American Institute of Mathematical Sciences

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December  2014, 7(6): 1363-1383. doi: 10.3934/dcdss.2014.7.1363

On bursting solutions near chaotic regimes in a neuron model

Feng Zhang 1, , Alice Lubbe 2, , Qishao Lu 3, and  Jianzhong Su 4,

1. 

North College of Beijing University of Chemical Technology, Hebei 065201, China
2. 

University of Texas at Arlington, Department of Mathematics, Box 19408, Arlington, TX 76019, United States
3. 

Beihang University, Department of Dynamics and Control, Beijing 100191, China
4. 

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

Received  April 2013 Revised  November 2013 Published  June 2014

In this paper, we use mathematical analysis to study the transition of dynamic behavior in a system of two synaptically coupled Hindmarsh-Rose (HR) neurons, based on its flow-induced Poincaré map. Numerical simulations have shown that the individual HR neuron has chaotic behavior, but neurons become regularized when coupled. Using a geometric method for dynamical systems, we begin with an investigation of the bifurcation structure of its fast subsystem. We show that the emergence of regular patterns out of chaos is due to a topological change in its underlying bifurcations. Then we focus on the transitional phase of coupling strength, where the bursting solutions need to pass near two homoclinic bifurcation points located on a branch of saddle points, and we study the flow-induced Poincaré maps. We observe that as the gap between the homoclinic points narrows, the image of the return map moves away from chaotic regions where winding numbers vary abruptly. That, along with Lyaponov exponent calculations, reveals the fine dynamics in the pathway across chaotic bursting behavior and regular bursting of coupled HR neurons as the synaptic coupling strength of the neurons increases. The main contribution of this paper is the mathematical description of the transition of synaptically coupled neurons from chaotic trajectories to regular burst phases using dynamical system tools such as Poincaré maps.
Keywords: square-wave bursting., bifurcation, chaotic behavior, Synaptically coupled neurons.
Mathematics Subject Classification: Primary: 34C15, 34C28, 92C20; Secondary: 74H65, 92B2.
Citation: Feng Zhang, Alice Lubbe, Qishao Lu, Jianzhong Su. On bursting solutions near chaotic regimes in a neuron model. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1363-1383. doi: 10.3934/dcdss.2014.7.1363
References:
[1]

H. D. I. Abarbanel, R. Huerta, M. I. Rabinovich, N. F. Rulkov, P. F. Rowat and A. I. Selverston, Synchronized action of synaptically coupled chaotic model neurons, Neural Computation, 8 (1996), 1567-1602. doi: 10.1162/neco.1996.8.8.1567.

[2]

V. Belykh, I. Belykh, E. Mosekilde and M. Colding-Jørgensen, Homoclinic bifurcations leading to bursting oscillations in cell models, European Physical Journal E, 3 (2000), 205-219. doi: 10.1007/s101890070012.

[3]

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

[4]

R. J. Butera, J. Rinzel and J. C. Smith, Models respiratory rhythm generation in the pre-Bötzinger complex. I. Bursting pacemaker neurons, J. Neurophysiol, 81 (1999), 382-397.

[5]

T. R. Chay and J. Keizer, Minimal model for membrane oscillations in the pancreatic beta-cell, Biophys. J., 42 (1983), 181-189. doi: 10.1016/S0006-3495(83)84384-7.

[6]

L. N. Cornelisse, W. J. J. M. Scheenen, W. J. H. Koopman, E. W. Roubos and S. C. A. M. Gielen, Minimal model for intracellular calcium oscillations and electrical bursting in melanotrope cells of Xenopus Laevis, Neural Comput., 13 (2000), 113-137.

[7]

M. Dhamala, V. K. Jirsa and M. Ding, Transitions to synchrony in coupled bursting neurons, Physical Review Letters, 92 (2004), 028101. doi: 10.1103/PhysRevLett.92.028101.

[8]

B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems, 1st edition, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898718195.

[9]

N. Fenichel, Geometric singular perturbation theory, J. D. E., 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9.

[10]

J. L. Hindmarsh and R. M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. R. Soc. Lond. B, 221 (1984), 87-102. doi: 10.1098/rspb.1984.0024.

[11]

E. M. Izhikevich, Neural Excitability, Spiking, and Bursting, I. J. B. C., 10 (2000), 1171-1266. doi: 10.1142/S0218127400000840.

[12]

E. Lee and D. Terman, Uniqueness and stability of periodic bursting solutions, J. Diff. Equ., 158 (1999), 48-78. doi: 10.1016/S0022-0396(99)80018-7.

[13]

S. Q. Ma, Z. Feng and Q. Lu, Dynamics and double hopf bifurcations of the Rose-Hindmarsh model with time delay, International Journal of Bifurcation and Chaos, 19 (2009), 3733-3751. doi: 10.1142/S0218127409025080.

[14]

G. S. Medvedev, Reduction of a model of an excitable cell to a one-dimensional map, Physica D, 202 (2005), 37-59. doi: 10.1016/j.physd.2005.01.021.

[15]

M. Pedersen and M. Sorensen, The effect of noise on beta-cell burst period, SIAM J. Appl. Math, 67 (2007), 530-542. doi: 10.1137/060655663.

[16]

J. Rinzel, A formal classification of bursting mechanisms in excitable systems, Proceedings of International Congress of Mathematics, 1 (1987), 1578-1593.

[17]

J. Rubin, Bursting induced by excitatory synaptic coupling in nonidentical conditional relaxation oscillators or square-wave bursters, Phys. Rev. E., 74 (2006), 021917, 15 pp. doi: 10.1103/PhysRevE.74.021917.

[18]

J. Rubin and D. Terman, Geometric singular perturbation analysis of neuronal dynamics, in Handbook of Dynamical Systems, Vol. 2, North Holland, Amsterdam, 2002, 93-146. doi: 10.1016/S1874-575X(02)80024-8.

[19]

N. F. Rulkov, Regularization of synchronized chaotic bursts, Physical Review Letters, 86 (2001), 183-186. doi: 10.1103/PhysRevLett.86.183.

[20]

A. Sherman, Contributions of modeling to understanding stimulus-secretion coupling in pancreatic $\beta$-cells, Amer. J. Physiol., 271 (1996), E362-E372.

[21]

A. Sherman and J. Rinzel, Rhythmogenic effects of weak electrotonic coupling in neuronal models, Proc. Natl. Acad. Sci., 89 (1992), 2471-2474. doi: 10.1073/pnas.89.6.2471.

[22]

D. Somers and N. Kopell, Rapid synchronization through fast threshold modulation, Biol. Cybern., 68 (1993), 393-407. doi: 10.1007/BF00198772.

[23]

J. Su, H. Perez and M. He, Regular bursting emerging from coupled chaotic neurons, Discrete and Continuous Dynamical Systems, supplemental issue, (2007), 946-955.

[24]

D. Terman, Chaotic spikes arising from a model of bursting in excitable membranes, J. Appl. Math., 51 (1991), 1418-1450. doi: 10.1137/0151071.

[25]

F. Zhang, W. Zhang, Q. Lu and J. Su, Transition mechanisms between periodic and chaotic bursting neurons, in Cognitive Neurodynamics (II), Springer Science+Media B., 2011, 247-251. doi: 10.1007/978-90-481-9695-1_38.

show all references

References:
[1]

H. D. I. Abarbanel, R. Huerta, M. I. Rabinovich, N. F. Rulkov, P. F. Rowat and A. I. Selverston, Synchronized action of synaptically coupled chaotic model neurons, Neural Computation, 8 (1996), 1567-1602. doi: 10.1162/neco.1996.8.8.1567.

[2]

V. Belykh, I. Belykh, E. Mosekilde and M. Colding-Jørgensen, Homoclinic bifurcations leading to bursting oscillations in cell models, European Physical Journal E, 3 (2000), 205-219. doi: 10.1007/s101890070012.

[3]

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

[4]

R. J. Butera, J. Rinzel and J. C. Smith, Models respiratory rhythm generation in the pre-Bötzinger complex. I. Bursting pacemaker neurons, J. Neurophysiol, 81 (1999), 382-397.

[5]

T. R. Chay and J. Keizer, Minimal model for membrane oscillations in the pancreatic beta-cell, Biophys. J., 42 (1983), 181-189. doi: 10.1016/S0006-3495(83)84384-7.

[6]

L. N. Cornelisse, W. J. J. M. Scheenen, W. J. H. Koopman, E. W. Roubos and S. C. A. M. Gielen, Minimal model for intracellular calcium oscillations and electrical bursting in melanotrope cells of Xenopus Laevis, Neural Comput., 13 (2000), 113-137.

[7]

M. Dhamala, V. K. Jirsa and M. Ding, Transitions to synchrony in coupled bursting neurons, Physical Review Letters, 92 (2004), 028101. doi: 10.1103/PhysRevLett.92.028101.

[8]

B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems, 1st edition, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898718195.

[9]

N. Fenichel, Geometric singular perturbation theory, J. D. E., 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9.

[10]

J. L. Hindmarsh and R. M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. R. Soc. Lond. B, 221 (1984), 87-102. doi: 10.1098/rspb.1984.0024.

[11]

E. M. Izhikevich, Neural Excitability, Spiking, and Bursting, I. J. B. C., 10 (2000), 1171-1266. doi: 10.1142/S0218127400000840.

[12]

E. Lee and D. Terman, Uniqueness and stability of periodic bursting solutions, J. Diff. Equ., 158 (1999), 48-78. doi: 10.1016/S0022-0396(99)80018-7.

[13]

S. Q. Ma, Z. Feng and Q. Lu, Dynamics and double hopf bifurcations of the Rose-Hindmarsh model with time delay, International Journal of Bifurcation and Chaos, 19 (2009), 3733-3751. doi: 10.1142/S0218127409025080.

[14]

G. S. Medvedev, Reduction of a model of an excitable cell to a one-dimensional map, Physica D, 202 (2005), 37-59. doi: 10.1016/j.physd.2005.01.021.

[15]

M. Pedersen and M. Sorensen, The effect of noise on beta-cell burst period, SIAM J. Appl. Math, 67 (2007), 530-542. doi: 10.1137/060655663.

[16]

J. Rinzel, A formal classification of bursting mechanisms in excitable systems, Proceedings of International Congress of Mathematics, 1 (1987), 1578-1593.

[17]

J. Rubin, Bursting induced by excitatory synaptic coupling in nonidentical conditional relaxation oscillators or square-wave bursters, Phys. Rev. E., 74 (2006), 021917, 15 pp. doi: 10.1103/PhysRevE.74.021917.

[18]

J. Rubin and D. Terman, Geometric singular perturbation analysis of neuronal dynamics, in Handbook of Dynamical Systems, Vol. 2, North Holland, Amsterdam, 2002, 93-146. doi: 10.1016/S1874-575X(02)80024-8.

[19]

N. F. Rulkov, Regularization of synchronized chaotic bursts, Physical Review Letters, 86 (2001), 183-186. doi: 10.1103/PhysRevLett.86.183.

[20]

A. Sherman, Contributions of modeling to understanding stimulus-secretion coupling in pancreatic $\beta$-cells, Amer. J. Physiol., 271 (1996), E362-E372.

[21]

A. Sherman and J. Rinzel, Rhythmogenic effects of weak electrotonic coupling in neuronal models, Proc. Natl. Acad. Sci., 89 (1992), 2471-2474. doi: 10.1073/pnas.89.6.2471.

[22]

D. Somers and N. Kopell, Rapid synchronization through fast threshold modulation, Biol. Cybern., 68 (1993), 393-407. doi: 10.1007/BF00198772.

[23]

J. Su, H. Perez and M. He, Regular bursting emerging from coupled chaotic neurons, Discrete and Continuous Dynamical Systems, supplemental issue, (2007), 946-955.

[24]

D. Terman, Chaotic spikes arising from a model of bursting in excitable membranes, J. Appl. Math., 51 (1991), 1418-1450. doi: 10.1137/0151071.

[25]

F. Zhang, W. Zhang, Q. Lu and J. Su, Transition mechanisms between periodic and chaotic bursting neurons, in Cognitive Neurodynamics (II), Springer Science+Media B., 2011, 247-251. doi: 10.1007/978-90-481-9695-1_38.

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