Splitting physics-informed neural networks for inferring the dynamics of integer- and fractional-order neuron models

TL;DR

This work introduces Splitting PINN, a framework that couples operator splitting with physics-informed neural networks to solve forward dynamics of neuron models, including fractional-order variants. By decomposing nonlinear systems into sub-problems and solving each with PINNs, the method achieves improved accuracy over vanilla PINN approaches, particularly for oscillatory and memory-influenced dynamics. A key contribution is the $L^1$ discretization for Caputo derivatives, enabling efficient and accurate simulation of fractional-order neuron models such as FO-HH. The authors demonstrate the approach on Leaky/Integrate-and-Fire, Izhikevich, Hodgkin–Huxley, and FO-HH models, highlighting memory effects on firing patterns and providing a pathway for applying split-PINN to other complex forward-dynamical systems. Overall, Splitting PINN offers a robust, scalable tool for neuroscientific modeling and computational science applications requiring accurate forward-solution of nonlinear, possibly memory-rich ODE systems.

Abstract

We introduce a new approach for solving forward systems of differential equations using a combination of splitting methods and physics-informed neural networks (PINNs). The proposed method, splitting PINN, effectively addresses the challenge of applying PINNs to forward dynamical systems and demonstrates improved accuracy through its application to neuron models. Specifically, we apply operator splitting to decompose the original neuron model into sub-problems that are then solved using PINNs. Moreover, we develop an $L^1$ scheme for discretizing fractional derivatives in fractional neuron models, leading to improved accuracy and efficiency. The results of this study highlight the potential of splitting PINNs in solving both integer- and fractional-order neuron models, as well as other similar systems in computational science and engineering.

Figures (22)

• Figure 1: Overview of the Splitting PINN : We first split the original system into sub-systems (sub-problems) for each sub-interval $[t^j, t^{j+1}]$. For each sub-interval, $x(t^j)$ is known, the sub-systems are solved using PINN and then the solutions are combined to obtain the approximate solution $x(t^{j+1})$. To evaluate the error, we obtain the reference solution, $x_{exact}(t^{j+1})$, by using a high-order numerical solver (for more details, see Appendix B). The algorithm proceeds until arriving at a given accuracy for each sub-interval.
• Figure 2: Schematic diagram for the Hodgkin-Huxley model hh1952. Left: ion channels on the membrane of the neuron. Right: simulated circuit with $I$ denoting the input current.
• Figure 3: Izhikevich model: Left: membrane potential versus time. Right: recovery variable versus time.
• Figure 4: Loss function of the Izhikevich model.
• Figure 5: HH model: comparison of splitting PINN results with the reference solution. (a) membrane potential; (b) activation variable of potassium channel; (c) activation variable of the sodium channel; (d) deactivation variable of the sodium channel. The inset plot shows the step function input current.

Theorems & Definitions (1)

• Definition 1