On the interpretation of Dirac $\delta$ pulses in differential equations for phase oscillators

TL;DR

This paper examines the use of Dirac delta functions in phase oscillator models, highlighting potential ambiguities and issues with common interpretations that affect the accuracy of modeling discontinuous phase dynamics.

Contribution

It clarifies the interpretation of Dirac delta functions in phase oscillator models and discusses the implications of different treatments on the system's behavior.

Findings

Common interpretations of delta functions can lead to unintended dynamics.

Careful treatment of delta functions is necessary for accurate phase modeling.

The canonical interpretation may not always produce the expected results.

Abstract

In this note we discuss the usage of the Dirac $\delta$ 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 ODE 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 behaviour in many cases.