Controlling chaos in random Boolean networks

– A variant of a simple method for chaos control is applied to achieve control in random Boolean networks (RBN). It is shown that a RBN in the chaotic phase can be forced to behave periodically if a certain quenched fraction γ of the automata is given a fixed state (the system variables) every τ time steps. An analytic relationship between γ and τ is derived and numerically tested. A simple theoretical approach to complex systems has been provided by the introduction of random Boolean networks (RBN), also called Kauffman nets [1]-[4]. First introduced by Kauffman, a set of N binary elements S(t) = (S1(t), . . . , SN (t)), with Si(t) ∈ Σ ≡ {0, 1} (i = 1, . . . , N), is updated by means of the following dynamic equations: Si(t+ 1) = Λi[Si1(t), Si2(t), . . . , SiK (t)] . (1) Such dynamical systems share some properties with cellular automata (CA), but here randomness is introduced at several levels. Each automaton is randomly connected with exactly K others which send inputs to it. Here Λi is a Boolean function also randomly chosen from a set FK of all the Boolean functions with connectivity K. An additional source of randomness is introduced through the random choice of the initial condition S(0) ≡ {Si(0)}, drawn from the set C(N) of Boolean N -strings. In spite of this random choice, the RBN exhibit a critical transition at Kc = 2. Two phases are observed: a frozen one, for K < Kc, and a chaotic phase for K > Kc [3], [4]. Here “chaos” is not the usual low-dimensional deterministic chaos but a phase where damage spreading takes place (i.e. propagation of changes caused by transient flips of a single unit). At the critical point, a small number of attractors (≈ O( √ N)) is observed which show high stability and low reachability among different attractors [3], [4]. These properties are clearly observed, for example, in the genome (for a recent reference, see [5]). This critical point was first estimated through numerical simulations [1], [2] and later analytically obtained by means of the so-called Derrida’s Annealed Approximation (DAA) [6], [7]. A simpler approximation, equivalent to DAA, has been introduced by Luque and Solé in ref. [8], and this latter one will also be used in this paper. c © Les Editions de Physique 598 EUROPHYSICS LETTERS Our aim here is to show how to control the chaotic phase in a random Boolean network by means of proportional pulses in the system variables. In recent years, chaos control [9] has been widely used in the analysis of many dynamical systems and biological implications have been suggested [10]. We will use a variant of the Güémez and Mat́ıas (GM) method [11]. This simple way of controlling chaos has been successfully applied to n-dimensional maps and also to discrete neural networks [12]. Though control of spatiotemporal chaos in coupled map lattice models has been reported [13], as far as we know this is the first example of control in complex dynamical systems with a discrete number of states. In this paper, we will consider the case of having a RBN with a distribution of connections [14] f(Ki) (Ki = 1, 2, . . . ,Km), i.e. ∑ f(Ki) = 1. The system has a mean connectivity given by 〈K〉 = ∑ Kif(Ki). Additionally, a bias p in the sampling of Boolean functions will be used, that is to say the probability p ≡ P [Λi(Si1(t), Si2(t), . . . , SiK (t)) = 1]. Now the underlying dynamical system has to be generalized to Si(t+ 1) = Λi[Si1(t), Si2(t), . . . , SiKi (t)] (2) (i.e. each Si receives Ki ∈ {1, 2, . . . ,Km} inputs) and the Boolean functions Λi are randomly chosen from the set