Skip to main content
SpringerLink
Log in

Burst-duration mechanism of in-phase bursting in inhibitory networks

  • L. P. Shilnikov-75
  • Special Issue
  • Published:
Regular and Chaotic Dynamics Aims and scope Submit manuscript

Abstract

We study the emergence of in-phase and anti-phase synchronized rhythms in bursting networks of Hodgkin-Huxley-type neurons connected by inhibitory synapses. We show that when the state of the individual neuron composing the network is close to the transition from bursting into tonic spiking, the appearance of the network’s synchronous rhythms becomes sensitive to small changes in parameters and synaptic coupling strengths. This bursting-spiking transition is associated with codimension-one bifurcations of a saddle-node limit cycle with homoclinic orbits, first described and studied by Leonid Pavlovich Shilnikov. By this paper, we pay tribute to his pioneering results and emphasize their importance for understanding the cooperative behavior of bursting neurons. We describe the burst-duration mechanism of inphase synchronized bursting in a network with strong repulsive connections, induced by weak inhibition. Through the stability analysis, we also reveal the dual property of fast reciprocal inhibition to establish in- and anti-phase synchronized bursting.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Rinzel, J., Bursting Oscillations in an Excitable Membrane Model, Ordinary and Partial Differential Equations (Dundee, 1984), B.D. Sleeman and R. J. Jarvis (Eds.), Lecture Notes in Math., vol. 1151, New York: Springer, 1985, pp. 304–316.

    Chapter  Google Scholar 

  2. Rinzel, J., A Formal Classification of Bursting Mechanisms in Excitable Systems, Mathematical Topics in Population Biology, Morphogenesis and Neuroscience, E. Teramoto and M. Yamaguti (Eds.), Lecture Notes in Biomath., vol. 71, Berlin: Springer, 1987, pp. 267–281.

    Google Scholar 

  3. Rinzel, J. and Ermentrout, B., Analysis of Neural Excitability and Oscillations, Methods of Neural Modeling: From Synapses to Networks, C. Koch and I. Segev (Eds.), MIT Press, 1989, pp. 135–169.

  4. Ermentrout, G.B. and Kopell, N., Parabolic Bursting in an Excitable System Coupled with a Slow Oscillation, SIAM J. Appl. Math., 1986, vol. 46, pp. 233–253.

    Article  MATH  MathSciNet  Google Scholar 

  5. Terman, D., Chaotic Spikes Arising from a Model of Bursting in Excitable Membranes, SIAM J. Appl. Math., 1991, vol. 51, pp. 1418–1450.

    Article  MATH  MathSciNet  Google Scholar 

  6. Wang, X. J., Genesis of Bursting Oscillations in the Hindmarsh-Rose Model and Homoclinicity to a Chaotic Saddle, Phys. D, 1993, vol. 62, nos. 1–4, pp. 263–274.

    Article  MATH  MathSciNet  Google Scholar 

  7. Bertram, R., Butte, M. J., Kiemel, T., and Sherman, A., Topological and Phenomenologial Classication of Bursting Oscillations, Bull. Math. Biol., 1995, vol. 57, pp. 413–439.

    MATH  Google Scholar 

  8. Izhikevich, E., Neural Excitability, Spiking and Bursting, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2000, vol. 10, pp. 1171–1266.

    Article  MATH  MathSciNet  Google Scholar 

  9. Belykh, V.N., Belykh, I.V., Colding-Joregensen, M., and Mosekilde, E. Homoclinic Bifurcations Leading to Bursting Oscillations in Cell Models, Eur. Phys. J. E. Soft Matter Biol. Phys., 2000, vol. 3, pp. 205–219.

    Article  Google Scholar 

  10. Doiron, B., Laing, C., and Longtin, A., Ghostbursting: A Novel Neuronal Burst Mechanism, Comput. Neurosci., 2002, vol. 12, pp. 5–25.

    Article  Google Scholar 

  11. Shilnikov, A. and Cymbalyuk, G., Homoclinic Bifurcations of Periodic Orbits en route from Tonic-Spiking to Bursting in Neuron Models, Regul. Chaotic Dyn., 2004, vol. 9, pp. 281–297.

    Article  MATH  MathSciNet  Google Scholar 

  12. Shilnikov, A., Calabrese, R., and Cymbalyuk, G., Mechanism of Bi-Stability: Tonic Spiking and Bursting in a Neuron Model, Phys. Rev. E, 2005, vol. 71, 056214, 9 p.

    Google Scholar 

  13. Shilnikov, A. and Cymbalyuk, G., Transition between Tonic-Spiking and Bursting in a Neuron Model via the Blue-Sky Catastrophe, Phys. Rev. Lett., 2005, vol. 94, 048101, 4 p.

    Google Scholar 

  14. Fröhlich, F. and Bazhenov, M., Coexistence of Tonic Firing and Bursting in Cortical Neurons, Phys. Rev. E, 2006, vol. 74, 031922, 7 p.

    Google Scholar 

  15. Chanell, P., Cymbalyuk, G., and Shilnikov, A., Origin of Bursting through Homoclinic Spike Adding in a Neuron Model, Phys. Rev. Lett., 2007, vol. 98, 134101, 4 p.

    Google Scholar 

  16. Wang, X.-J. and Rinzel, J., Alternating and Synchronous Rhythms in Reciprocally Inhibitory Model Neurons, Neural Comput., 1992, vol. 4, pp. 84–97.

    Article  Google Scholar 

  17. Van Vreeswijk, C., Abbott, L. F., and Bard Ermentrout, G., When Inhibition Not Excitation Synchronizes Neural Firing, Comput. Neurosci., 1994, vol. 1, pp. 313–321.

    Article  Google Scholar 

  18. Kopell, N. and Ermentrout, G. B., Mechanisms of Phase-Locking and Frequency Control, Handbook of Dynamical Systems, vol. 2, B. Fiedler (Ed.), Amsterdam: Elsevier, 2002, pp. 3–54.

    Chapter  Google Scholar 

  19. Elson, R. C., Selverston, A. I., Abarbanel, H.D. I., and Rabinovich, M. I., Inhibitory Synchronization of Bursting Inbiological Neurons: Dependence on Synaptic Time Constant, J. Neurophysiol., 2002, vol. 88, pp 1166–1176.

    Google Scholar 

  20. Golomb D. and Rinzel, J., Clustering in Globally Coupled Inhibitory Neurons, Phys. Rev. E, 1993, vol. 48, pp. 4810–4814.

    Article  Google Scholar 

  21. Somers, D. and Kopell, N., Rapid Synchronization through Fast Threshold Modulation, Biol. Cybernet., 1993, vol. 68, pp. 393–407.

    Article  Google Scholar 

  22. Sherman, A., Anti-Phase, Asymmetric, and Aperiodic Oscillations in Excitable Cells: 1. Coupled Bursters, Bull. Math. Biol., 1994, vol. 56, pp. 811–835.

    MATH  Google Scholar 

  23. Terman, D., Kopell, N., and Bose, A., Dynamics of Two Mutually Coupled Slow Inhibitory Neurons, Phys. D, 1998, vol. 117, pp. 241–275.

    Article  MATH  MathSciNet  Google Scholar 

  24. Rubin, J. and Terman, D., Synchronized Activity and Loss of Synchrony among Heterogeneous Conditional Oscillators, SIAM J. Appl. Dyn. Sys., 2002, vol. 1, pp. 146–174.

    Article  MATH  MathSciNet  Google Scholar 

  25. Lewis, T. and Rinzel, J., Dynamics of Spiking Neurons Connected by Both Inhibitory and Electrical Coupling, Comput. Neurosci., 2003, vol. 14, pp. 283–309.

    Article  Google Scholar 

  26. Rubin, J. and Terman, D., Geometric Singular Perturbation Analysis of Neuronal Dynamics, Handbook of Dynamical Systems, vol. 2, B. Fiedler (Ed.), Amsterdam: Elsevier, 2002, pp. 93–146.

    Google Scholar 

  27. Kopell, N. and Ermentrout, G. B., Chemical and Electrical Synapses Perform Complementary Roles in the Synchronization of Interneuronal Networks, Proc. Natl. Acad. Sci. USA, 2004, vol. 101, pp. 15482–15487.

    Article  Google Scholar 

  28. Bondarenko, V.E., Cymbalyuk, G. S., Patel, G., DeWeerth, S.P., and Calabrese, R. L., Bifurcation of Synchronous Oscillations into Torus in a System of Two Reciprocally Inhibitory Silicon Neurons: Experimental Observation and Modeling, Chaos, 2004, vol. 14, pp. 995–1003.

    Article  MATH  MathSciNet  Google Scholar 

  29. Bem, T. and Rinzel, J., Short Duty Cycle Distabilizes a Half-Center Oscillator, by Gap Junctions Can Restabilize the Anti-Phase Pattern, J. Neurophysiol., 2004, vol. 91, pp. 693–703.

    Article  Google Scholar 

  30. Cymbalyuk, G. S., Nikolaev, E. V., and Borisyuk, R.M., In-Phase and Anti-Phase Self-Oscillations in a Model of Two Electrically Coupled Pacemakers, Biol. Cybernet., 1994, vol. 71, pp. 153–160.

    Article  MATH  Google Scholar 

  31. Belykh, I., de Lange, E., and Hasler, M., Synchronization of Bursting Neurons: What Matters in the Network Topology, Phys. Rev. Lett., 2005, vol. 94, 188101, 4 p.

    Google Scholar 

  32. Izhikevich, E.M., Synchronization of Elliptic Bursters, SIAM Rev., 2001, vol. 43, no. 2, pp. 315–344.

    Article  MATH  MathSciNet  Google Scholar 

  33. Van Vreeswijk, C. and Hansel, D., Patterns of Synchrony in Neural Networks with Spike Adaptation, Neural Comput., 2001, vol. 13, pp. 959–992.

    Article  MATH  Google Scholar 

  34. Rabinovich, M. I., Varona, P., Selverston, A. I., and Abarbanel, H.D. I., Dynamical Principles in Neuroscience, Rev. Modern Phys., 2006, vol. 78, no. 4, pp. 1213–1265.

    Article  Google Scholar 

  35. Belykh, I. and Shilnikov, A., When Weak Inhibition Synchronizes Strongly Desynchronizing Networks of Bursting Neurons, Phys. Rev. Lett., 2008, vol. 101, 078102, 4 p.

    Google Scholar 

  36. Kopell, N., Toward a Theory of Modelling Central Pattern Generators, Neural Control of Rhythmic Movements in Vertebrates, A.H. Cohen, S. Rossignol, and S. Grillner (Eds.), New York: Wiley, 1987, pp. 369–413.

    Google Scholar 

  37. Getting, P.A., Emerging Principles Governing the Operation of Neural Networks, Annu. Rev. Neurosci., 1989, vol. 12, pp. 185–204.

    Article  Google Scholar 

  38. Marder, E. and Calabrese, R. L., Principles of Rhythmic Motor Pattern Generation, Physiol. Rev., 1996, vol. 76, no. 3, pp. 687–717.

    Google Scholar 

  39. Marder, E., Kopell, N., and Sigvardt, K., How Computation Aids in Understanding Biological Networks, Neurons, Networks, and Motor Behavior, P. S.G. Stein, A. Selverston, S. Grillner, and D. G. Stuart (Eds.), Cambridge: MIT Press, 1998, pp. 139–150.

    Google Scholar 

  40. Kristan, W. B., Calabrese, R. L., and Friesen, W.O., Neuronal Control of Leech Behavior, Progr. Neurobiol., 2005, vol. 76, pp. 279–327.

    Article  Google Scholar 

  41. Kristan, W. B., and Katz, P., Form and Function in Systems Neuroscience, Curr. Biol., 2006, vol. 16, R828–R831.

    Article  Google Scholar 

  42. Briggman, K.L. and Kristan, W. B., Multifunctional Pattern-Generating Circuits, Annu. Rev. Neurosci., 2008, vol. 31, pp. 271–294.

    Article  Google Scholar 

  43. Katz, P. S., Tritonia, http://www.scholarpedia.org/article/Tritonia.

  44. Brown, T.G., The Intrinsic Factors in the Act of Progerssion in the Mammal, Proc. R. Soc. Lond. B, 1911, vol. 84, pp. 308–319.

    Article  Google Scholar 

  45. Brown, T.G., On the Nature of the Fundamental Activity of the Nervous Ventres: Together with an Analysis of the Conditioning of Rhythmic Activity in Progression, and a Theory of the Evolution of Function in the Nervous System, J. Physiol., 1914, vol. 48, pp. 18–46.

    Google Scholar 

  46. Canavier, C.C., Baxter, D.A., Clark, J. W., and Byrne, J. H., Control of Multistability in Ring Circuits of Oscillators, Biol. Cybernet., 1999, vol. 80, pp. 87–102.

    Article  MATH  Google Scholar 

  47. Baxter, D.A., Lechner, H. A., Canavier, C. C., Butera, R. J., Franceschi, A. A., Clark, J.W., and Byrne, J. H., Coexisting Stable Oscillatory States in Single Cell and Multicellular Neuronal Oscillators, Oscillations in Neural Systems, D. S. Levine, V. R. Brown, V. T. Shirey (Eds.), Hillsdale, NJ: Erlbaum Associates, 1999, pp. 51–78.

    Google Scholar 

  48. Prinz, A. A., Bucher, D., and Marder, E., Similar Network Activity from Disparate Circuit Parameters, Nature Neurosci., 2004, vol. 7, pp. 1345–1352.

    Article  Google Scholar 

  49. Shilnikov, A., Gordon, R., and Belykh, I., Polyrhythmic Synchronization in Bursting Network Motifs, Chaos, 2008, vol. 18, 037120, 13 p.

    Article  MathSciNet  Google Scholar 

  50. Cymbalyuk, G. S., Gaudry, Q., Masino, M. A., and Calabrese, R. L., Bursting in Leech Heart Interneurons: Cell Autonomous and Network Based Mechanisms, J. Neurosci., 2002, vol. 22, pp. 10580–10592.

    Google Scholar 

  51. Tobin, A.-E. and Calabrese, R. L., Endogenous and Half-Center Bursting in Morphologically-Inspired Models of Leech Heart Interneurons, J. Neurophysiol., 2006, vol. 96, pp. 2089–2106.

    Article  Google Scholar 

  52. Jalil, S., Belykh, I., and Shilnikov, A., Fast Reciprocal Inhibition Can Synchronize Bursting Neurons, submitted for publication in Phys. Rev. E.

  53. De Lange, E. and Kopell, N., Fast Threshold Modulation, Scholarpedia

  54. Turaev, D.V. and Shilnikov, L.P., Blue Sky Catastrophe, Dokl. Math., 1995, vol. 51, pp. 404–407.

    MATH  Google Scholar 

  55. Shilnikov, A., Shilnikov, L.P., and Turaev, D.V., Blue Sky Catastrophe in Singularly Perturbed Systems, Mosc. Math. J., 2005, vol. 5, no. 1, pp. 269–282.

    MATH  MathSciNet  Google Scholar 

  56. Lukyanov, V. and Shilnikov, L.P., On Some Bifurcations of Dynamical Systems with Homoclinic Structures, Dokl. Akad. Nauk SSSR, 1978, vol. 243, no. 1, pp. 26–29 [Soviet Math. Dokl., 1978, vol. 19, 1314–1318].

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Georgia State University, 30 Pryor Street, Atlanta, GA, 30303, USA

    I. Belykh, S. Jalil & A. Shilnikov

  2. Neuroscience Institute, Georgia State University, 30 Pryor Street, Atlanta, GA, 30303, USA

    I. Belykh, S. Jalil & A. Shilnikov

Authors
  1. I. Belykh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Jalil
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Shilnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. Belykh.

Additional information

Dedicated to Leonid P. Shilnikov on the occasion of his 75th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Belykh, I., Jalil, S. & Shilnikov, A. Burst-duration mechanism of in-phase bursting in inhibitory networks. Regul. Chaot. Dyn. 15, 146–158 (2010). https://doi.org/10.1134/S1560354710020048

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1560354710020048

MSC2000 numbers

Key words

Search

Navigation