Analysis of continuous-time switching networks

Models of a number of biological systems, including gene regulation and neural networks, can be formulated as switching networks, in which the interactions between the variables depend strongly on thresholds. An idealized class of such networks in which the switching takes the form of Heaviside step functions but variables still change continuously in time has been proposed as a useful simplification to gain analytic insight. These networks, called here Glass networks after their originator, are simple enough mathematically to allow significant analysis without restricting the range of dynamics found in analogous smooth systems. A number of results have been obtained before, particularly regarding existence and stability of periodic orbits in such networks, but important cases were not considered. Here we present a coherent method of analysis that summarizes previous work and fills in some of the gaps as well as including some new results. Furthermore, we apply this analysis to a number of examples, including surprising long and complex limit cycles involving sequences of hundreds of threshold transitions. Finally, we show how the above methods can be extended to investigate aperiodic behaviour in specific networks, though a complete analysis will have to await new results in matrix theory and symbolic dynamics.