Bursting and two-parameter bifurcation in the Chay neuronal model

Lixia Duan; Zhuoqin Yang; Shenquan Liu; Dunwei Gong

doi:10.3934/dcdsb.2011.16.445

Article Contents

2011, Volume 16, Issue 2: 445-456. Doi: 10.3934/dcdsb.2011.16.445

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Bursting and two-parameter bifurcation in the Chay neuronal model

1.
College of Science, North China University of Technology, Beijing 100144
2.
School of Mathematics and System Sciences, Beihang University, Beijing 100191
3.
Department of Mathematics, South China University of Technology, Guangzhou 510641
4.
School of Information and Electrical Engineering, China University of Mining and Technology, Xuzhou, Jiangsu 221008

Received: December 31, 2009

Revised: January 31, 2011

Published: August 2011

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Abstract

In this paper, we study and classify the firing patterns in the Chay neuronal model by the fast/slow decomposition and the two-parameter bifurcations analysis. We show that the Chay neuronal model can display complex bursting oscillations, including the "fold/fold" bursting, the "Hopf/Hopf" bursting and the "Hopf/homoclinic" bursting. Furthermore, dynamical properties of different firing activities of a neuron are closely related to the bifurcation structures of the fast subsystem. Our results indicate that the codimension-2 bifurcation points and the related codimension-1 bifurcation curves of the fast-subsystem can provide crucial information to predict the existence and types of bursting with changes of parameters.

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Mathematics Subject Classification: Primary: 34C15; Secondary: 92C20.

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