PARABOLIC BURSTING REVISITED

Many excitable membrane systems display bursting oscillations, in whic h the membrane potential switches periodically between an active phase of rapid spiking and a silent phase of slow, quasi steady-state behav ior. A burster is called parabolic when the spike frequency is lower b oth at the beginning and end of the active phase. We show that classes of voltage-gated conductance equations can be reduced to the mathemat ical mechanism previously analyzed by Ermentrout and Kopell in [7]. Th e reduction uses a series of coordinate changes and shows that the mec hanism in [7] applies more generally than previously believed. The key hypothesis for the more general theory is that a certain slow periodi c 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 co nstant.