Encyclopedia of Computational Neuroscience

Living Edition
| Editors: Dieter Jaeger, Ranu Jung

Bifurcations Dynamics of Single Neurons and Small Networks

Living reference work entry
DOI: https://doi.org/10.1007/978-1-4614-7320-6_453-1

Definition

A bifurcation occurs when a system undergoes a qualitative change in its output as a result of a change in parameter. Under certain conditions, the voltage of a cell membrane can change from being at rest to becoming oscillatory as a result of a bifurcation. Oscillatory properties of small networks are often understood using bifurcation analysis.

Description

What Is a Bifurcation?

The word bifurcate is commonly used to denote a split, as in “up ahead, the road bifurcates into two parts.” The use of the term in this context implies that a driver traveling on this road can make a real-time choice of being able to take one or the other branch of the road when the bifurcation point is reached. In mathematics, and in its application to neuroscience, the term bifurcation has an entirely different contextual meaning. While a mathematical bifurcation is similar in that it involves different branches, the decision on which branch is chosen is largely made a priori and depends on the...

Keywords

Depression 
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Notes

Acknowledgment

This work was supported in part by NSF DMS 1122291.

References

  1. Belle M, Diekman C, Forger D, Piggins H (2009) Daily electrical silencing in the mammalian circadian clock. Science 326:281–284PubMedCrossRefGoogle Scholar
  2. Bose A, Manor Y, Nadim F (2001) Bistable oscillations arising from synaptic depression. SIAM J Appl Math 62:706–727CrossRefGoogle Scholar
  3. Ermentrout GB (1996) Type i membranes, phase resetting curves and synchrony. Neur Comp 8:979–1001CrossRefGoogle Scholar
  4. Ermentrout GB, Kopell N (1998) Fine structure of neural spiking and synchronization in the presence of conduction delays. Proc Natl Acad Sci 95:1259–1264PubMedCentralPubMedCrossRefGoogle Scholar
  5. Ermentrout GB, Terman D (2010) Mathematical foundations of neuroscience. Springer, New YorkCrossRefGoogle Scholar
  6. Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer, New YorkCrossRefGoogle Scholar
  7. Izhikevich E (2000) Neural excitability, spiking and bursting. Int J Bifur Chaos 10:1171–1266CrossRefGoogle Scholar
  8. Morris C, Lecar H (1981) Voltage oscillations in the barnacle giant muscle fiber. Biophys J 35:193–213PubMedCentralPubMedCrossRefGoogle Scholar
  9. Nusse H, Ott E, Yorke J (1994) Border-collision bifurcations: an explanation for observed bifurcation phenomena. Phys Rev E 49:1073–1077CrossRefGoogle Scholar
  10. Rinzel J (1985) Bursting oscillations in an excitable membrane model. In: Sleeman B, Jarvis R (eds) Ordinary and partial differential equations: proceedings of the 8th Dundee conference, vol 1151, Lecture notes in mathematics. Springer, New YorkCrossRefGoogle Scholar
  11. Rinzel J (1987) A formal classification of bursting mechanisms in excitable systems. In: Teramoto E, Yamaguti M (eds) Mathematical topics in population biology, morphogenesis and neurosciences, vol 71, Lecture notes in biomathematics. Springer, New YorkGoogle Scholar
  12. Rubin J, Shevtsova N, Ermentrout GB, Smith J, Rybak I (2009) Multiple rhythmic states in a model of the respiratory central pattern generator. J Neurophysiol 101:2146–2165PubMedCentralPubMedCrossRefGoogle Scholar
  13. Skinner F, Kopell N, Marder E (1994) Mechanisms for oscillation and frequency control in reciprocally inhibitory model neural networks. J Comput Neurosci 1:69–87PubMedCrossRefGoogle Scholar
  14. van Vreeswijk C, Abbott L, Ermentrout GB (1994) When inhibition, not excitation synchronizes neural firing. J Comp Neurosci 1:313–321CrossRefGoogle Scholar
  15. Zhang Y, Bose A, Nadim F (2009) The influence of the a-current on the dynamics of an oscillator-follower feed-forward inhibitory network. SIAM J Appl Dyn Syst 8:1564–1590PubMedCentralPubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNew Jersey Institute of TechnologyNewarkUSA