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

TL;DR

The paper addresses calcium dynamics in neurons by replacing the RyR channel's ODE-based open probability with a data-driven DNN within a PDE framework, forming a PDE-DNN hybrid that operates in a reduced 1D cylindrical geometry. The DNN learns a single scalar function to drive the open probability changes from inputs $(P,u,rac{du}{dt})$, drastically reducing model order while retaining essential dynamics. Training on ODE-generated data reproduces the original behavior but can inherit its physiologic inconsistencies, which motivates training on carefully designed artificial datasets to yield more realistic RyR behavior and tunable calcium waves. The resulting framework exhibits insensitivity to time-step size and enables easy adjustment of wave amplitude and duration through training data, offering a flexible and scalable approach to modeling calcium dynamics and potentially generalizable to other ion channels like voltage-gated types.

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.

Figures (14)

• Figure 1: Cross-section of the axon (It can also be viewed as a two-dimensional closed cell). The orange region is ER, denoted as $\Omega_e$. The blue region is cytosol, denoted as $\Omega_c$. There are no other organelles inside the axon. ER membrane is denoted as $\Gamma$, plasma membrane is denoted as $\partial\Omega$, where $\Omega = \Omega_e\cup\Gamma\cup \Omega_c$.
• Figure 2: The structure of the fully connected DNN model for the RyR channel. There are three inputs, one output, and three hidden layers. Among the inputs, $P_{n-1}$ is the open probability of the RyR channel at time $t_{n-1}$, $u_{n-1}$ is the calcium concentration in the cytosol at $t_{n-1}$, and similarly, $u_n$ is the Ca$^{2+}$ concentration at $t_n$. $(u_n - u_{n-1})/\Delta t$ approximates the time derivative of $u$ at $t_{n-1}$. The output $(P_n - P_{n-1})/\Delta t$ is an approximation of the time derivative of $P$ at $t_{n-1}$.
• Figure 3: Section \ref{['tset_ode']}. Training set from the ODE system. In all figures, the $x$-axis represents time, while the $y$-axis represents calcium concentration $u(t)$ and the open probability $P(t)$. From (a) to (c), the blue curves represent calcium signals, which fluctuate up and down over time with different magnitudes and durations; the green curves represent the corresponding open probability, ranging between 0 and 1. Figure (d) shows the collection of the entire training set, consisting of 26,000 pairs of $P(t)$ and $u(t)$.
• Figure 4: Section \ref{['tset_ode']}. Performance of the model on unseen data. (a) is an asymmetric calcium signal, and (b) is the response. (c) is a symmetric calcium signal, and (d) is its response. In both (b) and (d), the open probability $P$ generated by the neural network is shown in red, while the open probability from the ODE system is shown in green. The magnitudes of calcium signals are within the range of the training set.
• Figure 5: Section \ref{['tset_ode']}. Performance of the model on unseen data. (a) is a symmetric calcium signal, and (b) is the response. (c) is an asymmetric calcium signal, and (d) is its response. In both (b) and (d), the open probability $P$ generated by the neural network is shown in red, while the open probability from the ODE system is shown in green. The magnitudes of calcium signals are beyond the range of training set.