Bursting in a Subcritical Hopf Oscillator with a Nonlinear Feedback

TL;DR

This paper investigates how nonlinear feedback and time delay influence bursting behavior in a subcritical Hopf oscillator, revealing bifurcation scenarios and effects on spiking and burst duration.

Contribution

It introduces a detailed numerical analysis of bursting phenomena in a subcritical Hopf oscillator with nonlinear feedback and time delay, highlighting bifurcation changes and dynamical effects.

Findings

Time delay reduces interspike interval within bursts.

Nonlinear feedback induces various bifurcation scenarios.

Burst duration increases with feedback time delay.

Abstract

Bursting is a periodic transition between a quiescent state and a state of repetitive spiking. The phenomenon is ubiquitous in a variety of neurophysical systems. We numerically study the dynamical properties of a normal form of subcritical Hopf oscillator (at the bifurcation point) subjected to a nonlinear feedback. This dynamical system shows an infinite-period or a saddle-node on a limit cycle (SNLC) bifurcation for certain strengths of the nonlinear feedback. When the feedback is time delayed, the bifurcation scenario changes and the limit cycle terminates through a homoclinic or a saddle separatrix loop (SSL) bifurcation. This system when close to the bifurcation point exhibits various types of bursting phenomenon when subjected to a slow periodic external stimulus of an appropriate strength. The time delay in the feedback enhances the spiking rate i.e. reduces the interspike interval in a burst and also increases the width or the duration of a burst.