Variability of bursting patterns in a neuron model in the presence of noise

  • Paul Channell
  • Ibiyinka Fuwape
  • Alexander B. Neiman
  • Andrey L. ShilnikovEmail author


Spiking and bursting patterns of neurons are characterized by a high degree of variability. A single neuron can demonstrate endogenously various bursting patterns, changing in response to external disturbances due to synapses, or to intrinsic factors such as channel noise. We argue that in a model of the leech heart interneuron existing variations of bursting patterns are significantly enhanced by a small noise. In the absence of noise this model shows periodic bursting with fixed numbers of interspikes for most parameter values. As the parameter of activation kinetics of a slow potassium current is shifted to more hyperpolarized values of the membrane potential, the model undergoes a sequence of incremental spike adding transitions accumulating towards a periodic tonic spiking activity. Within a narrow parameter window around every spike adding transition, spike alteration of bursting is deterministically chaotic due to homoclinic bifurcations of a saddle periodic orbit. We have found that near these transitions the interneuron model becomes extremely sensitive to small random perturbations that cause a wide expansion and overlapping of the chaotic windows. The chaotic behavior is characterized by positive values of the largest Lyapunov exponent, and of the Shannon entropy of probability distribution of spike numbers per burst. The windows of chaotic dynamics resemble the Arnold tongues being plotted in the parameter plane, where the noise intensity serves as a second control parameter. We determine the critical noise intensities above which the interneuron model generates only irregular bursting within the overlapped windows.


Bursting Noise Neuron Model Chaos Transition Entropy Lyapunov exponent Homoclinic Bifurcation Spiking Adding Pattern 



The authors thank G. Cymbalyuk and R. Clewley for valuable discussions. We are grateful to anonymous reviewers for inspiring critique and useful suggestions. P.C. was supported by a fellowship through the GSU Brains and Behavior program; A.L.S. was supported by the GSU Brains and Behavior program and by the RFFI grant No 050100558. A.N. was supported in part by the Biomimetic Nanoscience and Nanotechnology program of Ohio University. I.F. was supported by the Faculty for the Future Fellowship awarded by the Schlumberger Foundation.


  1. Arnold, V. I., Afraimovich, V. S., Ilyashenko, Yu. S., & Shilnikov, L. P. (1994). Bifurcation theory. Dynamical systems. Encyclopaedia of mathematical sciences (Vol. V). New York: Springer.Google Scholar
  2. Bal, T., Nagy, F., & Moulins, M. (1988). The pyloric central pattern generator in crustacea: A set of conditional neural oscillators. Journal of Comparative Physiology A, 163(6), 715–727.CrossRefGoogle Scholar
  3. Belykh, V. N., Belykh, I. V., Colding-Jorgensen, M., & Mosekilde, E. (2000). Homoclinic bifurcations leading to bursting oscillations in cell models. The European Physical Journal E—Soft Matter, 3(3), 205–219.CrossRefGoogle Scholar
  4. Bertram, R., Butte, M. J., Kiemel, T., & Sherman, A. (1995). Topological and phenomenological classification of bursting oscillations. Bulletin of Mathematical Biology, 57(3), 413–439.Google Scholar
  5. Bulsara, A. R., Schieve, W. C., & Jacobs, E. W. (1990). Homoclinic chaos in systems perturbed by weak langevin noise. Physical Review A, 41(2), 668–681.CrossRefPubMedGoogle Scholar
  6. Carelli, P. V., Reyes, M. B., Sartorelli, J. C., & Pinto, R. D. (2005). Whole cell stochastic model reproduces the irregularities found in the membrane potential of bursting neurons. Journal of Neurophysiology, 94(2), 1169–1179.CrossRefPubMedGoogle Scholar
  7. Channell, P., Cymbalyuk, G., & Shilnikov, A. (2007a). Origin of bursting through homoclinic spike adding in a neuron model. Physical Review Letters, 98(13), 134101.CrossRefPubMedGoogle Scholar
  8. Channell, P., Cymbalyuk, G., & Shilnikov, A. L. (2007b). Applications of the poincare mapping technique to analysis of neuronal dynamics. Neurocomputing, 70, 10–12.Google Scholar
  9. Chay, T. R. (1985). Chaos in a three-variable model of an excitable cell. Physica D, 16(2), 233–242.CrossRefGoogle Scholar
  10. Chow, C. C., & White, J. A. (1996). Spontaneous action potentials due to channel fluctuations. Biophysical Journal, 71(6), 3013–3021.CrossRefPubMedGoogle Scholar
  11. Clewley, R., Soto-Trevino, C., & Nadim, F. (2009). Dominant ionic mechanisms explored in spiking and bursting using local low-dimensional reductions of a biophysically realistic model neuron. Journal of Computational Neuroscience, 26(1), 75–90.CrossRefPubMedGoogle Scholar
  12. 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, 159–166.CrossRefGoogle Scholar
  13. Cymbalyuk, G. S., & Shilnikov, A. L. (2005). Coexistence of tonic spiking oscillations in a leech neuron model. Journal of Computational Neuroscience, 18(3), 255–263.CrossRefPubMedGoogle Scholar
  14. Cymbalyuk, G. S., Gaudry, Q., Masino, M. A., & Calabrese, R. L. (2002). Bursting in leech heart interneurons: cell-autonomous and network-based mechanisms. Journal of Neuroscience, 22(24), 10580–10592.PubMedGoogle Scholar
  15. Deng, B., & Hines, G. (2002). Food chain chaos due to Shilnikov’s orbit. Chaos, 12(3), 533–538.CrossRefPubMedGoogle Scholar
  16. Elson, R. C., Huerta, R., Abarbanel, H. D., Rabinovich, H. D., & Selverston, A. I. (1999). Dynamic control of irregular bursting in an identified neuron of an oscillatory circuit. Journal of Neurophysiology, 82(1), 115–122.PubMedGoogle Scholar
  17. Elson, R. C., Selverston, A. I., Abarbanel, H. D. I., & Rabinovich, M. I. (2002). Dynamic control of irregular bursting in an identified neuron of an oscillatory circuit. Journal of Neurophysiology, 88, 1166–1182.PubMedGoogle Scholar
  18. Fan, Y. S., & Holden, A. V. (1995). Bifurcations, burstings, chaos and crises in the Rose-Hindmarsh model for neuronal activity. Chaos, Solitons and Fractals, 3, 439–449.CrossRefGoogle Scholar
  19. Fenichel, F. (1979). Geometric singular perturbation theory for ordinary differential equations. Journal of Differential Equations, 31, 53–98.CrossRefGoogle Scholar
  20. Feudel, U., Neiman, A., Pei, X., Wojtenek, W., Braun, H., Huber, M., et al. (2000). Homoclinic bifurcation in a Hodgkin-Huxley model of thermally sensitive neurons. Chaos, 10(1), 231–239.CrossRefPubMedGoogle Scholar
  21. Galan, R. F., Ermentrout, G. B., & Urban, N. N. (2008). Optimal time scale for spike-time reliability: Theory, simulations, and experiments. Journal of Neurophysiology, 99(1), 277–283.CrossRefPubMedGoogle Scholar
  22. Gavrilov, N. K., & Shilnikov, L. P. (1972). On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve. Mathematics of the USSR, Sbornik, 17(3), 467–485.CrossRefGoogle Scholar
  23. Goldobin, D. S., & Pikovsky, A. (2005). Synchronization and desynchronization of self-sustained oscillators by common noise. Physical Review E, 71, 045201.CrossRefGoogle Scholar
  24. Goldobin, D. S., & Pikovsky, A. (2006). Antireliability of noise-driven neurons. Physical Review E, 73, 061906.CrossRefGoogle Scholar
  25. Griffiths, R. E., & Pernarowski, M. C. (2006). Return map characterizations for a model of bursting with two slow variables. SIAM Journal on Applied Mathematics, 66(6), 1917–1948.CrossRefGoogle Scholar
  26. Gu, H., Yang, M., Li, L., Liu, Z., & Ren, W. (2002). Experimental observation of the stochastic bursting caused by coherence resonance in a neural pacemaker. NeuroReport, 13(13), 1657–1660.CrossRefPubMedGoogle Scholar
  27. Guckenheimer, J. (1996). Towards a global theory of singularly perturbed systems. Progress in Nonlinear Differential Equations and Their Applications, 19, 214–225.Google Scholar
  28. Hayashi, H., & Ishizuka, S. (1995). Chaotic responses of the hippocampal CA3 region to a mossy fiber stimulation in vitro. Brain Research, 686(2), 194–206.CrossRefPubMedGoogle Scholar
  29. Hill, A. A., Lu, J., Masino, M. A., Olsen, O. H., & Calabrese, R. L. (2001). A model of a segmental oscillator in the leech heartbeat neuronal network. Journal of Computational Neuroscience, 10(3), 281–302.CrossRefPubMedGoogle Scholar
  30. Hodgkin, A. L., & Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology, 117(4), 500–544.PubMedGoogle Scholar
  31. 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, Solitons and Fractals, 2, 349–269.CrossRefGoogle Scholar
  32. Izhikevich, E. M. (2000). Neural excitability, spiking and bursting. International Journal of Bifurcation and Chaos, 10, 1171–1266.CrossRefGoogle Scholar
  33. Izhikevich, E. M. (2007). Dynamical systems in neuroscience. The geometry of excitability and bursting. Cambridge: MIT.Google Scholar
  34. Jaeger, L., & Kantz, H. (1997). Homoclinic tangencies and non-normal jacobians—effects of noise in nonhyperbolic chaotic systems. Physica D, 105(1–3), 79–96.CrossRefGoogle Scholar
  35. Jones, C. K. R. T., & Kopell, N. (1994). Tracking invariant-manifolds with differential forms in singularly perturbed systems. Journal of Differential Equations, 108(1), 64–88.CrossRefGoogle Scholar
  36. Kramer M., Traub, R. D., & Kopell, N. J. (2008) New dynamics in cerebellar purkinje cells: Torus canards. Physics Review Letters, 101, 068103.CrossRefGoogle Scholar
  37. Kopell, N. (1988). Toward a theory of modelling central pattern generators. In A. H., Cohen, S., Rossingol, & S., Grillner (Eds.), Neural control of rhythmic movements in vertebrates (pp. 1–20). New York: Wiley.Google Scholar
  38. Kuske, R., & Baer, S. M. (2002). Asymptotic analysis of noise sensitivity in a neuronal burster. Bulletin of Mathematical Biology, 64(3), 447–481.CrossRefPubMedGoogle Scholar
  39. Mainen, Z. F., & Sejnowski, T. J. (1995). Reliability of spike timing in neocortical neurons. Science, 268(5216), 1503–1506.CrossRefPubMedGoogle Scholar
  40. Manwani, A., & Koch, C. (1999). Detecting and estimating signals in noisy cable structure, I: Neuronal noise sources. Neural Computation, 11(8), 1797–1829.CrossRefPubMedGoogle Scholar
  41. Marder, E., & Calabrese, R. L. (1996). Principles of rhythmic motor pattern generation. Physiological Reviews, 76(3), 687–717.PubMedGoogle Scholar
  42. Medvedev, G. M. (2005). Reduction of a model of an excitable cell to a one-dimensional map. Physica D, 202(1–2), 87–106.Google Scholar
  43. Medvedev, G. M. (2006). Transition to bursting via deterministic chaos. Physical Review Letters, 97, 048102.CrossRefPubMedGoogle Scholar
  44. Mira, C. (1987). Chaotic dynamics from the one-dimensional endomorphism to the two-dimensional diffeomorphism. Singapore: World Scientific.Google Scholar
  45. Pedersen, M. G., & Sorensen, M. P. (2007). The effect of noise of β-cell burst period. SIAM Journal on Applied Mathematics, 67, 530–542.CrossRefGoogle Scholar
  46. Pei, X., & Moss, F. (1996). Characterization of low-dimensional dynamics in the crayfish caudal photoreceptor. Nature, 379(6566), 618–621.CrossRefPubMedGoogle Scholar
  47. 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. Soviet Mathematics. Doklady, 1, 611–661.Google Scholar
  48. Rabinovich, M. I., Varona, P., Silverston, A. L., & Abarbanel, H. D. (2006). Dynamics principles in neuroscience. Reviews of Modern Physics, 78(4), 1213–1265.CrossRefGoogle Scholar
  49. Rinzel, J. (1985). Bursting oscillations in an excitable membrane model. Lecture Notes in Mathematics, 1151, 304–316.CrossRefGoogle Scholar
  50. Rinzel, J., & Ermentrout, B. (1998). Analysis of neural excitability and oscillations. In C., Koch & I., Segev (Eds.), Computational neuroscience (pp. 135–169). Cambridge: MIT.Google Scholar
  51. Rinzel, J., & Wang, X. J. (1995). Oscillatory and bursting properties of neurons. In M. Arbib (Ed.), The handbook of brain theory and neural networks (pp. 686–691). Cambridge: MIT.Google Scholar
  52. Rowat, P. (2007). Interspike interval statistics in the stochastic Hodgkin-Huxley model: Coexistence of gamma frequency bursts and highly irregular firing. Neural Computation, 19(5), 1215–1250.CrossRefPubMedGoogle Scholar
  53. Rowat, P. F., & Elson, R. C. (2004). State-dependent effects of Na channel noise on neuronal burst generation. Journal of Computational Neuroscience, 16(2), 87–112.CrossRefPubMedGoogle Scholar
  54. Schiff, S. J., Jerger, K., Duong, D. H., Chang, T., Spano, M. L., & Ditto, W. L. (1994). Controlling chaos in the brain. Nature, 370(6491), 615–620.CrossRefPubMedGoogle Scholar
  55. Sharkovsky, A. N., Kolyada, S. F., Sivak, A. G., & Fedorenko, V. V. (1997). Dynamics of one-dimensional maps. Mathematics and its applications (Vol. 407). Dordrecht: Kluwer.Google Scholar
  56. Shilnikov, A. L. (1993). On bifurcations of the Lorenz attractor in the Shimizu-Morioka model. Physica D, 62(1–4), 338–346.CrossRefGoogle Scholar
  57. Shilnikov, A., & Cymbalyuk, G. (2004). Homoclinic saddle-node orbit bifurcations en a route between tonic spiking and bursting in neuron models, invited paper. Regular & Chaotic Dynamics, 9, 281–297.CrossRefGoogle Scholar
  58. Shilnikov, A. L., Calabrese, R. L., & Cymbalyuk, G. (2005). Mechanism of bistability: Tonic spiking and bursting in a neuron model. Physical Review E, 71, 056214.CrossRefGoogle Scholar
  59. Shilnikov, A. L., & Cymbalyuk, G. (2005). Transition between tonic spiking and bursting in a neuron model via the blue-sky catastrophe. Physical Review Letters, 94(4), 048101.CrossRefPubMedGoogle Scholar
  60. Shilnikov, A. L., & Kolomiets, M. L. (2008). Methods of the qualitative theory for the Hindmarsh-Rose model: A case study. A tutorial. International Journal of Bifurcation and Chaos, 18(7), 1–32.Google Scholar
  61. Shilnikov, A. L., & Rulkov, N. F. (2003). Origin of chaos in a two-dimensional map modelling spiking-bursting neural activity. International Journal of Bifurcation and Chaos, 13(11), 3325–3340.CrossRefGoogle Scholar
  62. Shilnikov, A. L., & Rulkov, N. F. (2004). Subthreshold oscillations in a map-based neuron model. Physics Letters A, 328(2–3), 177–184.CrossRefGoogle Scholar
  63. Shilnikov, L. P., Shilnikov, A. L., Turaev, D., & Chua, L. O. (1998, 2001). Methods of qualitative theory in nonlinear dynamics (Vols. 1 and 2). Singapore: World Scientific.Google Scholar
  64. So, P., Ott, E., Schiff, S. J., Kaplan, D. T., Sauer, T., & Grebogi, C. (1996). Detecting unstable periodic orbits in chaotic experimental data. Physical Review Letters, 76(25), 4705–4708.CrossRefPubMedGoogle Scholar
  65. Steriade, M., Jones, E. G., & Llinas, R. R. (1990). Thalamic oscillations and signaling. New York: Wiley.Google Scholar
  66. Steriade, M., McCormick, D. A., & Sejnowski, T. J. (1993). Thalamocortical oscillations in the sleeping and aroused brain. Science, 262(5134), 679–685.CrossRefPubMedGoogle Scholar
  67. Su, J., Rubin, J., & Terman, D. (2004). Effects of noise on elliptic bursters. Nonlinearity, 17, 13300157.CrossRefGoogle Scholar
  68. Terman, D. (1991). Chaotic spikes arising from a model of bursting in excitable membranes. SIAM Journal on Applied Mathematics, 51(5), 1418–1450.CrossRefGoogle Scholar
  69. Terman, D. (1992). The transition from bursting to continuous spiking in excitable membrane models. Journal of Nonliear Science, 2(2), 135–182.CrossRefGoogle Scholar
  70. Tikhonov, A. N. (1948). On the dependence of solutions of differential equations from a small parameter. Matemati(̌c)eskij Sbornik, 22(64), 193–204.Google Scholar
  71. Wang, X. J. (1993). Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle. Physica D, 62(1–4), 263–274.CrossRefGoogle Scholar
  72. Yang, Z., Qishao, L., & Li, L. (2006). The genesis of period-adding bursting without bursting-chaos in the Chay model. Chaos, Solitons and Fractals, 27(3), 689–697.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Paul Channell
    • 1
  • Ibiyinka Fuwape
    • 2
    • 3
  • Alexander B. Neiman
    • 2
  • Andrey L. Shilnikov
    • 1
    • 4
    Email author
  1. 1.Department of Mathematics and StatisticsGeorgia State UniversityAtlantaUSA
  2. 2.Department of Physics and AstronomyOhio UniversityAthensUSA
  3. 3.Department of PhysicsFederal University, of TechnologyAkureNigeria
  4. 4.The Neuroscience InstituteGeorgia State UniversityAtlantaUSA

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