Complex Oscillations in the Delayed FitzHugh–Nagumo Equation

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• 作者：Maciej Krupa ; Jonathan D. Touboul
• 关键词：Delayed differential equations ; Slow–fast systems ; Mixed ; mode oscillations ; Bursting ; Chaos
• 刊名：Journal of Nonlinear Science
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：26
• 期：1
• 页码：43-81
• 全文大小：3,667 KB
• 参考文献：Baer, S.M., Erneux, T.: Singular Hopf bifurcation to relaxation oscillations. SIAM J. Appl. Math. 46(5), 721–739 (1986)MathSciNet CrossRef MATH
  Benoit, E., Callot, J., Diener, F., Diener, M., et al.: Chasse au canard (première partie). Collectanea Mathematica 32(1), 37–76 (1981)MathSciNet MATH
  Bernardo, L., Foster, R.: Oscillatory behavior in inferior olive neurons: mechanism, modulation, cell aggregates. Brain Res. Bull. 17, 773–784 (1986)CrossRef
  Brøns, M., Krupa, M., Wechselberger, M.: Mixed mode oscillations due to the generalized canard phenomenon. Fields Inst. Commun. 49, 39–63 (2006)
  Burke, J., Desroches, M., Barry, A.M., Kaper, T.J., Kramer, M.A., et al.: A showcase of torus canards in neuronal bursters. J. Math. Neurosci. 2(3), (2012). doi:10.​1186/​2190-8567-2-3
  Campbell, S., Stone, E., Erneux, T.: Delay induced canards in high speed machining. Dyn. Syst. 24(3), 373–392 (2009)MathSciNet CrossRef MATH
  Campbell, S.A.: Calculating centre manifolds for delay differential equations using maple. In: Delay Differential Equations, pp. 1–24. Springer (2009)
  Chiba, H.: Periodic orbits and chaos in fast-slow systems with Bogdanov–Takens type fold points. J. Differ. Equ. 250, 112–160 (2011)MathSciNet CrossRef MATH
  Corless, R., Gonnet, G., Hare, D., Jeffrey, D., Knuth, D.: On the lambertw function. Adv. Comput. Math. 5(1), 329–359 (1019–7168) (1996)
  Desroches, M., Guckenheimer, J., Krauskopf, B., Kuehn, C., Osinga, H., Wechselberger, M.: Mixed-mode oscillations with multiple time scales. SIAM Rev. 54(2), 211–288 (2012)MathSciNet CrossRef MATH
  Drover, J., Rubin, J., Su, J., Ermentrout, B.: Analysis of a canard mechanism by which excitatory synaptic coupling can synchronize neurons at low firing frequencies. SIAM J. Appl. Math. 65(1), 69–92 (2004)MathSciNet CrossRef MATH
  Dumortier, F., Roussarie, R.H.: Canard Cycles and Center Manifolds, vol. 577. American Mathematical Society, Providence (1996)
  Engelborghs, K., Luzyanina, T., Roose, D.: Numerical bifurcation analysis of delay differential equations using dde-biftool. ACM Trans. Math. Software (TOMS) 28(1), 1–21
  Engelborghs, K., Luzyanina, T., Samaey, G.: Dde-biftool v. 2.00: a matlab package for bifurcation analysis of delay differential equations. Technical Report TW-330, Department of Computer Science, K.U.Leuven, Leuven, Belgium (2001)
  Ermentrout, G., Terman, D.: Mathematical foundations neuroscience 35, (2010). doi:10.​1007/​978-0-387-87708-2
  Ermentrout, G.B., Wechselberger, M.: Canards, clusters, and synchronization in a weakly coupled interneuron model. SIAM J. Appl. Dyn. Syst. 8(1), 253–278 (2009)MathSciNet CrossRef MATH
  Faria, T., Magalhães, L.: Normal forms for retarded functional differential equations and applications to the bogdanov-takens singularity. J. Differ. Equ. 122, 201–224 (1995)CrossRef MATH
  Faria, T., Magalhães, L.T.: Normal forms for retarded functional differential equations with parameters and applications to hopf bifurcation. J. Differ. Equ. 122(2), 181–200 (1995)CrossRef MATH
  Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31(1), 53–98 (1979)MathSciNet CrossRef MATH
  FitzHugh, R.: Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biol. 17(4), 257–278 (0092–8240) (1955)
  Friart, G., Weicker, L., Danckaert, J., Erneux, T.: Relaxation and square-wave oscillations in a semiconductor laser with polarization rotated optical feedback. Opt. Express 22(6), 6905–6918 (2014)CrossRef
  Guckenheimer, J., Holmes, P.J.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, vol. 42. Applied Mathematical Sciences, Springer, Berlin (1983)CrossRef MATH
  Guckenheimer, J., Osinga, H.M.: The singular limit of a Hopf bifurcation. Disc. Contin. Dyn. Syst. Ser. A 32(8), 2805–2823 (2012)MathSciNet CrossRef MATH
  Hale, J., Lunel, S.: Introduction to Functional Differential Equations. Springer, Berlin (1993)CrossRef MATH
  Higuera, M., Knobloch, E., Vega, J.: Dynamics of nearly inviscid faraday waves in almost circular containers. Phys. D 201(1), 83–120 (2005)MathSciNet CrossRef MATH
  Hupkes, H.J., Sandstede, B.: Traveling pulse solutions for the discrete FitzHugh-Nagumo system. SIAM J. Appl. Dyn. Syst. 9(3), 827–882 (2010)MathSciNet CrossRef MATH
  Izhikevich, E.: Neural excitability, spiking, and bursting. Int. J. Bifurc. Chaos 10, 1171–1266 (2000)MathSciNet CrossRef MATH
  Jones, C.K.: Geometric singular perturbation theory. In: Dynamical systems, pp. 44–118. Springer (1995)
  Kandel, E.R., Schwartz, J.H., Jessell, T.M., et al.: Principles of Neural Science, vol. 4. McGraw-Hill, New York (2000)
  Koper, M.: Bifurcations of mixed-mode oscillations in a three-variable autonomous van der pol-duffing model with a cross-shaped phase diagram. Phys. D 80(1), 72–94 (1995)MathSciNet CrossRef MATH
  Kozyreff, G., Erneux, T.: Singular hopf bifurcation in a differential equation with large state-dependent delay. Proc. R. Soc. A Math. Phys. Eng. Sci. 470(2162), 20130,596 (2014)MathSciNet CrossRef
  Krupa, M., Touboul, J.: Canard explosion in delayed equations with multiple timescales. arXiv preprint arXiv:​1407.​7703 (2014)
  Kuznetsov, Y.: Elements of applied bifurcation theory 112, (1995). doi:10.​1007/​978-1-4757-3978-7
  Mallet-Paret, J., Nussbaum, R.D.: Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation. Annali di Matematica 145(1), 33–128 (1986)MathSciNet CrossRef MATH
  Mischler, S., Quininao, C., Touboul, J.: On a kinetic FitzHugh-Nagumo model of neuronal network. arXiv:​hal-01108872 (2015)
  Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50(10), 2061–2070 (1962)CrossRef
  Plant, R.E.: A FitzHugh differential-difference equation modeling recurrent neural feedback. SIAM J. Appl. Math. 40(1), 150–162 (1981)MathSciNet CrossRef MATH
  Rinzel, J.: A formal classification of bursting mechanisms in excitable systems. In: Mathematical Topics in Population Biology, Morphogenesis and Neurosciences, pp. 267–281. Springer (1987)
  Rinzel, J., Ermentrout, G.B.: Analysis of neural excitability and oscillations. In: Methods of Neuronal Modeling, pp. 251–292. MIT Press (1998)
  Rubin, J., Wechselberger, M.: Giant squid-hidden canard: the 3D geometry of the Hodgkin-Huxley model. Biol. Cybern. 97(1), 5–32 (2007)MathSciNet CrossRef MATH
  Sherman, A., Rinzel, J., Keizer, J.: Emergence of organized bursting in clusters of pancreatic beta-cells by channel sharing. Biophys. J. 54(3), 411–425 (1988)CrossRef
  Simpson, D.J., Kuske, R.: Mixed-mode oscillations in a stochastic, piecewise-linear system. Phys. D 240(14), 1189–1198 (2011)CrossRef MATH
  Stewart, I., Golubitsky, M., Pivato, M.: Symmetry groupoids and patterns of synchrony in coupled cell networks. SIAM J. Appl. Dyn. Syst. 2(4), 609–646 (2003)MathSciNet CrossRef MATH
  Stone, E., Campbell, S.A.: Stability and bifurcation analysis of a nonlinear dde model for drilling. J. Nonlinear Sci. 14(1), 27–57 (2004)MathSciNet CrossRef MATH
  Terman, D.: Chaotic spikes arising from a model of bursting in excitable membranes. SIAM J. Appl. Math. 51(5), 1418–1450 (1991)MathSciNet CrossRef MATH
  Touboul, J.: Limits and dynamics of stochastic neuronal networks with random delays. J. Stat. Phys 149(4), 569–597 (2012)MathSciNet CrossRef MATH
  Touboul, J., Hermann, G., Faugeras, O.: Noise-induced behaviors in neural mean field dynamics. SIAM J. Dyn. Syst. 11(1), 49–81 (2012)MathSciNet CrossRef MATH
  Touboul, J., Krupa, M., Desroches, M.: Noise-induced canard and mixed-mode oscillations in large stochastic networks with multiple timescales. arXiv preprint arXiv:​1302.​7159 (2013)
  Van der Pol, B.: On relaxation-oscillations. Lond Edinb Dublin Philos. Mag. J. Sci. 2(11), 978–992 (1926)CrossRef
  Weicker, L., Erneux, T., D’Huys, O., Danckaert, J., Jacquot, M., Chembo, Y., Larger, L.: Strongly asymmetric square waves in a time-delayed system. Phys. Rev. E 86(5), 055201 (2012)CrossRef
  Weicker, L., Erneux, T., D’Huys, O., Danckaert, J., Jacquot, M., Chembo, Y., Larger, L.: Slow–fast dynamics of a time-delayed electro-optic oscillator. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 371(1999), 20120459 (2013)MathSciNet CrossRef
  Zhang, W., Kirk, V., Sneyd, J., Wechselberger, M.: Changes in the criticality of hopf bifurcations due to certain model reduction techniques in systems with multiple timescales. The Journal of Mathematical Neuroscience (JMN) 1(1), 1–22 (2011)MathSciNet CrossRef
  
• 作者单位：Maciej Krupa (2) (3)
  Jonathan D. Touboul (1) (2)
  
  2. MYCENAE Laboratory, Inria Paris-Rocquencourt, Paris, France
  3. NEUROMATHCOMP Laboratory, Inria Sophia Antipolis-Méditeranée, Sophia Antipolis, France
  1. The Mathematical Neurosciences Team, Center for Interdisciplinary Research in Biology (CNRS UMR 7241, INSERM U1050, UPMC ED 158, MEMOLIFE PSL*), Paris, France
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Mathematical and Computational Physics
  Mechanics
  Applied Mathematics and Computational Methods of Engineering
  Economic Theory
  
• 出版者：Springer New York
• ISSN：1432-1467

文摘

Motivated by the dynamics of neuronal responses, we analyze the dynamics of the FitzHugh–Nagumo slow–fast system with delayed self-coupling. This system provides a canonical example of a canard explosion for sufficiently small delays. Beyond this regime, delays significantly enrich the dynamics, leading to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a delay-induced subcritical Bogdanov–Takens instability arising at the fold points of the S-shaped critical manifold. Underlying the transition from canard-induced to delay-induced dynamics is an abrupt switch in the nature of the Hopf bifurcation. Keywords Delayed differential equations Slow–fast systems Mixed-mode oscillations Bursting Chaos