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

Shyan-Shiou  Chen; Chang-Yuan  Cheng

doi:10.3934/dcdsb.2016.21.37

Article Contents

2016, Volume 21, Issue 1: 37-53. Doi: 10.3934/dcdsb.2016.21.37

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

Shyan-Shiou Chen^1, and
Chang-Yuan Cheng^2,

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

Received: January 31, 2013

Revised: April 30, 2015

Published: December 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 34K18, 34K17; Secondary: 34K60.

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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), 540-542.
[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), 510-521.
[3]	E. Benoit, J. L. Callot, F. Diener and D. M., Chasse au canard, Collect. Math., 32 (1981), 37-119.
[4]	S. A. Campbell and Y. Yuan, Zero singularities of codimension two and three in delay differential equations, Nonlinearity, 21 (2008), 2671-2691.doi: 10.1088/0951-7715/21/11/010.
[5]	A. M. Castelfranco and H. W. Stech, Periodic solutions in a model of recurrent neural feedback, SIAM J. Appl. Math., 47 (1987), 573-588.doi: 10.1137/0147039.
[6]	T. R. Chay, Chaos in a three-variable model of an excitable cell, Physica D: Nonlinear phenomena, 16 (1985), 233-242.doi: 10.1016/0167-2789(85)90060-0.
[7]	T. R. Chay, Glucose response to bursting-spiking pancreatic $\beta$-cells by a barrier kinetic model, Biol. Cybern., 52 (1985), 339-349.
[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), 815-827.doi: 10.1016/S0006-3495(85)83840-6.
[9]	T. R. Chay and J. Rinzel, Bursting, beating, and chaos in an excitable membrane model, Biophysical J., 47 (1985), 357-366.doi: 10.1016/S0006-3495(85)83926-6.
[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), 1350055, 21pp.doi: 10.1142/S0218127413500557.
[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), 1-19.doi: 10.1113/jphysiol.1971.sp009364.
[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), 81-102.doi: 10.1016/S0006-3495(77)85598-7.
[13]	K. Cooke and Z. Grossman, Discrete delay, distributed delay and stabilit switches, J. Math. Anal. Appl., 86 (1982), 592-627.doi: 10.1016/0022-247X(82)90243-8.
[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), e1000092.doi: 10.1371/journal.pcbi.1000092.
[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), 015017, 8pp.doi: 10.1063/1.2799471.
[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), 663-687.
[17]	R. J. Douglas, C. Koch, M. Mahowald, K. A. Martin and H. H. Suarez, Recurrent excitation in neocortical circuits, Science, 269 (1995), 981-985.doi: 10.1126/science.7638624.
[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), 1019-1030.
[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), 015115, 14pp.doi: 10.1063/1.2900015.
[20]	R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.doi: 10.1016/S0006-3495(61)86902-6.
[21]	J. K. Hale, Theory of Functional Differential Equations, Applied mathematical sciences. Springer-Verlag, 1977.
[22]	B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf bifurcation, Cambridge University Press, Cambridge, 1981.
[23]	J. L. Hindmarsh and R. M. Rose, A model of the nerve impluse using two first-order differential equations, Nature, 296 (1982), 162-164.
[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), 87-102.
[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), 500-544.
[26]	R. Katz and E. Pierrot-Deseilligny, Recurrent inhibition in humans, Progress in Neurobiology, 57 (1999), 325-355.doi: 10.1016/S0301-0082(98)00056-2.
[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), 72-94.doi: 10.1016/0167-2789(95)90061-6.
[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), 015106, 19pp.doi: 10.1063/1.2779859.
[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.
[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), 437-452.doi: 10.1142/S0218127411028490.
[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), 3733-3751.doi: 10.1142/S0218127409025080.
[32]	M. C. Mackey and U. an der Heiden, The dynamics of recurrent inhibition, J Math Biol, 19 (1984), 211-225.doi: 10.1007/BF00277747.
[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), 615-629.
[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), 250-257.doi: 10.1038/nn.3004.
[35]	J. S. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2071.doi: 10.1109/JRPROC.1962.288235.
[36]	V. Petrov, S. Scott and K. Showalter, Mixed mode oscillations in chemical systems, J. Chem. Phys., 97 (1992), p6191.doi: 10.1063/1.463727.
[37]	R. E. Plant, A FitzHugh differential-difference equation modeling recurrent neural feedback, SIAM Journal on applied mathematics, 40 (1981), 150-162.doi: 10.1137/0140012.
[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), 267-288.doi: 10.1098/rspb.1989.0049.
[39]	J. Rubin and M. Wechselberger, The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple time scales, Chaos, 18 (2008), 015105, 12pp.doi: 10.1063/1.2789564.
[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), 411-425.doi: 10.1016/S0006-3495(88)82975-8.
[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), 258-261.doi: 10.1038/382258a0.
[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), 985-998.doi: 10.1142/S0218127407017707.
[43]	T. Uchiyama and U. Windhorst, Effects of spinal recurrent inhibition on motoneuron short-term synchronization, Biol. Cybern., 96 (2007), 561-575.doi: 10.1007/s00422-007-0151-7.
[44]	U. Windhorst, On the role of recurrent inhibitory feedback in motor control}, Prog Neurobiol, 49 (1996), 517-587.doi: 10.1016/0301-0082(96)00023-8.
[45]	L. Zemanova, C. Zhou and J. Kurths, Structural and functional clusters of complex brain networks, Physica D, 224 (2006), 202-212.doi: 10.1016/j.physd.2006.09.008.
[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), 637-651.doi: 10.3934/dcdsb.2011.16.637.