On Non-Linear Dynamics and a Periodic Control Design Applied to the Potential of Membrane Action

Abstract:

The Fitzhugh-Nagumo (fn) mathematical model characterizes the action potential of the membrane. The dynamics of the Fitzhugh-Nagumo model have been extensively studied both with a view to their biological implications and as a test bed for numerical methods, which can be applied to more complex models. This paper deals with the dynamics in the (FH) model. Here, the dynamics are analyzed, qualitatively, through the stability diagrams to the action potential of the membrane. Furthermore, we also analyze quantitatively the problem through the evaluation of Floquet multipliers. Finally, the nonlinear periodic problem is controlled, based on the Chebyshev polynomial expansion, the Picard iterative method and on Lyapunov-Floquet transformation (L-F transformation).

• Periodical: Applied Mechanics and Materials (Volumes 5-6)
• Pages: 47-54
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.5-6.47
• Online since: October 2006
• Authors: Fabio R. Chavarette, N.J. Peruzzi, José Manoel Balthazar, H.A. Hermini
• Keywords: Action Potential, Fitzhugh-Nagumo Model, L-F Transformation, Nonlinear Dynamics

References

[1] A.L. Hodgkin and A.F. Huxley: J. Physiol., 116, (1952) 500-544.

[2] UCL Center. Nonlinear Dynamics, London. Available in <www. ucl. ac. uk/CNDA/> access in: 20 October (2005).

[3] J. Cronin: Mathematical aspects of Hodgkin-Huxley neural theory, Cambridge University Press, New York (1987).

[4] J. Wu, Y. Sun, L.P. Collis and R.B. Hill: Modeling, Simulation, Implementation, and Application of a Digital Voltage Clamp for Studying Excitable Tissues, in: The IASTED International Conference on Applied Modeling and Simulation, Cambridge, USA, November 4-6, (2002).

[5] R. Fitzhugh: Impulses and physiological states in models of nerve membrane, Biophys. J. 1, (1961), 445-466.

DOI: 10.1016/s0006-3495(61)86902-6

[6] H. Fukai, T. Nomura, S. Doi and S. Sato: Hopf bifurcations in multiple-parameter space of the Hodgkin-Huxley equations, I, II, Biol. Cybern. 82, (2000), pp.215-222; 223-229.

DOI: 10.1007/s004220050022

[7] J. Nagumo, S. Arimoto and S. Yoshizawa: An active pulse transmission like simulating nerve axon. Proc. IRE 50, (1962), 2061-(2070).

DOI: 10.1109/jrproc.1962.288235

[8] R. Fitzhugh: Thresholds and plateausin the Hodgkin-Huxley nerve equations, J. General Phys. 43, (1960), 867-896.

DOI: 10.1085/jgp.43.5.867

[9] S. Doi and S. Kumagai: Biophysical Neural Networks, R. R. Poznanski, ed., Mary Ann Lievert, Inc., Larchmont, (2001) 261-301.

[10] F.R. Chavarette, J. M. Balthazar, M. A. Ganazza and H. A. Hermini: 18 th Internat. Cong. of Mecha. Eng., (2005).

[11] F.R. Chavarette, J.M. Balthazar, M. Rafikov and H.A. Hermini: Int.J. of Bifurcation and Chaos. Submitted (2006).

[12] F.R. Chavarette, J. M. Balthazar and M. Rafikov: Submitted to Chaos, Solitions and Fractals, Submitted (2006).

[13] S.C. Sinha and P. Joseph: Control of General Dynamic Systems with Periodically Varying Parameters Via Lyapunov-Floquet Transformation, Journal of Dynamic, Measurement, and Control, Vol. 116, (1994), 650-658.

DOI: 10.1115/1.2899264

[14] S.C. Sinha and Der-Ho, Wu: An Efficient Computational Scheme for the Analysis of Periodic Systems, Journal of Sound and Vibration, 151(1), (1991), 91-117.

DOI: 10.1016/0022-460x(91)90654-3

[15] S.C. Sinha and E.A. Butcher: Symbolic Computation of Fundamental Solution Matrices for Linear Time-Periodic Dynamical Systems, Journal of Sound and Vibration, 206(1), (1997), 61-85.

DOI: 10.1006/jsvi.1997.1079

[16] R. Pandiyan R. and S.C. Sinha: Time-Varying Controller Synthesis for Nonlinear Systems Subject to Periodic Parametric Loadings, Journal of Vibration and Control, No. 7, (2001), 7390.

DOI: 10.1177/107754630100700105

[17] T. Kanamaru and J.M.T. Thompson: Introduction to Chaos and Nonlinear Dynamics (2005). Available in <http: /brain. cc. kogakuin. ac. jp/~kanamaru/Chaos/e/> access in: 20 October (2005).

[18] N.J. Peruzzi, J.M. Balthazar and B.R. Pontes: On a Control of a Non-ideal Mono-Rail System with Periodic coefficients, Proceedings of DETC'05, ASME 2005 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Long Beach, CA, USA, September 24 - 28, 2005, CD ROM, (2005).

DOI: 10.1115/detc2005-84726

[19] M.C.K. Khoo: Physiological Control Systems, IEEE Press Series in Biomedical Engineering, New York, (1999).

[20] J.M. Balthazar, D.T. Mook, H.I. Weber, M.L.R.F.R. Brasil, A. Fenili, D. Belato and J.L.P. Felix: Sixth conference on Dynamical Systems Theory and Applications, Poland, (2001), 2750.

[21] J.M. Balthazar, D.T. Mook, H.I. Weber, M.L.R.F.R. Brasil, A. Fenili, D. Belato and J.L.P. Felix: Meccanica, 38, (2003), 613-621.

DOI: 10.1023/a:1025877308510

[22] J.M. Balthazar, D.T. Mook, H.I. Weber, M.L.R.F.R. Brasil, A. Fenili, D. Belato, J.L.P. Felix and F.J. Garzeri, Dynamics Systems and Control, 22, (2004), 237-258.