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Adaptive observers for biophysical neuronal circuits

Thiago B. Burghi, Rodolphe Sepulchre

TL;DR

This article presents adaptive observers for online state and parameter estimation of a class of nonlinear systems motivated by biophysical models of neuronal circuits and presents an augmented adaptive observer for models with a nonlinearly parameterized internal dynamics.

Abstract

This paper presents adaptive observers for online state and parameter estimation of a class of nonlinear systems motivated by biophysical models of neuronal circuits. We first present a linear-in-the-parameters design that solves a classical recursive least squares problem. Then, building on this simple design, we present an augmented adaptive observer for models with a nonlinearly parameterized internal dynamics, the parameters of which we interpret as structured uncertainty. We present a convergence and robustness analysis based on contraction theory, and illustrate the potential of the approach in neurophysiological applications by means of numerical simulations.

Adaptive observers for biophysical neuronal circuits

TL;DR

This article presents adaptive observers for online state and parameter estimation of a class of nonlinear systems motivated by biophysical models of neuronal circuits and presents an augmented adaptive observer for models with a nonlinearly parameterized internal dynamics.

Abstract

This paper presents adaptive observers for online state and parameter estimation of a class of nonlinear systems motivated by biophysical models of neuronal circuits. We first present a linear-in-the-parameters design that solves a classical recursive least squares problem. Then, building on this simple design, we present an augmented adaptive observer for models with a nonlinearly parameterized internal dynamics, the parameters of which we interpret as structured uncertainty. We present a convergence and robustness analysis based on contraction theory, and illustrate the potential of the approach in neurophysiological applications by means of numerical simulations.

Paper Structure

This paper contains 33 sections, 10 theorems, 117 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. System class
  4. Problem statement
  5. Conductance-based single-neuron model
  6. Conductance-based neural network model
  7. Estimation of the output dynamics
  8. Challenges in neuronal model estimation
  9. Least squares and reduced-order observer
  10. RLS-based adaptive observer
  11. Estimation under uncertainty
  12. Estimating uncertain internal dynamics parameters
  13. Measurement errors
  14. Robustness to measurement errors
  15. Robustness to unstructured uncertainty
  16. ...and 18 more sections

Key Result

Lemma 1

Consider the neuronal model eq:single_neuron_cb-eq:gaussian_function, and assume $\mid u\mid \le \overline{u}$ for all $t \ge 0$. Let Whenever $v(0) \in [\underline{v},\overline{v}]$, $m_{\rm{ion}}(0) \in [0,1]$ and $h_{\rm{ion}}(0) \in [0,1]$, it follows that for all ${\rm{ion}} \in \mathcal{I}$ and all $t \ge 0$.

Figures (8)

  • Figure 1: Circuit representation of a neuron with voltage $v$ that is coupled though a synapse to a presynaptic neuron with voltage $v_p$.
  • Figure 2: Excitability in the HH model. A small current pulse causes no spike, while a larger current pulse causes a spike. HH parameter values are described in \ref{['tab:HH_nominal_bio_parameters']} and \ref{['sec:HH_parameters']}.
  • Figure 3: Cost function $V({\hat{\mu}}_{\textrm{Na}},w(0),T)$ given by \ref{['eq:HH_cost']} and its gradient.
  • Figure 4: Solutions of the HH predictor \ref{['eq:estimator_batch']} for two values of ${\hat{\mu}}_{\textrm{Na}}$ (red), compared to the solution of the true model displayed in Figure \ref{['fig:HH_spike']} (blue).
  • Figure 5: Estimated parameters of the adaptive observer \ref{['eq:adaptive_observer_int']}-\ref{['eq:adaptive_matrices_int']} in the estimation of the HH model when no measurement errors are present. Here, $\alpha = 0.1$ and $\beta = \gamma = 1$.
  • ...and 3 more figures

Theorems & Definitions (33)

  • Remark 1
  • Example 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • Example 2
  • Proposition 2
  • ...and 23 more