# Observers for Canonic Models of Neural Oscillators

D. Fairhurst; I. Tyukin; H. Nijmeijer; C. van Leeuwen

Mathematical Modelling of Natural Phenomena (2010)

- Volume: 5, Issue: 2, page 146-184
- ISSN: 0973-5348

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topFairhurst, D., et al. "Observers for Canonic Models of Neural Oscillators." Mathematical Modelling of Natural Phenomena 5.2 (2010): 146-184. <http://eudml.org/doc/197688>.

@article{Fairhurst2010,

abstract = {We consider the problem of state and parameter estimation for a class of nonlinear
oscillators 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 evoked
potential in neural membranes, including Hodgkin-Huxley, Hindmarsh-Rose, FitzHugh-Nagumo,
and Morris-Lecar models. We consider the problem of state and parameter reconstruction for
these models within the classical framework of observer design. This framework offers
computationally-efficient solutions to the problem of state and parameter reconstruction
of a system of nonlinear differential equations, provided that these equations are in the
so-called adaptive observer canonic form. We show that despite typical neural oscillators
being locally observable they are not in the adaptive canonic observer form. Furthermore,
we show that no parameter-independent diffeomorphism exists such that the original
equations of these models can be transformed into the adaptive canonic observer form. We
demonstrate, however, that for the class of Hindmarsh-Rose and FitzHugh-Nagumo models,
parameter-dependent coordinate transformations can be used to render these systems into
the adaptive observer canonical form. This allows reconstruction, at least partially and
up to a (bi)linear transformation, of unknown state and parameter values with exponential
rate of convergence. In order to avoid the problem of only partial reconstruction and at
the same time to be able to deal with more general nonlinear models in which the unknown
parameters enter the system nonlinearly, we present a new method for state and parameter
reconstruction for these systems. The method combines advantages of standard
Lyapunov-based design with more flexible design and analysis techniques based on the
notions of positive invariance and small-gain theorems. We show that this flexibility
allows to overcome ill-conditioning and non-uniqueness issues arising in this problem.
Effectiveness of our method is illustrated with simple numerical examples.},

author = {Fairhurst, D., Tyukin, I., Nijmeijer, H., van Leeuwen, C.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {Parameter estimation; adaptive observers; nonlinear parametrization; convergence; nonlinear systems; neural oscillators; parameter estimation},

language = {eng},

month = {3},

number = {2},

pages = {146-184},

publisher = {EDP Sciences},

title = {Observers for Canonic Models of Neural Oscillators},

url = {http://eudml.org/doc/197688},

volume = {5},

year = {2010},

}

TY - JOUR

AU - Fairhurst, D.

AU - Tyukin, I.

AU - Nijmeijer, H.

AU - van Leeuwen, C.

TI - Observers for Canonic Models of Neural Oscillators

JO - Mathematical Modelling of Natural Phenomena

DA - 2010/3//

PB - EDP Sciences

VL - 5

IS - 2

SP - 146

EP - 184

AB - We consider the problem of state and parameter estimation for a class of nonlinear
oscillators 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 evoked
potential in neural membranes, including Hodgkin-Huxley, Hindmarsh-Rose, FitzHugh-Nagumo,
and Morris-Lecar models. We consider the problem of state and parameter reconstruction for
these models within the classical framework of observer design. This framework offers
computationally-efficient solutions to the problem of state and parameter reconstruction
of a system of nonlinear differential equations, provided that these equations are in the
so-called adaptive observer canonic form. We show that despite typical neural oscillators
being locally observable they are not in the adaptive canonic observer form. Furthermore,
we show that no parameter-independent diffeomorphism exists such that the original
equations of these models can be transformed into the adaptive canonic observer form. We
demonstrate, however, that for the class of Hindmarsh-Rose and FitzHugh-Nagumo models,
parameter-dependent coordinate transformations can be used to render these systems into
the adaptive observer canonical form. This allows reconstruction, at least partially and
up to a (bi)linear transformation, of unknown state and parameter values with exponential
rate of convergence. In order to avoid the problem of only partial reconstruction and at
the same time to be able to deal with more general nonlinear models in which the unknown
parameters enter the system nonlinearly, we present a new method for state and parameter
reconstruction for these systems. The method combines advantages of standard
Lyapunov-based design with more flexible design and analysis techniques based on the
notions of positive invariance and small-gain theorems. We show that this flexibility
allows to overcome ill-conditioning and non-uniqueness issues arising in this problem.
Effectiveness of our method is illustrated with simple numerical examples.

LA - eng

KW - Parameter estimation; adaptive observers; nonlinear parametrization; convergence; nonlinear systems; neural oscillators; parameter estimation

UR - http://eudml.org/doc/197688

ER -

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