In cells, proteins are produced by DNA sequences. The rate of production is influenced by the presence of other proteins connected to the DNA sequence. We can represent such interaction by a regulatory network, a directed graph where the edge i → j exists if x_i affects the rate of change of x_j. For biological reasons, we expect x_i to appear in the rate of change of x_j through a sigmoidal function, H_{ij}.
In this talk, we first approximate each sigmoidal function H_{ij} by a step function. Each step function is defined by three parameters and we expect the parameter space to be high dimensional. In such high dimensional space, we want to determine which choice of parameters allow for oscillations in a given network. For this goal, we will use a combination of combinatorial tools and Morse graphs.
Once we have found parameters compatible with oscillations using step functions, we want to discuss the stability of such oscillations if the step functions are smoothened back into sigmoidals. For this, we will discuss tools developed in the context of machine learning.