Dynamical systems analysis of spike-adding mechanisms in transient bursts

Area of Science:

• Computational neuroscience
• Mathematical biology
• Dynamical systems theory

Background:

• Transient bursting in excitable cells, like neurons, is experimentally observed but lacks a clear theoretical explanation.
• Existing models often focus on periodic bursting, leaving transient behaviors less understood.

Purpose of the Study:

• To theoretically elucidate the underlying mechanism of transient bursting behavior in a simplified model of cellular excitability.
• To characterize the novel spike-adding mechanism responsible for transient bursting.

Main Methods:

• Analysis of a five-dimensional simplified model of after-depolarization exhibiting transient bursting.
• Application of one-parameter continuation of perturbed orbit segments as a boundary value problem.
• Utilizing geometric singular perturbation theory to explain the spike-adding mechanism.

Main Results:

• Identified a canard-like spike-adding transition responsible for transient bursting, distinct from mechanisms for periodic bursts.
• Demonstrated that unstable sheets of the critical manifold, arising from saddle equilibria in the singular limit, drive the spike-adding transition.
• Showed the transition involves a fast trajectory between two unstable saddle-type sheets of the slow manifold.

Conclusions:

• The spike-adding mechanism provides a theoretical framework for understanding transient bursting in excitable systems.
• Geometric singular perturbation theory offers insights into non-bifurcation mechanisms driving complex cellular dynamics.
• The presence of saddle equilibria significantly influences the nature of transient bursting dynamics.