TY - UNPB

T1 - Cross-currents between biology and mathematics on models of bursting

AU - Osinga, HM

AU - Sherman, Arthur

AU - Tsaneva-Atanasova, KT

N1 - 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).

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

M3 - Working paper and Preprints

BT - Cross-currents between biology and mathematics on models of bursting

ER -