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We consider the problem of state and parameter estimation for a wide class ofnonlinear oscillators. Observable variables are limited to a few components ofstate vector and an input signal. The problem of state and parameterreconstruction is viewed within the classical framework of observer design.This framework offers computationally-efficient solutions to the problem ofstate and parameter reconstruction of a system of nonlinear differentialequations, provided that these equations are in the so-called adaptive observercanonic form. We show that despite typical neural oscillators being locallyobservable they are not in the adaptive canonic observer form. Furthermore, weshow that no parameter-independent diffeomorphism exists such that the originalequations of these models can be transformed into the adaptive canonic observerform. We demonstrate, however, that for the class of Hindmarsh-Rose andFitzHugh-Nagumo models, parameter-dependent coordinate transformations can beused to render these systems into the adaptive observer canonical form. Thisallows reconstruction, at least partially and up to a (bi)lineartransformation, of unknown state and parameter values with exponential rate ofconvergence. In order to avoid the problem of only partial reconstruction andto deal with more general nonlinear models in which the unknown parametersenter 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 basedon the non-uniform small-gain theorems. Effectiveness of the method isillustrated with simple numerical examples.

Observers for canonic models of neural oscillators