Slow Manifold of a Neuronal Bursting Model

TL;DR

This paper introduces a novel Differential Geometry-based method to analytically determine the slow manifold in neuronal bursting models, overcoming limitations of traditional singular approximation techniques.

Contribution

A new approach adapted from Differential Geometry is proposed for analyzing the slow manifold in neuronal bursting models, demonstrated on the Hindmarsh-Rose model.

Findings

Successfully applied to the Hindmarsh-Rose model

Provides three equivalent methods for slow manifold determination

Overcomes limitations of singular approximation methods

Abstract

Comparing neuronal bursting models (NBM) with slow-fast autonomous dynamical systems (S-FADS), it appears that the specific features of a (NBM) do not allow a determination of the analytical slow manifold equation with the singular approximation method. So, a new approach based on Differential Geometry, generally used for (S-FADS), is proposed. Adapted to (NBM), this new method provides three equivalent manners of determination of the analytical slow manifold equation. Application is made for the three-variables model of neuronal bursting elaborated by Hindmarsh and Rose which is one of the most used mathematical representation of the widespread phenomenon of oscillatory burst discharges that occur in real neuronal cells.