  We consider the problem of state and parameter estimation for a wide class of
nonlinear oscillators. Observable variables are limited to a few components of
state vector and an input signal. The problem of state and parameter
reconstruction is viewed 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
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 non-uniform small-gain theorems. Effectiveness of the method is
illustrated with simple numerical examples.
