Algebraic models, inverse problems, and pseudomonomials from biology

Abstract

In contrast to the extensive use of linear algebra in biology, problems involving nonlinear polynomials—the field of algebraic biology—are more conspicuous. In this article, we highlight two biological problems where similar algebraic structures arise in very different contexts. Specifically, we will look at algebraic models of molecular networks, and combinatorial codes of place fields in neuroscience. Both of these topics involve inverse problems where the data is encoded with pseudomonomials, simple algebraic objects that do not seem to have been studied much on their own, but have gained considerable attention lately from their visibility in mathematical biology. Though this article is algebraic in nature, it is written for the general mathematician with a minimal algebra background assumed. It provides two distinctive features that have not yet appeared in the literature: (i) a survey of Boolean and logical modeling from a computational algebra perspective, and (ii) a unification of this topic with algebraic neuroscience by highlighting the role of pseudomonomials in both fields.