Modeling the modulation of neuronal bursting: a singularity theory approach

TL;DR

This paper introduces a mathematical framework combining singularity theory and geometric perturbations to model neuronal bursting modulation, revealing the role of high-codimension singularities in neuronal dynamics.

Contribution

It presents a novel approach using singularity theory to analyze and understand the modulation of neuronal bursting in conductance-based models.

Findings

Identification of a three-time scale bursting attractor

Relevance of winged cusp singularity in neuronal dynamics

General applicability to conductance-based models

Abstract

Exploiting the specific structure of neuron conductance-based models, the paper investigates the mathematical modeling of neuronal bursting modulation. The proposed approach combines singularity theory and geometric singular perturbations to capture the geometry of multiple time-scales attractors in the neighborhood of high-codimension singularities. We detect a three-time scale bursting attractor in the universal unfolding of the winged cusp singularity and discuss the physiological relevance of the bifurcation and unfolding parameters in determining a physiological modulation of bursting. The results suggest generality and simplicity in the organizing role of the winged cusp singularity for the global dynamics of conductance based models.