Slow Manifold of a Neuronal Bursting Model

Jean-Marc Ginoux (1) , Bruno Rossetto

1 PROTEE - Processus de Transfert et d'Echanges dans l'Environnement - EA 3819

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.

Domains: Dynamical Systems [math.DS]

Jean-Marc Ginoux, Bruno Rossetto. Slow Manifold of a Neuronal Bursting Model. M.A. Aziz-Alaoui & C. Bertelle. Emergent Properties in Natural and Artificial Dynamical Systems, Springer Berlin Heidelberg, pp.119-128, 2006, Understanding Complex Systems, 978-3-540-34822-1. ⟨10.1007/3-540-34824-7_6⟩. ⟨hal-01056181⟩