Recent advances in genomics have begun to allow particular gene regulation pathways to be worked out, but it has been realized since the 1960's that gene regulatory networks could in principle have arbitrarily complicated interaction structures and behaviour. One framework for the study of such systems in full generality has been to concentrate on structural classes and activation/repression switching dynamics by means of a 'hard switching' limit. The equations are continuous in time but the interactions between variables (protein concentrations) depend only on whether they are above or below a threshold, i.e. 'active' or 'inactive'. This simplified framework has been successfully applied to the analysis of dynamics in several real gene regulatory networks.

This class of networks ('Glass networks'), though not smooth, is mathematically tractable due to its piecewise-linear nature. Even in quite small (e.g. 4-gene) networks, there is a rich variety of stable dynamics, including multistability (crucial to cell differentiation, for example), periodic orbits (e.g. the cell cycle, circadian rhythms) and chaos. An analytic framework has been developed, involving reduction to explicitly calculated discrete maps on Poincaré sections. This allows proof of existence and stability of fixed point, periodic and, remarkably, even chaotic attractors. Some interesting mathematics arises, mainly in dynamical systems and linear algebra. Most of this work was originally done under some restrictive (and unrealistic) assumptions, such as equal decay rates and a single threshold per gene. The removal of these restrictions is an area of current research.