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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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