Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Cross-currents between biology and mathematics: The codimension of pseudo-plateau bursting

Pages: 2853 - 2877, Volume 32, Issue 8, August 2012      doi:10.3934/dcds.2012.32.2853

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Hinke M. Osinga - Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand (email)
Arthur Sherman - Laboratory of Biological Modeling, N.I.D.D.K. National Institutes of Health, 12A SOUTH DR MSC 5621, Bethesda, MD 20892-5621, United States (email)
Krasimira Tsaneva-Atanasova - Bristol Centre for Applied Nonlinear Mathematics, Department of Engineering Mathematics, University of Bristol, Queen’s Building, University Walk, Bristol BS8 1TR, United Kingdom (email)

Abstract: A great deal of work has gone into classifying bursting oscillations, periodic alternations of spiking and quiescence modeled by fast-slow systems. In such systems, one or more slow variables carry the fast variables through a sequence of bifurcations that mediate transitions between oscillations and steady states. A rigorous classification approach is to characterize the bifurcations found in the neighborhood of a singularity; a measure of the complexity of the bursting oscillation is then given by the smallest codimension of the singularities near which it occurs. Fold/homoclinic bursting, along with most other burst types of interest, has been shown to occur near a singularity of codimension three by examining bifurcations of a cubic Liénard system; hence, these types of bursting have at most codimension three. Modeling and biological considerations suggest that fold/homoclinic bursting should be found near fold/subHopf bursting, a more recently identified burst type whose codimension has not been determined yet. One would expect that fold/subHopf bursting has the same codimension as fold/homoclinic bursting, because models of these two burst types have very similar underlying bifurcation diagrams. However, no codimension-three singularity is known that supports fold/subHopf bursting, which indicates that it may have codimension four. We identify a three-dimensional slice in a partial unfolding of a doubly-degenerate Bodganov-Takens point, and show that this codimension-four singularity gives rise to almost all known types of bursting.

Keywords:  Fast-slow systems, bursting, bifurcation theory, ordinary differential equations.
Mathematics Subject Classification:  Primary: 70K70, 70K45; Secondary: 34C23.

Received: May 2011;      Revised: August 2011;      Available Online: March 2012.