Controlling chaos in random Boolean networks

Transcription

1 EUROPHYSICS LETTERS 20 March 1997 Europhys. Lett., 37 (9), pp (1997) Controlling chaos in random Boolean networks B. Luque and R. V. Solé Complex Systems Research Group, Departament de Fisica i Enginyeria Nuclear Universitat Politécnica de Catalunya - Sor Eulàlia d Anzizu s/n, Campus Nord Mòdul B Barcelona, Spain (received 21 October 1996; accepted in final form 7 February 1997) PACS j Fluctuation phenomena, random processes, and Brownian motion. Abstract. 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) = (S 1 (t),..., S N (t)), with S i (t) Σ {0, 1} (i = 1,..., N), is updated by means of the following dynamic equations: S i (t + 1) = Λ i [S i1 (t), S i2 (t),..., S ik (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 Λ i is a Boolean function also randomly chosen from a set F K 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) {S i (0)}, drawn from the set C(N) of Boolean N-strings. In spite of this random choice, the RBN exhibit a critical transition at K c = 2. Two phases are observed: a frozen one, for K < K c, and a chaotic phase for K > K c [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

2 598 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 Matías (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(k i ) (K i = 1, 2,..., K m ), i.e. f(k i ) = 1. The system has a mean connectivity given by K = K i f(k i ). Additionally, a bias p in the sampling of Boolean functions will be used, that is to say the probability p P [Λ i (S i1 (t), S i2 (t),..., S ik (t)) = 1]. Now the underlying dynamical system has to be generalized to S i (t + 1) = Λ i [S i1 (t), S i2 (t),..., S iki (t)] (2) (i.e. each S i receives K i {1, 2,..., K m } inputs) and the Boolean functions Λ i are randomly chosen from the set S(K m, p, Σ) = K m K i=1 F Ki,p (N, Σ). Here K m N is the maximum connectivity. Following the DAA, it can be shown [14] that the equation for the overlap evolution is now a rather more complicated one: [ a 12 (t + 1) = 2p(1 p) 1 + K m K i=1 And now the critical curve on the parameter space (p, K ) is K = G(p) = f(k i )a Ki 12 (t) ]. (3) 1 2p(1 p). (4) For K > G(p) a chaotic phase is reached, and a frozen one otherwise. For p = 1/2 it reduces to the standard RBN problem. Specifically, we want to stabilize a RBN in the chaotic phase by means of a perturbation of period τ applied to a quenched fraction γ of the total number of elements which is always the same through the whole time evolution. Our perturbation consists in giving this fraction of elements a certain fixed state (0 or 1) every τ time steps. In other words, a certain fraction γ is chosen and fixed (quenched set) at the beginning. If a given automaton belongs to the quenched set, and the randomly assigned value is, say, 0, then every τ steps this automaton will be forced to take such value. Following Derrida s annealed method, it can be seen that the dynamics of the overlapping a 12 between two annealed RBN is a 12 (t + i) = a K 12 (t + i 1) + {p2 + (1 p) 2 }(1 a K 12 (t + i 1)), i = 1, 2,..., τ 1, a 12 (t + τ) = A( K, p, τ; t) + (1 A( K, p, τ; t))γ, (5) where A( K, p, τ; t) = a K 12 (t + τ 1) + {p2 + (1 p) 2 }(1 a K 12 (t + τ 1)).

3 b. luque et al.: controlling chaos in random etc. 599 Fig. 1. Tree structure used in our annealed approximation. The unit S i is modified and this change can (or cannot) propagate through some of the K units which receive inputs from S i. This process is repeated using the annealed approach described in the text. Eventually a single change can percolate through the whole tree. The analysis of the marginal stability of the system with the fixed point a 12 = 1 gives the condition a 12 (t + τ) = [ K 2p(1 p)] τ (1 γ) < 1. (6) a 12 (t) a 12 This condition defines a critical curve in the space parameter p, K, τ, γ, K (1 γ) 1/τ = 1 2p(1 p). (7) In fact, this result states that the system behaves according to a new value of the mean (effective) connectivity, K eff K (1 γ) 1/τ defined by the first term in eq. (7): K eff is simply the average connectivity of the unperturbed network weighted by a factor that considers both the strength of the perturbation and its periodicity. For the case of null perturbation, γ = 0, we recover the critical curve K = (2p(1 p)) 1. If the perturbation is maximal, γ = 1, the system is always stable, at most with period τ. When the perturbation is applied every time step (τ = 1), the system has a fraction γ of elements frozen in the given state. This can be thought as a removal of connections in these elements, so the average connectivity in the system lowers from K to K (1 γ). This analytic result can be obtained in a different way [8]. Consider an annealed network with input connectivity K i. The output (average) connectivity K o will be the same (so we have K o = K i = K ). Now let us build up a tree representation of our annealed dynamics. We start from a single unit S i (t) S, which sends K outputs to other units, which in turn send K outputs each to other units. Here each tree level will correspond to the network states at different time steps (fig. 1). This tree gives us a picture of the possible paths followed by a change in S i. These paths are shown by arrows reaching a set of units {Si 1, S2 i,..., SKi i }. Intuitively, we know that in the frozen phase such a change does not propagate through the network (in other words, damage does not spread through the system). At the chaotic phase, a change in a single unit can generate an avalanche of changes through all the net. Let us consider a given tree and then take exactly the same tree, but with a single change in S i at t (i.e. a non-normalized D = 1 Hamming distance). Now two possible changes can be introduced: S i = 1 0 or S i = 0 1. This change (damage) can modify the state of one or more units. Let us consider the unit S j i. After one update, this unit will be in state Sj i = 1

4 600 EUROPHYSICS LETTERS Fig. 2. Control of a chaotic RBN. A system with N = 80 automata, average connectivity K = 3 and bias p = 0.62 (thus in the chaotic phase) is displayed. The control has been applied at t = 40 with a period τ = 2 and γ = As can be seen, the network reaches the stable state in a few time steps, and remains in the attractor of period L = 12 until the control is removed. with probability p (as defined previously) and a change in S i will modify S j i to S j i = 1 with probability 1 p. The complementary case is trivially obtained. So taking into account both cases, the probability of change in S j i is P (S j i = 0 1 S j i = 1 0) = 2p(1 p). If all outputs are considered (we have, on average, K outputs), at least one change will occur if K 2p(1 p) 1, which gives the critical condition for an unperturbed RBN. In our method, we also start from two annealed nets with identical dynamics. Now, however, the Hamming distance among them is the minimum one: a unit. In the ordered regime, such perturbation will disappear. In the chaotic one, it will be amplified. By considering the output neighbor trees, starting from the perturbed unit, we follow the propagation of this change. At each tree level we check if the distance is finite or zero. In this way, our approach can be understood in terms of percolation in Bethe s lattice with its intrinsic simplicity. If we are dealing with a network perturbed every τ time steps, we know that the same reasoning holds for the first τ 1 levels. That is, we would obtain the critical condition (2p(1 p) K ) τ 1 1 for the propagation of the damage until the (τ 1)-th level. In the τ-th level, the perturbation is applied. This means that now the propagation can only reach the following level through K (1 γ) branches, so the critical condition finally reads (2p(1 p) K ) τ (1 γ) 1, thus recovering the previous result (eq. (7)). An interesting connection between our result (7) and previous studies on chaos control in continuous dynamical systems [15] can be easily derived. For a given fixed pair ( K, p), we can use the linear approximation ln(1 γ) γ which leads to γ τ C, (8)

5 b. luque et al.: controlling chaos in random etc. 601 Fig. 3. Fig. 4. Fig. 3. Change in the length of the stable attractors for a perturbed RBN. The system shown has N = 500 automata, p = 0.6 and K = 3. The control is applied every τ = 2 time steps and the strength has been varied in intervals of size γ = A transient of length t trans = 500 is discarded and then the control is applied. After 500 time steps, we check if the control has been achieved. Fig. 4. Critical curve for perturbed chaotic RBN with varying K (dashed line). three different numerically estimated boundaries for three lattice sizes. Here we show where C ln(2 K p(1 p)) is a constant. An equivalent relation in the context of continuous dynamic systems has been derived by Matías and Güémez [15] in terms of the growth of perturbations as measured by the largest Lyapunov exponent. Some numerical simulations have been performed over different systems in order to test the obtained analytical results. In fig. 2 a system originally in the chaotic phase is shown. When the control is applied, the system sets in a periodic attractor as long as it is forced by the perturbation. When the control is removed, the system displays chaotic behavior again. The periodic attractor can be seen both in the temporal series of the number of elements with value 1 (fig. 2 a)) and in the pattern that appears in the automata (fig. 2b)). Figure 3 shows the change in the length of the stable attractor where the system settles down when the strength of the perturbation γ is increased. Statistically, one expects that length to decrease with an increasing perturbation, although it would be possible to obtain a different behavior in some particular cases. We have to keep in mind that, in some way, a higher γ implies a lower average connectivity in the system and this latter one means shorter attractors. The represented system has size N = 500, K = 3 and bias p = 0.6. After a transient period is discarded, the control is applied to the system, and some time steps later we measure if the system has reached a periodic attractor (see fig. 3). Although different values of γ have attractors of the same length, this does not mean that they are the same attractor at all. The first time that control is achieved is for γ c 0.5, in good agreement with the expected value γ c = This diagram is the RBN counterpart of the bifurcation diagram that could be recovered in a discrete chaotic system for a continuous variation of the control parameter. Finally, we represent in fig. 4 the numerical critical curve for T = 2, p = 0.7 and different system sizes N. This curve shows the first time that the system settles down to a stable attractor, that is the minimum value of γ necessary to achieve control in a system with a fixed average connectivity. Finally, let us mention that the γ c values leading to control in our previous examples show

6 602 EUROPHYSICS LETTERS that the fraction of units necessary to reach control is often rather high. However, the real genome shows a range of connectivities from high to low and so modifications on some units can generate a cascade of perturbations leading to control by using a small number of units. In a recent study [14] we showed that an exponential distribution of connectivities should be expected compatible with a critical (K c = 2) mean value. Under these conditions, the previous requirements could be strongly relaxed. It should be noted that the analysis of the gene interactions in real genomes has shown that a wide set of connectivities is present. Some few genes (as the so-called oncogenes [16]) are involved in the control of a large number of functional genes. The stability properties of the different cell types are basically determined by the outcome of gene interactions. Such stability could be the result of a self-organization to the critical point [17] and different mechanisms of stabilization have been proposed, as canalization and/or redundance [5]. Another possibility emerging from this study is that some mechanism of self-control could be incorporated into the genome dynamics. *** The authors would like to thank S. C. Manrubia, S. Kauffman, P. Miramontes, O. Miramontes, J. Delgado, J. Bascompte, B. Goodwin, L. M. de la Prida and an anonymous referee for useful comments. This work has been partially supported by a grant DGYCIT PB BL was supported by an Intercampus Grant. REFERENCES [1] Kauffman S. A., J. Theor. Biol., 22 (1969) 437. [2] Kauffman S. A., J. Theor. Biol., 44 (1974) 167. [3] Kauffman S. A., Physica D, 10 (1984) 145. [4] Kauffman S. A., Physica D, 42 (1990) 135. [5] Somogyi R. and Sniegoski C. A., Complexity, 1 (1996) 45. [6] Derrida B. and Pomeau Y., Europhys. Lett., 1 (1986) 45. [7] Derrida B. and Stauffer D., Europhys. Lett., 2 (1986) 739. [8] Luque B. and Solé R. V., to be published Phys. Rev. E, 55 (1997). [9] Shinbrot T., Grebogi C., Ott E. and Yorke J. A., Nature, 363 (1993) 411. [10] Doebeli M., Proc. R. Soc. London, Ser. B, 254 (1993) 281. [11] Güémez J. and Matías M. A., Phys. Lett. A, 181 (1993) 29. [12] Solé R. V. and Menéndez de la Prida L., Phys. Lett. A, 199 (1995) 65. [13] Gang H. and Zhilin Q., Phys. Rev. Lett., 72 (1994) 68. [14] Solé R. V. and Luque B., Phys. Lett. A, 196 (1995) 331. [15] Matías M. A. and Güémez J., Phys. Rev. E, 54 (1996) 198. [16] Singer M. and Berg P., Genes and Genomes (University Science Books, California) [17] Solé R. V., Manrubia S. C., Luque B., Delgado J. and Bascompte J., Complexity, 1 (1995) 13.