Observers for canonic models of neural oscillators

Mathematical Modelling of Natural Phenomena

Volumn 5, Issue 2, 2010, Pages 146-184

  Fairhurst, D ^a   Tyukin, I ^a,c,d   Nijmeijer, H ^b   Van Leeuwen, C ^c  

^a UNIVERSITY OF LEICESTER   (United Kingdom)

^b EINDHOVEN UNIVERSITY OF TECHNOLOGY   (Netherlands)

^c INST OF PHYS AND CHEMICAL RESEARCH   (Japan)

^d SAINT PETERSBURG ELECTROTECHNICAL UNIVERSITY LETI   (Russian Federation)

Author keywords

adaptive observers; convergence; neural oscillators; nonlinear parametrization; nonlinear systems; Parameter estimation

Indexed keywords

ADAPTIVE OBSERVER; CO-ORDINATE TRANSFORMATION; CONVERGENCE; DIFFEOMORPHISMS; EXPONENTIAL RATES; FITZHUGH-NAGUMO MODEL; FLEXIBLE DESIGNS; ILL-CONDITIONING; INPUT SIGNAL; MORRIS-LECAR MODEL; NEURAL MEMBRANES; NEURAL OSCILLATOR; NON-LINEAR MODEL; NON-LINEAR OSCILLATORS; NONLINEAR ORDINARY DIFFERENTIAL EQUATION; NONLINEAR PARAMETRIZATION; NONUNIQUENESS; NUMERICAL EXAMPLE; OBSERVER DESIGN; PARAMETER RECONSTRUCTION; PARAMETER VALUES; POSITIVE INVARIANCE; SMALL GAIN THEOREM; STATE VECTOR; SYSTEM OF NONLINEAR DIFFERENTIAL EQUATIONS; UNKNOWN PARAMETERS; UNKNOWN STATE;

APPROXIMATION THEORY; DESIGN; ESTIMATION; LINEAR TRANSFORMATIONS; MATHEMATICAL TRANSFORMATIONS; NONLINEAR ANALYSIS; NONLINEAR SYSTEMS; NUMERICAL METHODS; ORDINARY DIFFERENTIAL EQUATIONS; PARAMETER ESTIMATION;

NONLINEAR EQUATIONS;

EID: 77953710515     PISSN: 09735348     EISSN: 17606101     Source Type: Journal    
DOI: 10.1051/mmnp/20105206     Document Type: Review

References (38)

1. H.D.I. Abarbanel, D.R. Crevling, R. Farsian, M. Kostuk. Dynamical State and Parameter Estimation. SIAM J. Applied Dynamical Systems, 8 (2009), No. 4, 1341-1381.
2. P. Achard, E. Schutter. Complex parameter landscape for a comples neuron model. PLOS Computational Biology, 2 (2006), No. 7, 794-804.
3. G. Bastin, M. Gevers. Stable adaptive observers for nonlinear time-varying systems. IEEE Trans. on Automatic Control, 33 (1988), No. 7, 650-658. (Pubitemid 18665427)
4. R. Borisyuk, Y. Kazanovich. Oscillations and waves in the models of interactive neural populations. iosystems, 86 (2006), No. 1-3, 53-62. (Pubitemid 44466179)
5. D. Brewer, M. Barenco, R. Callard, M. Hubank, J. Stark. Fitting ordinary differential equations to short time course data. Philosophical Transactions of The Royal Society A, 366 2008), No. 1865, 519-544.
6. C. Cao, A.M. Annaswamy, A. Kojic. Parameter convergence in nonlinearly parametrized systems. IEEE Trans. on Automatic Control, 48 (2003), No. 3, 397-411.
7. R. FitzHugh. Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal, 1 (1961), 445-466.
8. A.N. Gorban. Basic types of coarse-graining. In A.N. Gorban, N. Kazantzis, I.G. Kevrekidis, H.C. Ottinger, and C. Theodoropoulos, editors. Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena, Springer, (2006), 117-176.
9. J.L. Hindmarsh, R.M. Rose. A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond., B 221 (1984), No. 1222, 87-102.
10. A.L. Hodgkin, A.F. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol., 117 (1952), 500-544.
11. A. Ilchman. Universal adaptive stabilization of nonlinear systems. Dyn. and Contr., (1997), No. 7, 199-213.
12. A. Isidori. Nonlinear control systems II. Springer-Verlag, second edition, 1999.
13. E. M. Izhikevich. Dynamical Systems in Neuroscience: the Geometry of Excitability and Bursting. MIT Press, 2007.
14. E. M. Izhikevich, G. M. Edelman. Large-scale model of mammalian thalamocortical systems. Proc. of Nat. Acad. Sci., 105 (2008), 3593-3598. (Pubitemid 351723597)
15. Y. Kazanovich, R. Borisyuk. An oscillatory neural model of multiple object tracking. Neural Computation, 18 (2006), No. 6, 1413-1440. (Pubitemid 43778738)
16. C. Koch. Biophysics of Computation. Information Processing in Signle Neurons. Oxford University Press, 2002.
17. G. Kreisselmeier. Adaptive obsevers with exponential rate of convergence. IEEE Trans. Automatic Control, AC-22 (1977), 2-8.
18. W. Lin, C. Qian. Adaptive control of nonlinearly parameterized systems: The smooth feedback case. IEEE Trans. Automatic Control, 47 (2002), No. 8, 1249-1266. (Pubitemid 34959759)
19. L. Ljung. System Identification: Theory for the User. Prentice-Hall, 1999.
20. L. Ljung. Perspectives in system identification. In Proceedings of the 17-th IFAC World Congress on Automatic Control, (2008), 7172-7184.
21. A. Loria, E. Panteley. Uniform exponential stability of linear time-varying systems: revisited. Systems and Control Letters, 47 (2007), No. 1, 13-24.
22. A.M. Lyapunov. The general problem of the stability of motion. Int. Journal of Control, 55 (1992), No. 3.
23. R. Marino. Adaptive observers for single output nonlinear systems. IEEE Trans. Automatic Control, 35 (1990), No. 9, 1054-1058.
24. R. Marino, P. Tomei. Global adaptive observers for nonlinear systems via filtered transformations. IEEE Trans. Automatic Control, 37 (1992), No. 8, 1239-1245.
25. R. Marino, P. Tomei. Adaptive observers with arbitrary exponential rate of convergence for nonlinear systems. IEEE Trans. Automatic Control, 40 (1995), No. 7, 1300-1304.
26. J. Milnor. On the concept of attractor. Commun. Math. Phys., 99 (1985), 177-195.
27. A. P. Morgan, K. S. Narendra. On the stability of nonautonomous differential equations x = [A+B(t)]x with skew symmetric matrix B(t). SIAM J. Control and Optimization, 37 (1977), No. 9, 1343-1354.
28. C. Morris, H. Lecar. Voltage oscillatins in the barnacle giant muscle fiber. Biophysics J., 35 (1981), 193-213. (Pubitemid 11079105)
29. K. S. Narendra, A. M. Annaswamy. Stable Adaptive systems. Prentice-Hall, 1989.
30. H. Nijmeijer, A. van der Schaft. Nonlinear Dynamical Control Systems. Springer-Verlag, 1990.
31. A. Prinz, C.P. Billimoria, E. Marder. Alternative to hand-tuning conductance-based models: Contruction and analysis of databases of model neurons. Journal of Neorophysiology, 90 (2003), 3998-4015. (Pubitemid 37541930)
32. I. Yu. Tyukin, D.V. Prokhorov, C. van Leeuwen. Adaptive algorithms in finite form for nonconvex parameterized systems with low-triangular structure. In Proceedings of the 8-th IFAC Workshop on Adaptation and Learning in Control and Signal Processing (ALCOSP 2004), (2004), 261-266.
33. I.Yu. Tyukin, D. V. Prokhorov, C. van Leeuwen. Adaptation and parameter estimation in systems with unstable target dynamics and nonlinear parametrization. IEEE Transactions on Automatic Control, 52 (2007), No. 9, 1543-1559. (Pubitemid 47456067)
34. I.Yu. Tyukin, D.V. Prokhorov, V.A. Terekhov. Adaptive control with nonconvex parameterization. IEEE Trans. on Automatic Control, 48 (2003), No. 4, 554-567.
35. I.Yu. Tyukin, E. Steur, H. Nijmeijer, C. van Leeuwen. Non-uniform small-gain theorems for systems with unstable invariant sets. SIAM Journal on Control and Optimization, 47 (2008), No. 2, 849-882.
36. I.Yu. Tyukin, E. Steur, H. Nijmeijer, C. van Leeuwen. Adaptive observers and parametric identification for systems in non-canonical adaptive observer form. (2009), preprint available at http://arxiv.org/abs/0903.2361.
37. W. van Geit, E. de Shutter, P. Achard. Automated neuron model optimization techniques: a review. Biol. Cybern, 99 (2008), 241-251.
38. V.B. Kazantsev, V.I. Nekorkin, V.I. Makarenko, R. Llinas. Self-referential phase reset based on inferior olive oscillator dynamics. Proceedings of National Academy of Science, 101 (2004), No. 52, 18183-18188. (Pubitemid 40054094)