Steady-state approximations for Hodgkin-Huxley cell models: Reduction of order for uterine smooth muscle cell model

by Shawn A. Means, Mathias W. Roesler, Amy S. Garrett, Leo Cheng, Alys R. Clark

Multi-scale mathematical bioelectrical models of organs such as the uterus, stomach or heart present challenges both for accuracy and computational tractability. These multi-scale models are typically founded on models of biological cells derived from the classic Hodkgin-Huxley (HH) formalism. Ion channel behaviour is tracked with dynamical variables representing activation or inactivation of currents that relax to steady-state dependencies on cellular membrane voltage. Timescales for relaxation may be orders of magnitude faster than companion ion channel variables or phenomena of physiological interest for the entire cell (such as bursting sequences of action potentials) or the entire organ (such as electromechanical coordination). Exploiting these time scales with steady-state approximations for relatively fast-acting systems is a well-known but often overlooked approach as evidenced by recent published models. We thus investigate feasibility of an extensive reduction of order for an HH-type cell model with steady-state approximations to the full dynamical activation and inactivation ion channel variables. Our effort utilises a published comprehensive uterine smooth muscle cell model that encompasses 19 ordinary differential equations and 105 formulations overall. The numerous ion channel submodels in the published model exhibit relaxation times ranging from order 10⁻¹ to 10⁵ milliseconds. Substitution of the faster dynamic variables with steady-state formulations demonstrates both an accurate reproduction of the full model and substantial improvements in time-to-solve, for test cases performed. Our demonstration here of an effective and relatively straightforward reduction method underlines the particular importance of considering time scales for model simplification before embarking on large-scale computations or parameter sweeps. As a preliminary complement to more intensive reduction of order methods such as parameter sensitivity and bifurcation analysis, this approach can rapidly and accurately improve computational tractability for challenging multi-scale organ modelling efforts.