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.

翻译：传统上，神经元钙动力学采用偏微分方程（PDEs）和常微分方程（ODEs）进行建模。其中，PDE部分侧重于反应-扩散过程，而ODE部分则处理细胞或细胞器膜上离子通道的传输问题。然而，由于生物过程固有的复杂性和未知因素，解析确定离子通道的基础方程极具挑战性。因此，我们采用深度神经网络（DNNs）来建模离子通道的开放概率——这一任务若使用ODE方法处理可能十分复杂。该技术还减少了建模开放概率所需的未知参数数量。当使用有效数据进行训练时，相同的神经网络架构可适用于不同的离子通道，如钠、钾和钙通道。此外，基于给定数据，我们可以构建更具生理合理性且可定制的DNN模型。随后，我们将DNN模型整合到具有内质网的神经元钙动力学系统中，从而形成了一个结合PDE与DNN的混合模型。数值结果展示了PDE-DNN模型的灵活性和优势。