On the preservation of limit cycles in Boolean networks under different updating schemes Gonzalo A. Ruz 1 , Marco Montalva 1 and Eric Goles 1 1 Facultad de Ingenier´ ıa y Ciencias, Universidad Adolfo Ib´ a˜ nez, Av. Diagonal Las Torres 2640, Santiago, Chile gonzalo.ruz@uai.cl Abstract Boolean networks under different deterministic updating schemes are analyzed. It is direct to show that fixed points are invariant against changes in the updating scheme, never- theless, it is still an open problem to fully understand what happens to the limit cycles. In this paper, a theorem is pre- sented which gives a sufficient condition for a Boolean net- work not to share the same limit cycle under different up- dating modes. We show that the hypotheses of the theorem are sharp, in the sense that if any of these hypotheses do not hold, then shared limit cycles may appear. We find that the connectivity of the network is an important factor as well as the Boolean functions in each node, in particular the XOR functions. Introduction Boolean networks were introduced by S. Kauffman (Kauff- man, 1969) and R. Thomas (Thomas, 1973) as a mathemat- ical model of gene regulatory networks. It has been used to model, for example, the floral morphogenesis of Arabidop- sis thaliana (Mendoza and Alvarez-Buylla, 1998), the fis- sion yeast cell cycle (Davidich and Bornholdt, 2008; Goles et al., 2013), and the budding yeast cell cycle (Li et al., 2004; Goles et al., 2013). Formally, let x = {x 1 ,...,x n } be a finite set with x i ∈{0, 1} for i =1,...,n. Let N =(G, F, π) be a Boolean network, where G =(V,E) is a digraph; V being the set of n nodes and E the set of edges. F is a Boolean function, F : {0, 1} n →{0, 1} n composed of n local functions f i : {0, 1} n →{0, 1}. Each local function f i depends only on the variables belonging to the neighborhood V - (i)= {j ∈ V |(j, i) ∈ E}. The inde- gree of vertex i is |V - (i)|, and π is an arbitrary order to up- date the nodes π : {1,...,n}→{1,...,n}. For example, the parallel or synchronous updating mode (or scheme) has π(i)=1 (every node is updated at the same time), whereas, for the sequential one, π is a permutation. A combination of the parallel and the sequential updating mode is the block- sequential where the set of nodes, for a given sequence, is partitioned into blocks. The nodes in a same block are up- dated in parallel, but blocks follow each other sequentially. Overall, there are an exponential number of updates. In fact, if the network has n nodes, the number of updates is given by Demongeot et al. (2008): T n = n-1  k=0  n k  T k , T 0 =1. Without loss of generality, f i (x)= f i (x 1 ,...,x n ) will be used sometimes, although it should be clear to the reader that the local function really depends only on the variables in the neighborhood. Since the updating schemes are repeated periodically and the hypercube is a finite set, the dynamics of the network converges to attractors which are fixed points, i.e vectors such that x i = f i (x) for any i, or limit cycles, defined by x t+p i = x t i for i = {1,...,n}, where p> 1 is the period. In this paper we will consider limit cycles that have non-constant values (a constant node does not change its value during the limit cycle) within it 1 . One of the first to compare updating modes was F. Robert (Robert, 1986) for the parallel and sequential update. More recently, the robustness of such networks related to changes in the updating modes have been studied in Goles and Sali- nas (2008), where the authors prove that networks with monotonic loops 2 can not share limit cycles between the parallel and the sequential update. Furthermore, a first step to understand the different updates was done in Elena (2009); numerical experiments, under small threshold net- works (n =3) were carried out in order to exhibit the dif- ferent dynamics for every updating mode. Also, theoretical tools were developed in order to classify dynamics under different updating modes as well as to build efficient algo- rithms (Aracena et al., 2009; Montalva, 2011). In Goles and Noual (2012) a theoretical study of the dynamics of disjunc- tive networks under all updating schedules was presented. In Ruz and Goles (2013), results from reverse engineering syn- thesizing threshold Boolean networks with predefined limit cycles, showed that shared limit cycles, of length two, from parallel to sequential updates, were obtained for networks with indegree 3 and indegree 5. 1 If there is a constant node, one may consider a new network, smaller than the original one, with non-constant nodes. 2 A loop is a self connected node.