Qualitative methods for case study of the Hindmarch--Rose model

M. Kolomiets ; A. Shilnikov

Russian journal of nonlinear dynamics, Tome 6 (2010) no. 1, pp. 23-52.

Résumé

We demonstrate that bifurcations of periodic orbits underlie the dynamics of the Hindmarsh–Rose model and other square-wave bursting models of neurons of the Hodgkin–Huxley type. Such global bifurcations explain in-depth the transitions between the tonic spiking and bursting oscillations in a model. We show that a modified Hindmarsh–Rose model can exhibit the blue sky bifurcation, and a bistability of the coexisting tonic spiking and bursting activities.

Détail

Keywords: Hindmarsh–Rose model, dynamics, bistability, tonic spiking, bursting.
Mots-clés : neuron, bifurcations, blue sky catastrophe

Bibliographie

[1] Hindmarsh J. L., Rose R. M., “A model of neuronal bursting using three coupled first-order differential equations”, Proc. R. Soc. Lond. Ser. B Biol. Sci., 221 (1984), 87–102 | DOI

[2] Holden A. V., Fan Y. S., “From simple to simple bursting oscillatory behavior via intermittent chaos in the Rose–Hindmarsh model for neuronal activity”, Chaos Solitons Fractals, 2 (1992), 221–236 | DOI | MR | Zbl

[3] Wang X.-J., “Genesis of bursting oscillations in the Hindmarsh–Rose model and homoclinicity to a chaotic saddle”, Phys. D, 62 (1993), 263–274 | DOI | MR | Zbl

[4] Huerta R., Rabinovich M., Abarbanel H., Bazhenov M., “Spike-train bifurcation scaling in two coupled chaotic neurons”, Phys. Rev. E, 55 (1997), R2108–R2110 | DOI

[5] Izhikevich E. M., “Which model to use for cortical spiking neurons?”, IEEE Transactions on Neural Networks, 15:5 (2004), 1063–1070 | DOI

[6] Rosenblum M. G., Pikovsky A. S., “Controlling synchronization in an ensemble of globally coupled oscillators”, Phys. Rev. Lett., 92 (2004), 114102, 4 pp. | DOI

[7] Belykh I., de Lange E., Hasler M., “Synchronization of bursting neurons: What matters in the network topology”, Phys. Rev. Lett., 94 (2005), 188101, 4 pp. | DOI

[8] Belykh V., Belykh I., Mosekilde E., “The hyperbolic Plykin attractor can exist in neuron models”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 3567–3578 | DOI | MR | Zbl

[9] S. Coombses and P. Bressloff (eds.), Bursting: The genesis of rhythm in the nervous system, World Scientific, Singapore, 2005 | MR

[10] 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 (1952), 500–544

[11] Kopell N., “Toward a theory of modelling central pattern generators”, Neural control of rhythmic movements in vertebrates, eds. A. H. Cohen, S. Rossignol, S. Grillner, Wiley, New York, 1988, 369–414

[12] Rinzel J., “Bursting oscillations in an excitable membrane model”, Ordinary and Partial Differential Equations (Dundee, 1984), Lecture Notes in Math., 1151, Springer, Berlin, 1985, 304–316 | MR

[13] Rinzel J., Ermentrout B., “Analysis of neural excitability and oscillations”, Methods in Neuronal Modeling: From Synapses to Networks, eds. C. Koch, I. Segev, MIT Press, Cambridge, MA, 1989, 135–169

[14] Wang X.-J., Rinzel J., “Oscillatory and bursting properties of neurons”, The handbook of brain theory and neural networks, eds. M. Arbib, MIT Press, Cambridge, MA, 1995, 686–691

[15] Terman D., “The transition from bursting to continuous spiking in excitable membrane model”, J. Nonlinear Sci., 2 (1992), 133–182 | DOI | MR

[16] Bertram R., “A computational study of the effects of serotonin on a molluscan burster neuron”, Biol. Cybernet., 69 (1993), 257–267 | DOI | Zbl

[17] Bertram R., Butte M. J., Kiemel T., Sherman A., “Topological and phenomenological classification of bursting oscillations”, Bull. Math. Biol., 57:3 (1995), 413–439 | Zbl

[18] Izhikevich E., “Neural excitability, spiking and bursting”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10:6 (2000), 1171–1266 | DOI | MR | Zbl

[19] Canavier C. C., Baxter D. A., Clark L., Byrne J., “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 (1993), 2252–2257

[20] Butera R. J., “Multirhythmic bursting”, Chaos, 8 (1998), 274–284 | DOI | MR | Zbl

[21] Cymbalyuk G., Gaudry Q., Masino M. A., Calabrese R. L., “A model of a segmental oscillator in the leech heartbeat neuronal network”, J. Neurosci., 22 (2002), 10580–10587

[22] Bazhenov M., Timofeev I., Steriade M., Sejnowski T. J., “Spiking-bursting activity in the thalamic reticular nucleus initiates sequences of spindle oscillations in thalamic networks”, J. Neurophysiol., 84 (2000), 1076–1087

[23] Shilnikov A., Calabrese R., Cymbalyuk G., “Mechanism of bistability: Tonic spiking and bursting in a neuron model”, Phys. Rev. E, 71 (2005), 056214, 9 pp. | DOI | MR

[24] Cymbalyuk G., Shilnikov A. L., “Co-existent tonic spiking modes in a leech neuron model”, J. Comput. Neurosci., 18:3 (2005), 255–263 | DOI | MR

[25] Shilnikov A. L., Calabrese R., Cymbaluyk G., “How a neuron model can demonstrate coexistence of tonic spiking and bursting?”, Neurocomputing, 65–66 (2005), 869–875 | DOI

[26] Frohlich F., Bazhenov M., “Coexistence of tonic firing and bursting in cortical neurons”, Phys. Rev. E, 74 (2006), 031922, 7 pp. | DOI

[27] Guckenheimer J., Gueron S., Harris-Warrick R. .M., “Mapping the dynamics of a bursting neuron”, Philos. Trans. R. Soc. Lond. B Biol. Sci., 341 (1993), 345–359 | DOI

[28] Kuznetsov Yu. A., Rinaldi S., “Remarks on food chain dynamics”, Math. Biosci., 134 (1996), 1–33 | DOI | MR | Zbl

[29] Belykh V. N., Belykh I. V., Colding-Joregensen M., Mosekilde E., “Homoclinic bifurcations leading to bursting oscillations in cell models”, Eur. Phys. J. E Soft Matter, 3 (2000), 205–219 | DOI

[30] Feudel U., Neiman A., Pei X., Wojtenek W., Braun H., Huber M., Moss F., “Homoclinic bifurcation in a Hodgkin–Huxley model of thermally sensitive neurons”, Chaos, 10 (2000), 231–239 | DOI | MR | Zbl

[31] Deng B., Hines G., “Food chain chaos due to Shilnikov's orbit”, Chaos, 12 (2002), 533–538 | DOI | MR | Zbl

[32] Shilnikov A., Cymbaluyk G., “Homoclinic bifurcations of periodic orbits en a route from tonic spiking to bursting in neuron models”, Regul. Chaotic Dyn., 9:3 (2004), 281–297 | DOI | MR | Zbl

[33] Shilnikov A., Cymbalyuk G., “Transition between tonic-spiking and bursting in a neuron model via the blue-sky catastrophe”, Phys. Rev. Lett., 94 (2005), 048101, 4 pp. | DOI

[34] Channell P., Cymbalyuk G., Shilnikov A. L., “Origin of bursting through homoclinic spike adding in a neuron model”, Phys. Rev. Lett., 98 (2007), 134101, 4 pp. | DOI

[35] Channell P., Cymbalyuk G., Shilnikov A. L., “Applications of the Poincare mapping technique to analysis of neuronal dynamics”, Neurocomputing, 70 (2007), 10–12

[36] Chay T. R., Keizer J., “Minimal model for membrane oscillations in the pancreatic beta-cell”, Biophys. J., 42:2 (1983), 181–190 | DOI

[37] Chay T. R., “Chaos in a three-variable model of an excitable cell”, Phys. D, 16 (1985), 233–242 | DOI | Zbl

[38] Bertram R., Sherman A., “Dynamical complexity and temporal plasticity in pancreatic beta-cells”, J. Biosci., 25 (2000), 197–209

[39] Hill A., Lu J., Masino M., Olsen O., Calabrese R. L., “A model of a segmental oscillator in the leech heartbeat neuronal network”, J. Comput. Neurosci., 10 (2001), 281–302 | DOI

[40] Gu H., Yang M., Li L., Liu Z., Ren W., “Dynamics of autonomous stochastic resonance in neural period adding bifurcation scenarios”, Phys. Lett. A, 319 (2003), 89–96 | MR | Zbl

[41] Yang Z., Lu Q., Li L., “The genesis of period-adding bursting without bursting-chaos in the Chay model”, Chaos Solitons Fractals, 27 (2006), 689–697 | DOI | MR | Zbl

[42] Best J., Borisyuk A., Rubin J., Terman D., Wechselberger M., “The dynamic range of bursting in a model respirator pacemaker network”, SIAM J. Appl. Dyn. Syst., 4 (2005), 1107–1139 | DOI | MR | Zbl

[43] Andronov A. A., Leontovich E. A., “Nekotorye sluchai zavisimosti predelnykh tsiklov ot parametra”, Uchenye zapiski GGU, 1939, no. 6, 3–24 | MR

[44] Andronov A. A., Leontovich E. A., Gordon I. I., Maier A. G., Teoriya bifurkatsii dinamicheskikh sistem na ploskosti, Nauka, M., 1967, 487 pp. | MR

[45] Andronov A. A., Vitt A. A., Khaikin S. E., Teoriya kolebanii, 2-e izd., Fizmatgiz, M., 1959, 916 xe pp.

[46] Gradstein I. S., “On behavior of solutions of systems of linear differential equations degenerating in the limit”, Dokl. Akad. Nauk SSSR, 53 (1946), 391–394 ; Градштейн И. С., “О поведении решений системы линейных дифференциальных уравнений с постоянными коэффициентами, вырождающихся в пределе”, Изв. АН СССР. Сер. матем., 13:3 (1949), 253–280 | MR | Zbl

[47] Tikhonov A. N., “O zavisimosti reshenii differentsialnykh uravnenii ot malogo parametra”, Matem. sb., 22(64):2 (1948), 193–204 | Zbl

[48] Pontryagin L. S., Rodygin L. V., “Periodicheskoe reshenie odnoi sistemy obyknovennykh differentsialnykh uravnenii s malym parametrom pri proizvodnykh”, Dokl. AN SSSR, 132:3 (1960), 537–540 | MR | Zbl

[49] Fenichel N., “Geometric singular perturbation theory for ordinary differential equations”, J. Differential Equations, 31 (1979), 53–98 | DOI | MR | Zbl

[50] Mischenko E. F., Rozov N. Kh., Differential equations with small parameters and relaxation oscillations, Plenum Press, New York, 1980, 230 pp. | MR

[51] Mischenko E. F., Kolesov Yu. S., Kolesov A. Yu., Rozov N. Kh., Asymptotic methods in singularly perturbed systems, Monogr. in Contemp. Math., Consultants Bureau, New York, 1994, 281 pp. | MR

[52] Arnold V. I., Afraimovich V. S., Ilyashenko Yu. S., Shilnikov L. P., “Teoriya bifurkatsii”, Dinamicheskie sistemy–5, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 5, eds. V. I. Arnold, VINITI, M., 1986, 5–218 | MR

[53] Jones C. K. R. T., Kopell N., “Tracking invariant manifolds with differential forms in singularly perturbed systems”, J. Differential Equations, 108 (1994), 64–88 | DOI | MR | Zbl

[54] Guckenheimer J., “Towards a global theory of singularly perturbed dynamical systems”, Nonlinear dynamical systems and chaos (Groningen, 1995), Progr. Nonlinear Differential Equations Appl., 19, Birkhauser, Basel, 1996, 213–225 | MR | Zbl

[55] Doiron B., Laing C., Longtin A., Maler L., “Ghostbursting: A novel neuronal burst mechanism”, J. Comput. Neurosci., 12 (2002), 5–15 | DOI

[56] Laing C. R., Doiron B., Longtin A., Noonan L., Turner R. W., Maler L., “Type I burst excitability”, J. Comput. Neurosci., 14 (2003), 329–335 | DOI

[57] Rowat P. F., Elson R. C., “State-dependent effects of $Na$ channel noise on neuronal burst generation”, J. Comput. Neurosci., 16 (2004), 87–112 | DOI

[58] Shilnikov L. P., Turaev D. V., “O katastrofakh golubogo neba”, Dokl. RAN, 342 (1995), 596–599 | MR

[59] Shilnikov L. P., Turaev D. V., “On simple bifurcations leading to hyperbolic attractors”, Comput. Math. Appl., 34 (1997), 441–457 | MR

[60] Abraham R. H., “Chaostrophes, intermittency, and noise”, Chaos, fractals, and dynamics (Univ. Guelph, Ont., 1981/1983), Lect. Notes Pure Appl. Math., 98, Dekker, New York, 1985, 3–22 | MR

[61] Shilnikov L., Turaev D., “A new simple bifurcation of a periodic orbit of “blue sky catastrophe” type”, Methods of qualitative theory of differential equations and related topics, Dedicated to the memory of E. A. Leontovich-Andronova, Amer. Math. Soc. Transl. Ser. 2, 200, AMS, Providence, RI, 2000, 165–188 | MR | Zbl

[62] Gavrilov N., Shilnikov A., “Example of a blue sky catastrophe”, Methods of qualitative theory of differential equations and related topics, Dedicated to the memory of E. A. Leontovich-Andronova, Amer. Math. Soc. Transl. Ser. 2, 200, AMS, Providence, RI, 2000, 99–105 | MR | Zbl

[63] Shilnikov A. L., Shilnikov L. P., Turaev D. V., “On some mathematical topics in classical synchronization: A tutorial”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14:7 (2004), 2143–2160 | DOI | MR | Zbl

[64] Shilnikov L., Shilnikov A., Turaev D., Chua L., Methods of qualitative theory in nonlinear dynamics, P. 1, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, 4, World Scientific, River Edge, NJ, 1998, 392 pp. | MR | Zbl

[65] Shilnikov L., Shilnikov A., Turaev D., Chua L., Methods of qualitative theory in nonlinear dynamics, P. 2, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, 5, World Scientific, River Edge, NJ, 2001, 393–957 | MR | Zbl

[66] Shilnikov A., Shilnikov L., Turaev D., “Blue sky catastrophe in singularly perturbed systems”, Mosc. Math. J., 5:1 (2005), 269–282 | MR | Zbl

[67] Ermentrout B., Kopell N., “Multiple pulse interactions and averaging in systems of coupled neural oscillators”, J. Math. Biol., 29 (1991), 195–217 | DOI | MR | Zbl

[68] Guckenheimer J., Willms A., “Asymptotic analysis of subcritical Hopf-homoclinic bifurcation”, Phys. D, 139 (2000), 195–216 | DOI | MR | Zbl

[69] Callot J.-L., Diener F., Diener M., “Le probleme de la “chasse au canard””, C. R. Acad. Sci. Paris Ser. 1, 286 (1978), 1059–1061 | MR

[70] Benoit E., Callot J.-L., Diener F., Diener M., “Chasse au canard”, Collect. Math., 32 (1981), 37–119 | MR | Zbl

[71] Krupa M., Szmolyan P., “Extending geometric singular perturbation theory to nonhyperbolic points – fold and canard points in two dimensions”, SIAM J. Math. Anal., 33 (2001), 286–314 | DOI | MR | Zbl

[72] Dumortier F., Roussarie R., Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121, no. 577, AMS, 1996, 100 pp. | MR

[73] Cymbalyuk G. S., Calabrese R. L., “A model of slow plateau-like oscillations based upon the fast $Na+$ current in a window mode”, Neurocomputing, 38–40 (2001), 159–166 | DOI

[74] Kuznetsov Yu. A., Elements of applied bifurcation theory, Appl. Math. Sci., 112, 2nd updated ed., Springer, New York, NY, 1998, 616 pp. | MR | Zbl

[75] ftp://ftp.cwi.nl/pub/CONTENT

[76] Deng B., “Food chain chaos with canard explosion”, Chaos, 14:4 (2004), 1083–1092 | DOI | MR | Zbl

[77] Shilnikov A. L., Rulkov N. F., “Origin of chaos in a two-dimensional map modeling spiking-bursting neural activity”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13:11 (2003), 3325–3340 | DOI | MR | Zbl

[78] Shilnikov A. L., Rulkov N. F., “Subthreshold oscillations in a map-based neuron model”, Phys. Lett. A, 328 (2004), 177–184 | DOI | Zbl

[79] Medvedev G. M., “Transition to bursting via deterministic chaos”, Phys. Rev. Lett., 97 (2006), 048102, 4 pp. | DOI

[80] Shilnikov A. L., “Bifurkatsii i khaos v sisteme Marioka–Shimitsu”, Metody kachestvennoi teorii differentsialnykh uravnenii, Mezhvuz. tematich. sb. nauch. tr., GGU, Gorkii, 1986, 180–193 | MR

[81] Shilnikov A. L., “On bifurcations of the Lorenz attractor in the Shimizu–Marioka system”, Phys. D, 62 (1993), 338–346 | DOI | MR

[82] Deng B., “Homoclinic twisting bifurcation and cusp horseshoe maps”, J. Dynam. Differential Equations, 5 (1993), 417–467 | DOI | MR | Zbl

[83] Robinson C., “Homoclinic bifurcation to a transitive attractor of Lorenz type”, Nonlinearity, 2:4 (1989), 495–518 | DOI | MR | Zbl

[84] Rychlik M., “Lorenz attractor through Shilnikov type bifurcation 1”, Ergodic Theory Dynam. Systems, 10:4 (1990), 793–821 | DOI | MR | Zbl

[85] Shilnikov A. L., Shilnikov L. P., Turaev D. V., “Normal forms and Lorenz attractors”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 3:5 (1993), 1123–1139 | DOI | MR

[86] Leontovich E. A., “O rozhdenii predelnykh tsiklov ot separatrisy”, Dokl. AN SSSR, 78:4 (1951), 641–644

[87] Shilnikov L. P., “Ob odnom sluchae suschestvovaniya schetnogo mnozhestva periodicheskikh dvizhenii”, Dokl. AN SSSR, 160:3 (1965), 558–561

[88] Nozdracheva V. P., “Bifurkatsii negruboi petli separatrisy”, Differents. uravn., 18:9 (1982), 1551–1558 | MR | Zbl

[89] Chow S.-B., Deng B., Fiedler B., “Homoclinic bifurcations of resonant eigenvalues”, J. Dynam. Differential Equations, 2:2 (1990), 177–244 | DOI | MR | Zbl

[90] Shilnikov A. L., Turaev D. V., Singular orbit-flip bifurcation in slow-fast systems, gotovitsya k pechati

[91] Lukyanov V. I., Shilnikov L. P., “O nekotorykh bifurkatsiyakh dinamicheskikh sistem s gomoklinicheskimi strukturami”, Dokl. AN SSSR, 243:1 (1978), 26–29 | MR

[92] Gavrilov N. K., Shilnikov L. P., “O trekhmernykh dinamicheskikh sistemakh, blizkikh k sistemam s negruboi gomoklinicheskoi krivoi 2”, Matem. sb., 90(132):1 (1973), 139–156 | MR | Zbl

[93] Belykh I., Shilnikov A., “When weak inhibition synchronizes strongly desynchronizing networks of bursting neurons”, Phys. Rev. Lett., 101 (2008), 078102, 4 pp. | DOI

[94] Shilnikov A. L., Gordon R., Belykh I. V., “Polyrhythmic synchronization in bursting network motifs”, Chaos, 18 (2008), 037120, 13 pp. | DOI | MR