Fig. 1.

Emergence of complex dynamics in simple signaling networks. (a) The units constituting the network, which are located on the nodes of a one-dimensional lattice, have bidirectional nearest-neighbor connections. (b) A number k_eN of additional unidirectional links is established between pairs of randomly selected units, where k_e is the mean excess connectivity and N is the number of units in the system. At time t = 0, we assign to each unit i = 1,...,N a state σ_i (0) randomly chosen from the set {0,1} and a Boolean function F_i; (Eq. 1). This Boolean function (or rule) determines the way in which the inputs are processed. Each unit effectively processes two inputs, one unit corresponding to the average state of its neighbors and one unit corresponding to its own state. With probability η, a unit “reads” a random Boolean variable instead of the state of a neighbor, where the parameter η quantifies the intensity of the noise. Note that the noise does not alter the state of the units but only the value read by its neighbor. At each subsequent time step, each unit updates its state synchronously according to its Boolean function. (c–e) Time evolution of systems comprising 512 units with F_i = 232 for all units, η = 0.1, and k_e = 0.15 (c), k_e = 0.45 (d), and k_e = 0.90 (e). Red indicates σ_i(t) = 1, and yellow indicates σ_i(t) = 0. The time evolution for systems starting from the same initial configuration and using the same sequence of random numbers is shown. Thus, the difference in the dynamics is uniquely due to the different number of long-distance links. For k_e = 0.15, the system quickly evolves toward a configuration with several clusters in which all of the units are in the same state. The boundaries of these clusters drift because of the noise, but the state of the system S(t) is quite stable, and the dynamics are close to Brownian noise. In contrast, for k_e = 0.90, a large stable cluster develops and the state of the system changes only when some units change state because of the effect of the noise. This process yields white-noise dynamics. For k_e = 0.45, clusters are formed, but they are no longer stable, in contrast to what happens for small k_e. In this case, information propagates through the random links, which can lead to a change in the state of one or more units inside a cluster. Our results suggest that because these long-range connections exist on all length scales, they lead to long-range correlations in the dynamics and the observed 1/f behavior (Fig. 3b).