Stability and Hopf bifurcation in leech heart interneuron model

Document Type : Research Article

Authors

• F. Parvizi ¹
• M. Razvan ²
• Y. Alipour Fakhri ¹

¹ Department of Mathematics, Payame Noor University(PNU), Tehran, Iran.

² Department of Mathematics, Faculty of Mathematical Sciences, University of Sharif, Tehran, Iran.

10.22067/ijnao.2022.75087.1101

Abstract

This article investigates the activity regimes of a realistic neuron model (as a slow-fast system). The authors study this model using the dynam-ical systems theory, for example, qualitative theory methods of slow-fast systems. The authors obtain the stability conditions of equilibria in leech heart interneurons under defined pharmacological conditions and following Hodgkin–Huxley formalism. Although in neuronal models, the membrane is usually considered  capacitance as a fixed parameter, the membrane ca-pacitance parameter is assumed as a control parameter to guarantee the existence of Hopf bifurcation using the Routh–Hurwitz criteria. The au-thors investigate the transition mechanism between the silent phase and tonic spiking mode. Furthermore, some simulations are provided using XPPAUT software for analytical results.

Keywords

• Stability
• Hopf bifurcation
• Routh–Hurwitz criteria

20.1001.1.24236977.2023.13.2.2.5

Main Subjects

• None of the above, but in this section

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References

[1] Bertram, R., Butte, M.J., Kiemel, T. and Sherman, A. Topological and phenomenologial classification of bursting oscillations, Bull. Math. Biol. 57(3) (1995) 413–439.
[2] Calabrese, R.L., Nadim, F. and Olsen, Ø.H. Heartbeat control in the medicinal leech: a model system for understanding the origin, coordi-nation, and modulation of rhythmic motor patterns, J. Neurobiol. 27(3) (1995) 390–402.
[3] Cymbalyuk, G.S. and Calabrese, R.L. A model of slow plateau-like oscil-lations based upon the fast Na+ current in a window mode, Neurocom-puting 38 (2001) 159–166.
[4] Cymbalyuk, G.S., Gaudry, Q., Masino, M.A., and Calabrese, R.J. Burst-ing in leech heart interneurons: cell-autonomous and network-based mechanisms, J. Neurosci. 22 (24) (2002) 10580–10592. Neurosci. 22, 10580 (2002).

[5] Cymbalyuk, G. and Shilnikov, A. Coexistence of tonic spiking oscilla-tions in a leech neuron model, J. Comput. Neurosci. 18(3) (2005) 255–263.
[6] Gentet, L.J., Stuart G.J., and Clements J.D. Direct measurement of specific membrane capacitance in neurons, Biophys. J. 79 (2000) 314–320.
[7] Grimnes, S. and Martinsen, Ø.G. Alpha-dispersion in human tissue, J. Phys.: Conf. Ser. 224(1) (2010) 012073.
[8] Guckenheimer, J. and Holmes, P.J. Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Vol. 42. Springer Science & Business Media, 2013.
[9] Hill, J., Lu, M., Masino, O. Olsen, R. and Calabrese, L. A model of a segmental oscillator in the leech heartbeat neuronal network, J. Comput. Neuroscience. 10 (2001) 281–302.
[10] Howell, B., Medina, L.E. and Grill, W.M. Effects of frequency-dependent membrane capacitance on neural excitability, J. Neural Eng. 12 (2015) 056015.
[11] Izhikevich, E. Neural excitability, spiking and bursting, Int. J. Bifurc. Chaos 10 (6) (2000) 1171–1266.
[12] Kolomiets, M. L. and Shilnikov, A. L. Poincarè return maps in neu-ral dynamics: three examples, International Conference on Difference Equations and Applications, pp. 45–57. Springer, Cham, 2019.
[13] Liu, W. Criterion of Hopf bifurcation without using eigenvalues, J. Math. Anal. Appl. 182(1) (1994) 250–256.
[14] Malashchenko, T., Shilnikov, A. and Cymbalyuk, G. Bistability of burst-ing and silence regimes in a model of a leech heart interneuron, Phys. Rev. E 84(4) (2011) 041910.
[15] McIntyre, C.C., Richardson A.G., and Grill, W.M. Modeling the ex-citability of mammalian nerve fibers: influence of after potentials on the recovery cycle J. Neurophysiol. 87 (2002) 995–1006.
[16] McNeal, D. R. Analysis of a model for excitation of myelinated nerve IEEE Trans. Biomed. Eng. 23 (1976) 329–337 .
[17] Monfared, Z. and Dadi, Z. Analysing panel flutter in supersonic flow by Hopf bifurcation, Iranian Journal of Numerical Analysis and Optimiza-tion 4(2) (2014) 1–14.
[18] Opdyke, C.A. and Calabrese, R.L. A persistent sodium current con-tributes to oscillatory activity in heart interneurons of the medicinal leech, J. Comp. Physiol. 175. (1994) 781–789.

[19] Shilnikov, A.L. and Cymbalyuk G. Transition between tonic spiking and bursting in a neuron model via the blue-sky catastrophe, Phys. Rev. Lett. 94 (4) (2005) 048101.
[20] Süli E. Numerical solution of ordinary differential equations, Mathemat-ical Institute, University of Oxford, 2010.
[21] Wanga Q., Duana, Z., Fengc, Z., Chena, G. and Lu, Q. Synchronization transition in gap-junction-coupled leech neurons, Phys. A: Stat. Mech. Appl. 387 (2008) 4404–4410.