Methods of the qualitative theory for the Hindmarsh-Rose model: A case study. A tutorial

International Journal of Bifurcation and Chaos

Volumn 18, Issue 8, 2008, Pages 2141-2168

  Shilnikov, Andrey ^a   Kolomiets, Marina ^b  

^a GEORGIA STATE UNIVERSITY   (United States)

^b State Agricultural Academy   (Russian Federation)

Author keywords

Bifurcation; Bursting; Neuron models; Slow fast dynamics; Tonis spiking; Transition

Indexed keywords

MATHEMATICAL TECHNIQUES;

BURSTING; NEURON MODEL; SLOW-FAST DYNAMICS; TONIS SPIKING; TRANSITION;

BIFURCATION (MATHEMATICS);

EID: 54949130175     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127408021634     Document Type: Article

References (97)

1. Abraham, R. H. [1985] "Chaostrophes, intermittency, and noise," in Chaos, Fractals, and Dynamics, Conf. Univ. Guelph/Can. 1981 and 1983, Lecture Notes Pure Appl. Math. 98, pp. 3-22.
2. Andronov, A. A. & Leontovich, E. A. [1937] "Some cases of dependence of limit cycles on a parameter," Uchenye zapiski Gorkovskogo Universiteta 6, 3-24.
3. Andronov, A. A., Vitt, A. A. & Khaikin, S. E. [1966] Theory of Oscillations, International Series of Monographs in Physics, Vol. 4 (Pergamon Press, Oxford).
4. Andronov, A. A., Leontovich, E. A., Gordon, I. E. & Maier, A. G. [1971] The Theory of Bifurcations of Dynamical Systems on a Plane, Israel Program of Scientific Translation, Jerusalem.
5. Arnold, V. I., Afrajmovich, V. S., Ilyashenko, Yu. S. & Shil'nikov, L. P. [1984] Bifurcation Theory, Dynamical Systems V, Encyclopaedia of Mathematical Sciences (Springer-Verlag).
6. Bazhenov, M., Timofeev, I., Steriade, M. & Sejnowski, T. J. [2000] "Spiking-bursting activity in the thalamic reticular nucleus initiates sequences of spindle oscillations in thalamic networks," J. Neurophysiol. 84, 1076-1087.
7. Belykh, V. N., Belykh, I. V., Colding-Joregensen, M. & Mosekilde, E. [2000] "Homoclinic bifurcations leading to bursting oscillations in cell models," Eur. Phys. J. E 3, 205-214.
8. Belykh, V., Belykh, I. & Mosekilde, E. [2005a] "The hyperbolic Plykin attractor can exist in neuron models," Int. J. Bifurcation and Chaos 15, 1112-1130.
9. Belykh, I., de Lange, E. & Easier, M. [2005b] "Synchronization of bursting neurons: what matters in the network topology," Phys. Rev, Lett. 94, 188101-188105.
10. Belykh, I. V. & Shilnikov, A. [2008] "When weak inhibition synchronizes strongly desynchronizing network of bursting neurons," Phys. Rev, Lett., in press.
11. Benoit, E., Callot, J.-L., Diener, F. & Diener, M. [1981] "Chasse au canard," Collect. Math. 32, 37-119.
12. Bertram, R. [1993] "A computational study of the effects of serotonin on a molluscan burster neuron," Biol. Cybern. 69, 257-267.
13. Bertram, R., Butte, M. J., Kiemel, T. & Sherman, A. [1995] "Topological and phenomenological classification of bursting oscillations," Bull. Math. Biol. 57, 413-439.
14. Bertram, R. & Sherman, A. [2000] "Dynamical complexity and temporal plasticity in pancreatic beta-cells," J. Biosci. 25, 197-209.
15. Best, J., Borisyuk, A, Rubin, J., Terman, D. & Wechselberger, M. [2005] "The dynamic range of bursting in a model respirator pacemaker network," SIAM J. Appl. Dyn. Syst. 4, 1107-1139.
16. Butera, R. [1998] "Multirhythmic bursting," Chaos 8, 274-282.
17. Callot, J.-L., Diener, F. & Diener, M. [1978] "Le probléme de la chasse au canard," C. R. Acad. Sci. 286, 1059.
18. Canavier, C. C., Baxter, D. A., Clark, L. & Byrne, J. [1993] "Nonlinear dynamics in a model neuron provide a novel mechanism for transient synaptic inputs to produce long-term alterations of postsynaptic activity" J. Neurophysiol. 69, 2252-2270.
19. Channell, P., Cymbalyuk, G. & Shilnikov, A. L. [2007a] "Applications of the Poincare mapping technique to analysis of neuronal dynamics," Neurocomputing 70, 10-12.
20. Channell, P., Cymbalyuk, G. & Shilnikov, A. L. [2007b] "Origin of bursting through homoclinic spike adding in a neuron model," Phys. Rev. Lett. 98, 134101-134105.
21. Chay, T. R. & Keizer, J. [1983] "Minimal model for membrane oscillations in the pancreatic beta-cell," Biophys. J. 42, 181-190.
22. Chay, T. R. [1985] "Chaos in a three-variable model of an excitable cell," Physica D 16, 233-242.
23. Chow, S.-B., Deng, B. & Fiedler, B. [1990] "Homoclinic bifurcations of resonant eigenvalues," J. Dyn. Diff. Eqs. 2, 177-245.
24. Coombses, S. & Bressloff, P. (eds.) [2005] Bursting: The Genesis of Rhythm in the Neurvous System (World Scientific, Singapore).
25. Cymbalyuk, G. S. & Calabrese, R. L. [2001] "A model of slow plateau-like oscillations based upon the fast Na+ current in a window mode," Neurocomputing 38-40, 159-166.
26. Cymbalyuk, G., Gaudry, Q., Masino, M. A. & Calabrese, It. L. [2002] "A model of a segmental oscillator in the leech heartbeat neuronal network," J. Neurosci. 22, 10580-10587.
27. Cymbalyuk, G. & Shilnikov, A. L. [2005] "Co-existent tonic spiking modes in a leech neuron model," J. Comput. Neurosci. 18, 255-263.
28. Deng, B. [1993] "Homoclinic twisting bifurcation and cusp horseshoe maps," J. Dyn. Diff. Eqs. 5, 417-467.
29. Deng, B. & Hines, G. [2002] "Food chain chaos due to Shilnikov's orbit," Chaos 12, 533-538.
30. Deng, B. [2004] "Food chain chaos with canard explosion," Chaos 14.
31. Doiron, B., Laing, C. & Longtin, A. [2002] "Ghostbursting: A novel neuronal burst mechanism," Comp. Neurosci. 12, 5-15.
32. Dumortier, F. & Roussarie, R. [1996] "Canard cycles and center manifolds," Mem. Amer. Math. Soc. 121, 101-121.
33. Ermentrout, B. & Kopell, N. [1991] "Multiple pulse interactions and averaging in systems of coupled neural oscillators," J. Math. Biol. 29, 195-217.
34. Fenichel, N. [1979] "Geometric singular perturbation theory for ordinary differential equations," J. Diff. Eqs. 31, 53-98.
35. Feudel, U., Neiman, A., Pei, X., Wojtenek, W., Braun, II., Huber, M. & Moss, F. [2000] "Homoclinic bifurcation in a Hodgkin-Huxley model of thermally sensitive neurons," Chaos 10, 231-239.
36. Frohlich, F. & Bazhenov, M. [2006] "Coexistence of tonic firing and bursting in cortical neurons," Phys. Rev. E 74, 031922-031929.
37. Gavrilov, N. K. & Shilnikov, L. P. [1973] "On threedimensional dynamical systems close to systems with a structurally unstable homoclinic curve," Math. USSR-Sb. 19, 139-150.
38. Gavrilov, N. K. & Shilnikov, A. L. [2000] "Example of a blue sky catastrophe," in Methods of Qualitative Theory of Differential Equations and Related Topics, AMS Transl. Series II, 200, pp. 99-105.
39. Gradstein, I. S. [1946] "On behaviour of solutions of systems of linear differential equations degenerating in the limit," Dokl. Acad. Sci. URSS 53, 391-394.
40. Gu, H., Yang, M., Li, L., Liu, Z. & Ren, W. [2003] "Dynamics of autonomous stochastic resonance in neural period adding bifurcation scenarios," Phys. Lett. A 319, 89-96.
41. Guckenheimer, J., Gueron, S. & Harris-Warrick, R. M. [1993] "Mapping the dynamics of a bursting neuron," Philos, Trans. Royal Soc. Land. Biol. Sci. 341, 345-359.
42. Guckenheimer, J. [1996] "Towards a global theory of singularly perturbed systems," in Progr. Nonlin. Diff. Eqs. Their Appl. 19, 214-225.
43. Guckenheimer, J. & Willms, A. [2000] "Analysis of a subcritical Hopf-homoclinic bifurcation," Physica D 139, 195-216.
44. Hill, A., Lu, J., Masino, M., Olsen, O. & Calabrese, R. L. [2001] "A model of a segmental oscillator in the leech heartbeat neuronal network," J. Com.put. Neurosci. 10, 281-302.
45. Hindmarsh, J. L. & Rose, R. M. [1984] "A model of neuronal bursting using three coupled first-order differential equations," Proc. Roy. Soc. Land. B 221, 87.
46. Hodgkin, A. L. & Huxley A. F. [1952] "A quantitative description of membrane current and its application to conduction and excitation in nerve," J. Physiol. 117, 500-544.
47. Holden, A. V. & Fan, Y. S. [1992] "From simple to simple bursting oscillatory behaviour via intermittent chaos in the Rose-Hindmarsh model for neuronal activity," Chaos Solit. Fract. 2, 221-230.
48. Huerta, R., Rabinovich, M., Abarbanel, H. & Bazhenov. M. [1997] "Spike-train bifurcation scaling in two coupled chaotic neurons," Phys. Rev. E 55, 2108-2110.
49. Izhikevich, E. M. [2000] "Neural excitability, spiking and bursting," Int. J. Bifurcation and Chaos 10, 1171-1266.
50. Izhikevich, E. M. [2004] "Which model to use for cortical spiking neurons?" IEEE Trans. Neural Networks 12, 1063-1107.
51. Jones, C. K. R. T. & Kopell, N. [1994] "Tracking invariant manifolds with differential forms in singularly perturbed systems," J. Diff. Eqs. 108, 64-88.
52. Kopell, N. [1998] "Toward a theory of modelling central pattern generators," in Neural Control of Rhythmic Movements in Vertebrates, eds. Cohen, A. H., Rossignol, S. & Grillner, S. (Wiley NY), pp. 23-33.
53. Kramer, M. A., Traub," R. D. & Kopell, N. [2008] New Dynamics in Cerebeller Purkinje Cells: Torus Canards.
54. 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.
55. Kuznetsov, Yu. A. & Rinaldi, S. [1996] "Remarks on food chain dynamics," Math. Bio sciences 134, 1-33.
56. Kuznetsov, Yu. A. [1998] Elements of Applied Bifurcation Theory, 2nd updated ed., Applied Mathematical Sciences, Vol. 112 (Springer, NY). CONTENTS is available at ftp://ftp.cwi.nl/pub/CONTENT.
57. Laing, C. R., Doiron, B., Longtin, A., Noonan, L., Turner, R. W. & Maler, L. [2003] "Type I burst excitability," J. Comput. Neurosci. 14, 329-335.
58. Leontovich, E. A. [1951] "On birth of limit cycles from separatrices," DAN SSSR 74, 641-644.
59. Lukyanov, V. & Shilnikov, L. P. [1978] "On some bifurcations of dynamical systems with homoclinic structures," Sov. Math. Dokl. 19, 1314-1318.
60. Malaschenko, T., Shilnikov, A. L. & Cymbalyuk, G. [2008] "Subtheshold oscillations in a neuron model," J. Comp. Neurosci., to be submitted.
61. Medvedev, G. M. [2006] "Transition to bursting via deterministic chaos," Phys. Rev. Lett. 97, 048102.
62. Mischenko, E. F. & Rozov, N. Kh. [1980] Differential Equations with Small Parameters and Relaxation Oscillations (Plenum Press).
63. Mischenko, E. F., Kolesov, Yu. S., Kolesov, A. Yu. & Rozov, N. Kh. [1984] Asymptotic Methods in Singularly Perturbed Systems, Monographs in Contemporary Mathematics (Consultants Bureau, NY).
64. Nozdracheva, V. P. [1982] "Bifurcation of a noncoarse separatrix loop," Diff. Eqs. 18, 1098-1104.
65. Pontryagin, L. S. & Rodygin. L. V. [1960] "Periodic solution of a system of ordinary differential equations with a small parameter in the terms containing derivatives," Sov. Math. Dokl. 1, 611-661.
66. Rinzel, J. [1985] "Bursting oscillations in an excitable membrane model," in Ordinary and Partial Differential Equations, Lecture Notes in Mathematics, Vol. 1151, pp. 304-320.
67. Rinzel, J. & Ermentrout, B. [1989] "Analysis of neural excitability and oscillations," in Methods of Neural Modeling: From Synapses to Networks, eds. Koch, C. & Segev, I. (MIT Press), pp. 135-169.
68. Robinson, C. [1989] "Homoclinic bifurcation to a transitive attractor of Lorenz type," Nonlinearity 2, 495-518.
69. Rosenblum, M. G. & Pikovsky, A. S. [2004] "Controlling synchronization in an ensemble of globally coupled oscillators," Phys. Rev. Lett. 92, 114102-114106.
70. Rowat, P. F. & Elson, R. C. [2004] "State-dependent effects of Na channel noise on neuronal burst generation," J. Comput. Neurosci. 16, 87-112.
71. Rychlik, M. [1990] "Lorenz attractor through Shilnikov type bifurcation," Erg. Th. Dyn. Syst. 10, 793-821.
72. Shilnikov, A. L. [1986] "Bifurcations and chaos in the Shimizu-Marioka system," [In Russian], in Methods and Qualitative Theory of Differential Equations, Gorky State University, pp. 180-193;
73. [1991] English translation in Selecta Mathematica Sovietica 10, 105-117.
74. Shilnikov, A. L. [1993] "On bifurcations of the Lorenz attractor in the Shimizu-Marioka system," Physica D 62, 338-346.
75. Shilnikov, A. L., Shilnikov, L. P. & Turaev, D. V. [1993] "Normal forms and Lorenz attractors," Int. J. Bifurcation and Chaos 3, 1123-1139.
76. Shilnikov, A. L. & Rulkov, N. F. [2003] "Origin of chaos in a two-dimensional map modelling spiking-bursting neural activity" Int. J. Bifurcation and Chaos 13, 3325-3340.
77. Shilnikov, A. L. & Cymbaluyk, G. [2004] "Homoclinic saddle-node orbit bifurcations en route between tonic spiking and bursting in neuron models. Invited paper," Regular & Chaotic Dyn. 9, 1-34.
78. Shilnikov, A. L. & Rulkov, N. F. [2004] "Subthreshold oscillations in a map-based neuron model," Phys. Lett. A 328, 177-184.
79. Shilnikov, A. L., Shilnikov, L. P. & Turaev, D. V. [2004] "Mathematical aspects of classical synchronization theory, Tutorial," Int. J. Bifurcation and Chaos 14, 2143-2160.
80. Shilnikov, A. L., Calabrese, R. L. & Gymbaluyk, G. [2005a] "How a neuron model can demonstrate coexistence of tonic spiking and bursting?" Neurocomputing 65-66, 869-875.
81. Shilnikov, A. L., Calabrese, R. L. & Gymbalyuk, G. [2005b] "Mechanism of bi-stability: tonic spiking and bursting in a neuron model," Phys. Rev. E 71, 205-214.
82. Shilnikov, A. L. & Cymbalyuk. G. [2005] "Transition between tonic-spiking and bursting in a neuron model via the blue-sky catastrophe," Phys. Rev. Lett. 94, 048101-048105.
83. Shilnikov, A. L., Shilnikov, L. P. & Turaev, D. V. [2005c] "Blue sky catastrophe in singularly perturbed systems," Moscow Math. J. 5, 205-218.
84. Shilnikov, A. L. & Kolomiets, M. L. [2008] "Bursting manifolds of neuronal models," Int. J. Bifurcation and Chaos, to be submitted.
85. Shilnikov, A. L., Gordon, R. & Belykh, I. V. [2008] "Polyrhytmic synchronization in bursting network motifs," J. Chaos 18, in press.
86. Shilnikov, A. L. & Turaev, D. V. [2008] "Singular orbitflip bifurcation in slow-fast systems," in preparation.
87. Shilnikov, L. P. [1965] "Case of the existence of a denumerable set of periodic motions," Sov. Math. Dokl. 6, 163-166.
88. Shilnikov, L. P. & Turaev, D. V. [1997] "On simple bifurcations leading to hyperbolic attractors," Comput. Math. Appl. 34, 441-457.
89. Shilnikov, L. P., Shilnikov, A. L., Turaev, D. V. & Chua, L. [1998] Methods of Qualitative Theory in Nonlinear Dynamics. Part I (World Scientific, Singapore).
90. Shilnikov, L. P. & Turaev, D. V. [2000] "A new simple bifurcation of a periodic orbit of blue sky catastrophe type," in Methods of Qualitative Theory of Differential Equations and Related Topics, AMS Transl. Series II, 200, pp. 165-188.
91. Shilnikov, L. P., Shilnikov, A. L., Turaev, D. V. & Chua, L. [2001] Methods of Qualitative Theory in Nonlinear Dynamics. Part II (World Scientific, Singapore).
92. Terman, D. [1992] "The transition from bursting to continuous spiking in an excitable membrane model," J. Nonlin. Sci. 2, 133-182.
93. Tikhonov, A. N. [1948] "On the dependence of solutions of differential equations from a small parameter," Mat. Sb. 2, 193-204.
94. Turaev. D. V. & Shilnikov, L. P. [1995] "Blue sky catastrophes," Dokl. Math. 51, 404-407.
95. Wang, X. J. [1993] "Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle," Physica 62, 263-274.
96. Wang, X. J. & Rinzel, J. [1995] "Oscillatory and bursting properties of neurons," in The Handbook of Brain Theory and Neural Networks, ed. Arbib, M. (MIT Press), pp. 686-691.
97. Yang, Z., Lu, Q. & Li. L. [2006] "The genesis of periodadding bursting without bursting-chaos in the Chay model chaos," Solit. Fract. 27, 689-697.