TY - JOUR
T1 - Existence and stability of limit cycles in a macroscopic neuronal population model
AU - Rodrigues, S.
AU - Gonçalves, J.
AU - Terry, J.R.
PY - 2007
Y1 - 2007
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?eid=2-s2.0-34547608099&partnerID=MN8TOARS
U2 - 10.1016/j.physd.2007.06.010
DO - 10.1016/j.physd.2007.06.010
M3 - Article
SN - 0167-2789
VL - 233
SP - 39
EP - 65
JO - Physica D - Nonlinear Phenomena
JF - Physica D - Nonlinear Phenomena
IS - 1
ER -