  Several studies have shown that bursting neurons can encode information in
the number of spikes per burst: As the stimulus varies, so does the length of
individual bursts. The represented stimuli, however, vary substantially among
different sensory modalities and different neurons. The goal of this paper is
to determine which kind of stimulus features can be encoded in burst length,
and how those features depend on the mathematical properties of the underlying
dynamical system. We show that the initiation and termination of each burst is
triggered by specific stimulus features whose temporal characteristsics are
determined by the types of bifurcations that initiate and terminate firing in
each burst. As only a few bifurcations are possible, only a restricted number
of encoded features exists. Here we focus specifically on describing parabolic,
square-wave and elliptic bursters. We find that parabolic bursters, whose
firing is initiated and terminated by saddle-node bifurcations, behave as
prototypical integrators: Firing is triggered by depolarizing stimuli, and
lasts for as long as excitation is prolonged. Elliptic bursters, contrastingly,
constitute prototypical resonators, since both the initiating and terminating
bifurcations possess well-defined oscillation time scales. Firing is therefore
triggered by stimulus stretches of matching frequency and terminated by a
phase-inversion in the oscillation. The behavior of square-wave bursters is
somewhat intermediate, since they are triggered by a fold bifurcation of cycles
of well-defined frequency but are terminated by a homoclinic bifurcation
lacking an oscillating time scale. These correspondences show that stimulus
selectivity is determined by the type of bifurcations. We also demonstrate that
additional biological properties that do not modify the bifurcation structure
play a minor role in stimulus encoding.
