A rational extended thermodynamic model for dopamine diffusion and uptake

In this work, a hyperbolic model is proposed within the framework of Rational Extended Thermodynamics to study dopamine diffusion in the striatum. The model incorporates a nonlinear uptake mechanism described by Michaelis-Menten kinetics. While it has the main characteristics of the classical parabolic Nicholson model, it also introduces key features typical of hyperbolic systems. Analytical solutions are derived for some particular cases, enabling a direct comparison between the hyperbolic and parabolic formulations. The analytical solutions obtained through the Laplace Transform method, is validated with numerical simulations carried out using the Taylor-Galerkin Finite Element Method (TG-FEM). These two different approaches provide a comprehensive insight of the model and highlight the differences with the parabolic case. Finally, the propagation of acceleration waves is studied to analyze the system's response to specific perturbations. The temporal evolution of such disturbances is examined in detail, and the critical time at which derivative discontinuities evolve into shock waves is identified.