynamical systems analysis of spike-adding mechanisms in transient bursts. J Math Neurosci 2:7

    Article  PubMed  Google Scholar 

  • Organ L, Kiss IZ, Hudson JL (2003) Bursting oscillations during metal electrodissolution: experiments and model. J Phys Chem B 107:6648–6659

    Article  CAS  Google Scholar 

  • Pernarowski M, Miura RM, Kevorkian J (1992) Perturbation techniques for models of bursting electrical activity in pancreatic beta-cells. SIAM J Appl Math 52:1627–1650

    Article  Google Scholar 

  • Plant RE (1981) Bifurcation and resonance in a model for bursting nerve cells. J Math Biol 11:15–32

    Article  PubMed  CAS  Google Scholar 

  • Reinker S (2004) Stochastic resonance in thalamic neurons and resonant neuron models. Dissertation, University of British Columbia

  • Rinzel J (1987) Mathematical topics in population biology, morphogenesis and neurosciences. In: Teramoto E, Yamaguti M (eds) Lecture notes in biomathematics, vol 71. Springer, Berlin, pp 267–281

  • Shishkova MA (1973) Examination of one system of differential equations with a small parameter in highest derivatives. Dokl Akad Nauk SSSR 209:576–579

    Google Scholar 

  • Szmolyan P, Wechselberger M (2001) Canards in R3. J Diff Equ 177:419–453

    Article  Google Scholar 

  • Terman D (1991) Chaotic spikes arising from a model of bursting in excitable membranes. SIAM J Appl Math 51(5):1418–1450

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Tsaneva-Atanasova K, Osinga HM, Rieß T, Sherman A (2010) Full system bifurcation analysis of endocrine bursting models. J Theor Biol 264:1133–1146

    Article  PubMed  Google Scholar 

  • Xiujing H, Bo J, Qinsheng B (2009) Symmetric bursting of focus-focus type in the controlled Lorenz system with two time scales. Phys Lett A 373:3643–3649

    Google Scholar 

  • Ying J, Qin-Sheng B (2010) Symmetric bursting behaviour in non-smooth Chua’s circuit. Chin Phys B 19(8):080510

    Google Scholar 

  • Zaks MA, Sailer X, Schimansky-Geier L (2005) Noise induced complexity: from sub-threshold oscillations to spiking in coupled excitable systems. Chaos 15:026117

    Google Scholar 

  • Zhao G, Hou Z, Xin H (2005) Canard explosion and coherent biresonance in the rate oscillation of CO oxidation on platinum surface. J Phys Chem A 109:8515–8519

    Article  PubMed  CAS  Google Scholar 

Download references

Acknowledgments

The authors thank the reviewers for their helpful comments. This research was in part supported by a grant from IPM (No. 9092128).

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Authors and Affiliations

  1. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran

    A. H. Abbasian & H. Fallah

  2. Department of Mathematics and Computer Sciences, Amirkabir University of Technology, P.O. Box: 15875-4413, Tehran, Iran

    H. Fallah

  3. Department of Mathematical Sciences, Sharif University of Technology, P. O. Box: 11155-9415, Tehran, Iran

    M. R. Razvan

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  1. A. H. Abbasian
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  2. H. Fallah
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  3. M. R. Razvan
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Correspondence to H. Fallah.

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Abbasian, A.H., Fallah, H. & Razvan, M.R. Symmetric bursting behaviors in the generalized FitzHugh–Nagumo model. Biol Cybern 107, 465–476 (2013). https://doi.org/10.1007/s00422-013-0559-1

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  • DOI: https://doi.org/10.1007/s00422-013-0559-1

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