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Observers for Canonic Models of NeuralOscillators

Published online by Cambridge University Press:  10 March 2010

D. Fairhurst ,
I. Tyukin ,
H. Nijmeijer  and
C. van Leeuwen
D. Fairhurst
Affiliation:
Department of Mathematics, University of Leicester, University Road, LE1 7RH, UK
I. Tyukin*
Affiliation:
Department of Mathematics, University of Leicester, University Road, LE1 7RH, UK RIKEN (Institute for Physical and Chemical Research) Brain Science Institute, 2-1, Hirosawa, Wako-shi, Saitama, 351-0198, Japan Deptartment of Automation and Control Processes, St-Petersburg State University of Electrical Engineering, Prof. Popova str. 5, 197376, Russia
H. Nijmeijer
Affiliation:
Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513 , 5600 MB, Eindhoven, The Netherlands
C. van Leeuwen
Affiliation:
RIKEN (Institute for Physical and Chemical Research) Brain Science Institute, 2-1, Hirosawa, Wako-shi, Saitama, 351-0198, Japan
*
* Corresponding author. E-mail:[email protected]
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Abstract

We consider the problem of state and parameter estimation for a class of nonlinearoscillators defined as a system of coupled nonlinear ordinary differential equations.Observable variables are limited to a few components of state vector and an input signal.This class of systems describes a set of canonic models governing the dynamics of evokedpotential in neural membranes, including Hodgkin-Huxley, Hindmarsh-Rose, FitzHugh-Nagumo,and Morris-Lecar models. We consider the problem of state and parameter reconstruction forthese models within the classical framework of observer design. This framework offerscomputationally-efficient solutions to the problem of state and parameter reconstructionof a system of nonlinear differential equations, provided that these equations are in theso-called adaptive observer canonic form. We show that despite typical neural oscillatorsbeing locally observable they are not in the adaptive canonic observer form. Furthermore,we show that no parameter-independent diffeomorphism exists such that the originalequations of these models can be transformed into the adaptive canonic observer form. Wedemonstrate, however, that for the class of Hindmarsh-Rose and FitzHugh-Nagumo models,parameter-dependent coordinate transformations can be used to render these systems intothe adaptive observer canonical form. This allows reconstruction, at least partially andup to a (bi)linear transformation, of unknown state and parameter values with exponentialrate of convergence. In order to avoid the problem of only partial reconstruction and atthe same time to be able to deal with more general nonlinear models in which the unknownparameters enter the system nonlinearly, we present a new method for state and parameterreconstruction for these systems. The method combines advantages of standardLyapunov-based design with more flexible design and analysis techniques based on thenotions of positive invariance and small-gain theorems. We show that this flexibilityallows to overcome ill-conditioning and non-uniqueness issues arising in this problem.Effectiveness of our method is illustrated with simple numerical examples.

Keywords

Parameter estimationadaptive observersnonlinear parametrizationconvergencenonlinear systemsneural oscillators
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 2: Mathematics and neuroscience , 2010 , pp. 146 - 184
Copyright
© EDP Sciences, 2010

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References

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