Existence and stability of limit cycles in a macroscopic neuronal population

S Rodrigues, J Goncalves, JR Terry

Research output: Contribution to journalArticle (Academic Journal)peer-review

15 Citations (Scopus)

Abstract

We present rigorous results concerning the existence and stability of limit cycles in a macroscopic model of neuronal activity. The specific model we consider is developed from the Ki set methodology, popularized by Walter Freeman. In particular we focus on a specific reduction of the KII sets, denoted RKII sets. We analyse the unfolding of supercritical Hopf bifurcations via consideration of the normal forms and centre manifold reductions. Subsequently we analyse the global stability of limit cycles on a region of parameter space and this is achieved by applying a new methodology termed Global Analysis of Piecewise Linear Systems. The analysis presented may also be used to consider coupled systems of this type. A number of macroscopic mean-field approaches to modelling human EEG may be considered as coupled RKII networks. Hence developing a theoretical understanding of the onset of oscillations in models of this type has important implications in clinical neuroscience, as limit cycle oscillations have been demonstrated to be critical in the onset of certain types of epilepsy.
Translated title of the contributionExistence and stability of limit cycles in a macroscopic neuronal population
Original languageEnglish
Pages (from-to)39 - 65
Number of pages27
JournalPhysica D: Nonlinear Phenomena
Volume233 (1)
DOIs
Publication statusPublished - Sep 2007

Bibliographical note

Publisher: Elsevier BV

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