Abstract
We review how a conjectural codimension-four unfolding of the full family of cubic Liénard equations helped to identify the central singularity as an excellent candidate for the organizing center that unifies different types of spiking action potentials of excitable cells. This point of view and the subsequent numerical investigation of the respective bifurcation diagrams led, in turn, to new insight on how this codimension-four unfolding manifests itself as a sequence of bifurcation diagrams on the surface of a sphere.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bazykin, A.D., Kuznetsov, Y.A., Khibnik, A.I.: Bifurcation Diagrams of Planar Dynamical Systems. Research Computing Center, Pushchino (in Russian) (1985, preprint)
Bertram, R., Butte, M.J., Kiemel, T., Sherman, A.: Topological and phenomenological classification of bursting oscillations. Bull. Math. Biol. 57 (3), 413–439 (1995)
Dangelmayr, G., Guckenheimer, J.: On a four parameter family of planar vector fields. Arch. Ration. Mech. 97, 321–352 (1987)
Dhooge, A., Govaerts, W., Kuznetsov, Y.A.: Matcont: a Matlab package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Softw. 29 (2), 141–164 (2003). Available via http://www.matcont.ugent.be/
Doedel, E.J.: Auto: Continuation and Bifurcation Software for Ordinary Differential Equations. With major contributions from A.R. Champneys, T. F. Fairgrieve, Yu. A. Kuznetsov, B. E. Oldeman, R. C. Paffenroth, B. Sandstede, X. J. Wang and C. Zhang (2007). Available via http://cmvl.cs.concordia.ca/
Dumortier, F., Roussarie, R., Sotomayor, J.: Generic 3-parameters families of planar vector fields, unfoldings of saddle, focus and elliptic singularities with nilpotent linear parts. In: Dumortier, F., Roussarie, R., Sotomayor, J., Zoladek, H. (eds.) Bifurcations of Planar Vector Fields: Nilpotent Singularities and Abelian Integrals. Lecture Notes in Mathematics, vol. 1480, pp. 1–164. Springer, Berlin (1991)
Golubitsky, M., Josić, K., Kaper, T.J.: An unfolding theory approach to bursting in fast-slow systems. In: Broer, H.W., Krauskopf, B., Vegter, G. (eds.) Global Analysis of Dynamical Systems, pp. 277–308. Institute of Physics Publishing, Bristol (2001)
Guckenheimer, J., Malo, S.: Computer-generated proofs of phase portraits for planar systems. Int. J. Bifurcation Chaos 6 (5), 889–892 (1996)
Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117 (4), 500–544 (1952)
Hoppensteadt, F.C., Izhikevich, E.M.: Weakly Connected Neural Networks. Springer, New York (1997)
Khibnik, A.I., Krauskopf, B., Rousseau, C.: Global study of a family of cubic Liénard equations. Nonlinearity 11 (6), 1505–1519 (1998)
LeBeau, A.P., Robson, A.B., McKinnon, A.E., Sneyd, J.: Analysis of a reduced model of corticotroph action potentials. J. Theor. Biol. 192 (3), 319–339 (1998)
Malo, S.: Rigorous computer verification of planar vector field structure. Ph.D. thesis, Cornell University (1994)
Osinga, H.M., Sherman, A., Tsaneva-Atanasova, K.T.: Cross-currents between biology and mathematics: the codimension of pseudo-plateau bursting. Discrete Continuous Dyn. Syst. Ser. A 32 (8), 2853–2877 (2012)
Rinzel, J.: Bursting oscillations in an excitable membrane model. In: Sleeman, B.D., Jarvis, R.D. (eds.) Ordinary and Partial Differential Equations. Lecture Notes in Mathematics, vol. 1151, pp. 304–316. Springer, New York (1985)
Rinzel, J.: A formal classification of bursting mechanisms in excitable systems. In: Gleason, A.M. (ed.) Proceedings of the International Congress of Mathematicians, vols. 1, 2, pp. 1578–1593. American Mathematical Society, Providence RI (1987); also (with slight differences) In: Teramoto, E., Yamaguti, M. (eds.) Mathematical Topics in Population Biology, Morphogenesis and Neuroscience. Lecture Notes in Biomathematics, vol. 71, pp. 267–281. Springer, Berlin (1987)
Stern, J.V., Osinga, H.M., LeBeau, A., Sherman, A.: Resetting behavior in a model of bursting in secretory pituitary cells: distinguishing plateaus from pseudo-plateaus. Bull. Math. Biol. 70 (1), 68–88 (2008)
Tsaneva-Atanasova, K., Sherman, A., van Goor, F., Stojilkovic, S.: Mechanism of spontaneous and receptor-controlled electrical activity in pituitary somatotrophs: experiments and theory. J. Neurophysiol. 98 (1), 131–144 (2007)
van der Pol, B.: A theory of the amplitude of free and forced triode vibrations. Radio Rev. 1, 701–710 (1920)
van der Pol, B.: On relaxation oscillations. Lond. Edinb. Dublin Philos. Mag. Ser. 7 2, 978–992 (1926)
van Goor, F., Li, Y., Stojilkovic, S.: Paradoxical role of large-conductance calcium-activated K+ BK channels in controlling action potential-driven Ca2+ entry in anterior pituitary cells. J. Neurosci. 21 (16), 5902–5915 (2001)
Wang, X., Kooij, R.E.: Limit cycles in a cubic system with a cusp. SIAM J. Math. Anal. 23 (6), 1609–1622 (1992)
Acknowledgements
The work presented here is quite directly related to work of and with Christiane Rousseau, and it is a pleasure to have this opportunity to thank her for explicit and implicit support and encouragement during many years. She was instrumental in getting us into unfoldings on spheres and compactifications of phase spaces, techniques that we keep using throughout our work. We have been enjoying meeting Christiane in many different places, including regularly during our visits to Montréal of course. We also thank our co-authors Alexander Khibnik, Arthur Sherman, and Krasimira Tsaneva-Atanasova, who have been great companions in this unfolding adventure.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Krauskopf, B., Osinga, H.M. (2016). A Codimension-Four Singularity with Potential for Action. In: Toni, B. (eds) Mathematical Sciences with Multidisciplinary Applications. Springer Proceedings in Mathematics & Statistics, vol 157. Springer, Cham. https://doi.org/10.1007/978-3-319-31323-8_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-31323-8_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-31321-4
Online ISBN: 978-3-319-31323-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)