  Interaction via pulses is common in many natural systems, especially
neuronal. In this article we study one of the simplest possible systems with
pulse interaction: a phase oscillator with delayed pulsatile feedback. When the
oscillator reaches a specific state, it emits a pulse, which returns after
propagating through a delay line. The impact of an incoming pulse is described
by the oscillator's phase reset curve (PRC). In such a system we discover an
unexpected phenomenon: for a sufficiently steep slope of the PRC, a periodic
regular spiking solution bifurcates with several multipliers crossing the unit
circle at the same parameter value. The number of such critical multipliers
increases linearly with the delay and thus may be arbitrary large. This
bifurcation is accompanied by the emergence of numerous "jittering" regimes
with non-equal interspike intervals (ISIs). Each of these regimes corresponds
to a periodic solution of the system with a period roughly proportional to the
delay. The number of different "jittering" solutions emerging at the
bifurcation point increases exponentially with the delay. We describe the
combinatorial mechanism that underlies the emergence of such a variety of
solutions. In particular, we show how a periodic solution exhibiting several
distinct ISIs can imply the existence of multiple other solutions obtained by
rearranging of these ISIs. We show that the theoretical results for phase
oscillators accurately predict the behavior of an experimentally implemented
electronic oscillator with pulsatile feedback.
