Partial Reset in Pulse-Coupled Oscillators

Abstract

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 present and analyze a class of pulse-coupled oscillators where the reset 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; at the same time the description in terms of a partial reset enables a rigorous mathematical investigation of the collective network dynamics. The partial reset acts as a desynchronization mechanism. For N 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 c...