Hybrid PDE-Deep Neural Network Model for Calcium Dynamics in Neurons

TL;DR

This paper introduces a hybrid PDE-DNN model for neuronal calcium dynamics, employing neural networks to model ion channel probabilities, which simplifies the modeling process and enhances flexibility compared to traditional methods.

Contribution

The paper presents a novel hybrid PDE-DNN approach that models ion channel open probabilities with neural networks, reducing complexity and allowing for adaptable, physiologically plausible models.

Findings

The hybrid model effectively captures calcium dynamics in neurons.

Neural networks can be trained to model different ion channels with shared architecture.

Numerical results demonstrate the model's flexibility and advantages.

Abstract

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.