# Learning the Hodgkin-Huxley Model with Operator Learning Techniques

arXiv: https://arxiv.org/abs/2406.02173 · HTML: https://mlsanity.com/paper/2406.02173/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 54 references that are in the map.
- First posted on arXiv 4 Jun 2024.

## Builds on (most cited first, 19 of 19)

- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [Sequential Deep Operator Networks (S-DeepONet) for Predicting Full-field Solutions Under Time-dependent Loads](https://mlsanity.com/paper/2306.08218/) (2023)
- [Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations](https://mlsanity.com/paper/2112.08125/) (2021)
- [Nonlinear Approximation via Compositions](https://mlsanity.com/paper/1902.10170/) (2019)
- [Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularities](https://mlsanity.com/paper/2010.12217/) (2020)
- [Spiking Neural Operators for Scientific Machine Learning](https://mlsanity.com/paper/2205.10130/) (2022)
- [Linking Gaussian Process regression with data-driven manifold embeddings for nonlinear data fusion](https://mlsanity.com/paper/1812.06467/) (2018)
- [Discovering dynamical features of Hodgkin-Huxley-type model of physiological neuron using artificial neural network](https://mlsanity.com/paper/2203.14138/) (2022)
- [Splitting physics-informed neural networks for inferring the dynamics of integer- and fractional-order neuron models](https://mlsanity.com/paper/2304.13205/) (2023)

## Built on by (most cited first, 10 of 10)

- [Learning High-dimensional Ionic Model Dynamics Using Fourier Neural Operators](https://mlsanity.com/paper/2505.14039/) (2025)
- [NOBLE -- Neural Operator with Biologically-informed Latent Embeddings to Capture Experimental Variability in Biological Neuron Models](https://mlsanity.com/paper/2506.04536/) (2025)
- [AMORE: Adaptive Multi-Output Operator Network for Stiff Chemical Kinetics](https://mlsanity.com/paper/2510.12999/) (2025)
- [HyperSBINN: A Hypernetwork-Enhanced Systems Biology-Informed Neural Network for Efficient Drug Cardiosafety Assessment](https://mlsanity.com/paper/2408.14266/) (2024)
- [Learning cardiac activation and repolarization times with operator learning](https://mlsanity.com/paper/2505.08631/) (2025)
- [Learning Hybrid Biophysical Neuron Models with Neural ODEs](https://mlsanity.com/paper/2606.16693/) (2026)
- [A model learning framework for inferring the dynamics of transmission rate depending on exogenous variables for epidemic forecasts](https://mlsanity.com/paper/2410.11545/) (2024)
- [Translation Invariance of Neural Operators for the FitzHugh-Nagumo Model](https://mlsanity.com/paper/2603.17523/) (2026)
- [RTS Smoother-Guided Learning of Physics-Based Neural Differential Models](https://mlsanity.com/paper/2607.15180/) (2026)
- [Retracing Hodgkin and Huxley: State Recovery Does Not Certify Mechanism](https://mlsanity.com/paper/2609.34566/) (2026)

## Abstract (arXiv)

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.