Bifurcation control of Fitzhugh-Nagumo models

Year 2018, Volume: 22 Issue: Special, 375 - 391, 05.10.2018

Reşat Özgür Doruk Hamza Ihnısh

Abstract

A theoretical bifurcation control strategy is presented for a single Fitzhugh-Nagumo (FN) type neuron. The bifurcation conditions are tracked for varying parameters of the individual FN neurons. A MATLAB package called as MATCONT is utilized for this purpose and all parameters of the neuron is analyzed one-by-one. Analysis by MATCONT revealed five Hopf (H) and one Limit-Point/Saddle Point (LP) bifurcation. The Hopf type of bifurcations are controlled by a washout filter supported by projective control theory. Washout filters are designed as first and second order. First order washout filter which is also physically applicable appeared to be more advantageous than the second order version. It appeared that, the LP case could not be stabilized by the aid of a washout filter. To solve this issue, a nonlinear controller is proposed. The only drawback associated with that is its inability to keep the original equilibrium point. Simulations are also provided to validate the research done.

Keywords

Fitzhugh-Nagumo neurons, Bifurcation; Washout filter; Projective control; Nonlinear control

References

• [1] Hodgkin, A. L., Huxley, A. F., 1952. Propagation of electrical signals along giant nerve fibres, Proceedings of the Royal Society of London. Series B, Biological Sciences, 177–183.
• [2] Fitzhugh, R., 1960. Thresholds and plateaus in the hodgkin-huxley nerve equations, The Journal of general physiology, 43(5), 867–896.
• [3] Morris, C., Lecar, H., 1981. Voltage oscillations in the barnacle giant muscle fiber., Biophysical journal, 35(1), 193.
• [4] FitzHugh, R., 1961. Impulses and physiological states in theoretical models of nerve membrane, Biophysical journal, 1(6), 445.
• [5] Binczak, S., Kazantsev, V., Nekorkin, V., Bilbault, J., 2003. Experimental study of bifurcations in modified fitzhugh-nagumo cell, Electronics Letters, 39(13), 1.
• [6] Gaiko, V. A., 2011. Multiple limit cycle bifurcations of the fitzhugh–nagumo neuronal model, Nonlinear Analysis: Theory, Methods & Applications, 74(18), 7532–7542.
• [7] Izhikevich, E. M., FitzHugh, R., 2006. Fitzhughnagumo model, Scholarpedia, 1(9), 1349.
• [8] Rocsoreanu, C., Georgescu, A., Giurgiteanu, N., 2012. The FitzHugh-Nagumo model: bifurcation and dynamics, volume 10, Springer Science & Business Media.
• [9] Sweers, G., Troy, W. C., 2003. On the bifurcation curve for an elliptic system of fitzhugh–nagumo type, Physica D: Nonlinear Phenomena, 177(1), 1–22.
• [10] Tanabe, S., Pakdaman, K., 2001. Dynamics of moments of fitzhugh-nagumo neuronal models and stochastic bifurcations, Physical Review E, 63(3), 031911.
• [11] Wang, Q., Lu, Q., Chen, G., Duan, L., et al., 2009. Bifurcation and synchronization of synaptically coupled fhn models with time delay, Chaos, Solitons & Fractals, 39(2), 918–925.
• [12] Crawford, J. D., 1991. Introduction to bifurcation theory, Reviews of Modern Physics, 63(4), 991.
• [13] Hassard, B. D., Kazarinoff, N. D., Wan, Y.-H., 1981. Theory and applications of Hopf bifurcation, volume 41, CUP Archive.
• [14] Kuznetsov, Y. A., 2006. Andronov-hopf bifurcation, Scholarpedia, 1(10), 1858.
• [15] Marsden, J. E., McCracken, M., 2012. The Hopf bifurcation and its applications, volume 19, Springer Science & Business Media.
• [16] Rinzel, J., Keaner, J. P., 1983. Hopf bifurcation to repetitive activity in nerve, SIAM Journal on Applied Mathematics, 43(4), 907–922.
• [17] Kuznetsov, Y. A., 2006. Saddle-node bifurcation, Scholarpedia, 1(10), 1859.
• [18] Zhou, T., 2013. Saddle-node bifurcation, in Encyclopedia of Systems Biology, 1889–1889, Springer.
• [19] Chen, G., Moiola, J. L., Wang, H. O., 2000. Bifurcation control: theories, methods, and applications, International Journal of Bifurcation and Chaos, 10(03), 511–548.
• [20] Doruk, R. O., 2010. Feedback controlled electrical nerve stimulation: A computer simulation, Computer methods and programs in biomedicine, 99(1), 98–112.
• [21] Hassouneh, M. A., Lee, H.-C., Abed, E. H., 2004. Washout filters in feedback control: Benefits, limitations and extensions, in American Control Conference, 2004. Proceedings of the 2004, volume 5, 3950–3955, IEEE.
• [22] Chen, D., Wang, H. O., Chen, G., 1998. Anti-control of hopf bifurcations through washout filters, in Decision and Control, 1998. Proceedings of the 37th IEEE Conference on, volume 3, 3040–3045, IEEE.
• [23] Abed, E. H., Fu, J.-H., 1986. Local feedback stabilization and bifurcation control, i. hopf bifurcation, Systems & Control Letters, 7(1), 11–17.
• [24] Balanov, A. G., Janson, N. B., Schöll, E., 2004. Control of noise-induced oscillations by delayed feedback, Physica D: Nonlinear Phenomena, 199(1), 1–12.
• [25] Aqil, M., Hong, K.-S., Jeong, M.-Y., 2012. Synchronization of coupled chaotic fitzhugh–nagumo systems, Communications in Nonlinear Science and Numerical Simulation, 17(4), 1615–1627.
• [26] Luo, X. S., Zhang, B., Qin, Y. H., et al., 2010. Controlling chaos in space-clamped fitzhugh–nagumo neuron by adaptive passive method, Nonlinear Analysis: Real World Applications, 11(3), 1752–1759.
• [27] Mishra, D., Yadav, A., Ray, S., Kalra, P. K., 2006. Controlling synchronization of modified fitzhughnagumo neurons under external electrical stimulation, NeuroQuantology, 4(1).
• [28] Rajasekar, S., Murali, K., Lakshmanan, M., 1997. Control of chaos by nonfeedback methods in a simple electronic circuit system and the fitzhugh-nagumo equation, Chaos, Solitons & Fractals, 8(9), 1545–
• [29] Vaidyanathan, S., 2015. Adaptive control of the fitzhugh-nagumo chaotic neuron model, International Journal of PharmTech Research, 8(6), 117–127.
• [30] Zhang, T., Wang, J., Fei, X., Deng, B., 2007. Synchronization of coupled fitzhugh–nagumo systems via mimo feedback linearization control, Chaos, Solitons & Fractals, 33(1), 194–202.
• [31] Liu, J., West, M., 2001. Combined parameter and state estimation in simulation-based filtering, in Sequential Monte Carlo methods in practice, 197–223, Springer.
• [32] Dochain, D., 2003. State and parameter estimation in chemical and biochemical processes: a tutorial, Journal of process control, 13(8), 801–818.
• [33] Evensen, G., 2009. The ensemble kalman filter for combined state and parameter estimation, IEEE Control Systems, 29(3), 83–104.
• [34] Ding, F., 2014. Combined state and least squares parameter estimation algorithms for dynamic systems, Applied Mathematical Modelling, 38(1), 403–412.
• [35] Johnson, M. L., Faunt, L. M., 1992. [1] parameter estimation by least-squares methods, Methods in enzymology, 210, 1–37.
• [36] Strejc, V., 1980. Least squares parameter estimation, Automatica, 16(5), 535–550.
• [37] Sharman, K., 1988. Maximum likelihood parameter estimation by simulated annealing, in Acoustics, Speech, and Signal Processing, 1988. ICASSP-88., 1988 International Conference on, 2741–2744, IEEE.
• [38] Rauch, H. E., Striebel, C., Tung, F., 1965. Maximum likelihood estimates of linear dynamic systems, AIAA journal, 3(8), 1445–1450.
• [39] Ghahramani, Z., Hinton, G. E., 1996. Parameter estimation for linear dynamical systems, Technical report, Technical Report CRG-TR-96-2, University of Totronto, Dept. of Computer Science.
• [40] Isidori, A., 2013. Nonlinear control systems, Springer Science & Business Media.
• [41] MEDANIC , J., USKOKOVIC , Z., 1983. The design of optimal output regulators for linear multivariable systems with constant disturbances, International Journal of Control, 37(4), 809–830.
• [42] Nguyen, N. T., 2018. Least-squares parameter identification, in Model-Reference Adaptive Control, 125–149, Springer.
• [43] Doruk, R. O., Zhang, K., 2018. Fitting of dynamic recurrent neural network models to sensory stimulusresponse data, Journal of Biological Physics, .
• [44] Asai, Y., Nomura, T., Sato, S., Tamaki, A., Matsuo, Y., Mizukura, I., Abe, K., 2003. A coupled oscillator model of disordered interlimb coordination in patients with parkinson’s disease, Biological Cybernetics, 88(2), 152–162.
• [45] Nana, L., 2009. Bifurcation analysis of parametrically excited bipolar disorder model, Communications in Nonlinear Science and Numerical Simulation, 14(2), 351–360.
• [46] Nagumo, J., Arimoto, S., Yoshizawa, S., 1962. An active pulse transmission line simulating nerve axon, Proceedings of the IRE, 50(10), 2061–2070.
• [47] Andronov, A. A., 1971. Theory of bifurcations of dynamic systems on a plane, volume 554, Israel Program for Scientific Translations [available from the US Dept. of Commerce, National Technical Information Service, Springfield, Va.].
• [48] Dhooge, A., Govaerts, W., Kuznetsov, Y. A., 2003. Matcont: a matlab package for numerical bifurcation analysis of odes, ACM Transactions on Mathematical Software (TOMS), 29(2), 141–164.
• [49] Govaerts, W., Kuznetsov, Y. A., Sautois, B., 2006. Matcont, Scholarpedia, 1(9), 1375.
• [50] WISE, K., Deylami, F., 1991. Approximating a linear quadratic missile autopilot design using an output feedback projective control, in AIAA Guidance, Navigation and Control Conference, New Orleans, LA, 114–122.
• [51] Wise, K. A., Nguyen, T., 1992. Optimal disturbance rejection in missile autopilot design using projective controls, IEEE Control Systems, 12(5), 43–49.