Fitzhugh-Nagumo Modelleri İçin Çatallanma Denetimi
Bu yazıda tekil Fitzhugh-Nagumo (FN) nöron modelleri için teorik bir çatallanma denetim çalışması sunulmaktadır. Değişmekte olan parametreler için çatallanma analizleri MATLAB üzerinde çalışan MATCONT uygulaması ile yapılmıştır. Söz konusu analizde 5 Hopf (H) ve 1 adette Sınır Noktası/Eyer Dü˘gümü (LP) olgusuna rastlanmıştır. Hopf tipi çatallanmalar izdüşümsel denetim ile desteklenmiş arındırma süzgeçleri kullanılarak sağlanmıştır. Arındırma süzgeçleri birinci ve ikinci derece olarak uygulanmıştır. Birinci derece süzgeç ikinci dereceye göre daha avantajlı oldu˘gu anlaşılmıştır. Birinci derece süzgeç hem daha uygulanabilir olmakta hem de daha hızlı davranmaktadır. LP türü çatallanmalar için derecesinden bağımsız olarak arındırma süzgecinden yapılan çıktı geri beslemesi başarılı olamamakta ve bu nedenle birini derece süzgecle beraber birde zar potansiyelinden ek bir geri besleme alınmaktadır. Bunun dezavantajı süzgecin yüksek geçirgen niteli˘ginin bozulmasına neden olmakta ve LP denge noktasının korunmasına olanak vermemektedir. Bu soruna çözüm olması için doğrusal olmayan bir denetleyici tasarımıda gösterilmektedir. Bunun tek dezavantajı orjinal denge noktaları korunamaktadır. Sonuçlar benzetimlerle desteklenmektedir.
Bifurcation Control of Fitzhugh-Nagumo Models
A theoretical bifurcation control strategy is presented for a single Fitzhugh-Nagumo (FN) type neuron. The bifurcation conditions are tracked for varying parametersof the individual FN neurons. A MATLAB package called as MATCONT is utilizedfor this purpose and all parameters of the neuron is analyzed one-by-one. Analysis byMATCONT 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 projectivecontrol theory. Washout filters are designed as first and second order. First order washoutfilter which is also physically applicable appeared to be more advantageous than thesecond order version. It appeared that, the LP case could not be stabilized by the aid of awashout filter. To solve this issue, a nonlinear controller is proposed. The only drawbackassociated with that is its inability to keep the original equilibrium point. Simulations arealso provided to validate the research done.
___
- [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–
1558.
- [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.
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.