On the interpretation of Dirac δ pulses in differential equations for phase oscillators

In this note, we discuss the usage of the Dirac δ function in models of phase oscillators with pulsatile inputs. Many authors use a product of the delta function and the phase response curve in the right-hand side of an ordinary differential equation to describe the discontinuous phase dynamics in such systems. We point out that this notation has to be treated with care as it is ambiguous. We argue that the presumably most canonical interpretation does not lead to the intended behavior in many cases.