Periodicity in piecewise-linear switching networks with delay

Gene regulatory networks and neural networks can be modeled by piecewise-linear switching systems of differential equations, known as Glass networks. These biological networks exhibit delays in regulatory activity, for example, transcription, translation and spatial transport in gene networks, and transmission delays in neural networks. Such delays may be significant in determining their dynamical behavior. Here Glass networks with a discrete delay are introduced and analyzed. Fixed points away from thresholds are straightforward to identify, even in the presence of delays, so the focus of this work is on cyclic patterns of switching. Under a condition that ensures an unambiguous pattern of switching, it is shown by means of a fractional linear mapping that delayed Glass networks have a periodic orbit for all positive finite delays. Furthermore, an algorithm is presented to locate the periodic orbit for a given cycle, to determine whether the periodic orbit is locally asymptotically stable, and to check if it is unique. In addition, the complete dynamics of the two-dimensional delayed Glass network is provided: if there is a cycle of four orthants, then there exists a unique globally stable limit cycle; whereas if there is a black wall, then across the wall there exists a unique limit cycle that is globally stable with respect to the associated orthants. This behavior is in contrast to the non-delayed case, in which spiralling approach to fixed points on threshold boundaries can occur.