Stability of functions in Boolean models of gene regulatory networks

Boolean networks are used to model large nonlinear systems such as gene regulatory networks. We will present results that can be used to understand how the choice of functions affects the network dynamics. The so called bias-map and its fixed points depict much of the function's dynamical role in the network. We define the concept of stabilizing functions and show that many Post and canalizing functions are also stabilizing functions. Boolean networks constructed using the same type of stabilizing functions are always stable regardless of the average in-degree of network functions. We derive the number of all stabilizing functions and find it to be much larger than the number of Post and canalizing functions. We also discuss the implementation of functions and apply the presented results to biological data that give an approximation of the distribution of regulatory functions in eucaryotic cells. We find that the obtained theoretical results on the number of active genes are biologically plausible. Finally, based on the presented results, we discuss why canalizing and Post regulatory functions seem to be common in cells.     

From topology to dynamics in biochemical networks

Abstract formulations of the regulation of gene expression as random Boolean switching networks have been studied extensively over the past three decades. These models have been developed to make statistical predictions of the types of dynamics observed in biological networks based on network topology and interaction bias, p. For values of mean connectivity chosen to correspond to real biological networks, these models predict disordered dynamics. However, chaotic dynamics seems to be absent from the functioning of a normal cell. While these models use a fixed number of inputs for each element in the network, recent experimental evidence suggests that several biological networks have distributions in connectivity. We therefore study randomly constructed Boolean networks with distributions in the number of inputs, K, to each element. We study three distributions: delta function, Poisson, and power law (scale free). We analytically show that the critical value of the interaction bias parameter, p, above which steady state behavior is observed, is independent of the distribution in the limit of the number of elements N--> infinity. We also study these networks numerically. Using three different measures (types of attractors, fraction of elements that are active, and length of period), we show that finite, scale-free networks are more ordered than either the Poisson or delta function networks below the critical point. Thus the topology of scale-free biochemical networks, characterized by a wide distribution in the number of inputs per element, may provide a source of order in living cells. (c) 2001 American Institute of Physics.     

The phase diagram of random Boolean networks with nested canalizing functions

We obtain the phase diagram of random Boolean networks with nested canalizing functions. Using the annealed approximation, we obtain the evolution of the number b _t of nodes with value one, and the network sensitivity λ, and compare with numerical simulations of quenched networks. We find that, contrary to what was reported by Kauffman et al. [Proc. Natl. Acad. Sci. 101, 17102 (2004)], these networks have a rich phase diagram, were both the “chaotic" and frozen phases are present, as well as an oscillatory regime of the value of b _t . We argue that the presence of only the frozen phase in the work of Kauffman et al. was due simply to the specific parametrization used, and is not an inherent feature of this class of functions. However, these networks are significantly more stable than the variant where all possible Boolean functions are allowed.     

Boolean networks with variable number of inputs (K)

We studied a random Boolean network model with a variable number of inputs K per element. An interesting feature of this model, compared to the well-known fixed-K networks, is its higher orderliness. It seems that the distribution of connectivity alone contributes to a certain amount of order. In the present research, we tried to disentangle some of the reasons for this unexpected order. We also studied the influence of different numbers of source elements (elements with no inputs) on the network's dynamics. An analysis carried out on the networks with an average value of K=2 revealed a correlation between the number of source elements and the dynamic diversity of the network. As a diversity measure we used the number of attractors, their lengths and similarity. As a quantitative measure of the attractors' similarity, we developed two methods, one taking into account the size and the overlapping of the frozen areas, and the other in which active elements are also taken into account. As the number of source elements increases, the dynamic diversity of the networks does likewise: the number of attractors increases exponentially, while their similarity diminishes linearly. The length of attractors remains approximately the same, which indicates that the orderliness of the networks remains the same. We also determined the level of order that originates from the canalizing properties of Boolean functions and the propagation of this influence through the network. This source of order can account only for one-half of the frozen elements; the other half presumably freezes due to the complex dynamics of the network. Our work also demonstrates that different ways of assigning and redirecting connections between elements may influence the results significantly. Studying such systems can also help with modeling and understanding a complex organization and self-ordering in biological systems, especially the genetic ones.     

振幅耦合动态网络中相邻结点间的相同步

研究用适当的量化指标来刻画动态网络的相同步,为此定义了新的量化指标:相邻结点的网络平均锁相值和网络平均相频差.动态网络结点选择的是多旋转中心的Lorenz混沌振子,对Lorenz系统进行柱面坐标变换,用振幅耦合方法构造动态网络.分别对星形网络和小世界网络进行了仿真计算,结果表明随着耦合强度的增大,网络中相邻结点的两个系统之间存在相同步现象,而且相同步行为与定义的量化指标之间存在较准确的对应关系.     

Hierarchical synchronization in complex networks with heterogeneous degrees

We study synchronization behavior in networks of coupled chaotic oscillators with heterogeneous connection degrees. Our focus is on regimes away from the complete synchronization state, when the coupling is not strong enough, when the oscillators are under the influence of noise or when the oscillators are nonidentical. We have found a hierarchical organization of the synchronization behavior with respect to the collective dynamics of the network. Oscillators with more connections (hubs) are synchronized more closely by the collective dynamics and constitute the dynamical core of the network. The numerical observation of this hierarchical synchronization is supported with an analysis based on a mean field approximation and the master stability function.     

Activities and sensitivities in boolean network models

We study how the notions of importance of variables in Boolean functions as well as the sensitivities of the functions to changes in these variables impact the dynamical behavior of Boolean networks. The activity of a variable captures its influence on the output of the function and is a measure of that variable's importance. The average sensitivity of a Boolean function captures the smoothness of the function and is related to its internal homogeneity. In a random Boolean network, we show that the expected average sensitivity determines the well-known critical transition curve. We also discuss canalizing functions and the fact that the canalizing variables enjoy higher importance, as measured by their activities, than the noncanalizing variables. Finally, we demonstrate the important role of the average sensitivity in determining the dynamical behavior of a Boolean network.     

Stable irregular dynamics in complex neural networks

Irregular dynamics in multidimensional systems is commonly associated with chaos. For infinitely large sparse networks of spiking neurons, mean field theory shows that a balanced state of highly irregular activity arises under various conditions. Here we analytically investigate the microscopic irregular dynamics in finite networks of arbitrary connectivity, keeping track of all individual spike times. For delayed, purely inhibitory interactions we demonstrate that any irregular dynamics that characterizes the balanced state is not chaotic but rather stable and convergent towards periodic orbits. These results highlight that chaotic and stable dynamics may be equally irregular.     

On the constructive role of noise in stabilizing itinerant trajectories in chaotic dynamical systems

This work aims at studying dynamical models of neural networks, which exhibit phase transitions between states of various complexities. We use the biologically motivated KIII model, which has demonstrated excellent performance as a robust dynamical memory device. KIII is a high-dimensional dynamical system with extremely fragmented boundaries between limit cycles, tori, fixed points, and chaotic attractors. We study the role of additive noise in the development of itinerant trajectories. Noise not only stabilizes aperiodic trajectories, but there is an optimum noise level with highly itinerant behavior. We speculate that the previously found optimum classification performance of KIII as a function of the noise level, also identified as chaotic resonance, is related to chaotic itinerant oscillations among various ordered states.     

Dynamical motifs: building blocks of complex dynamics in sparsely connected random networks

Spatiotemporal network dynamics is an emergent property of many complex systems that remains poorly understood. We suggest a new approach to its study based on the analysis of dynamical motifs-small subnetworks with periodic and chaotic dynamics. We simulate randomly connected neural networks and, with increasing density of connections, observe the transition from quiescence to periodic and chaotic dynamics. This transition is explained by the appearance of dynamical motifs in the structure of these networks. We also observe domination of periodic dynamics in simulations of spatially distributed networks with local connectivity and explain it by the absence of chaotic and the presence of periodic motifs in their structure.     

Chaos in symmetric phase oscillator networks

Phase-coupled oscillators serve as paradigmatic models of networks of weakly interacting oscillatory units in physics and biology. The order parameter which quantifies synchronization so far has been found to be chaotic only in systems with inhomogeneities. Here we show that even symmetric systems of identical oscillators may not only exhibit chaotic dynamics, but also chaotically fluctuating order parameters. Our findings imply that neither inhomogeneities nor amplitude variations are necessary to obtain chaos; i.e., nonlinear interactions of phases give rise to the necessary instabilities.     

Recurrence network analysis of experimental signals from bubbly oil-in-water flows

Based on the signals from oil–water two-phase flow experiment, we construct and analyze recurrence networks to characterize the dynamic behavior of different flow patterns. We first take a chaotic time series as an example to demonstrate that the local property of recurrence network allows characterizing chaotic dynamics. Then we construct recurrence networks for different oil-in-water flow patterns and investigate the local property of each constructed network, respectively. The results indicate that the local topological statistic of recurrence network is very sensitive to the transitions of flow patterns and allows uncovering the dynamic flow behavior associated with chaotic unstable periodic orbits.     

Tristable and multiple bistable activity in complex random binary networks of two-state units

We study complex networks of stochastic two-state units. Our aim is to model discrete stochastic excitable dynamics with a rest and an excited state. Both states are assumed to possess different waiting time distributions. The rest state is treated as an activation process with an exponentially distributed life time, whereas the latter in the excited state shall have a constant mean which may originate from any distribution. The activation rate of any single unit is determined by its neighbors according to a random complex network structure. In order to treat this problem in an analytical way, we use a heterogeneous mean-field approximation yielding a set of equations generally valid for uncorrelated random networks. Based on this derivation we focus on random binary networks where the network is solely comprised of nodes with either of two degrees. The ratio between the two degrees is shown to be a crucial parameter. Dependent on the composition of the network the steady states show the usual transition from disorder to homogeneously ordered bistability as well as new scenarios that include inhomogeneous ordered and disordered bistability as well as tristability. The various steady states differ in their spiking activity expressed by a state dependent spiking rate. Numerical simulations agree with analytic results of the heterogeneous mean-field approximation.     

Evolution of canalizing Boolean networks

Boolean networks with canalizing functions are used to model gene regulatory networks. In order to learn how such networks may behave under evolutionary forces, we simulate the evolution of a single Boolean network by means of an adaptive walk, which allows us to explore the fitness landscape. Mutations change the connections and the functions of the nodes. Our fitness criterion is the robustness of the dynamical attractors against small perturbations. We find that with this fitness criterion the global maximum is always reached and that there is a huge neutral space of 100% fitness. Furthermore, in spite of having such a high degree of robustness, the evolved networks still share many features with “chaotic” networks.     

Dynamics of Transiently Chaotic Neural Network and Its Application to Optimization

Through adding a nonlinear self-feedback term in the evolution equations of nerual network,we introduced a transiently chaotic neural network model.In order to utilize the transiently chaotic dynamics mechanism in optimization problem efficiently,we have analyzed the dynamical pocedure of the transiently chaotic neural network model and studied the function of the crucial bifurcation parameter which governs the chaotic behavior of the system.Based on the dynamical analysis of the transiently chaotic neural network model,Chaotic annealing algorithm is also examined and improved.As an example,we applied chaotic annealing method to the traveling salesman problem and obtained good results.     

Comparison of the influences of nodal dynamics on network synchronized regions

A new piecewise linear unified chaotic (PLUC) system is firstly presented, and then its fundamental dynamical behaviors are analyzed. This modified chaotic system, as well as the unified chaotic (UC) one, is taken as network nodal oscillators for investigating the difference of influences of nodal dynamics on the bifurcation of network synchronized regions. It is found that beyond the greatly similar bifurcation modes between PLUC and UC networks, the synchronized regions in PLUC networks are far narrower at almost each parameter a than those in UC networks for most of inner coupling matrices, indicating the PLUC node makes the network more difficult to synchronization. Our numerical investigations show that this phenomenon is closely related with nodal dynamical properties, such as the boundary of attractors, the largest Lyapunov exponent and Lyapunov dimension.     

Understanding the enhanced synchronization of delay-coupled networks with fluctuating topology

We study the dynamics of networks with coupling delay, from which the connectivity changes over time. The synchronization properties are shown to depend on the interplay of three time scales: the internal time scale of the dynamics, the coupling delay along the network links and time scale at which the topology changes. Concentrating on a linearized model, we develop an analytical theory for the stability of a synchronized solution. In two limit cases, the system can be reduced to an “effective” topology: in the fast switching approximation, when the network fluctuations are much faster than the internal time scale and the coupling delay, the effective network topology is the arithmetic mean over the different topologies. In the slow network limit, when the network fluctuation time scale is equal to the coupling delay, the effective adjacency matrix is the geometric mean over the adjacency matrices of the different topologies. In the intermediate regime, the system shows a sensitive dependence on the ratio of time scales, and on the specific topologies, reproduced as well by numerical simulations. Our results are shown to describe the synchronization properties of fluctuating networks of delay-coupled chaotic maps.     

Effects of average degree of network on an order-disorder transition in opinion dynamics

We have investigated the influence of the average degree \langle k \rangle of network on the location of an order--disorder transition in opinion dynamics. For this purpose, a variant of majority rule (VMR) model is applied to Watts--Strogatz (WS) small-world networks and Barab\'{a}si--Albert (BA) scale-free networks which may describe some non-trivial properties of social systems. Using Monte Carlo simulations, we find that the order--disorder transition point of the VMR model is greatly affected by the average degree \langle k \rangle of the networks; a larger value of \langle k \rangle results in a more ordered state of the system. Comparing WS networks with BA networks, we find WS networks have better orderliness than BA networks when the average degree \langle k \rangle is small. With the increase of \langle k \rangle, BA networks have a more ordered state. By implementing finite-size scaling analysis, we also obtain critical exponents \beta/\nu, \gamma/\nu and 1/\nu for several values of average degree \langle k \rangle. Our results may be helpful to understand structural effects on order--disorder phase transition in the context of the majority rule model.     

Robust dynamical effects in traffic and chaotic maps on trees

In the dynamic processes on networks collective effects emerge due to the couplings between nodes, where the network structure may play an important role. Interaction along many network links in the nonlinear dynamics may lead to a kind of chaotic collective behavior. Here we study two types of well-defined diffusive dynamics on scale-free trees: traffic of packets as navigated random walks, and chaotic standard maps coupled along the network links. We show that in both cases robust collective dynamic effects appear, which can be measured statistically and related to non-ergodicity of the dynamics on the network. Specifically, we find universal features in the fluctuations of time series and appropriately defined return-time statistics.