# 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 and 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. [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. [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. [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] J. Rinzel, A formal classification of bursting mechanisms in excitable systems, Proc. of Inter. Cong. of Math, (1987), 1578-1593. [6] E. M. Izhikevich, Neural Excitability, Spiking, and Bursting, Int. J. of Bifurcation Chaos, 10 (2000), 1171-1266. doi: 10.1142/S0218127400000840. [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. [8] A. Sherman, Contributions of modeling to understanding stimulus-secretion coupling in pancreatic $\beta$-cells, Amer. J. Physiol., 271 (1996), E362-E372. [9] D. Terman, Chaotic spikes arising from a model of bursting in excitable membranes, J. Appl. Math., 51 (1991), 1418-1450. [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. [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. [12] J. Rubin and D. Terman, Geometric singular perturbation analysis of neuronal dynamics, Handbook of Dynamical Systems, Elsevier Science, 2 (2002), 93-146. [13] D. Somers and N. Kopell, Rapid synchronization through fast threshold modulation, Biol. Cybern., 68 (1993), 393-407. doi: 10.1007/BF00198772. [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. [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. [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. [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. [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. [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. [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. [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. [22] N. F. Rulkov, Regularization of synchronized chaotic bursts, Phys. Rev. Letters, 86 (2001), 183-186. doi: 10.1103/PhysRevLett.86.183. [23] B. Ermentrout, "Simulating, Analyzing, and Animating Dynamical Systems," SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898718195. [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. [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. [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. [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.

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##### 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. [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. [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. [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] J. Rinzel, A formal classification of bursting mechanisms in excitable systems, Proc. of Inter. Cong. of Math, (1987), 1578-1593. [6] E. M. Izhikevich, Neural Excitability, Spiking, and Bursting, Int. J. of Bifurcation Chaos, 10 (2000), 1171-1266. doi: 10.1142/S0218127400000840. [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. [8] A. Sherman, Contributions of modeling to understanding stimulus-secretion coupling in pancreatic $\beta$-cells, Amer. J. Physiol., 271 (1996), E362-E372. [9] D. Terman, Chaotic spikes arising from a model of bursting in excitable membranes, J. Appl. Math., 51 (1991), 1418-1450. [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. [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. [12] J. Rubin and D. Terman, Geometric singular perturbation analysis of neuronal dynamics, Handbook of Dynamical Systems, Elsevier Science, 2 (2002), 93-146. [13] D. Somers and N. Kopell, Rapid synchronization through fast threshold modulation, Biol. Cybern., 68 (1993), 393-407. doi: 10.1007/BF00198772. [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. [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. [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. [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. [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. [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. [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. [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. [22] N. F. Rulkov, Regularization of synchronized chaotic bursts, Phys. Rev. Letters, 86 (2001), 183-186. doi: 10.1103/PhysRevLett.86.183. [23] B. Ermentrout, "Simulating, Analyzing, and Animating Dynamical Systems," SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898718195. [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. [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. [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. [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.
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