  Traditionally, calcium dynamics in neurons are modeled using partial
differential equations (PDEs) and ordinary differential equations (ODEs). The
PDE component focuses on reaction-diffusion processes, while the ODE component
addresses transmission via ion channels on the cell's or organelle's membrane.
However, analytically determining the underlying equations for ion channels is
highly challenging due to the complexity and unknown factors inherent in
biological processes. Therefore, we employ deep neural networks (DNNs) to model
the open probability of ion channels, a task that can be intricate when
approached with ODEs. This technique also reduces the number of unknowns
required to model the open probability. When trained with valid data, the same
neural network architecture can be used for different ion channels, such as
sodium, potassium, and calcium. Furthermore, based on the given data, we can
build more physiologically reasonable DNN models that can be customized.
Subsequently, we integrated the DNN model into calcium dynamics in neurons with
endoplasmic reticulum, resulting in a hybrid model that combines PDEs and DNNs.
Numerical results are provided to demonstrate the flexibility and advantages of
the PDE-DNN model.
