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Systems biology and systems neurophysiology in particular have recently emerged as powerful tools for a number of key applications in the biomedical sciences. Nevertheless, such models are often based on complex combinations of multiscale (and possibly multiphysics) strategies that require ad hoc computational strategies and pose extremely high computational demands. Recent developments in the field of deep neural networks have demonstrated the possibility of formulating nonlinear, universal approximators to estimate solutions to highly nonlinear and complex problems with significant speed and accuracy advantages in comparison with traditional models. After synthetic data validation, we use so-called physically constrained neural networks (PINN) to simultaneously solve the biologically plausible Hodgkin-Huxley model and infer its parameters and hidden time-courses from real data under both variable and constant current stimulation, demonstrating extremely low variability across spikes and faithful signal reconstruction. The parameter ranges we obtain are also compatible with prior knowledge. We demonstrate that detailed biological knowledge can be provided to a neural network, making it able to fit complex dynamics over both simulated and real data.