Efficient Implicit Runge-Kutta Methods for Fast-Responding Ligand-Gated Neuroreceptor Kinetic Models

Abstract

Neurophysiological models of the brain typically utilize systems of ordinary differential equations to simulate single-cell electrodynamics. To accurately emulate neurological treatments and their physiological effects on neurodegenerative disease, models that incorporate biologically-inspired mechanisms, such as neurotransmitter signalling, are necessary. Additionally, applications that examine populations of neurons, such as multiscale models, can demand solving hundreds of millions of these systems at each simulation time step. Therefore, robust numerical solvers for biologically-inspired neuron models are vital. To address this requirement, we evaluate the numerical accuracy and computational efficiency of three L-stable implicit Runge-Kutta methods when solving kinetic models of the ligand-gated glutamate and gamma-aminobutyric acid (GABA) neurotransmitter receptors. Efficient implementations of each numerical method are discussed, and numerous performance metrics including accuracy, simulation time steps, execution speeds, Jacobian calculations, and LU factorizations are evaluated to identify appropriate strategies for solving these models. Comparisons to popular explicit methods are presented and highlight the advantages of the implicit methods. In addition, we show a machine-code compiled implicit Runge-Kutta method implementation that possesses exceptional accuracy and superior computational efficiency.