Physics-informed neural network approach to intracellular calcium oscillations: Dynamics and sensitivity analysis of $ IP_{3}R $ signaling

Intracellular calcium ($ Ca^{2+} $) signaling regulates critical physiological processes ranging from neurotransmission to gene expression. These dynamics are driven by the release of $ Ca^{2+} $ from the endoplasmic reticulum (ER) via Inositol 1, 4, 5-trisphosphate receptors ($ IP_{3}R $), resulting in complex oscillatory patterns. Traditional mathematical modeling of these non-linear dynamics often encounters computational bottlenecks: the stiffness of the governing equations requires expensive adaptive time-stepping, making large-scale parameter sweeps and inverse modeling computationally prohibitive. In this study, we propose a Physics-Informed Neural Network (PINN) framework to simulate $ IP_{3} $-mediated calcium oscillations. By training a feedforward neural network to minimize the residuals of the governing non-linear Ordinary Differential Equations (ODEs), we approximate the system dynamics governed by channel release ($ J_{channel} $), pump uptake ($ J_{pump} $), and leak fluxes ($ J_{leak} $). Our results demonstrate that the neural network approach effectively captures the phase-space behavior and temporal evolution of the system. This method offers a continuous, mesh-agnostic framework tailored for comprehensive sensitivity analysis, providing a robust tool for analyzing stiff biological systems.