Hyperbolic Plykin attractor can exist in neuron models

V. Belykh, I. Belykh, Erik Mosekilde

Research output: Contribution to journalJournal articleResearchpeer-review


Strange hyperbolic attractors are hard to find in real physical systems. This paper provides the first example of a realistic system, a canonical three-dimensional (3D) model of bursting neurons, that is likely to have a strange hyperbolic attractor. Using a geometrical approach to the study of the neuron model, we derive a flow-defined Poincare map giving ail accurate account of the system's dynamics. In a parameter region where the neuron system undergoes bifurcations causing transitions between tonic spiking and bursting, this two-dimensional map becomes a map of a disk with several periodic holes. A particular case is the map of a disk with three holes, matching the Plykin example of a planar hyperbolic attractor. The corresponding attractor of the 3D neuron model appears to be hyperbolic (this property is not verified in the present paper) and arises as a result of a two-loop (secondary) homoclinic bifurcation of a saddle. This type of bifurcation, and the complex behavior it can produce, have not been previously examined.
Original languageEnglish
JournalInternational Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Issue number11
Pages (from-to)3567-3578
Publication statusPublished - 2005


  • neuron model
  • geometrical approach
  • Plykin attractor


Dive into the research topics of 'Hyperbolic Plykin attractor can exist in neuron models'. Together they form a unique fingerprint.

Cite this