- Previous Article
- DCDS-S Home
- This Issue
-
Next Article
Random attractor for stochastic reversible Schnackenberg equations
On bursting solutions near chaotic regimes in a neuron model
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 |
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. |
[1] |
Jianzhong Su, Humberto Perez-Gonzalez, Ming He. Regular bursting emerging from coupled chaotic neurons. Conference Publications, 2007, 2007 (Special) : 946-955. doi: 10.3934/proc.2007.2007.946 |
[2] |
Jiaoyan Wang, Jianzhong Su, Humberto Perez Gonzalez, Jonathan Rubin. A reliability study of square wave bursting $\beta$-cells with noise. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 569-588. doi: 10.3934/dcdsb.2011.16.569 |
[3] |
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 and Continuous Dynamical Systems - B, 2011, 16 (2) : 637-651. doi: 10.3934/dcdsb.2011.16.637 |
[4] |
Linghai Zhang. Wave speed analysis of traveling wave fronts in delayed synaptically coupled neuronal networks. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2405-2450. doi: 10.3934/dcds.2014.34.2405 |
[5] |
Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026 |
[6] |
Anastasiia Panchuk, Frank Westerhoff. Speculative behavior and chaotic asset price dynamics: On the emergence of a bandcount accretion bifurcation structure. Discrete and Continuous Dynamical Systems - B, 2021, 26 (11) : 5941-5964. doi: 10.3934/dcdsb.2021117 |
[7] |
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 |
[8] |
Jibin Li, Fengjuan Chen. Exact travelling wave solutions and their dynamical behavior for a class coupled nonlinear wave equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 163-172. doi: 10.3934/dcdsb.2013.18.163 |
[9] |
Qixiang Wen, Shenquan Liu, Bo Lu. Firing patterns and bifurcation analysis of neurons under electromagnetic induction. Electronic Research Archive, 2021, 29 (5) : 3205-3226. doi: 10.3934/era.2021034 |
[10] |
Maria Francesca Carfora, Enrica Pirozzi. Stochastic modeling of the firing activity of coupled neurons periodically driven. Conference Publications, 2015, 2015 (special) : 195-203. doi: 10.3934/proc.2015.0195 |
[11] |
Jorge Duarte, Cristina Januário, Nuno Martins. A chaotic bursting-spiking transition in a pancreatic beta-cells system: observation of an interior glucose-induced crisis. Mathematical Biosciences & Engineering, 2017, 14 (4) : 821-842. doi: 10.3934/mbe.2017045 |
[12] |
Thai Son Doan, Martin Rasmussen, Peter E. Kloeden. The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 875-887. doi: 10.3934/dcdsb.2015.20.875 |
[13] |
Roberta Sirovich, Laura Sacerdote, Alessandro E. P. Villa. Cooperative behavior in a jump diffusion model for a simple network of spiking neurons. Mathematical Biosciences & Engineering, 2014, 11 (2) : 385-401. doi: 10.3934/mbe.2014.11.385 |
[14] |
Zhuoqin Yang, Tingting Guan. Bifurcation analysis of complex bursting induced by two different time-scale slow variables. Conference Publications, 2011, 2011 (Special) : 1440-1447. doi: 10.3934/proc.2011.2011.1440 |
[15] |
Lixia Duan, Dehong Zhai, Qishao Lu. Bifurcation and bursting in Morris-Lecar model for class I and class II excitability. Conference Publications, 2011, 2011 (Special) : 391-399. doi: 10.3934/proc.2011.2011.391 |
[16] |
Francesca Alessio, Vittorio Coti Zelati, Piero Montecchiari. Chaotic behavior of rapidly oscillating Lagrangian systems. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 687-707. doi: 10.3934/dcds.2004.10.687 |
[17] |
A.V. Borisov, A.A. Kilin, I.S. Mamaev. Reduction and chaotic behavior of point vortices on a plane and a sphere. Conference Publications, 2005, 2005 (Special) : 100-109. doi: 10.3934/proc.2005.2005.100 |
[18] |
Julia Calatayud, Juan Carlos Cortés, Marc Jornet. On the random wave equation within the mean square context. Discrete and Continuous Dynamical Systems - S, 2022, 15 (2) : 409-425. doi: 10.3934/dcdss.2021082 |
[19] |
Timothy J. Lewis. Phase-locking in electrically coupled non-leaky integrate-and-fire neurons. Conference Publications, 2003, 2003 (Special) : 554-562. doi: 10.3934/proc.2003.2003.554 |
[20] |
B. Fernandez, P. Guiraud. Route to chaotic synchronisation in coupled map lattices: Rigorous results. Discrete and Continuous Dynamical Systems - B, 2004, 4 (2) : 435-456. doi: 10.3934/dcdsb.2004.4.435 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]