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    Analytical Properties and Solutions of the FitzHugh – Rinzel Model

    2019, Vol. 15, no. 1, pp.  3-12

    Author(s): Zemlyanukhin A. I., Bochkarev A. V.

    The FitzHugh – Rinzel model is considered, which differs from the famous FitzHugh – Nagumo model by the presence of an additional superslow dependent variable. Analytical properties of this model are studied. The original system of equations is transformed into a third-order nonlinear ordinary differential equation. It is shown that, in the general case, the equation does not pass the Painlevé test, and the general solution cannot be represented by Laurent series. Using the singular manifold method in terms of the Schwarzian derivative, an exact particular solution in the form of a kink is constructed, and restrictions on the coefficients of the equation necessary for the existence of such a solution are revealed. An asymptotic solution is obtained that shows good agreement with the numerical one. This solution can be used to verify the results in a numerical study of the FitzHugh – Rinzel model.
    Keywords: neuron, FitzHugh – Rinzel model, singular manifold, exact solution, asymptotic solution
    Citation: Zemlyanukhin A. I., Bochkarev A. V., Analytical Properties and Solutions of the FitzHugh – Rinzel Model, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 1, pp.  3-12
    DOI:10.20537/nd190101


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    References
    [1] Hodgkin, A. L. and Huxley, A. F., “A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve”, J. Physiol., 117:4 (1952), 500–544  crossref
    [2] FitzHugh, R., “Impulses and Physiological States in Theoretical Models of Nerve Membrane”, Biophys. J., 1:6 (1961), 445–466  crossref
    [3] Nagumo, J., Arimoto, S., and Yoshizawa, S., “An Active Pulse Transmission Line Simulating Nerve Axon”, Proc. of the IRE, 50:10 (1962), 2061–2070  crossref
    [4] FitzHugh, R., “A Kinetic Model for the Conductance Changes in Nerve Membranes”, J. Cell. Comp. Physiol., 66:suppl. 2 (1965), 111–117  crossref
    [5] Mornev, O. A., Tsyganov, I. M., Aslanidi, O. V., and Tsyganov, M. A., “Beyond the Kuramoto – Zel'dovich Theory: Steadily Rotating Concave Spiral Waves and Their Relation to the Echo Phenomenon”, JETP Lett., 77:6 (2003), 270–275  mathnet  crossref  mathscinet  adsnasa; Pis'ma v Zh. Èksper. Teoret. Fiz., 77:6 (2003), 319–325 (Russian)
    [6] Llibre, J. and Vidal, C., “Periodic Solutions of a Periodic FitzHugh – Nagumo System”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25:13 (2015), 1550180, 6 pp.  crossref  mathscinet  zmath
    [7] Tsuji, Sh., Ueta, T., Kawakami, H., and Aihara, K., “A Design Method of Bursting Using Two-Parameter Bifurcation Diagrams in FitzHugh – Nagumo Model”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14:7 (2004), 2241–2252  crossref  mathscinet  zmath
    [8] Gelens, L., Anderson, G. A., and Ferrell, J. E., Jr., “Spatial Trigger Waves: Positive Feedback Gets You a Long Way”, Mol. Biol. Cell, 25:22 (2014), 3486–3493  crossref
    [9] Cheng, X., and Ferrell, J. E., Jr., “Apoptosis Propagates through the Cytoplasm As Trigger Waves”, Science, 361:6402 (2018), 607–612  crossref  adsnasa
    [10] Kudryashov, N. A., “Asymptotic and Exact Solutions of the FitzHugh – Nagumo Model”, Regul. Chaotic Dyn., 23:2 (2018), 152–160  mathnet  crossref  mathscinet  zmath  adsnasa
    [11] Kudryashov, N. A., Rybka, R. B., and Sboev, A. G., “Analytical Properties of the Perturbed FitzHugh – Nagumo Model”, Appl. Math. Lett., 76 (2018), 142–147  crossref  mathscinet  zmath  elib
    [12] Rinzel, J., “A Formal Classification of Bursting Mechanisms in Excitable Systems”, Mathematical Topics in Population Biology, Morphogenesis and Neurosciences, Lecture Notes in Biomath., 71, eds. E. Teramoto, M. Yamaguti, Springer, Berlin, 1987, 267–281  crossref  mathscinet
    [13] Belykh, V. N. and Pankratova, E. V., “Chaotic Synchronization in Ensembles of Coupled Neurons Modeled by the FitzHugh – Rinzel System”, Radiophys. Quantum El., 49:11 (2006), 910–921  crossref  elib; Izv. Vyssh. Uchebn. Zaved. Radiofizika, 49:11 (2006), 1002–1014 (Russian)
    [14] Wojcik, J. and Shilnikov, A., Voltage Interval Mappings for an Elliptic Bursting Model, 2013, arXiv: 1310.5232 [nlin.CD]  mathscinet
    [15] Conte, R., “Singularities of Differential Equations and Integrability”, Introduction to Methods of Complex Analysis and Geometry for Classical Mechanics and Nonlinear Waves (Chamonix, 1993), eds. D. Benest, C. Froeschle, Frontières, Gif-sur-Yvette, 1994, 49–143  mathscinet  zmath
    [16] Weiss, J., Tabor, M., and Carnevale, G., “The Painlevé Property for Partial Differential Equations”, J. Math. Phys., 24:3 (1983), 522–526  crossref  mathscinet  zmath  adsnasa
    [17] Weiss, J., “The Painlevé Property for Partial Differential Equations: 2. Bäcklund Transformation, Lax Pairs, and the Schwarzian Derivative”, J. Math. Phys., 24:6 (1983), 1405–1413  crossref  mathscinet  zmath  adsnasa
    [18] Kudryashov, N. A., “From Singular Manifold Equations to Integrable Evolution Equations”, J. Phys. A, 27:7 (1994), 2457–2470  crossref  mathscinet  zmath  adsnasa



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