Codimension-two homoclinic bifurcations underlying spike adding in the Hindmarsh-Rose Burster

SIAM Journal on Applied Dynamical Systems

Volumn 11, Issue 3, 2012, Pages 939-962

  Linaro, Daniele ^a,b   Champneys, Alan ^c   Desroches, Mathieu ^d   Storace, Marco ^a  

^a UNIVERSITY OF GENOA   (Italy)

^b UNIVERSITY OF ANTWERP   (Belgium)

^c UNIVERSITY OF BRISTOL   (United Kingdom)

^d INRIA ROCQUENCOURT   (France)

Author keywords

Bifurcation analysis; Codimension two homoclinic degeneracies; Hindmarsh Rose model; Period adding

Indexed keywords

ACTION POTENTIALS; BIFURCATION ANALYSIS; CODIMENSION-TWO; EXCITABLE SYSTEMS; GLOBAL BIFURCATIONS; HINDMARSH ROSE; HINDMARSH-ROSE MODEL; HOMOCLINIC; HOMOCLINIC BIFURCATIONS; LOBE-SHAPED; LOCAL ANALYSIS; PARAMETER PLANE; PARAMETER REGIMES; PATH FOLLOWING; PERIOD ADDING; SINGULAR LIMITS; SINGULAR PERTURBATIONS; SLOW MANIFOLDS; TURNING POINTS; UNSTABLE MANIFOLD;

COMPUTER SIMULATION; DYNAMICAL SYSTEMS;

BIFURCATION (MATHEMATICS);

EID: 84866434344     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/110848931     Document Type: Article

References (41)

1. L. A. Belyakov, A case of the generation of a periodic motion with homoclinic curves, Mat. Zametki, 15 (1974), pp. 336-341.
2. L. A. Belyakov, The bifurcation set in a system with a homoclinic saddle curve, Mat. Zametki, 28 (1980), pp. 910-916.
3. L. A. Belyakov, Bifurcation of systems with homoclinic curve of a saddle-focus with saddle quantity zero, Math. Notes, 36 (1984), pp. 838-843.
4. V. N. Belykh, I. V. Belykh, M. Colding-Jorgensen, and E. Mosekilde, Homoclinic bifurcations leading to the emergence of bursting oscillations in cell models, Eur. Phys. J. E, 3 (2000), pp. 205-219.
5. A. R. Champneys, V. Kirk, E. Knobloch, B. E. Oldeman, and J. D. M. Rademacher, Unfolding a tangent equilibrium-to-periodic heteroclinic cycle, SIAM J. Appl. Dyn. Syst., 8 (2009), pp. 1261-1304.
6. A. R. Champneys and Yu. A. Kuznetsov, Numerical detection and continuation of codimension-2 homoclinic bifurcations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 4 (1994), pp. 785-822.
7. A. R. Champneys, V. Kirk, E. Knobloch, B. E. Oldeman, and J. Sneyd, When Shil'nikov meets Hopf in excitable systems, SIAM J. Appl. Dyn. Syst., 6 (2007), pp. 663-693.
8. P. Channell, G. Cymbalyuk, and A. Shilnikov, Origin of bursting through homoclinic spike adding in a neuron model, Phys. Rev. Lett., 98 (2007), 134101.
9. O. De Feo, G. M. Maggio, and M. P. Kennedy, The Colpitts oscillator: Families of periodic solutions and their bifurcations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), pp. 935-958.
10. E. de Lange and M. Hasler, Predicting single spikes and spike patterns with the Hindmarsh-Rose model, Biol. Cybernet., 99 (2008), pp. 349-360.
11. M. Desroches and M. R. Jeffrey, Canards and curvature: The smallness of ? in slow-fast dynamics, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), pp. 2404-2421.
12. M. Desroches, B. Krauskopf, and H. M. Osinga, Numerical continuation of canard orbits in slow-fast dynamical systems, Nonlinearity, 23 (2010), pp. 739-765.
13. E. J. Doedel, B. Krauskopf, and H. M. Osinga, Global bifurcations of the Lorenz manifold, Nonlinearity, 19 (2006), pp. 2947-2972.
14. E. J. Doedel and B. E. Oldeman, AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations, Concordia University, Montreal, Quebec, Canada, 2009.
15. N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), pp. 53-98.
16. J. M. Gonźalez-Miranda, Complex bifurcation structures in the Hindmarsh-Rose neuron model, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), pp. 3071-3083.
17. J. Guckenheimer and C. Kuehn, Computing slow manifolds of saddle type, SIAM J. Appl. Dyn. Syst., 8 (2009), pp. 854-879.
18. J. L. Hindmarsh and R. M. Rose, A model of the nerve impulse using two first-order differential equations, Nature, 296 (1982), pp. 162-164.
19. J. L. Hindmarsh and R. M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. Roy. Soc. London Ser. B Biol. Sci., 221 (1984), pp. 87-102.
20. A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. London, 117 (1952), pp. 500-544.
21. A. J. Homburg and B. Krauskopf, Resonant homoclinic flip bifurcations, J. Dynam. Differential Equations, 12 (2000), pp. 807-850.
22. A. J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, in Handbook of Dynamical Systems III, F. Takens, H. Broer, and B. Hasselblatt, eds., Elsevier, New York, 2010, pp. 379-524.
23. G. Innocenti and R. Genesio, On the dynamics of chaotic spiking-bursting transition in the Hindmarsh-Rose neuron, Chaos, 19 (2009), 023124.
24. E. M. Izhikevich, Neural excitability, spiking and bursting, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), pp. 1171-1266.
25. B. Krauskopf, H. M. Osinga, and J. Gaĺan-Vioque, Numerical Continuation Methods for Dynamical Systems. Path Following and Boundary Value Problems, Springer-Verlag, New York, 2007.
26. Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed., Springer-Verlag, New York, 2004.
27. Y. A. Kuznetsov, O. De Feo, and S. Rinaldi, Belyakov homoclinic bifurcations in a tritrophic food chain model, SIAM J. Appl. Math., 62 (2001), pp. 462-487.
28. D. Linaro, Nonlinear Dynamical Systems: Applications to Biological and Electronic Oscillators, Ph.D. thesis, University of Genoa, Genoa, Italy, 2011.
29. E. Mosekilde, B. Lading, S. Yanchuk, and Yu. Maistrenko, Bifurcation structure of a model of bursting pancreatic cells, BioSystems, 63 (2001), pp. 3-13.
30. B. E. Oldeman, B. Krauskopf, and A. R. Champneys, Death of period-doublings: Locating the homoclinic-doubling cascade, Phys. D, 146 (2000), pp. 100-120.
31. H. M. Osinga and K. T. Tsaneva-Atanasova, Dynamics of plateau bursting depending on the location of its equilibrium, J. Neuroendocrinology, 22 (2010), pp. 1301-1314.
32. P. C. Rech, Dynamics of a neuron model in different two-dimensional parameter-spaces, Phys. Lett. A, 375 (2011), pp. 1461-1464.
33. B. Sandstede, Constructing dynamical systems having homoclinic bifurcation points of codimension two, J. Dynam. Differential Equations, 9 (1997), pp. 269-288.
34. A. Shilnikov, R. L. Calabrese, and G. Cymbalyuk, Mechanism of bistability: Tonic spiking and bursting in a neuron model, Phys. Rev. E, 71 (2005), 056214.
35. A. Shilnikov and M. Kolomiets, Methods of the qualitative theory for the Hindmarsh-Rose model: A case study. A tutorial, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), pp. 2141-2168.
36. L. Shilnikov, The existence of a denumerable set of periodic motions in four-dimensional space in an extended neighborhood of a saddle-focus, Soviet Math. Dokl., 8 (1967), pp. 54-58.
37. L. Shilnikov, On the birth of a periodic motion from a trajectory bi-asymptotic to an equilibrium state of the saddle type, Soviet Math. Sbornik, 35 (1968), pp. 240-264.
38. L. Shilnikov and D. Turaev, Methods of Qualitative Theory of Differential Equations and Related Topics, Amer. Math. Soc. Transl. Ser. 2, AMS, Providence, RI, 2000.
39. M. Storace, D. Linaro, and E. de Lange, The Hindmarsh-Rose neuron model: Bifurcation analysis and piecewise-linear approximations, Chaos, 18 (2008), 033128.
40. D. Terman, Chaotic spikes arising from a model of bursting in excitable membranes, SIAM J. Appl. Math., 51 (1991), pp. 1418-1450.
41. D. Terman, The transition from bursting to continuous spiking in excitable membrane models, J. Nonlinear Sci., 2 (1992), pp. 135-182.