Fast-slow bursters in the unfolding of a high codimension singularity and the ultra-slow transitions of classes

TL;DR

This paper introduces a model combining fast and slow subsystems, using a high codimension singularity unfolding, to replicate and analyze various bursting behaviors and their transitions in physical and biological systems.

Contribution

It presents a novel framework that integrates high codimension singularity unfolding with slow modulation to produce and transition between diverse bursting classes.

Findings

Model reproduces nearly all bursting classes predicted for planar fast systems.

Transitions between bursting classes can be controlled via ultra-slow parameter modulation.

The framework predicts possible bursting pattern transitions in complex systems.

Abstract

Bursting is a phenomenon found in a variety of physical and biological systems. For example, in neuroscience, bursting is believed to play a key role in the way information is transferred in the nervous system. In this work, we propose a model that, appropriately tuned, can display several types of bursting behaviors. The model contains two subsystems acting at different timescales. For the fast subsystem we use the planar unfolding of a high codimension singularity. In its bifurcation diagram, we locate paths that underly the right sequence of bifurcations necessary for bursting. The slow subsystem steers the fast one back and forth along these paths leading to bursting behavior. The model is able to produce almost all the classes of bursting predicted for systems with a planar fast subsystems. Transitions between classes can be obtained through an ultra-slow modulation of the model's parameters. A detailed exploration of the parameter space allows predicting possible transitions. This provides a single framework to understand the coexistence of diverse bursting patterns in physical and biological systems or in models.