  In a series of two papers, we investigate the mechanisms by which complex
oscillations are generated in a class of nonlinear dynamical systems with
resets modeling the voltage and adaptation of neurons. This first paper
presents mathematical analysis showing that the system can support bursts of
any period as a function of model parameters. In continuous dynamical systems
with resets, period-incrementing structures are complex to analyze. In the
present context, we use the fact that bursting patterns correspond to periodic
orbits of the adaptation map that governs the sequence of values of the
adaptation variable at the resets. Using a slow-fast approach, we show that
this map converges towards a piecewise linear discontinuous map whose orbits
are exactly characterized. That map shows a period-incrementing structure with
instantaneous transitions. We show that the period-incrementing structure
persists for the full system with non-constant adaptation, but the transitions
are more complex. We investigate the presence of chaos at the transitions.
