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Stability and bifurcation analysis of coupled Fitzhugh-Nagumo oscillators. (English) Zbl 1184.92009

Suzuki, Masakazu (ed.) et al., The joint conference of ASCM 2009 and MACIS 2009. 9th international conference on Asian symposium on computer mathematics and 3rd international conference on mathematical aspects of computer and information sciences, Fukuoka, Japan, December 14–17, 2009. Selected papers. Fukuoka: Kyushu University, Faculty of Mathematics. COE Lecture Note 22, 356-359 (2009).
Summary: Neurons are the central biological objects in understanding how the brain works. The famous Hodgkin-Huxley model, which describes how action potentials of a neuron are initiated and propagated, consists of four coupled nonlinear differential equations. Because these equations are difficult to deal with, there also exist several simplified models, of which many exhibit polynomial-like nonlinearity. Examples of such models are the Fitzhugh-Nagumo (FHN) model, the Hindmarsh-Rose (HR) model, the Morris-Lecar (ML) model and the Izhikevich model.
We first prescribe the biologically relevant parameter ranges for the FHN model and subsequently study the dynamical behaviour of coupled neurons on small networks of two or three nodes. To do this, we use a computational real algebraic geometry method called the discriminant variety (DV) method [see D. Lazard and F. Rouiller, J. Symb. Comput. 42, No. 6, 636–667 (2007; Zbl 1156.14044)] to perform the stability and bifurcation analysis of these small networks. A time series analysis of the FHN model can be found elsewhere in related work.
For the entire collection see [Zbl 1181.68015].

MSC:

92C20 Neural biology
55R80 Discriminantal varieties and configuration spaces in algebraic topology
68W30 Symbolic computation and algebraic computation

Citations:

Zbl 1156.14044

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