Dynamics in piecewise linear and continuous models of complex switching networks

Activities of genes are controlled in a combinatorial fashion by the concentrations of chemical called transcription factors. We model this type of network by piecewise linear differential equations formed by embedding a logical switching network in a differential equation. We generate continuous nonlinear equations by replacing the step function discontinuities in the piecewise linear equations by sigmoidal control functions. As the sigmoidal functions become steep, the continuous equations approach piecewise linear differential equations. We carry out numerical studies of the continuous and piecewise linear equations for a 4-dimensional example with particularly interesting and complex behavior, showing that the dynamics in the continuous equation approaches those in the piecewise linear equation as the sigmoids become steep.