Computation of the topological entropy in chaotic biophysical bursting models for excitable cells. (English) Zbl 1106.92012

Summary: One of the interesting complex behaviors in many cell membranes is bursting, in which a rapid oscillatory state alternates with phases of relative quiescence. Although there is an elegant interpretation of many experimental results in terms of nonlinear dynamical systems, the dynamics of bursting models is not completely described.
We study the dynamical behavior of two specific three-variable models from the literature that replicate chaotic bursting. With results from symbolic dynamics, we characterize the topological entropy of one-dimensional maps that describe the salient dynamics of the attractors. The analysis of the variation of this important numerical invariant with the parameters of the systems allows us to quantify the complexity of the phenomenon and to distinguish different chaotic scenarios. This work provides an example of how our understanding of physiological models can be enhanced by the theory of dynamical systems.


92C20 Neural biology
37N25 Dynamical systems in biology
92C37 Cell biology
37B40 Topological entropy
37B10 Symbolic dynamics
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
Full Text: DOI EuDML


[1] I. Atwater, C. M. Dawson, A. Scott, G. Eddlestone, and E. Rojas, “Oscillatory behavior in electrical activity from pancreatic \beta -cell,” Hormone and Metabolic Research, vol. 10, pp. 100-107, 1980.
[2] T. R. Chay, “Chaos in a three-variable model of an excitable cell,” Physica D, vol. 16, no. 2, pp. 233-242, 1985. · Zbl 0582.92007 · doi:10.1016/0167-2789(85)90060-0
[3] T. R. Chay and J. Keizer, “Minimal model for membrane oscillations in the pancreatic \beta -cell,” Biophysical Journal, vol. 42, no. 2, pp. 181-190, 1983.
[4] J. Duarte and J. Sousa Ramos, “Topological entropy as a measure of chaos in forced excitable systems,” International Journal of Pure and Applied Mathematics, vol. 4, no. 2, pp. 165-180, 2003. · Zbl 1029.37009
[5] R. FitzHugh, “Impulses and physiological states in theoretical models of nerve membrane,” Biophysical Journal, vol. 1, pp. 445-466, 1961.
[6] J. Guckenheimer, R. Harris-Warrick, J. Peck, and A. Willms, “Bifurcation, bursting, and spike frequency adaptation,” Journal of Computational Neuroscience, vol. 4, no. 3, pp. 257-277, 1997. · Zbl 0888.92010 · doi:10.1023/A:1008871803040
[7] B.-L. Hao and W.-M. Zheng, Applied Symbolic Dynamics and Chaos, vol. 7 of Directions in Chaos, World Scientific, New Jersey, 1998. · Zbl 0914.58017
[8] J. L. Hindmarsh and R. M. Rose, “A model of neuronal bursting using three coupled 1st order differential equations,” Proceedings of the Royal Society of London, vol. B221, pp. 87-102, 1984. · doi:10.1098/rspb.1984.0024
[9] A. L. Hodgkin and A. F. Huxley, “A quantitative description of membrane current and its application to conduction and excitation in nerve,” The Journal of Physiology, vol. 117, no. 4, pp. 500-544, 1952.
[10] E. M. Izhikevich, “Neural excitability, spiking and bursting,” International Journal of Bifurcation and Chaos, vol. 10, no. 6, pp. 1171-1266, 2000. · Zbl 1090.92505 · doi:10.1142/S0218127400000840
[11] E. M. Izhikevich, “Resonance and selective communication via bursts in neurons having subthreshold oscillations,” BioSystems, vol. 67, no. 1-3, pp. 95-102, 2002. · doi:10.1016/S0303-2647(02)00067-9
[12] E. M. Izhikevich, N. S. Desai, E. C. Walcott, and F. C. Hoppensteadt, “Bursts as a unit of neural information: selective communication via resonance,” Trends in Neuroscience, vol. 26, no. 3, pp. 161-167, 2003. · doi:10.1016/S0166-2236(03)00034-1
[13] J. P. Lampreia and J. Sousa Ramos, “Computing the topological entropy of bimodal maps,” in European Conference on Iteration Theory (Caldes de Malavella, 1987), pp. 431-437, World Scientific, New Jersey, 1989.
[14] J. P. Lampreia and J. Sousa Ramos, “Symbolic dynamics of bimodal maps,” Portugaliae Mathematica, vol. 54, no. 1, pp. 1-18, 1997. · Zbl 0877.58020
[15] J. E. Lisman, “Bursts as a unit of neural information: making unreliable synapses reliable,” Trends in Neuroscience, vol. 20, no. 1, pp. 38-43, 1997. · doi:10.1016/S0166-2236(96)10070-9
[16] Georgi S. Medvedev, “Reduction of a model of an excitable cell to a one-dimensional map,” Physica D, vol. 202, no. 1-2, pp. 37-59, 2005. · Zbl 1144.92307 · doi:10.1016/j.physd.2005.01.021
[17] J. Milnor and W. Thurston, “On iterated maps of the interval,” in Dynamical Systems (College Park, Md, 1986-1987), vol. 1342 of Lecture Notes in Math., pp. 465-563, Springer, Berlin, 1988. · Zbl 0664.58015
[18] C. Mira, Chaotic Dynamics. From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism, World Scientific, Singapore, 1987. · Zbl 0641.58002
[19] M. Misiurewicz and W. Szlenk, “Entropy of piecewise monotone mappings,” Studia Mathematica, vol. 67, no. 1, pp. 45-63, 1980. · Zbl 0445.54007
[20] P. F. Pinsky and J. Rinzel, “Intrinsic and network rhythmogenesis in a reduced Traub model for CA3 neurons,” Journal of Computational Neuroscience, vol. 1, no. 1-2, pp. 39-60, 1994. · doi:10.1007/BF00962717
[21] M. I. Rabinovich and H. D. Abarbanel, “The role of chaos in neural systems,” Neuroscience, vol. 87, no. 1, pp. 5-14, 1998. · doi:10.1016/S0306-4522(98)00091-8
[22] X.-J. Wang, “Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle,” Physica D, vol. 62, no. 1-4, pp. 263-274, 1993. · Zbl 0783.58053 · doi:10.1016/0167-2789(93)90286-A
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.