  Pulse-coupled threshold units serve as paradigmatic models for a wide range
of complex systems. When the state variable of a unit crosses a threshold, the
unit sends a pulse that is received by other units, thereby mediating the
interactions. At the same time, the state variable of the sending unit is
reset. Here we study pulse-coupled models with a reset that may be partial only
and is mediated by a partial reset function. Such a partial reset characterizes
intrinsic physical or biophysical features of a unit (e.g., resistive coupling
between dendrite and soma of compartmental neurons) and at the same time makes
possible a rigorous mathematical investigation of the collective network
dynamics.
  The partial reset acts as a desynchronization mechanism. For all-to-all
pulse-coupled oscillators an increase in the strength of the partial reset
causes a sequence of desynchronizing bifurcations from the fully synchronous
state via states with large clusters of synchronized units through states with
smaller clusters to complete asynchrony. We analytically derive sufficient and
necessary conditions for the existence and stability of periodic cluster states
on the local dynamics of the oscillators and on the partial reset function to
reveal the mechanism underlying the desynchronization transition. We show that
the entire sequence may occur due to arbitrarily small changes of the partial
reset and is robust against structural perturbations.
