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Analysis of large-amplitude pulses in short time intervals: application to neuron interactions. (English) Zbl 1189.37099
Summary: This paper deals with the analysis of a nonlinear dynamical system which characterizes the axons interaction and is based on a generalization of FitzHugh-Nagumo system. The parametric domain of stability is investigated for both the linear and third-order approximation. A further generalization is studied in presence of high-amplitude (time-dependent) pulse. The corresponding numerical solution for some given values of parameters are analyzed through the wavelet coefficients, showing both the sensitivity to local jumps and some unexpected inertia of neuron’s as response to the high-amplitude spike.

37N25 Dynamical systems in biology
92B20 Neural networks for/in biological studies, artificial life and related topics
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[1] S. El Boustani, M. Pospischil, M. Rudolph-Lilith, and A. Destexhe, “Activated cortical states: experiments, analyses and models,” Journal of Physiology Paris, vol. 101, no. 1-3, pp. 99-109, 2007.
[2] R. Fitzhugh, “Impulses and physiological states in theoretical models of nerve membrane,” Biophysical Journal, vol. 1, pp. 455-466, 1961.
[3] N. V. Georgiev, “Identifying generalized Fitzhugh-Nagumo equation from a numerical solution of Hodgkin-Huxley model,” Journal of Applied Mathematics, no. 8, pp. 397-407, 2003. · Zbl 1077.37524
[4] J. Guckenheimer and I. S. Labouriau, “Bifurcation of the Hodgkin and Huxley equations: a new twist,” Bulletin of Mathematical Biology, vol. 55, no. 5, pp. 937-952, 1993. · Zbl 0778.92007
[5] B. Hassard, “Bifurcation of periodic solutions of the Hodgkin-Huxley model for the squid giant axon,” Journal of Theoretical Biology, vol. 71, no. 3, pp. 401-420, 1978.
[6] A. L. Hodgkin and A. F. Huxley, “Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo,” The Journal of Physiology, vol. 116, pp. 449-472, 1952.
[7] A. L. Hodgkin and A. F. Huxley, “The components of membrane conductance in the giant axon of Loligo,” The Journal of Physiology, vol. 116, no. 4, pp. 473-496, 1952.
[8] A. L. Hodgkin and A. F. Huxley, “The dual effect of membrane potential on sodium conductance in the giant axon of Loligo,” The Journal of Physiology, vol. 116, pp. 497-506, 1952.
[9] J. Nagumo, S. Arimoto, and S. Yoshizawa, “An active pulse transmission line simulating nerve axon,” Proceedings of the IRE, vol. 50, no. 10, pp. 2061-2070, 1962.
[10] J. Rinzel and R. N. Miller, “Numerical calculation of stable and unstable periodic solutions to the Hodgkin-Huxley equations,” Mathematical Biosciences, vol. 49, no. 1-2, pp. 27-59, 1980. · Zbl 0429.92014
[11] J. Wang, L. Chen, and X. Fei, “Analysis and control of the bifurcation of Hodgkin-Huxley model,” Chaos, Solitons and Fractals, vol. 31, no. 1, pp. 247-256, 2007. · Zbl 1140.37370
[12] G. Mattioli and M. Scalia, “Modelling hodgkin-huxley neurons interaction,” in Proceedings of the International Conference on Computational Science and Its Applications (ICCSA ’09), vol. 5592 of Lecture Notes in Computer Science, pp. 745-751, 2009. · Zbl 05576772
[13] C. Cattani and J. Rushchitsky, Wavelet and Wave Analysis as Applied to Materials with Micro or Nanostructure, vol. 74 of Series on Advances in Mathematics for Applied Sciences, World Scientific, Hackensack, NJ, USA, 2007. · Zbl 1152.74001
[14] C. Cattani, “Haar wavelet-based technique for sharp jumps classification,” Mathematical and Computer Modelling, vol. 39, no. 2-3, pp. 255-278, 2004. · Zbl 1046.94504
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