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 Systems Biology
Emergence of complex dynamics in a simple model of signaling networks

Edited by Nancy J. Kopell, Boston University, Boston, MA (received for review July 6, 2004)
Abstract
Various physical, social, and biological systems generate complex fluctuations with correlations across multiple time scales. In physiologic systems, these longrange correlations are altered with disease and aging. Such correlated fluctuations in living systems have been attributed to the interaction of multiple control systems; however, the mechanisms underlying this behavior remain unknown. Here, we show that a number of distinct classes of dynamical behaviors, including correlated fluctuations characterized by 1/f scaling of their power spectra, can emerge in networks of simple signaling units. We found that, under general conditions, complex dynamics can be generated by systems fulfilling the following two requirements, (i) a “smallworld” topology and (ii) the presence of noise. Our findings support two notable conclusions. First, complex physiologiclike signals can be modeled with a minimal set of components; and second, systems fulfilling conditions i and ii are robust to some degree of degradation (i.e., they will still be able to generate 1/f dynamics).
Complex systems are typically composed of interacting units that communicate information and are able to process and withstand a broad range of stresses (1–4). In physiology, freerunning healthy systems typically generate complex output signals that have longrange correlations [i.e., a 1/f decay of the power spectra for low frequencies (**, 5–7)]. Deviations from the 1/f pattern have been associated with disease or aging in various contexts (3, 8).
Despite its practical and fundamental interest (9), the origin of such correlated dynamics remains an unsolved problem (4). Until recently, attention has focused primarily on the complexity of the specific physiologic subsystems or on the nature of the nonlinear interactions between them (10–12). In particular, Boolean variables (which can take one of two values, 0 or 1) and Boolean functions have been extensively used to model the state and dynamics of complex systems (see ref. 12 for an introduction). The reason such a “simplistic” description may be appropriate arises from the fact that Boolean variables provide good approximations to the nonlinear functions encountered in many control systems (10, 13–15). Random Boolean networks (RBNs) were proposed by Kauffman (10) as models of genetic regulatory networks, and they have also been studied in other contexts (13, 14). Wolfram (15), in contrast, proposed that cellular automata models, which are a class of ordered Boolean networks with identical units, may explain the realword complexity. Neither of these two classes of models has been shown to generate the complex dynamics with 1/f fluctuations observed in healthy physiologic systems.
Here, we propose a modeling approach (Fig. 1a ) that departs from traditional approaches in that we pay special attention to the topology of the network of interactions (4) and the role of noise (16). Our model is rooted in the following two considerations that are observed frequently in realworld systems. (i) The units in the system are connected mostly locally but also with some longrange connections, giving rise to socalled “smallworld” topology (17, 18); and (ii) the interaction between the units is affected by “noisy” communication and/or by noisy stimuli (19–23). We demonstrate that simple rules, such as the majority rule, are able to generate signal with complex fluctuations under simple, but physiologically relevant, conditions.
Methods
The Model. We placed the Boolean units comprising the network on the nodes of a onedimensional ring and established bidirectional nearestneighbor connections (Fig. 1). Then, we added k_{e}N additional unidirectional links (where k_{e} is the mean excess connectivity and N is the number of units in the system) between pairs of randomly selected units. Hence, each unit had a set of links through which incoming signals arrive and that the unit then processes. During the system evolution, the state of each neighbor was replaced by a random value with probability η, which parameterizes the intensity of the noise.
We assigned to each unit i = 1,..., N a Boolean function F_{i} , which determined the way that the states of the neighbors and its own state were processed. We restricted our study to Boolean functions that had only the following two “effective” inputs: the state of the unit and the average state of all other neighbors. This restriction yielded 64 unique symmetric Boolean functions (see Figs. 2 and 711 and Supporting Methods, which are published as supporting information on the PNAS web site), and it had the advantage of permitting a topologyindependent implementation of the Boolean functions, thus enabling a systematic study of the effect of different rules on the dynamics of the system.
Quantification of the Dynamical Behavior of the System. We started all of our numerical simulations with a random initial configuration and let the system evolve synchronously according to the rules of the model. We defined the state S(t) of the system as the sum of the states σ _{i} of all of the Boolean units as follows: We recorded S(t) during the course of the simulation (see Fig. 3 a–c ) and quantified the complexity of the time series generated in terms of its autocorrelation function (3). We applied the detrended fluctuationanalysis method (5), which quantifies longrange time correlations in the dynamical output of a system by means of a single scaling exponent α (Fig. 3d ). Brownian noise corresponds to α = 1.5, whereas uncorrelated white noise corresponds to α = 0.5. For many physiologic signals, one observes α ≈ 1, corresponding to 1/f behavior, which can be seen as a “trade off” between the two previous cases (3).
Results
RBNs. The RBN model corresponds to a completely random network with randomly selected Boolean rules for the units. As shown in Fig. 4a , we find whitenoise dynamics for essentially any pair of values of k_{e} and η within the ranges considered, suggesting that, even in the presence of noise, a system of random Boolean functions cannot generate 1/f dynamics. This result is not unexpected because the random collection of Boolean functions comprising the system prevents the development of any order or predictability in the dynamics.
CellularAutomata Models with SmallWorld Topology and Noise. We systematically study the 64 symmetric rules (see Figs. 2 and 711 and Supporting Methods) for different pairs of values of k_{e} and η. Some of the rules have parallels to physiologically meaningful dynamics. Rule 232 is a majority rule; that is, each unit will be active at the next time step only if the majority of its neighbors is active presently. Rule 50 is a threshold rule with refractory time period; that is, whenever the inputs of the neighbors surpass a certain value, a unit becomes active in the next time step and then will be inactive for at least one time step. The 64 symmetric rules lead to three qualitatively distinct phase spaces (Fig. 4 b–d ). Rule 232, the majority rule, which is representative of the first type of phase space, displays three distinct types of dynamical behavior (Fig. 4b ). For small k_{e} , we find mostly Brownianlike scaling. For large k_{e} , we find mostly whitenoise dynamics. Of greatest interest, for intermediate values of k_{e} and for a broad range of values of the intensity of the noise η, we find 1/f fluctuations.
Rule 50, which is representative of a second type of phase space, displays fewer types of dynamical behavior. In particular, we find only a narrow range of noise intensities (with a weak dependence on k_{e} ) for which the dynamics display 1/f correlations. For η ≳ 0.1, the dynamics become uncorrelated (Fig. 4c ). Rule 160, which is representative the third type of phase space, displays whitenoise dynamics for all values of k_{e} and η (Fig. 4d ).
Note that for k_{e} = 0 (i.e., when the network is a onedimensional lattice), the model is not able to generate 1/f dynamics. This result implies that, in the context of the model, the existence of longrange connections (i.e., the smallworld topology achieved by making k_{e} ≠ 0) is an essential ingredient for the emergence of 1/f dynamics.
Robustness of the Findings. To determine the generality of the results presented above, one needs to address the questions of how these findings are affected by (i) changes in the topology of the network and (ii) “errors” in the implementation of the rules by the units.
Concerning i, we note that the network topologies that have been considered span the cases of ordered onedimensional lattices, smallworld networks, and random graphs (24). However, all considered networks are composed of units with approximately the same degree, i.e., the same number of connections. To investigate the role of the distribution of number of connections, we also study networks that span the range of empirically observed degree distribution: a δ distribution, an exponential distribution, and a power law distribution. The last case corresponds to the socalled scalefree networks (18). Notably, we find that the picture of the phase space shown in Fig. 4b does not get altered by these changes in the degree distribution (see Figs. 7, 8, 9, 10, 11 and Supporting Methods).
To address ii, we systematically explore the dynamical behaviors in the phase space defined by (k_{e} ,η) for systems composed of units operating according to either rule 232 or a randomly selected rule. Notably, we find that, with as many as onefourth of all units operating according to random Boolean functions, the model still displays a rich phase space including white, 1/f, and Brownian noise (Fig. 5). This finding holds even if, instead of using random Boolean functions, we consider a single Boolean function (Fig. 6).
Discussion
Our results are notable for various reasons. First, they demonstrate that a model of signaling networks is able to generate a broad range of behaviors that are reminiscent of those observed in physiologic systems. Second, we show, for a rather general class of models, that various dynamical behaviors can emerge only under restrictive, but physiologically relevant, assumptions; namely, the system must have a smallworld topology, and noise must be present.
An interesting aspect of our results is that some of the rules that we consider have plausible physiologic interpretations. For example, rule 232 is a rule in which a unit changes its state to that of the majority of the incoming inputs. A majority rule appears to be operative in the central nervous system, where multiple fibers (excitatory or inhibitory) converge onto a single neuron. Action potentials converging in a neuron summate to bring the neuron to the threshold for firing (25). A majority rule also appears to be operative in the baroreflex control of the cardiovascular system. The baroreflex is a feedback loop that controls heart rate continuously by modulating the degree of sympathetic and parasympathetic nervous system input to the sinus node of the heart. Changes in heart rate values are determined by whichever input is dominant at the moment (25).
Our results demonstrate that complex fluctuations are present even when a fraction of the units obeys randomly chosen Boolean functions, suggesting that systems composed of units operating according to the majority will be robust to the removal or failure of units. Also, because this complex dynamics does not depend on a scalefree topology, these systems do not necessarily display the vulnerability to targeted attack observed in scalefree networks (26). In biological systems, this robustness could support the “physiologic reserve” enabling an organism to overcome age or diseaserelated loss of system components.
Additionally, our findings raise the intriguing possibility that the interactions within physiologic systems and their degradation with aging and pathology may be symbolically mapped as a “walk” on the phase space (k_{e} ,η). According to this model, 1/f dynamics similar to that found for healthy physiology are generated when noise intensity and connectivity reside in a well defined range (27). A loss of complexity with a breakdown of longrange correlations would be anticipated when these parameters assume values outside this range (Fig. 4). Support for this formulation comes from analysis of heart rate dynamics with aging and disease, where connectivity or coupling among system components is likely to be degraded (3, 4, 8). Similarly, evidence suggests that decreased social connectedness and the corresponding decrease in noisy stimulation may be associated with increased cardiac mortality and decreased functional recovery from stroke or dementia (28, 29).
In a related way, the development of complex, adaptive dynamics during the maturation of the organism may be accounted for, at least in part, by the evolution of appropriate connections (see refs. 30 and 31 for empirical evidence). Our model predicts that the fraction of nonlocal connections has an optimal range of values; hence, an excessive number of certain types of inputs may also degrade functionality. Thus, our results are consistent with empirical evidence suggesting that development and maintenance of healthy function may require adjusting the number of connections. Last, the model may also provide a robust way to generate fluctuations that closely resemble physiologic signals, which could be implemented in medical devices, such as mechanical ventilators (32, 33) and hormone infusion pumps (34).
Acknowledgments
We thank A. Arenas, J. J. Collins, L. Glass, R. Guimera, I. Henry, C.C. Lo, G. Moody, J. M. Ottino, C.K. Peng, C. J. Perez, M. SalesPardo, H. E. Stanley, and G. Weisbuch for stimulating discussions. L.A.N.A. was supported by a Searle Leadership Fund Award and National Institutes of Health National Institute of General Medical Sciences Award K25. A.L.G. was supported by National Institutes of Health National Center for Research Resources Grant P41 RR13622 and the G. Harold and Leila Y. Mathers Foundation. L.A.L. was supported by National Institutes of Health Grants AG04390 and AG08812, and he holds the Irving and Edyth S. Usen and Family Chair in Geriatric Medicine at the Hebrew Rehabilitation Center for Aged. A.D.G. was supported by the Spanish Ministerio de Educación, Cultura, y Deporte Dirección General de Enseñanza Superior Grants BFM20000626 and BFM200308258 and European Commission Future and Emerging Technologies Open Project “Coevolution and SelfOrganization in Dynamical Networks” (COSIN) IST200133555.
Footnotes

↵ † To whom correspondence should be addressed at: Department of Chemical and Biological Engineering, McCormick School of Engineering and Applied Science, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208. Email: amaral{at}northwestern.edu.

This paper was submitted directly (Track II) to the PNAS office.

Abbreviation: RBN, random Boolean network.

↵ ** A powerlawdecaying power spectrum S(f) ∝ f ^{β} is the signature of a signal with powerlawdecaying correlations. The case β = 2 corresponds to a Brownian noise, whereas β = 0 corresponds to a completely uncorrelated “white” noise. The intermediate case, S(f) ∝ 1/f, is a “compromise” between the smalltimescale roughness but largetimescale smoothness of white noise and the smalltimescale smoothness but largetimescale roughness of Brownian noise.
 Copyright © 2004, The National Academy of Sciences
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