Feedback Identification of conductance-based models

TL;DR

The paper addresses consistent parameter identification for nonlinear, excitably oscillatory conductance-based neuronal dynamics under input-additive noise by applying the Prediction Error Method in a closed-loop setting. It leverages the exponentially contracting inverse dynamics induced by high-gain output feedback, akin to voltage-clamp, to justify applying classical nonlinear identification techniques to neuronal systems. A rigorous consistency result is proven under standard assumptions, and the framework supports both fixed-kinetics and library-based ion channel kinetics, with parameter recovery formulas for c, g_j, and ν_j. Numerical examples using Hodgkin-Huxley and Connor-Stevens–type models illustrate convergence of estimates under realistic noise levels, validating the approach and its connection to the voltage-clamp paradigm for neuronal system identification.

Abstract

This paper applies the classical prediction error method (PEM) to the estimation of nonlinear discrete-time models of neuronal systems subject to input-additive noise. While the nonlinear system exhibits excitability, bifurcations, and limit-cycle oscillations, we prove consistency of the parameter estimation procedure under output feedback. Hence, this paper provides a rigorous framework for the application of conventional nonlinear system identification methods to discrete-time stochastic neuronal systems. The main result exploits the elementary property that conductance-based models of neurons have an exponentially contracting inverse dynamics. This property is implied by the voltage-clamp experiment, which has been the fundamental modeling experiment of neurons ever since the pioneering work of Hodgkin and Huxley.

Figures (10)

• Figure 1: Schematic representation of a neuronal system.
• Figure 2: Left: Time constant functions in the Hodgkin-Huxley model. Right: Activation functions in the Hodgkin-Huxley model.
• Figure 3: Simulated state trajectories of the Hodgkin-Huxley model (Example \ref{['ex:HH']}) for $i_{\text{app}}(t) = 10 \; \mathrm{\upmu A/cm^2}$.
• Figure 4: The voltage-clamp experiment: electrodes are used to inject the current $i_{\text{app}}(t)$ and measure the voltage $v(t)$ of the neuronal membrane. The amplifiers are ideal differential amplifiers, and $\bar{g}_e$ models the electrode conductance. When $\bar{\gamma}\gg 1$, this implements the feedback law \ref{['eq:CT_output_fb']} with $\gamma = \bar{\gamma}\bar{g}_e$.
• Figure 5: Block-diagram of the system \ref{['eq:DT_CL_model']}.

Theorems & Definitions (6)

• proof
• proof
• proof
• proof
• proof
• proof