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September  2011, 16(2): 445-456. doi: 10.3934/dcdsb.2011.16.445

Bursting and two-parameter bifurcation in the Chay neuronal model

Lixia Duan 1, , Zhuoqin Yang 2, , Shenquan Liu 3, and  Dunwei Gong 4,

1. 

College of Science, North China University of Technology, Beijing 100144, China
2. 

School of Mathematics and System Sciences, Beihang University, Beijing 100191, China
3. 

Department of Mathematics, South China University of Technology, Guangzhou 510641, China
4. 

School of Information and Electrical Engineering, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China

Received  January 2010 Revised  February 2011 Published  June 2011

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.
Keywords: neuronal model, bursting., Bifurcation, fast-slow dynamical analysis.
Mathematics Subject Classification: Primary: 34C15; Secondary: 92C2.
Citation: Lixia Duan, Zhuoqin Yang, Shenquan Liu, Dunwei Gong. Bursting and two-parameter bifurcation in the Chay neuronal model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 445-456. doi: 10.3934/dcdsb.2011.16.445
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show all references

References:
[1]

J. Neurosci., 22 (2002), 9053-9062. Google Scholar

[2]

Network: Comput. Neural. Syst., 14 (2003), 103-9118. Google Scholar

[3]

E. M. Izhikevich, et.al., \url{http://www.scholarpedia.org/article/Bursting}., ().   Google Scholar

[4]

in "Ordinary and Partial Differential Equations, Lecture Notes in Mathematics" (eds. B.D. Sleeman, R.J. Jarvis), Springer, Berlin, 1151 (1985), 304-316.  Google Scholar

[5]

Bull. Math. Biol., 57 (1995), 413-439. Google Scholar

[6]

Biol. Cybern., 71 (1994), 281-291. doi: 10.1007/BF00239616.  Google Scholar

[7]

Int. J. of Bifurcation and Chaos, 10 (2000), 1171-1266. doi: 10.1142/S0218127400000840.  Google Scholar

[8]

Physica D, 62 (1993), 263-274. doi: 10.1016/0167-2789(93)90286-A.  Google Scholar

[9]

Bull. Math. Biol., 62 (2000), 695-715. doi: 10.1006/bulm.2000.0172.  Google Scholar

[10]

Chaos, Solitons and Fractals, 18 (2003), 759-773. doi: 10.1016/S0960-0779(03)00027-4.  Google Scholar

[11]

Z. Phys. Chem., 216 (2002), 487-97. doi: 10.1524/zpch.2002.216.4.487.  Google Scholar

[12]

Eur. Phys. J. E, 3 (2000), 205-219. doi: 10.1007/s101890070012.  Google Scholar

[13]

Biol. Cybern., 71 (1994), 417-431. doi: 10.1007/BF00198918.  Google Scholar

[14]

Phys. Lett. A, 319 (2003), 89-96. doi: 10.1016/j.physleta.2003.09.077.  Google Scholar

[15]

Int. J. Bifurcation Chaos, 14 (2004), 4143-4159. doi: 10.1142/S0218127404011892.  Google Scholar

[16]

Sci. China Ser. G-Phys Mech. Astron., 6 (2008), 687-698. Google Scholar

[17]

in "The Genesis of Rhythm in the Nervous System. Bursting" (eds. S. Coombes and P. C. Bressloff), World Scientific Publish, (2005), 89-122. doi: 10.1142/9789812703231_0004.  Google Scholar

[18]

Discrete and Continuous Dynamical Systems Series B, 13 (2010), 503-516.  Google Scholar

[19]

Discrete and Continuous Dynamical Systems Series B, 12 (2009), 151-168. doi: 10.3934/dcdsb.2009.12.151.  Google Scholar

[20]

Neurocomp, 72 (2008), 341-351. doi: 10.1016/j.neucom.2008.01.019.  Google Scholar

[21]

Sci. China Ser. E-Tech. Sci., 52 (2009), 771-781. Google Scholar

[22]

Physica D, 16 (1985), 233-242. doi: 10.1016/0167-2789(85)90060-0.  Google Scholar

[23]

3rd edition, Springer-Verlag, New York, 2004.  Google Scholar

[24]

G. B. Ermentrout, "Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students," 1st, edition, ().  doi: 10.1137/1.9780898718195.  Google Scholar

[25]

, Y. A. Kuznetsov and V. V. Levitin,, ftp.cwi.nl/pub/CONTENT., ().   Google Scholar

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