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HYPERBOLIC PLYKIN ATTRACTOR CAN EXIST IN NEURON MODELS

Mathematics Department, Volga State Academy, 5, Nesterov st., Nizhny Novgorod 603 600, Russia

Laboratory of Nonlinear Systems, Swiss Federal Institute of Technology Lausanne (EPFL), EPFL-IC-ISC-LANOS, Station 14, 1015 Lausanne, Switzerland

Author for correspondence. Permanent address: Department of Mathematics and Statistics, Georgia State University, 30 Pryor Street, Atlanta, GA 30303, USA.

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ERIK MOSEKILDE

Department of Physics, The Technical University of Denmark, 2800 Kongens Lyngby, Denmark

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https://doi.org/10.1142/S0218127405014222Cited by:27 (Source: Crossref)

Abstract

Strange hyperbolic attractors are hard to find in real physical systems. This paper provides the first example of a realistic system, a canonical three-dimensional (3D) model of bursting neurons, that is likely to have a strange hyperbolic attractor. Using a geometrical approach to the study of the neuron model, we derive a flow-defined Poincaré map giving an accurate account of the system's dynamics. In a parameter region where the neuron system undergoes bifurcations causing transitions between tonic spiking and bursting, this two-dimensional map becomes a map of a disk with several periodic holes. A particular case is the map of a disk with three holes, matching the Plykin example of a planar hyperbolic attractor. The corresponding attractor of the 3D neuron model appears to be hyperbolic (this property is not verified in the present paper) and arises as a result of a two-loop (secondary) homoclinic bifurcation of a saddle. This type of bifurcation, and the complex behavior it can produce, have not been previously examined.

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References

V. N. Belykh, N. F. Pedersen and O. H. Soerensen, Phys. Rev. B 16, 4860 (1977). Crossref, Web of Science, Google Scholar
V. N. Belykh, A smooth dynamical system having a unique saddle equilibrium point may generate the Plykin attractor, Abstr. Int. Conf. Differential Equations and Dynamical Systems (Steklov Math. Institute Press, 2000) pp. 13–15. Google Scholar
V. N. Belykhet al., Europ. Phys. J. E 3, 205 (2000). Crossref, Web of Science, Google Scholar
N. S. Gavrilov and L. P. Shilnikov, Math. USSR Sbornik 88, 467 (1972). Google Scholar
M. Golubitsky, K. Josic and T. Kaper, Festschrift Dedicated to Floris Takens, Global Analysis of Dynamical Systems (2001) pp. 277–308. Google Scholar
S. V. Gonchenko, L. P. Shilnikov and D. V. Turaev, Int. J. Chaos 6, 1 (1996). Web of Science, Google Scholar
J. Guckenheimer and P. Holmes , Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , Applied Mathematical Sciences 42 ( Springer-Verlag , NY , 1990 ) . Google Scholar
J. Guckenheimer and R. A. Oliva, SIAM J. Appl. Dyn. Syst. 1, 105 (2002). Crossref, Web of Science, Google Scholar
J. L. Hindmarsh and R. M. Rose, Proc. Roy. Soc. London B 221, 87 (1984). Crossref, Web of Science, Google Scholar
A. Homburg, H. Kokubu and M. Krupa, Erg. Th. Dyn. Syst. 14, 667 (1994). Crossref, Web of Science, Google Scholar
T. J. Hunt and R. S. Mac Kay, Nonlinearity 16, 1499 (2003). Crossref, Web of Science, Google Scholar
E. M. Izhikevich, Int. J. Bifurcation and Chaos 10, 1171 (2000). Link, Web of Science, Google Scholar
S. E. Newhouse, Topology 13, 918 (1974). Google Scholar
I. M. Ovsyannikov and L. P. Shilnikov, Math. USSR Sbornik 73, 415 (1992). Crossref, Web of Science, Google Scholar
J. Palis and C. Pugh, Fifty Problems in Dynamical Systems, Lecture Notes in Mathematics 486 (1974) pp. 34–353. Google Scholar
R. Plykin, Math. USSR Sbornik 23, 233 (1974). Crossref, Google Scholar
J. Rinzel, Mathematical Topics in Population Biology, Morphogenesis, and Neurosciences, Lecture Notes in Biomathematics 71, eds. E. Teramoto and M. Yamaguti (Springer-Verlag, Berlin, 1987) pp. 251–291. Google Scholar
J. Rinzel and B. Ermentrout, Methods in Neuronal Modeling, eds. C. Koch and I. Segev (MIT Press, Cambridge, MA, 1989) pp. 251–291. Google Scholar
L. P. Shilnikov, Bifurcation Theory and Turbulence, Nonlinear and Turbulent Processes in Physics 3 (Harwood Academic Publ., 1984) pp. 1627–1635. Google Scholar
L. P. Shilnikov et al. , Methods of Qualitative Theory in Nonlinear Dynamics (Part I) , World Scientific Series on Nonlinear Science, Series A 4 ( 1998 ) . Crossref, Google Scholar
L. P. Shilnikov et al. , Methods of Qualitative Theory in Nonlinear Dynamics (Part II) , World Scientific Series on Nonlinear Science, Series A 5 ( 2001 ) . Crossref, Google Scholar
A. L. Shilnikov, L. P. Shilnikov and D. V. Turaev, Int. J. Bifurcation and Chaos 14, 2143 (2004). Link, Web of Science, Google Scholar
A. L. Shilnikov, R. Calabrese and G. Cymbalyuk, Phys. Rev. E 71, 056214 (2005). Crossref, Web of Science, Google Scholar
A. L. Shilnikov and G. Cymbalyuk, Phys. Rev. Lett. 94, 048101 (2005). Crossref, Web of Science, Google Scholar
D. Terman, SIAM J. Appl. Math. 51, 1418 (1991). Crossref, Web of Science, Google Scholar
D. Terman, J. Nonlin. Sci. 2, 133 (1992). Google Scholar
D. V. Turaev and L. P. Shilnikov, Soviet Math. Dokl. 342, 596 (1995). Google Scholar
X.-J. Wang, Physica D 62, 263 (1993). Crossref, Web of Science, Google Scholar