Learning the Hodgkin–Huxley model with operator learning techniques

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 $L^2$ error as low as 1.4% in learning the solutions of the Hodgkin–Huxley ionic model.