[Paper Review] Bursting in a Subcritical Hopf Oscillator with a Nonlinear Feedback

G. Sethia, Abhijit Sen|ArXiv.org|Mar 24, 2006

stochastic dynamics and bifurcation12 references3 citations

TL;DR

This paper proposes a subcritical Hopf oscillator with nonlinear time-delayed feedback to model neuronal bursting dynamics. By tuning feedback strength and time delay, the system exhibits either parabolic (SNLC/SNLC) or square-wave (SN/SSL) bursting, where time delay enhances spiking frequency and burst duration, especially in the bistable regime induced by sufficient 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.

Motivation & Objective

• To model neuronal bursting dynamics using a minimal mathematical framework that captures essential bifurcation mechanisms.
• To investigate how time-delayed nonlinear feedback alters bifurcation scenarios in a subcritical Hopf oscillator near the bifurcation point.
• To explore the emergence of different bursting types (parabolic and square-wave) through control of feedback strength and time delay.
• To understand how time delay enhances spiking rate and burst duration, relevant for reliable neuronal signaling.
• To identify the transition between monostable and bistable regimes induced by time delay, enabling distinct bursting patterns.

Proposed method

• A normal form of the subcritical Hopf oscillator is used, with the bifurcation parameter set to zero (μ = 0), placing the system at the critical point.
• A quadratic nonlinear feedback term is introduced, with time delay τ to model signal propagation effects in neural systems.
• The governing equation is expressed in complex form: ẋ(t) = (i(ω + b|z|²) + |z|² - |z|⁴)z(t) - kz²(t - τ), with z = x + iy.
• Bifurcation analysis is performed using XPPAUT and AUTO for τ = 0, and DDE-BIFTOOL for time-delayed systems.
• The system is driven by a slow periodic external stimulus (ε = 0.02, Ω = 0.01) to simulate physiological bursting.
• Bifurcation diagrams and limit cycle frequency analysis are used to identify SNLC (saddle-node on limit cycle) and SSL (saddle separatrix loop) bifurcations.

Experimental results

Research questions

• RQ1 How does time-delayed nonlinear feedback alter the bifurcation structure of a subcritical Hopf oscillator?
• RQ2 What types of bursting dynamics (parabolic vs. square-wave) emerge from the interplay of feedback strength and time delay?
• RQ3 How does time delay affect the spiking rate and burst duration in the resulting bursting patterns?
• RQ4 What is the role of the bistable regime—induced by sufficient time delay—in enabling square-wave bursting?
• RQ5 Can the model generate controllable bursting with tunable spike frequency and burst width using only feedback parameters?

Key findings

• For τ = 0, the system exhibits a saddle-node on a limit cycle (SNLC) bifurcation at k_c ≈ 0.42506, leading to parabolic bursting when driven by a slow stimulus.
• With finite time delay (τ > 0), the bifurcation structure changes: the stable limit cycle branch extends beyond k_c, creating a bistable region.
• When τ < τ_t (threshold delay), the system remains monostable; increasing τ enhances spiking rate and burst width without inducing bistability.
• For τ ≥ τ_t (e.g., τ = 0.5), the system enters a bistable regime where the onset of spiking occurs via SNLC and termination via SSL, producing square-wave bursting.
• The spike rate increases and burst duration widens with increasing time delay, particularly in the bistable regime, due to slower dynamics near the SSL.
• The frequency of limit cycles tends to zero as the SNLC bifurcation is approached, confirming the infinite-period nature of the transition.