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January  2016, 21(1): 37-53. doi: 10.3934/dcdsb.2016.21.37

## Delay-induced mixed-mode oscillations in a 2D Hindmarsh-Rose-type model

 1 Department of Mathematics, National Taiwan Normal University, 88, Ting-chou Rd., Sec. 4, Taipei 116, Taiwan 2 Department of Applied Mathematics, National Pingtung University, Pingtung, Taiwan

Received  February 2013 Revised  May 2015 Published  November 2015

In this study, we investigate a Hindmarsh-Rose-type model with the structure of recurrent neural feedback. The number of equilibria and their stability for the model with zero delay are reviewed first. We derive conditions for the existence of a Hopf bifurcation in the model and derive equations for the direction and stability of the bifurcation with delay as the bifurcation parameter. The ranges of parameter values for the existence of a Hopf bifurcation and the system responses with various levels of delay are obtained. When a Hopf bifurcation due to delay occurs, canard-like mixed-mode oscillations (MMOs) are produced at the parameter value for which either the fold bifurcation of cycles or homoclinic bifurcation occurs in the system without delay. This behavior can be found in a planar system with delay but not in a planar system without delay. Therefore, the results of this study will be helpful for determining suitable parameters to represent MMOs with a simple system with delay.
Citation: Shyan-Shiou Chen, Chang-Yuan Cheng. Delay-induced mixed-mode oscillations in a 2D Hindmarsh-Rose-type model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 37-53. doi: 10.3934/dcdsb.2016.21.37
##### References:
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Chay, Chaos in a three-variable model of an excitable cell,, Physica D: Nonlinear phenomena, 16 (1985), 233. doi: 10.1016/0167-2789(85)90060-0. Google Scholar [7] T. R. Chay, Glucose response to bursting-spiking pancreatic $\beta$-cells by a barrier kinetic model,, Biol. Cybern., 52 (1985), 339. Google Scholar [8] T. R. Chay and J. Keizer, Theory of the effect of extracellular potassium on oscillations in the pancreatic beta-cell,, Biophysical J., 48 (1985), 815. doi: 10.1016/S0006-3495(85)83840-6. Google Scholar [9] T. R. Chay and J. Rinzel, Bursting, beating, and chaos in an excitable membrane model,, Biophysical J., 47 (1985), 357. doi: 10.1016/S0006-3495(85)83926-6. Google Scholar [10] S. S. Chen, C. Y. Cheng and Y.-R. Lin, Application of a two-dimensional Hindmarsh-Rose type model for bifurcation analysis,, Int. J. Bifurcation Chaos, 23 (2013). doi: 10.1142/S0218127413500557. Google Scholar [11] J. A. Connor and C. F. Stevens, Inward and delayed outward membrane currents in isolated neural somata under voltage clamp,, J. Physiol., 213 (1971), 1. doi: 10.1113/jphysiol.1971.sp009364. Google Scholar [12] J. A. Connor, D. Walter and R. McKown, Neural repetitive firing: Modifications of the Hodgkin-Huxley axon suggested by experimental results from crustacean axons,, Biophysical Journal, 18 (1977), 81. doi: 10.1016/S0006-3495(77)85598-7. Google Scholar [13] K. Cooke and Z. Grossman, Discrete delay, distributed delay and stabilit switches,, J. Math. Anal. Appl., 86 (1982), 592. doi: 10.1016/0022-247X(82)90243-8. Google Scholar [14] G. Deco, V. K. Jirsa, P. A. Robinson, M. Breakspear and K. Friston, The dynamic brain: from spiking neurons to neural masses and cortical fields,, PLOS computational biology, 4 (2008). doi: 10.1371/journal.pcbi.1000092. Google Scholar [15] M. Desroches, B. Krauskopf and H. M. Osinga, Mixed-mode oscillations and slow manifolds in the self-coupled Fitzhugh-Nagumo system,, Chaos, 18 (2008). doi: 10.1063/1.2799471. Google Scholar [16] M. Dichter and W. A. Spencer, Penicillin-induced interictal discharges from the cat hippocampus, II. mechanisms underlying orgin and restriction,, J. Neurophyiol., 32 (1969), 663. Google Scholar [17] R. J. Douglas, C. Koch, M. Mahowald, K. A. Martin and H. H. Suarez, Recurrent excitation in neocortical circuits,, Science, 269 (1995), 981. doi: 10.1126/science.7638624. Google Scholar [18] V. Dragoi and M. Sur, Dynamic properties of recurrent inhibition in primary visual cortex: Contrast and orientation dependence of contextual effects,, J Neurophysiol, 83 (2000), 1019. Google Scholar [19] I. Erchova and D. J. McGonigle, Rhythms of the brain: An examination of mixed mode oscillation approaches to the analysis of neurophysiological data,, Chaos, 18 (2008). doi: 10.1063/1.2900015. Google Scholar [20] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophys. J., 1 (1961), 445. doi: 10.1016/S0006-3495(61)86902-6. Google Scholar [21] J. K. Hale, Theory of Functional Differential Equations,, Applied mathematical sciences. Springer-Verlag, (1977). Google Scholar [22] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf bifurcation,, Cambridge University Press, (1981). Google Scholar [23] J. L. Hindmarsh and R. M. Rose, A model of the nerve impluse using two first-order differential equations,, Nature, 296 (1982), 162. Google Scholar [24] J. L. Hindmarsh and R. M. Rose, A model of neuronal bursting using three coupled first order differential equaions,, Proc. R. Soc. Lond. B Biol. Sci., 221 (1984), 87. Google Scholar [25] A. L. Hodgkin and A. F. Huxley, A qualitative description of membrane current and its application to conduction and excitation in nerve,, J. Physiol., 117 (1952), 500. Google Scholar [26] R. Katz and E. Pierrot-Deseilligny, Recurrent inhibition in humans,, Progress in Neurobiology, 57 (1999), 325. doi: 10.1016/S0301-0082(98)00056-2. Google Scholar [27] M. T. M. Koper, Bifurcations of mixed-mode oscillations in a three-variable autonomous Van der Pol-Duffing model with a cross-shaped phase diagram,, Physica D, 80 (1995), 72. doi: 10.1016/0167-2789(95)90061-6. Google Scholar [28] M. Krupa, N. Popović, N. Kopell and H. G. Rotstein, Mixed-mode oscillations in a three time-scale model for the dopaminergic neuron,, Chaos, 18 (2008). doi: 10.1063/1.2779859. Google Scholar [29] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory (3rd ed.),, Number 112 in Applied Mathematical Sciences. Springer Verlag New York, (2004). doi: 10.1007/978-1-4757-3978-7. Google Scholar [30] S. Ma and Z. S. Feng, Fold-hopf bifurcations of the Rose-Hindmarsh model with time delay,, Int. J. Bifurcation and Chaos, 21 (2011), 437. doi: 10.1142/S0218127411028490. Google Scholar [31] S. Ma, Z. S. Feng and Q. S. Lu, Dynamics and double hopf bifurcations of the Rose-Hindmarsh model with time delay,, Int. J. Bifurcation and Chaos, 19 (2009), 3733. doi: 10.1142/S0218127409025080. Google Scholar [32] M. C. Mackey and U. an der Heiden, The dynamics of recurrent inhibition,, J Math Biol, 19 (1984), 211. doi: 10.1007/BF00277747. Google Scholar [33] B. Mattei, A. Schmied, R. Mazzocchio, B. Decchi, A. Rossi and J. P. Vedel, Pharmacologically induced enhancement of recurrent inhibition in humans: Effects on motoneurone discharge patterns,, J Physiol, 548 (2003), 615. Google Scholar [34] A. Mitra, S. S. Mitra and R. W. Tsien, Heterogeneous reallocation of presynaptic efficacy in recurrent excitatory circuits adapting to inactivity,, Nature Neuroscience, 15 (2012), 250. doi: 10.1038/nn.3004. Google Scholar [35] J. S. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon,, Proc. IRE, 50 (1962), 2061. doi: 10.1109/JRPROC.1962.288235. Google Scholar [36] V. Petrov, S. Scott and K. Showalter, Mixed mode oscillations in chemical systems,, J. Chem. Phys., 97 (1992). doi: 10.1063/1.463727. Google Scholar [37] R. E. Plant, A FitzHugh differential-difference equation modeling recurrent neural feedback,, SIAM Journal on applied mathematics, 40 (1981), 150. doi: 10.1137/0140012. Google Scholar [38] R. M. Rose and J. L. Hindmarsh, The assembly of ionic currents in a thalamic neuron I. the three-dimensional model,, Proc. R. Soc. Lond. B, 237 (1989), 267. doi: 10.1098/rspb.1989.0049. Google Scholar [39] J. Rubin and M. Wechselberger, The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple time scales,, Chaos, 18 (2008). doi: 10.1063/1.2789564. Google Scholar [40] A. Sherman, J. Rinzel and J. Keizer, Emergence of organized bursting in clusters of pancreatic beta-cells by channel sharing,, Biophysical J., 54 (1988), 411. doi: 10.1016/S0006-3495(88)82975-8. Google Scholar [41] K. J. Stratford, K. Tarczy-Hornoch, K. A. Martin, N. J. Bannister and J. J. Jack, Excitatory synaptic inputs to spiny stellate cells in cat visual cortex,, Nature, 382 (1996), 258. doi: 10.1038/382258a0. Google Scholar [42] S. Tsuji, T. Ueta, H. Kawakami, H. Fujii and K. Aihara, Bifurcations in two-dimensional Hindmarsh-Rose type model,, Int. J. Bifurcation and Chaos, 17 (2007), 985. doi: 10.1142/S0218127407017707. Google Scholar [43] T. Uchiyama and U. Windhorst, Effects of spinal recurrent inhibition on motoneuron short-term synchronization,, Biol. Cybern., 96 (2007), 561. doi: 10.1007/s00422-007-0151-7. Google Scholar [44] U. Windhorst, On the role of recurrent inhibitory feedback in motor control},, Prog Neurobiol, 49 (1996), 517. doi: 10.1016/0301-0082(96)00023-8. Google Scholar [45] L. Zemanova, C. Zhou and J. Kurths, Structural and functional clusters of complex brain networks,, Physica D, 224 (2006), 202. doi: 10.1016/j.physd.2006.09.008. Google Scholar [46] F. Zhang, W. Zhang, P. Meng and J. Z. Su, Bifurcation analysis of bursting solutions of two Hindmarsh-Rose neurons with joint electrical and synaptic coupling,, Discrete and Continuous Dynamical Systems-B, 16 (2011), 637. doi: 10.3934/dcdsb.2011.16.637. Google Scholar

show all references

##### References:
 [1] P. Andersen, J. C. Eccles and Y. Loyning, Recurrent inhibition in the hippocampus with identification of the inhibitory cell and its synapses,, Nature, 198 (1963), 540. Google Scholar [2] M. C. Ashby and J. T. Isaac, Maturation of a recurrent excitatory neocortical circuit by experience-dependent unsilencing of newly formed dendritic spines,, Neuron, 70 (2011), 510. Google Scholar [3] E. Benoit, J. L. Callot, F. Diener and D. M., Chasse au canard,, Collect. Math., 32 (1981), 37. Google Scholar [4] S. A. Campbell and Y. Yuan, Zero singularities of codimension two and three in delay differential equations,, Nonlinearity, 21 (2008), 2671. doi: 10.1088/0951-7715/21/11/010. Google Scholar [5] A. M. Castelfranco and H. W. Stech, Periodic solutions in a model of recurrent neural feedback,, SIAM J. Appl. Math., 47 (1987), 573. doi: 10.1137/0147039. Google Scholar [6] T. R. Chay, Chaos in a three-variable model of an excitable cell,, Physica D: Nonlinear phenomena, 16 (1985), 233. doi: 10.1016/0167-2789(85)90060-0. Google Scholar [7] T. R. Chay, Glucose response to bursting-spiking pancreatic $\beta$-cells by a barrier kinetic model,, Biol. Cybern., 52 (1985), 339. Google Scholar [8] T. R. Chay and J. Keizer, Theory of the effect of extracellular potassium on oscillations in the pancreatic beta-cell,, Biophysical J., 48 (1985), 815. doi: 10.1016/S0006-3495(85)83840-6. Google Scholar [9] T. R. Chay and J. Rinzel, Bursting, beating, and chaos in an excitable membrane model,, Biophysical J., 47 (1985), 357. doi: 10.1016/S0006-3495(85)83926-6. Google Scholar [10] S. S. Chen, C. Y. Cheng and Y.-R. Lin, Application of a two-dimensional Hindmarsh-Rose type model for bifurcation analysis,, Int. J. Bifurcation Chaos, 23 (2013). doi: 10.1142/S0218127413500557. Google Scholar [11] J. A. Connor and C. F. Stevens, Inward and delayed outward membrane currents in isolated neural somata under voltage clamp,, J. Physiol., 213 (1971), 1. doi: 10.1113/jphysiol.1971.sp009364. Google Scholar [12] J. A. Connor, D. Walter and R. McKown, Neural repetitive firing: Modifications of the Hodgkin-Huxley axon suggested by experimental results from crustacean axons,, Biophysical Journal, 18 (1977), 81. doi: 10.1016/S0006-3495(77)85598-7. Google Scholar [13] K. Cooke and Z. Grossman, Discrete delay, distributed delay and stabilit switches,, J. Math. Anal. Appl., 86 (1982), 592. doi: 10.1016/0022-247X(82)90243-8. Google Scholar [14] G. Deco, V. K. Jirsa, P. A. Robinson, M. Breakspear and K. Friston, The dynamic brain: from spiking neurons to neural masses and cortical fields,, PLOS computational biology, 4 (2008). doi: 10.1371/journal.pcbi.1000092. Google Scholar [15] M. Desroches, B. Krauskopf and H. M. Osinga, Mixed-mode oscillations and slow manifolds in the self-coupled Fitzhugh-Nagumo system,, Chaos, 18 (2008). doi: 10.1063/1.2799471. Google Scholar [16] M. Dichter and W. A. Spencer, Penicillin-induced interictal discharges from the cat hippocampus, II. mechanisms underlying orgin and restriction,, J. Neurophyiol., 32 (1969), 663. Google Scholar [17] R. J. Douglas, C. Koch, M. Mahowald, K. A. Martin and H. H. Suarez, Recurrent excitation in neocortical circuits,, Science, 269 (1995), 981. doi: 10.1126/science.7638624. Google Scholar [18] V. Dragoi and M. Sur, Dynamic properties of recurrent inhibition in primary visual cortex: Contrast and orientation dependence of contextual effects,, J Neurophysiol, 83 (2000), 1019. Google Scholar [19] I. Erchova and D. J. McGonigle, Rhythms of the brain: An examination of mixed mode oscillation approaches to the analysis of neurophysiological data,, Chaos, 18 (2008). doi: 10.1063/1.2900015. Google Scholar [20] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophys. J., 1 (1961), 445. doi: 10.1016/S0006-3495(61)86902-6. Google Scholar [21] J. K. Hale, Theory of Functional Differential Equations,, Applied mathematical sciences. Springer-Verlag, (1977). Google Scholar [22] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf bifurcation,, Cambridge University Press, (1981). Google Scholar [23] J. L. Hindmarsh and R. M. Rose, A model of the nerve impluse using two first-order differential equations,, Nature, 296 (1982), 162. Google Scholar [24] J. L. Hindmarsh and R. M. Rose, A model of neuronal bursting using three coupled first order differential equaions,, Proc. R. Soc. Lond. B Biol. Sci., 221 (1984), 87. Google Scholar [25] A. L. Hodgkin and A. F. Huxley, A qualitative description of membrane current and its application to conduction and excitation in nerve,, J. Physiol., 117 (1952), 500. Google Scholar [26] R. Katz and E. Pierrot-Deseilligny, Recurrent inhibition in humans,, Progress in Neurobiology, 57 (1999), 325. doi: 10.1016/S0301-0082(98)00056-2. Google Scholar [27] M. T. M. Koper, Bifurcations of mixed-mode oscillations in a three-variable autonomous Van der Pol-Duffing model with a cross-shaped phase diagram,, Physica D, 80 (1995), 72. doi: 10.1016/0167-2789(95)90061-6. Google Scholar [28] M. Krupa, N. Popović, N. Kopell and H. G. Rotstein, Mixed-mode oscillations in a three time-scale model for the dopaminergic neuron,, Chaos, 18 (2008). doi: 10.1063/1.2779859. Google Scholar [29] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory (3rd ed.),, Number 112 in Applied Mathematical Sciences. Springer Verlag New York, (2004). doi: 10.1007/978-1-4757-3978-7. Google Scholar [30] S. Ma and Z. S. Feng, Fold-hopf bifurcations of the Rose-Hindmarsh model with time delay,, Int. J. Bifurcation and Chaos, 21 (2011), 437. doi: 10.1142/S0218127411028490. Google Scholar [31] S. Ma, Z. S. Feng and Q. S. Lu, Dynamics and double hopf bifurcations of the Rose-Hindmarsh model with time delay,, Int. J. Bifurcation and Chaos, 19 (2009), 3733. doi: 10.1142/S0218127409025080. Google Scholar [32] M. C. Mackey and U. an der Heiden, The dynamics of recurrent inhibition,, J Math Biol, 19 (1984), 211. doi: 10.1007/BF00277747. Google Scholar [33] B. Mattei, A. Schmied, R. Mazzocchio, B. Decchi, A. Rossi and J. P. Vedel, Pharmacologically induced enhancement of recurrent inhibition in humans: Effects on motoneurone discharge patterns,, J Physiol, 548 (2003), 615. Google Scholar [34] A. Mitra, S. S. Mitra and R. W. Tsien, Heterogeneous reallocation of presynaptic efficacy in recurrent excitatory circuits adapting to inactivity,, Nature Neuroscience, 15 (2012), 250. doi: 10.1038/nn.3004. Google Scholar [35] J. S. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon,, Proc. IRE, 50 (1962), 2061. doi: 10.1109/JRPROC.1962.288235. Google Scholar [36] V. Petrov, S. Scott and K. Showalter, Mixed mode oscillations in chemical systems,, J. Chem. Phys., 97 (1992). doi: 10.1063/1.463727. Google Scholar [37] R. E. Plant, A FitzHugh differential-difference equation modeling recurrent neural feedback,, SIAM Journal on applied mathematics, 40 (1981), 150. doi: 10.1137/0140012. Google Scholar [38] R. M. Rose and J. L. Hindmarsh, The assembly of ionic currents in a thalamic neuron I. the three-dimensional model,, Proc. R. Soc. Lond. B, 237 (1989), 267. doi: 10.1098/rspb.1989.0049. Google Scholar [39] J. Rubin and M. Wechselberger, The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple time scales,, Chaos, 18 (2008). doi: 10.1063/1.2789564. Google Scholar [40] A. Sherman, J. Rinzel and J. Keizer, Emergence of organized bursting in clusters of pancreatic beta-cells by channel sharing,, Biophysical J., 54 (1988), 411. doi: 10.1016/S0006-3495(88)82975-8. Google Scholar [41] K. J. Stratford, K. Tarczy-Hornoch, K. A. Martin, N. J. Bannister and J. J. Jack, Excitatory synaptic inputs to spiny stellate cells in cat visual cortex,, Nature, 382 (1996), 258. doi: 10.1038/382258a0. Google Scholar [42] S. Tsuji, T. Ueta, H. Kawakami, H. Fujii and K. Aihara, Bifurcations in two-dimensional Hindmarsh-Rose type model,, Int. J. Bifurcation and Chaos, 17 (2007), 985. doi: 10.1142/S0218127407017707. Google Scholar [43] T. Uchiyama and U. Windhorst, Effects of spinal recurrent inhibition on motoneuron short-term synchronization,, Biol. Cybern., 96 (2007), 561. doi: 10.1007/s00422-007-0151-7. Google Scholar [44] U. Windhorst, On the role of recurrent inhibitory feedback in motor control},, Prog Neurobiol, 49 (1996), 517. doi: 10.1016/0301-0082(96)00023-8. Google Scholar [45] L. Zemanova, C. Zhou and J. Kurths, Structural and functional clusters of complex brain networks,, Physica D, 224 (2006), 202. doi: 10.1016/j.physd.2006.09.008. Google Scholar [46] F. Zhang, W. Zhang, P. Meng and J. Z. Su, Bifurcation analysis of bursting solutions of two Hindmarsh-Rose neurons with joint electrical and synaptic coupling,, Discrete and Continuous Dynamical Systems-B, 16 (2011), 637. doi: 10.3934/dcdsb.2011.16.637. Google Scholar
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