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Parameter Estimation of the Network of FitzHugh-Nagumo Neurons Based on the Speed-Gradient and Filtering

Aleksandra Rybalko, Alexander Fradkov

TL;DR

The paper tackles parameter identification in networks of diffusively coupled FitzHugh–Nagumo neurons when only membrane potentials are observable and measurement noise is present. It introduces a transformation to a linear regression form using a filter-differentiator, then applies a speed-gradient adaptive law to estimate the true FHN parameters from the regressor, with convergence guaranteed under a persistent excitation condition. A Lyapunov-based analysis provides sufficient conditions for asymptotic convergence, and simulations with a 5-node network demonstrate large improvements in parameter accuracy and controllable convergence behavior through the gain matrix. The approach is scalable to larger networks and has potential applications in nervous system modeling, including EEG-informed brain modeling, with future work addressing robustness to disturbances and time-varying parameters.

Abstract

The paper addresses the problem of parameter estimation (or identification) in dynamical networks composed of an arbitrary number of FitzHugh-Nagumo neuron models with diffusive couplings between each other. It is assumed that only the membrane potential of each model is measured, while the other state variable and all derivatives remain unmeasured. Additionally, potential measurement errors in the membrane potential due to sensor imprecision are considered. To solve this problem, firstly, the original FitzHugh-Nagumo network is transformed into a linear regression model, where the regressors are obtained by applying a filter-differentiator to specific combinations of the measured variables. Secondly, the speed-gradient method is applied to this linear model, leading to the design of an identification algorithm for the FitzHugh-Nagumo neural network. Sufficient conditions for the asymptotic convergence of the parameter estimates to their true values are derived for the proposed algorithm. Parameter estimation for a network of five interconnected neurons is demonstrated through computer simulation. The results confirm that the sufficient conditions are satisfied in the numerical experiments conducted. Furthermore, the algorithm's capabilities for adjusting the identification accuracy and time are investigated. The proposed approach has potential applications in nervous system modeling, particularly in the context of human brain modeling. For instance, EEG signals could serve as the measured variables of the network, enabling the integration of mathematical neural models with empirical data collected by neurophysiologists.

Parameter Estimation of the Network of FitzHugh-Nagumo Neurons Based on the Speed-Gradient and Filtering

TL;DR

The paper tackles parameter identification in networks of diffusively coupled FitzHugh–Nagumo neurons when only membrane potentials are observable and measurement noise is present. It introduces a transformation to a linear regression form using a filter-differentiator, then applies a speed-gradient adaptive law to estimate the true FHN parameters from the regressor, with convergence guaranteed under a persistent excitation condition. A Lyapunov-based analysis provides sufficient conditions for asymptotic convergence, and simulations with a 5-node network demonstrate large improvements in parameter accuracy and controllable convergence behavior through the gain matrix. The approach is scalable to larger networks and has potential applications in nervous system modeling, including EEG-informed brain modeling, with future work addressing robustness to disturbances and time-varying parameters.

Abstract

The paper addresses the problem of parameter estimation (or identification) in dynamical networks composed of an arbitrary number of FitzHugh-Nagumo neuron models with diffusive couplings between each other. It is assumed that only the membrane potential of each model is measured, while the other state variable and all derivatives remain unmeasured. Additionally, potential measurement errors in the membrane potential due to sensor imprecision are considered. To solve this problem, firstly, the original FitzHugh-Nagumo network is transformed into a linear regression model, where the regressors are obtained by applying a filter-differentiator to specific combinations of the measured variables. Secondly, the speed-gradient method is applied to this linear model, leading to the design of an identification algorithm for the FitzHugh-Nagumo neural network. Sufficient conditions for the asymptotic convergence of the parameter estimates to their true values are derived for the proposed algorithm. Parameter estimation for a network of five interconnected neurons is demonstrated through computer simulation. The results confirm that the sufficient conditions are satisfied in the numerical experiments conducted. Furthermore, the algorithm's capabilities for adjusting the identification accuracy and time are investigated. The proposed approach has potential applications in nervous system modeling, particularly in the context of human brain modeling. For instance, EEG signals could serve as the measured variables of the network, enabling the integration of mathematical neural models with empirical data collected by neurophysiologists.

Paper Structure

This paper contains 7 sections, 4 theorems, 35 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. FHN neural network model
  3. Identification of FHN neural network model
  4. Simulation results
  5. Conclusion
  6. Identification using the speed-gradient method based on the integral objective function
  7. Proof of Theorem \ref{['th']}

Key Result

Theorem 1

If the vector function of the FHN neural network model's observed values FHN$\bm z(\bm x) = (x_1 \quad x_2 \quad x_3 \quad x_4 \quad 1)^{\mathrm{T}}$ satisfies the persistent excitation (PE) condition PE and $\sigma < \varepsilon b/r$, then the identification law theta(t) guarantees that $\bm \theta

Figures (6)

  • Figure 1: The topology of the considered network of five FHN models.
  • Figure 2: Dependence between the integration interval length, $L$, and the smallest eigenvalue of the considered matrix, $\bm M_L$.
  • Figure 3: Parameter estimation errors, $\theta_i - \theta_i^*$, of the network of the FHN models \ref{['F']} with couplings defined as on Fig. \ref{['network_structure']} and \ref{['network_params']}, parameter values \ref{['params']} and the set of initial data \ref{['initial_data']}, obtained with the algorithm \ref{['theta(t)']}.
  • Figure 4: The norm of estimation error, $\|\bm \theta - \bm\theta^*\|$, of the network of the FHN models \ref{['F']} with couplings defined as on Fig. \ref{['network_structure']} and \ref{['network_params']}, parameter values \ref{['params']} and the set of initial data \ref{['initial_data']}, obtained with the algorithm \ref{['theta(t)']}, in case of $\bm \Gamma = g\mathrm{\bm I}$, $g = 0.1, 1, 5, 10$.
  • Figure 5: The norm of estimation error, $\|\theta_i - \theta_i^*\|$, of the network of the FHN models \ref{['F']} with couplings defined as on Fig. \ref{['network_structure']} and \ref{['network_params']}, parameter values \ref{['params']} and the set of initial data \ref{['initial_data']}, obtained with the algorithm \ref{['theta(t)']}, in case of $\Gamma = g\mathrm{I}$, $g = 0.0001, 0.001, 0.01, 0.1$.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Remark 4
  • Theorem 2
  • proof
  • Proposition 3
  • Proposition 4
  • proof