Skip to main content

Bursting Oscillations

  • Chapter
  • First Online:
Mathematical Foundations of Neuroscience

Part of the book series: Interdisciplinary Applied Mathematics ((IAM,volume 35))

  • 7364 Accesses

Abstract

Many neurons exhibit much more complicated firing patterns than simple repetitive firing. A common mode of firing in many neurons and other excitable cells is bursting oscillations. This is characterized by a silent phase of near-steady-state resting behavior alternating with an active phase of rapid, spikelike oscillations. Examples of bursting behavior are shown in Fig. 5.1. Note that bursting arises in neuronal structures throughout the central nervous system. Bursting activity in certain thalamic cells, for example, is implicated in the generation of sleep rhythms, whereas patients with parkinsonian tremor exhibit increased bursting activity in neurons within the basal ganglia. Cells involved in the generation of respiratory rhythms within the pre-Botzinger complex also display bursting oscillations.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 99.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. S. M. Baer, T. Erneux, and J. Rinzel. The slow passage through a hopf bifurcation: Delay memory effects, and resonances. SIAM J. Appl. Math., 49:55–71, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  2. R. Bertram, M. J. Butte, T. Kiemel, and A. Sherman. Topological and phenomenological classification of bursting oscillations. Bull. Math. Biol., 57:413–439, 1995.

    MATH  Google Scholar 

  3. S. Coombes and P. C. Bressloff. Bursting: The Genesis of Rhythm in the Nervous System. World Scientific, Singapore, 2005.

    Book  MATH  Google Scholar 

  4. A. Destexhe and P. Gaspard. Bursting oscillations from a homoclinic tangency in a time delay system. Phys. Lett. A, 173:386–391, 1993.

    Article  Google Scholar 

  5. G. B. Ermentrout and N. Kopell. Frequency plateaus in a chain of weakly coupled oscillators. I. SIAM J. Math. Anal., 15(2):215–237, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Hertz, A. Krogh, and R. Palmer. Introduction To The Theory of Neural Computation, Volume I. Perseus Books, New York, 1991.

    Google Scholar 

  7. X. Huang, W. C. Troy, Q. Yang, H. Ma, C. R. Laing, S. J. Schiff, and J. Y. Wu. Spiral waves in disinhibited mammalian neocortex. J. Neurosci., 24:9897–9902, 2004.

    Article  Google Scholar 

  8. E. M. Izhikevich. Phase equations for relaxation oscillators. SIAM J. Appl. Math., 60(5):1789–1804 (electronic), 2000.

    Article  MathSciNet  MATH  Google Scholar 

  9. L. Lapicque. Recherches quantitatives sur l’excitation electrique des nerfs traitee comme une plarization. J. Physiol. Gen., 9:620–635, 1907.

    Google Scholar 

  10. P. C. Matthews, R. E. Mirollo, and S. H. Strogatz. Dynamics of a large system of coupled nonlinear oscillators. Phys. D, 52(2-3):293–331, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  11. W. Rall and J. Rinzel. Branch input resistance and steady attenuation for input to one branch of a dendritic neuron model. Biophys. J., 13:648–688, 1973.

    Article  Google Scholar 

  12. L. Ren and G. B. Ermentrout. Monotonicity of phaselocked solutions in chains and arrays of nearest-neighbor coupled oscillators. SIAM J. Math. Anal., 29(1):208–234 (electronic), 1998.

    Article  MathSciNet  MATH  Google Scholar 

  13. M. E. Rush and J. Rinzel. The potassium A-current, low firing rates and rebound excitation in Hodgkin–Huxley models. Bull. Math. Biol., 57:899–929, 1995.

    MATH  Google Scholar 

  14. S. H. Strogatz. Nonlinear Dynamics and Chaos. Addison-Wesley, Readings, MA, 1984.

    Google Scholar 

  15. S. H. Strogatz. From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Phys. D, 143(1-4):1–20, 2000. Bifurcations, patterns and symmetry.

    Article  MathSciNet  MATH  Google Scholar 

  16. D. Ulrich. Dendritic resonance in rat neocortical pyramidal cells. J. Neurophysiol., 87:2753–2759, 2002.

    Google Scholar 

  17. L. Ren and G. B. Ermentrout. Monotonicity of phaselocked solutions in chains and arrays of nearest-neighbor coupled oscillators. SIAM J. Math. Anal., 29(1):208–234 (electronic), 1998.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to G. Bard Ermentrout .

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Ermentrout, G.B., Terman, D.H. (2010). Bursting Oscillations. In: Mathematical Foundations of Neuroscience. Interdisciplinary Applied Mathematics, vol 35. Springer, New York, NY. https://doi.org/10.1007/978-0-387-87708-2_5

Download citation

Publish with us

Policies and ethics