title: Parabolic bursting revisited.
abstract: Many excitable membrane systems display bursting oscillations, in which the membrane potential switches periodically between an active phase of rapid spiking and a silent phase of slow, quasi steady-state behavior. A burster is called parabolic when the spike frequency is lower both at the beginning and end of the active phase. We show that classes of voltage-gated conductance equations can be reduced to the mathematical mechanism previously analyzed by Ermentrout and Kopell in [7]. The reduction uses a series of coordinate changes and shows that the mechanism in [7] applies more generally than previously believed. The key hypothesis for the more general theory is that a certain slow periodic orbit must stay close to a curve of degenerate homoclinic points for the fast system, at least during the active phase. We do not require that the slow system have a periodic orbit when the voltage is held constant.
authors.affiliation: Boston University Department of Mathematics, MA 02215, USA.
authors.forename: C
authors.initials: C
authors.lastname: Soto-Treviño
authors.forename: N
authors.initials: N
authors.lastname: Kopell
authors.forename: D
authors.initials: D
authors.lastname: Watson
mesh_terms: D000818:Animals; D002468:Cell Physiological Phenomena; D004553:Electric Conductivity; D008433:Mathematics; D008564:Membrane Potentials; D008954:Models, Biological; D009474:Neurons / Q000502:physiology; D009991:Oscillometry