# American Institute of Mathematical Sciences

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
##### References:
 [1] T. R. Chay and J. Keizer, Minimal model for membrane oscillations in the pancreatic beta-cell, Biophys. J., 42 (1983), 181-195. doi: 10.1016/S0006-3495(83)84384-7.  Google Scholar [2] 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 Comp., 13 (2000), 113-137. doi: 10.1162/089976601300014655.  Google Scholar [3] J. L. Hindmarsh and R. M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. Rroyal. Soc. Lond. B, 221 (1984), 87-102. doi: 10.1098/rspb.1984.0024.  Google Scholar [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. Google Scholar [5] J. Rinzel, A formal classification of bursting mechanisms in excitable systems, Proc. of Inter. Cong. of Math, (1987), 1578-1593.  Google Scholar [6] E. M. Izhikevich, Neural Excitability, Spiking, and Bursting, Int. J. of Bifurcation Chaos, 10 (2000), 1171-1266. doi: 10.1142/S0218127400000840.  Google Scholar [7] R. Bertram, M. J. Butte, T. Kiemel and A. Sherman, Topological and phenomenological classification of bursting oscillations, Bull. Math. Biol., 57 (1995), 413-439. Google Scholar [8] A. Sherman, Contributions of modeling to understanding stimulus-secretion coupling in pancreatic $\beta$-cells, Amer. J. Physiol., 271 (1996), E362-E372. Google Scholar [9] D. Terman, Chaotic spikes arising from a model of bursting in excitable membranes, J. Appl. Math., 51 (1991), 1418-1450.  Google Scholar [10] 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.  Google Scholar [11] V. N. Belykh, I. V. Belykh, M. Colding-Jørgensen and E. Mosekilde, Homoclinic bifurcations leading to the emergence of bursting oscillations in cell models, Euro. Phys. J. E, 3 (2000), 205-219. doi: 10.1007/s101890070012.  Google Scholar [12] J. Rubin and D. Terman, Geometric singular perturbation analysis of neuronal dynamics, Handbook of Dynamical Systems, Elsevier Science, 2 (2002), 93-146.  Google Scholar [13] D. Somers and N. Kopell, Rapid synchronization through fast threshold modulation, Biol. Cybern., 68 (1993), 393-407. doi: 10.1007/BF00198772.  Google Scholar [14] M. Dhamala, V. K. Jirsa and M. Ding, Transitions to synchrony in coupled bursting neurons, Phys. Rev. Letters, 2 (2004), 028101. doi: 10.1103/PhysRevLett.92.028101.  Google Scholar [15] J. Rubin, Bursting induced by excitatory synaptic coupling in nonidentical conditional relaxation oscillators or square-wave bursters, Phys. Rev. E, 74 (2006), 201917. doi: 10.1103/PhysRevE.74.021917.  Google Scholar [16] T. Bem, Y. L. Feuvre, J. Rinzel and P. Meyrand, Eleltrical coupling induces bistability of rhythms in networks of inhibitory spiking neurons, Euro. J. Neuroscience, 22 (2005), 2661-2668. doi: 10.1111/j.1460-9568.2005.04405.x.  Google Scholar [17] T. J. Lewis and J. Rinzel, Dynamics of spiking neurons connected by both inhibitory and electrical coupling, J. of Comp. Neuroscience, 14 (2003), 283-309. doi: 10.1023/A:1023265027714.  Google Scholar [18] F. G. Kazanci and B. Ermentrout, Pattern formation in an array of oscillators with electrical and chemical coupling, SIAM J. Appl. Math, 67 (2007), 512-529. doi: 10.1137/060661041.  Google Scholar [19] B. Pfeuty, G. Mato, D. Golomb and D. Hansel, The combined effects of inhibitory and electrical synapses in synchrony, Neural Comp., 17 (2005), 633-670. doi: 10.1162/0899766053019917.  Google Scholar [20] 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 Comp., 8 (1996), 1567-1602. doi: 10.1162/neco.1996.8.8.1567.  Google Scholar [21] J. Su, H. Perez and M. He, Regular bursting emerging from coupled chaotic neurons, Discrete and Continuous Dynamical Systems, supplemental issue (2007), 946-955.  Google Scholar [22] N. F. Rulkov, Regularization of synchronized chaotic bursts, Phys. Rev. Letters, 86 (2001), 183-186. doi: 10.1103/PhysRevLett.86.183.  Google Scholar [23] B. Ermentrout, "Simulating, Analyzing, and Animating Dynamical Systems," SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898718195.  Google Scholar [24] E. Lee and D. Terman, Uniqueness and stability of periodic bursting solutions, J. of Diff. Equations, 158 (1999), 48-78. doi: 10.1016/S0022-0396(99)80018-7.  Google Scholar [25] 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.  Google Scholar [26] 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.  Google Scholar [27] S. Ma, Z. S. Feng and Q. S. Lu, A two-parameter geometrical criteria for delay differential equations, Discrete and Continuous Dynamical Systems-B, 9 (2008), 397-413. Google Scholar

show all references

##### References:
 [1] T. R. Chay and J. Keizer, Minimal model for membrane oscillations in the pancreatic beta-cell, Biophys. J., 42 (1983), 181-195. doi: 10.1016/S0006-3495(83)84384-7.  Google Scholar [2] 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 Comp., 13 (2000), 113-137. doi: 10.1162/089976601300014655.  Google Scholar [3] J. L. Hindmarsh and R. M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. Rroyal. Soc. Lond. B, 221 (1984), 87-102. doi: 10.1098/rspb.1984.0024.  Google Scholar [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. Google Scholar [5] J. Rinzel, A formal classification of bursting mechanisms in excitable systems, Proc. of Inter. Cong. of Math, (1987), 1578-1593.  Google Scholar [6] E. M. Izhikevich, Neural Excitability, Spiking, and Bursting, Int. J. of Bifurcation Chaos, 10 (2000), 1171-1266. doi: 10.1142/S0218127400000840.  Google Scholar [7] R. Bertram, M. J. Butte, T. Kiemel and A. Sherman, Topological and phenomenological classification of bursting oscillations, Bull. Math. Biol., 57 (1995), 413-439. Google Scholar [8] A. Sherman, Contributions of modeling to understanding stimulus-secretion coupling in pancreatic $\beta$-cells, Amer. J. Physiol., 271 (1996), E362-E372. Google Scholar [9] D. Terman, Chaotic spikes arising from a model of bursting in excitable membranes, J. Appl. Math., 51 (1991), 1418-1450.  Google Scholar [10] 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.  Google Scholar [11] V. N. Belykh, I. V. Belykh, M. Colding-Jørgensen and E. Mosekilde, Homoclinic bifurcations leading to the emergence of bursting oscillations in cell models, Euro. Phys. J. E, 3 (2000), 205-219. doi: 10.1007/s101890070012.  Google Scholar [12] J. Rubin and D. Terman, Geometric singular perturbation analysis of neuronal dynamics, Handbook of Dynamical Systems, Elsevier Science, 2 (2002), 93-146.  Google Scholar [13] D. Somers and N. Kopell, Rapid synchronization through fast threshold modulation, Biol. Cybern., 68 (1993), 393-407. doi: 10.1007/BF00198772.  Google Scholar [14] M. Dhamala, V. K. Jirsa and M. Ding, Transitions to synchrony in coupled bursting neurons, Phys. Rev. Letters, 2 (2004), 028101. doi: 10.1103/PhysRevLett.92.028101.  Google Scholar [15] J. Rubin, Bursting induced by excitatory synaptic coupling in nonidentical conditional relaxation oscillators or square-wave bursters, Phys. Rev. E, 74 (2006), 201917. doi: 10.1103/PhysRevE.74.021917.  Google Scholar [16] T. Bem, Y. L. Feuvre, J. Rinzel and P. Meyrand, Eleltrical coupling induces bistability of rhythms in networks of inhibitory spiking neurons, Euro. J. Neuroscience, 22 (2005), 2661-2668. doi: 10.1111/j.1460-9568.2005.04405.x.  Google Scholar [17] T. J. Lewis and J. Rinzel, Dynamics of spiking neurons connected by both inhibitory and electrical coupling, J. of Comp. Neuroscience, 14 (2003), 283-309. doi: 10.1023/A:1023265027714.  Google Scholar [18] F. G. Kazanci and B. Ermentrout, Pattern formation in an array of oscillators with electrical and chemical coupling, SIAM J. Appl. Math, 67 (2007), 512-529. doi: 10.1137/060661041.  Google Scholar [19] B. Pfeuty, G. Mato, D. Golomb and D. Hansel, The combined effects of inhibitory and electrical synapses in synchrony, Neural Comp., 17 (2005), 633-670. doi: 10.1162/0899766053019917.  Google Scholar [20] 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 Comp., 8 (1996), 1567-1602. doi: 10.1162/neco.1996.8.8.1567.  Google Scholar [21] J. Su, H. Perez and M. He, Regular bursting emerging from coupled chaotic neurons, Discrete and Continuous Dynamical Systems, supplemental issue (2007), 946-955.  Google Scholar [22] N. F. Rulkov, Regularization of synchronized chaotic bursts, Phys. Rev. Letters, 86 (2001), 183-186. doi: 10.1103/PhysRevLett.86.183.  Google Scholar [23] B. Ermentrout, "Simulating, Analyzing, and Animating Dynamical Systems," SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898718195.  Google Scholar [24] E. Lee and D. Terman, Uniqueness and stability of periodic bursting solutions, J. of Diff. Equations, 158 (1999), 48-78. doi: 10.1016/S0022-0396(99)80018-7.  Google Scholar [25] 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.  Google Scholar [26] 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.  Google Scholar [27] S. Ma, Z. S. Feng and Q. S. Lu, A two-parameter geometrical criteria for delay differential equations, Discrete and Continuous Dynamical Systems-B, 9 (2008), 397-413. Google Scholar
 [1] Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026 [2] Qixiang Wen, Shenquan Liu, Bo Lu. Firing patterns and bifurcation analysis of neurons under electromagnetic induction. Electronic Research Archive, , () : -. doi: 10.3934/era.2021034 [3] 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 [4] 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 [5] 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 [6] 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 [7] Elena Bonetti, Giovanna Bonfanti, Riccarda Rossi. Analysis of a model coupling volume and surface processes in thermoviscoelasticity. Discrete & Continuous Dynamical Systems, 2015, 35 (6) : 2349-2403. doi: 10.3934/dcds.2015.35.2349 [8] Ming-Jong Yao, Yu-Chun Wang. Theoretical analysis and a search procedure for the joint replenishment problem with deteriorating products. Journal of Industrial & Management Optimization, 2005, 1 (3) : 359-375. doi: 10.3934/jimo.2005.1.359 [9] Xue Qiao, Zheng Wang, Haoxun Chen. Joint optimal pricing and inventory management policy and its sensitivity analysis for perishable products: Lost sale case. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021079 [10] Hyukjin Lee, Cheng-Chew Lim, Jinho Choi. Joint backoff control in time and frequency for multichannel wireless systems and its Markov model for analysis. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1083-1099. doi: 10.3934/dcdsb.2011.16.1083 [11] Ghalip Abdukerim, Eziz Tursun, Yating Yang, Xiao Li. Uyghur morphological analysis using joint conditional random fields: Based on small scaled corpus. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 823-836. doi: 10.3934/dcdss.2019055 [12] Shubo Zhao, Ping Liu, Mingchao Jiang. Stability and bifurcation analysis in a chemotaxis bistable growth system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1165-1174. doi: 10.3934/dcdss.2017063 [13] Dan Liu, Shigui Ruan, Deming Zhu. Bifurcation analysis in models of tumor and immune system interactions. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 151-168. doi: 10.3934/dcdsb.2009.12.151 [14] Miaoran Yao, Yongxin Zhang, Wendi Wang. Bifurcation analysis for an in-host Mycobacterium tuberculosis model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2299-2322. doi: 10.3934/dcdsb.2020324 [15] Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367 [16] Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031 [17] Juan Su, Bing Xu, Lan Zou. Bifurcation analysis of an enzyme-catalyzed reaction system with branched sink. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6783-6815. doi: 10.3934/dcdsb.2019167 [18] Liming Cai, Jicai Huang, Xinyu Song, Yuyue Zhang. Bifurcation analysis of a mosquito population model for proportional releasing sterile mosquitoes. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6279-6295. doi: 10.3934/dcdsb.2019139 [19] Juping Ji, Lin Wang. Bifurcation and stability analysis for a nutrient-phytoplankton model with toxic effects. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3073-3081. doi: 10.3934/dcdss.2020135 [20] Jun Zhou. Bifurcation analysis of a diffusive plant-wrack model with tide effect on the wrack. Mathematical Biosciences & Engineering, 2016, 13 (4) : 857-885. doi: 10.3934/mbe.2016021

2020 Impact Factor: 1.327