Revealing bistability in neurological disorder models by solving parametric polynomial systems geometrically

Abstract

Understanding the mechanisms of the brain is a common theme for both computational neuroscience and artificial intelligence. Machine learning technique, like artificial neural network, has been benefiting from a better understanding of the neuronal network in human brains. In the study of neurons, mathematical modeling plays a vital role. In this paper, we analyze the important phenomenon of bistability in neurological disorders modeled by ordinary differential equations in virtue of our recently developed method for solving bi-parametric polynomial systems. Unlike the algebraic symbolic approach, our numeric method solves parametric systems geometrically. With respect to the classical bifurcation analysis approach, our method naturally has good initial points thanks to the critical point technique in real algebraic geometry. Special heuristic strategies are proposed for addressing the multiscale problem of parameters and variables occurring in biological models, as well as taking into account the fact that the variables representing concentrations are non-negative. Comparing with its symbolic algebraic counterparts, one merit of this geometrical method is that it may compute smaller boundaries.