On the Number of Attractors of Positive and Negative Boolean Automata Circuits

Abstract

Abstract. In line with fields of theoretical computer science and biology that study Boolean automata networks often seen as models of regulation networks, we present some results concerning the dynamics of networks whose underlying interaction graphs are circuits, that is, Boolean automata circuits. In the context of biological regulation, former studies have highlighted the importance of circuits on the asymptotic dynamical behaviour of the biological networks that contain them. Our work focuses on the number of attractors of Boolean automata circuits. We prove how to obtain formally the exact value of the total number of attractors of a circuit of arbitrary size n as well as, for every positive integer p, the number of its attractors of period p depending on whether the circuit has an even or an odd number of inhibitions. As a consequence, we obtain that both numbers depend only on the parity of the number of inhibitions and not on their distribution along the circuit. Keywords: Discrete dynamical system, formal neural network, positive and negative circuit, asymptotic behaviour, attractor. IntroductionThe theme of this article is set in the general framework of complex dynamical systems and, more precisely, that of regulation or neural networks modeled by means of discrete mathematical tools. Since McCulloch and Pitts [1] proposed threshold Boolean automata networks to represent neural networks formally in 1943 and later, at the end of the 1960's, Kauffman [2] and Thomas [3] introduced the first models of genetic regulation networks, many other studies based on the same or different formalisms were carried out on the theoretical properties of such networks. One of the main motivations of many of them was to better understand those emergent dynamical behaviours that networks display and that cannot be explained or predicted by a simple analysis of the local interactions existing between the components of the networks. Amongst the studies published in this context since the end of the 1990's, one may cite [4,5,6,7,8]. Earlier on, Hopfield [9,10] emphasised the notions of memory and learning and Goles et al. revealed in [11,12,13] interesting dynamical properties of some particular networks. Further, later works by Thomas and Kauffman [14,15] yielded conjectures and gave rise to problematics that are still relevant in the field of regulation networks beyond the particular definition of the models one may choose to use. For instance, Thomas highlighted the importance of specific patterns on the dynamics of discrete regulation networks and Kauffman gave an approximation of the number of different possible asymptotic behaviours of Boolean networks.From the point of view of theoretical biology as well as that of theoretical computer science, it seems to be of great interest to address the question of the number of attractors in the dynamics of a network. Close to the 16th Hilbert problem concerning the number of limit cycles of dynamical systems [16], this question has already been considered in some works [17,1...

“Immunetworks”, intersecting circuits and dynamics

“…Network B has three fixed points amongst which only one has a non empty attraction basin of size 13. Networks C and D both have circuits as interaction graphs and in agreement with results in [7,15], they both only have attractors without attraction basins.…”

“…In similar lines, let us highlight here that it is easy to derive from results in [7,15] that positive circuits have at least twice as many attractors as negative circuits of the same size. A result of this is that fixing the state of an underlying positive circuit of size n in a network (either by intersecting it with another circuit or by submitting it to a fixed input resulting from a source node such as a micro-RNA as we will see later on, for instance) potentially has much more drastic consequences than fixing the state of an underlying negative circuit of size n, in terms of the reduction in the number of possible attractors of the network.…”

“…Studies performed in [16,17,18] showed that, for most networks, the maximum number of attractors is obtained with the parallel update schedule. Indeed, in [19], the analysis concerning the dynamics of circuits exploiting results of [9] shows that sequentialising the updating of a network tends to make it equivalent to a network whose circuits are smaller and thus whose attractors are smaller and less numerous (see [7,8,15] for a combinatorial study of the dynamical behaviours of circuits).…”

“…Thus, for random networks of given connectivity, the expected number of attractors may be derived from (i) the probability of there being a circuit of a given size ℓ in their interaction graph and (ii) the exact formulae (given in [7,15]) for the number of attractors of positive and negative isolated circuits of size ℓ.…”

“…A huge mathematical effort has been done to identify the sources of the attractor multiplicity, known to be closely related to the number of positive circuits underlying regulatory networks [4,15,22,23,57,75,76,77,78] as well as to understand the causes of attractor uniqueness. Both these problems are of high interest if we want to explain the number of differentiated functions (around 300 for the human being [57]) as well as the presence of some unique functions that are devoted to one tissue.…”