Cross-currents between biology and mathematics on models of bursting

HM Osinga, Arthur Sherman, KT Tsaneva-Atanasova

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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. The most rigorous approach is to characterize the bifurcations found in the neighborhood of a singularity. 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. 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, we are unable to determine a codimension-three singularity that supports fold/subHopf bursting. Furthermore, we believe that it is not possible to find a codimension-three singularity that gives rise to all known types of bursting. Instead, we identify a three-dimensional slice that contains all known types of bursting in a partial unfolding of a doubly-degenerate Bodganov–Takens point, which has codimension four.
Original languageEnglish
Publication statusUnpublished - 2011

Bibliographical note

Sponsorship: The research of H.M.O. was supported by an EPSRC Advanced Research Fellowship grant, that of A.S. by the Intramural Research Program, NIDDK, NIH, and that K.T. T.-A. by an EPSRC grant (EP/I018638/1).

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