September  2011, 16(2): 637-651. doi: 10.3934/dcdsb.2011.16.637

Bifurcation analysis of bursting solutions of two Hindmarsh-Rose neurons with joint electrical and synaptic coupling

1. 

College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, China, China

2. 

Department of Dynamics and Control, Beihang University, Beijing 100191, China

3. 

The University of Texas at Arlington, Department of Mathematics, Box 19408, Arlington, TX 76019

Received  May 2010 Revised  December 2010 Published  June 2011

In this paper, we investigate the dynamic behavior of a system of two coupled Hindmarsh-Rose (HR) neurons, based on bifurcation analysis of its fast subsystem. The individual HR neuron has chaotic behavior, but they can become regularized when coupled through synaptic coupling or joint electrical-synaptic coupling. Through numerical methods we first investigate the bifurcation structure of its fast subsystem. We show that the emerging of periodic patterns of neurons is related to topological changes of its underlying bifurcations. The Lyaponov exponent calculations further reveal the pathway from chaotic bursting behavior to regular bursting of HR neurons. Finally, we include both electrical and synaptic coupling in the system, and numerically calculate the time dynamics. Even though electrical couplings (or gap junctions) usually does not regularize chaotic trajectories, but joint coupling has been more effective than synaptic coupling alone in producing stable rhythms. The main contribution of this paper is that we provide a mathematical description for transitions of neuron dynamics from chaotic trajectories to regular bursting when synaptic and electrical-synaptic coupling strengthens, using bifurcation analysis.
Citation: Feng Zhang, Wei Zhang, Pan Meng, Jianzhong Su. Bifurcation analysis of bursting solutions of two Hindmarsh-Rose neurons with joint electrical and synaptic coupling. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 637-651. doi: 10.3934/dcdsb.2011.16.637
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show all references

References:
[1]

Biophys. J., 42 (1983), 181-195. doi: 10.1016/S0006-3495(83)84384-7.  Google Scholar

[2]

Neural Comp., 13 (2000), 113-137. doi: 10.1162/089976601300014655.  Google Scholar

[3]

Proc. Rroyal. Soc. Lond. B, 221 (1984), 87-102. doi: 10.1098/rspb.1984.0024.  Google Scholar

[4]

J. Neurophysiol, 81 (1999), 382-397. Google Scholar

[5]

Proc. of Inter. Cong. of Math, (1987), 1578-1593.  Google Scholar

[6]

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

[7]

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

[8]

Amer. J. Physiol., 271 (1996), E362-E372. Google Scholar

[9]

J. Appl. Math., 51 (1991), 1418-1450.  Google Scholar

[10]

Proc. Natl. Acad. Sci., 89 (1992), 2471-2474. doi: 10.1073/pnas.89.6.2471.  Google Scholar

[11]

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

[12]

Elsevier Science, 2 (2002), 93-146.  Google Scholar

[13]

Biol. Cybern., 68 (1993), 393-407. doi: 10.1007/BF00198772.  Google Scholar

[14]

Phys. Rev. Letters, 2 (2004), 028101. doi: 10.1103/PhysRevLett.92.028101.  Google Scholar

[15]

Phys. Rev. E, 74 (2006), 201917. doi: 10.1103/PhysRevE.74.021917.  Google Scholar

[16]

Euro. J. Neuroscience, 22 (2005), 2661-2668. doi: 10.1111/j.1460-9568.2005.04405.x.  Google Scholar

[17]

J. of Comp. Neuroscience, 14 (2003), 283-309. doi: 10.1023/A:1023265027714.  Google Scholar

[18]

SIAM J. Appl. Math, 67 (2007), 512-529. doi: 10.1137/060661041.  Google Scholar

[19]

Neural Comp., 17 (2005), 633-670. doi: 10.1162/0899766053019917.  Google Scholar

[20]

Neural Comp., 8 (1996), 1567-1602. doi: 10.1162/neco.1996.8.8.1567.  Google Scholar

[21]

Discrete and Continuous Dynamical Systems, supplemental issue (2007), 946-955.  Google Scholar

[22]

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[23]

SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898718195.  Google Scholar

[24]

J. of Diff. Equations, 158 (1999), 48-78. doi: 10.1016/S0022-0396(99)80018-7.  Google Scholar

[25]

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[26]

SIAM J. Appl. Math., 67 (2007), 530-542. doi: 10.1137/060655663.  Google Scholar

[27]

Discrete and Continuous Dynamical Systems-B, 9 (2008), 397-413. Google Scholar

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