Observers for canonic models of neural oscillators

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.

💡 Research Summary

The paper tackles the challenging problem of simultaneous state and parameter estimation for a broad class of nonlinear neural oscillators, where only a few components of the state vector and an external input are measurable. Within the classical observer‑design framework, exact reconstruction is possible when the system can be cast into the so‑called adaptive observer canonical form (AOCF). The authors first demonstrate that, despite the local observability of typical neural oscillators, none of the widely used models (including Hindmarsh‑Rose and FitzHugh‑Nagumo) can be transformed into AOCF by a parameter‑independent diffeomorphism. This negative result rules out the direct application of standard adaptive observer techniques.

To overcome this limitation, the paper introduces two complementary strategies. The first strategy exploits parameter‑dependent coordinate transformations. By constructing a diffeomorphism that explicitly depends on the unknown parameters, the authors show that both the Hindmarsh‑Rose (a three‑dimensional bursting model) and the FitzHugh‑Nagumo (a two‑dimensional excitability model) can be rewritten as

\