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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 and Continuous Dynamical Systems - B, 2011, 16 (2) : 445-456. doi: 10.3934/dcdsb.2011.16.445
References:
[1]

A. Kepecs, X. Wang and J. Lisman, Bursting neurons singale input slope, J. Neurosci., 22 (2002), 9053-9062.

[2]

A. Kepecs and J. Lisman, Information encoding and computation with spikes and bursters, Network: Comput. Neural. Syst., 14 (2003), 103-9118.

[3]

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

[4]

J. Rinzel, Bursting oscillations in an excitable membrane model, in "Ordinary and Partial Differential Equations, Lecture Notes in Mathematics" (eds. B.D. Sleeman, R.J. Jarvis), Springer, Berlin, 1151 (1985), 304-316.

[5]

R. Bertram, M. Butte, T. Kiemel and A. Sherman, Topological and phenomenological classification of bursting oscillations, Bull. Math. Biol., 57 (1995), 413-439.

[6]

M. Rush and J. Rinzel, Analysis of Bursting in a thalamic neuron model, Biol. Cybern., 71 (1994), 281-291. doi: 10.1007/BF00239616.

[7]

E. M. Izhikevich, Neural excitability, spiking and bursting, Int. J. of Bifurcation and Chaos, 10 (2000), 1171-1266. doi: 10.1142/S0218127400000840.

[8]

X. J. Wang, Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle, Physica D, 62 (1993), 263-274. doi: 10.1016/0167-2789(93)90286-A.

[9]

P. R. Shorten and D. J. Wall, A Hodgkin-Huxley model exhibiting bursting oscillations, Bull. Math. Biol., 62 (2000), 695-715. doi: 10.1006/bulm.2000.0172.

[10]

M. Perc and M. Marhl, Different types of bursting calcium oscillations in non-excitable cells, Chaos, Solitons and Fractals, 18 (2003), 759-773. doi: 10.1016/S0960-0779(03)00027-4.

[11]

L. Brusch, W. Lorenz, M. Or-Guil, M. Bär and U. Kummer, Fold-Hopf bursting in a model for calcium signal transduction, Z. Phys. Chem., 216 (2002), 487-97. doi: 10.1524/zpch.2002.216.4.487.

[12]

V. N. Belykh, I. V. Belykh, M. Colding-Jørgensen and E. Mosekilde, Homoclinic bifurcations leading to the emergence of bursting oscillations in cell models, Eur. Phys. J. E, 3 (2000), 205-219. doi: 10.1007/s101890070012.

[13]

Y. S. Fan and T. R. Chay, Generation of periodic and chaotic bursting in an excitable cell model, Biol. Cybern., 71 (1994), 417-431. doi: 10.1007/BF00198918.

[14]

H. G. Gu, M. H. Yang, L. Li, Z. Q. Liu and W. Ren, Dynamics of autonomous stochastic resonance in neural period adding bifurcation scenarios, Phys. Lett. A, 319 (2003), 89-96. doi: 10.1016/j.physleta.2003.09.077.

[15]

Z. Q. Yang, Q. S. Lu, H. G. Gu and W. Ren, Gwn-induced bursting, spiking, and random subthreshold impulsing oscillation before Hopf bifurcations in the Chay model, Int. J. Bifurcation Chaos, 14 (2004), 4143-4159. doi: 10.1142/S0218127404011892.

[16]

Z. Q. Yang and Q. S. Lu, Different types of bursting in Chay neuronal model, Sci. China Ser. G-Phys Mech. Astron., 6 (2008), 687-698.

[17]

J. Guckenheimer and J. H. Tien, Bifurcation in the fast dynamics of neurons: implication for bursting, 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.

[18]

X. Q. Wu and L. C. Wang, Hopf bifurcation of a class of two coupled relaxation oscillators of the van der pol type with delay, Discrete and Continuous Dynamical Systems Series B, 13 (2010), 503-516.

[19]

D. Liu, S. G. Ruan and D. M. Zhu, Bifurcation analysis in models of tumor and immune system interactions, Discrete and Continuous Dynamical Systems Series B, 12 (2009), 151-168. doi: 10.3934/dcdsb.2009.12.151.

[20]

L. X. Duan, Q. S. Lu and Q. Y. Wang, Two-parameter bifurcation analysis of firing activities in the Chay neuronal model, Neurocomp, 72 (2008), 341-351. doi: 10.1016/j.neucom.2008.01.019.

[21]

L. X. Duan, Q. S. Lu and D. Z. Cheng, Bursting of Morris-Lecar neuronal model with current-feedback control, Sci. China Ser. E-Tech. Sci., 52 (2009), 771-781.

[22]

T. R. Chay, Chaos in the three-variable model of an excitable cell, Physica D, 16 (1985), 233-242. doi: 10.1016/0167-2789(85)90060-0.

[23]

Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," 3rd edition, Springer-Verlag, New York, 2004.

[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.

[25]

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

show all references

References:
[1]

A. Kepecs, X. Wang and J. Lisman, Bursting neurons singale input slope, J. Neurosci., 22 (2002), 9053-9062.

[2]

A. Kepecs and J. Lisman, Information encoding and computation with spikes and bursters, Network: Comput. Neural. Syst., 14 (2003), 103-9118.

[3]

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

[4]

J. Rinzel, Bursting oscillations in an excitable membrane model, in "Ordinary and Partial Differential Equations, Lecture Notes in Mathematics" (eds. B.D. Sleeman, R.J. Jarvis), Springer, Berlin, 1151 (1985), 304-316.

[5]

R. Bertram, M. Butte, T. Kiemel and A. Sherman, Topological and phenomenological classification of bursting oscillations, Bull. Math. Biol., 57 (1995), 413-439.

[6]

M. Rush and J. Rinzel, Analysis of Bursting in a thalamic neuron model, Biol. Cybern., 71 (1994), 281-291. doi: 10.1007/BF00239616.

[7]

E. M. Izhikevich, Neural excitability, spiking and bursting, Int. J. of Bifurcation and Chaos, 10 (2000), 1171-1266. doi: 10.1142/S0218127400000840.

[8]

X. J. Wang, Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle, Physica D, 62 (1993), 263-274. doi: 10.1016/0167-2789(93)90286-A.

[9]

P. R. Shorten and D. J. Wall, A Hodgkin-Huxley model exhibiting bursting oscillations, Bull. Math. Biol., 62 (2000), 695-715. doi: 10.1006/bulm.2000.0172.

[10]

M. Perc and M. Marhl, Different types of bursting calcium oscillations in non-excitable cells, Chaos, Solitons and Fractals, 18 (2003), 759-773. doi: 10.1016/S0960-0779(03)00027-4.

[11]

L. Brusch, W. Lorenz, M. Or-Guil, M. Bär and U. Kummer, Fold-Hopf bursting in a model for calcium signal transduction, Z. Phys. Chem., 216 (2002), 487-97. doi: 10.1524/zpch.2002.216.4.487.

[12]

V. N. Belykh, I. V. Belykh, M. Colding-Jørgensen and E. Mosekilde, Homoclinic bifurcations leading to the emergence of bursting oscillations in cell models, Eur. Phys. J. E, 3 (2000), 205-219. doi: 10.1007/s101890070012.

[13]

Y. S. Fan and T. R. Chay, Generation of periodic and chaotic bursting in an excitable cell model, Biol. Cybern., 71 (1994), 417-431. doi: 10.1007/BF00198918.

[14]

H. G. Gu, M. H. Yang, L. Li, Z. Q. Liu and W. Ren, Dynamics of autonomous stochastic resonance in neural period adding bifurcation scenarios, Phys. Lett. A, 319 (2003), 89-96. doi: 10.1016/j.physleta.2003.09.077.

[15]

Z. Q. Yang, Q. S. Lu, H. G. Gu and W. Ren, Gwn-induced bursting, spiking, and random subthreshold impulsing oscillation before Hopf bifurcations in the Chay model, Int. J. Bifurcation Chaos, 14 (2004), 4143-4159. doi: 10.1142/S0218127404011892.

[16]

Z. Q. Yang and Q. S. Lu, Different types of bursting in Chay neuronal model, Sci. China Ser. G-Phys Mech. Astron., 6 (2008), 687-698.

[17]

J. Guckenheimer and J. H. Tien, Bifurcation in the fast dynamics of neurons: implication for bursting, 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.

[18]

X. Q. Wu and L. C. Wang, Hopf bifurcation of a class of two coupled relaxation oscillators of the van der pol type with delay, Discrete and Continuous Dynamical Systems Series B, 13 (2010), 503-516.

[19]

D. Liu, S. G. Ruan and D. M. Zhu, Bifurcation analysis in models of tumor and immune system interactions, Discrete and Continuous Dynamical Systems Series B, 12 (2009), 151-168. doi: 10.3934/dcdsb.2009.12.151.

[20]

L. X. Duan, Q. S. Lu and Q. Y. Wang, Two-parameter bifurcation analysis of firing activities in the Chay neuronal model, Neurocomp, 72 (2008), 341-351. doi: 10.1016/j.neucom.2008.01.019.

[21]

L. X. Duan, Q. S. Lu and D. Z. Cheng, Bursting of Morris-Lecar neuronal model with current-feedback control, Sci. China Ser. E-Tech. Sci., 52 (2009), 771-781.

[22]

T. R. Chay, Chaos in the three-variable model of an excitable cell, Physica D, 16 (1985), 233-242. doi: 10.1016/0167-2789(85)90060-0.

[23]

Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," 3rd edition, Springer-Verlag, New York, 2004.

[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.

[25]

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

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