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Learning the Hodgkin-Huxley Model with Operator Learning Techniques

Edoardo Centofanti, Massimiliano Ghiotto, Luca F. Pavarino

TL;DR

This work investigates learning the operator that maps a time-varying applied current to the Hodgkin–Huxley transmembrane potential using three operator-learning architectures: DeepONet, Fourier Neural Operator (FNO), and Wavelet Neural Operator (WNO). By comparing encoding strategies, kernel representations, and discretizations, the study shows that FNO achieves the best mean relative $L^2$ error of about $1.4\%$ (with $k_{\max}=16$ Fourier modes), while DeepONet attains roughly $2.2\%$ and WNO about $3.3\%$, highlighting the trade-offs between accuracy, flexibility, and computational cost. The results demonstrate the viability of neural-operator approaches for stiff, threshold-driven ionic dynamics and motivate applying these methods to broader physiological models in neuroscience and cardiology. Overall, the work provides a principled comparison of operator-learning paradigms for a classical nonlinear ODE system and offers guidance for selecting architectures when handling time-dependent forcing in excitable-cell models.

Abstract

We construct and compare three operator learning architectures, DeepONet, Fourier Neural Operator, and Wavelet Neural Operator, in order to learn the operator mapping a time-dependent applied current to the transmembrane potential of the Hodgkin- Huxley ionic model. The underlying non-linearity of the Hodgkin-Huxley dynamical system, the stiffness of its solutions, and the threshold dynamics depending on the intensity of the applied current, are some of the challenges to address when exploiting artificial neural networks to learn this class of complex operators. By properly designing these operator learning techniques, we demonstrate their ability to effectively address these challenges, achieving a relative L2 error as low as 1.4% in learning the solutions of the Hodgkin-Huxley ionic model.

Learning the Hodgkin-Huxley Model with Operator Learning Techniques

TL;DR

This work investigates learning the operator that maps a time-varying applied current to the Hodgkin–Huxley transmembrane potential using three operator-learning architectures: DeepONet, Fourier Neural Operator (FNO), and Wavelet Neural Operator (WNO). By comparing encoding strategies, kernel representations, and discretizations, the study shows that FNO achieves the best mean relative error of about (with Fourier modes), while DeepONet attains roughly and WNO about , highlighting the trade-offs between accuracy, flexibility, and computational cost. The results demonstrate the viability of neural-operator approaches for stiff, threshold-driven ionic dynamics and motivate applying these methods to broader physiological models in neuroscience and cardiology. Overall, the work provides a principled comparison of operator-learning paradigms for a classical nonlinear ODE system and offers guidance for selecting architectures when handling time-dependent forcing in excitable-cell models.

Abstract

We construct and compare three operator learning architectures, DeepONet, Fourier Neural Operator, and Wavelet Neural Operator, in order to learn the operator mapping a time-dependent applied current to the transmembrane potential of the Hodgkin- Huxley ionic model. The underlying non-linearity of the Hodgkin-Huxley dynamical system, the stiffness of its solutions, and the threshold dynamics depending on the intensity of the applied current, are some of the challenges to address when exploiting artificial neural networks to learn this class of complex operators. By properly designing these operator learning techniques, we demonstrate their ability to effectively address these challenges, achieving a relative L2 error as low as 1.4% in learning the solutions of the Hodgkin-Huxley ionic model.
Paper Structure (16 sections, 1 theorem, 25 equations, 8 figures, 1 table)

This paper contains 16 sections, 1 theorem, 25 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Hodgkin-Huxley model
  3. Operator Learning
  4. Deep Operator Networks
  5. Neural Operator
  6. Fourier Neural Operator
  7. Wavelet Neural Operator
  8. Results
  9. Dataset, loss and optimizer
  10. DeepONet results
  11. FNO results
  12. WNO results
  13. Training Comparison
  14. Conclusions
  15. Acknowledgements
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\sigma$ be a continuous non-polynomial function, $\mathcal{X}$ is a Banach space, $\mathcal{K}_1\subset \mathcal{X}$, $\Omega\subset \mathbb{R}^{d}$ are two compact sets in $\mathcal{X}$ and $\mathbb{R}^d$ respectively, $\mathcal{A}$ is a compact set in $C(\mathcal{K}_1)$, G a nonlinear continu holds for all $f\in \mathcal{A}$ and $y\in \Omega$. In this context, $C(\mathcal{K})$ is the Banach

Figures (8)

  • Figure 1: Different $I_{app}$ pulses lead to different responses from the system. A pulse with an under-threshold height leads to zero peaks and a negligible perturbation of the system from its equilibrium state. Once the threshold value is crossed (around $2.21\, \mu A/cm^2$ with these parameters), the system responds with a single peak or more peaks depending on the input current.
  • Figure 2: Schematic DeepONet architecture. The outputs of the branch and the trunk networks, here represented as forward neural networks, are multiplied through a scalar product in order to approximate the operator $G$.
  • Figure 3: Schematic FNO architecture.
  • Figure 4: Schematic WNO architecture.
  • Figure 5: (\ref{['fig:don_scatter']}) $L^2$ train and test loss vs network width for DeepONet. All the branch nets considered have a depth of 4, while all the trunks considered have a depth of 3. The width is indicated on the $x$ axis and it is the same for all the layers and the output layer. A width of 700 neurons is the best among the ones explored for our problem. The training error is indicated with a blue line, while the test error is indicated with an orange line. (\ref{['fig:fno_scatter']}) $L^2$ train and test loss vs Fourier modes for the Fourier transform employed in the FNO. 16 and 32 modes resulted in a better performing trained network for our problem with our configuration. The training error is indicated with a blue line, while the test error is indicated with an orange line. (\ref{['fig:wno_scatter']}) $L^2$ train and test loss vs $R_{\theta_t}$ width for WNO. $R_{\theta_t}$ is a tensor of trainable parameters in the wavelet space. The width is indicated on the $x$ axis and it is the same for all the levels of the wavelet transform. In our numerical tests, a width of 64 has resulted in the best approximation error. The training error seems to improve with a superlinear rate when the width is increased. The training error is indicated with a blue line, while the test error is indicated with an orange line.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Theorem 1: Universal Approximation Theorem for Operator
  • Definition 1: Encoder
  • Definition 2: Reconstructor or Decoder
  • Definition 3: Neural Operator
  • Definition 4: Integral Kernel Operators
  • Definition 5: Single Hidden Layer Update
  • Definition 6: Fourier transform and anti-transform
  • Definition 7