Robustness of Oscillatory Behavior in Correlated Networks

Understanding network robustness against failures of network units is useful for preventing large-scale breakdowns and damages in real-world networked systems. The tolerance of networked systems whose functions are maintained by collective dynamical behavior of the network units has recently been analyzed in the framework called dynamical robustness of complex networks. The effect of network structure on the dynamical robustness has been examined with various types of network topology, but the role of network assortativity, or degree–degree correlations, is still unclear. Here we study the dynamical robustness of correlated (assortative and disassortative) networks consisting of diffusively coupled oscillators. Numerical analyses for the correlated networks with Poisson and power-law degree distributions show that network assortativity enhances the dynamical robustness of the oscillator networks but the impact of network disassortativity depends on the detailed network connectivity. Furthermore, we theoretically analyze the dynamical robustness of correlated bimodal networks with two-peak degree distributions and show the positive impact of the network assortativity.     

Collective Motion in a Network of Self-Propelled Agent Systems

Collective motions of animals that move towards the same direction is a conspicuous feature in nature. Such groups of animals are called a self-propelled agent (SPA) systems. Many studies have been focused on the synchronization of isolated SPA systems. In real scenarios, different SPA systems are coupled with each other forming a network of SPA systems. For example, a flock of birds and a school of fish show predator-prey relationships and different groups of birds may compete for food. In this work, we propose a general framework to study the collective motion of coupled self-propelled agent systems. Especially, we study how three different connections between SPA systems: symbiosis, predator-prey, and competition influence the synchronization of the network of SPA systems. We find that a network of SPA systems coupled with symbiosis relationship arrive at a complete synchronization as all its subsystems showing a complete synchronization; a network of SPA systems coupled by predator-prey relationship can not reach a complete synchronization and its subsystems converges to different synchronized directions; and the competitive relationship between SPA systems could increase the synchronization of each SPA systems, while the network of SPA systems coupled by competitive relationships shows an optimal synchronization for small coupling strength, indicating that small competition promotes the synchronization of the entire system.     

Synchronizing a multicellular system by external input: an artificial control strategy

MOTIVATION: Although there are significant advances on elucidating the collective behaviors on biological organisms in recent years, the essential mechanisms by which the collective rhythms arise remain to be fully understood, and further how to synchronize multicellular networks by artificial control strategy has not yet been well explored. RESULTS: A control strategy is developed to synchronize gene regulatory networks in a multicellular system when spontaneous synchronization cannot be achieved. We first construct an impulsive control system to model the process of periodically injecting coupling substances with constant or random impulsive control amounts into the common extracellular medium, and further study its effects on the dynamics of individual cells. We derive the threshold of synchronization induced by the periodic substance input. Therefore, we can synchronize the multicellular network to a specific collective behavior by changing the frequency and amplitude of the periodic stimuli. Moreover, a two-stage scheme is proposed to facilitate the synchronization in this paper. We show that the presence of the external input may also initiate different dynamics. The multicellular network of coupled repressilators is used to show the effectiveness of the proposed method. The results not only provide a perspective to understand the interactions between external stimuli and intrinsic physiological rhythms, but also may lead to development of realistic artificial control strategy and medical therapy. AVAILABILITY: CONTACT: aihara@sat.t.u-tokyo.ac.jp.     

Dynamical modeling of drug effect using hybrid systems

Drug discovery today is a complex, expensive, and time-consuming process with high attrition rate. A more systematic approach is needed to combine innovative approaches in order to lead to more effective and efficient drug development. This article provides systematic mathematical analysis and dynamical modeling of drug effect under gene regulatory network contexts. A hybrid systems model, which merges together discrete and continuous dynamics into a single dynamical model, is proposed to study dynamics of the underlying regulatory network under drug perturbations. The major goal is to understand how the system changes when perturbed by drugs and give suggestions for better therapeutic interventions. A realistic periodic drug intake scenario is considered, drug pharmacokinetics and pharmacodynamics information being taken into account in the proposed hybrid systems model. Simulations are performed using MATLAB/SIMULINK to corroborate the analytical results.     

Neural-space generalization of a topological transformation

An investigation is performed to assess the generalization capability found in neural network paradigms to approximate a 2-dimensional coordinate (topological) transformation. A developed strategy uses the example to give a physical meaning to what is meant by generalization. The example shows how to use a neural network paradigm to generalize a two-degree of freedom topological transformation from cartesian end-point coordinates to corresponding joint angle coordinates based only on examples of the mapping. The importance of this example is that it provides a clear understanding of how and what a neural network is actually communications and brings a theoretical idea to a useful understanding. When examples characterize the topology, a collective generalization property begins to emerge and the network learns the topology. If the network is presented with additional examples of the transformation, the network can generate the corresponding joint angles to any unseen position, that is, by generalization. It is also significant that the network's generalization property emerges from the network based on very few training examples! Further, the networks power exists with very few neurons. Results suggest the use of the paradigm's generalization capability to provide solutions to unknown or intractable algorithms for applications.     

Pseudo-Lyapunov exponents and predictability of Hodgkin-Huxley neuronal network dynamics

We present a numerical analysis of the dynamics of all-to-all coupled Hodgkin-Huxley (HH) neuronal networks with Poisson spike inputs. It is important to point out that, since the dynamical vector of the system contains discontinuous variables, we propose a so-called pseudo-Lyapunov exponent adapted from the classical definition using only continuous dynamical variables, and apply it in our numerical investigation. The numerical results of the largest Lyapunov exponent using this new definition are consistent with the dynamical regimes of the network. Three typical dynamical regimes—asynchronous, chaotic and synchronous, are found as the synaptic coupling strength increases from weak to strong. We use the pseudo-Lyapunov exponent and the power spectrum analysis of voltage traces to characterize the types of the network behavior. In the nonchaotic (asynchronous or synchronous) dynamical regimes, i.e., the weak or strong coupling limits, the pseudo-Lyapunov exponent is negative and there is a good numerical convergence of the solution in the trajectory-wise sense by using our numerical methods. Consequently, in these regimes the evolution of neuronal networks is reliable. For the chaotic dynamical regime with an intermediate strong coupling, the pseudo-Lyapunov exponent is positive, and there is no numerical convergence of the solution and only statistical quantifications of the numerical results are reliable. Finally, we present numerical evidence that the value of pseudo-Lyapunov exponent coincides with that of the standard Lyapunov exponent for systems we have been able to examine.     

On the robustness of complex heterogeneous gene expression networks

We analyze a continuous gene expression model on the underlying topology of a complex heterogeneous network. Numerical simulations aimed at studying the chaotic and periodic dynamics of the model are performed. The results clearly indicate that there is a region in which the dynamical and structural complexity of the system avoid chaotic attractors. However, contrary to what has been reported for Random Boolean Networks, the chaotic phase cannot be completely suppressed, which has important bearings on network robustness and gene expression modeling.     

Finding quasi-optimal network topologies for information transmission in active networks

This work clarifies the relation between network circuit (topology) and behaviour (information transmission and synchronization) in active networks, e.g. neural networks. As an application, we show how one can find network topologies that are able to transmit a large amount of information, possess a large number of communication channels, and are robust under large variations of the network coupling configuration. This theoretical approach is general and does not depend on the particular dynamic of the elements forming the network, since the network topology can be determined by finding a Laplacian matrix (the matrix that describes the connections and the coupling strengths among the elements) whose eigenvalues satisfy some special conditions. To illustrate our ideas and theoretical approaches, we use neural networks of electrically connected chaotic Hindmarsh-Rose neurons.     

Identification of potential targets in biological signalling systems through network perturbation analysis

The network-based representation and analysis of biological systems contributes to a greater understanding of their structures and functions at different levels of complexity. These techniques can also be used to identify potential novel therapeutic targets based on the characterisation of vulnerable or highly influential network components. There is a need to investigate methods for estimating the impact of molecular perturbations. The prediction of high-impact or critical targets can aid in the identification of novel strategies for controlling the level of activation of specific, therapeutically relevant genes or proteins. Here, we report a new computational strategy for the analysis of the vulnerability of cellular signalling networks based on the quantitative assessment of the impact of large-scale, dynamic perturbations. To show the usefulness of this methodology, two complex signalling networks were analysed: the caspase-3 and the adenosine-regulated calcium signalling systems. This allowed us to estimate and rank the perturbation impact of the components defining these networks. Testable hypotheses about how these targets could modify the dynamic operation of the systems are provided. In the case of the caspase-3 system, the predictions and rankings were in line with results obtained from previous experimental validations of computational predictions generated by a relatively more computationally complex technique. In the case of the adenosine-regulated calcium system, we offer new testable predictions on the potential effect of different targets on the control of calcium flux. Unlike previous methods, the proposed approach provides perturbation-specific scores for each network component. The proposed perturbation assessment methodology may be applied to other systems to gain a deeper understanding of their dynamic operation and to assist the discovery of new therapeutic targets and strategies.     

Non-monotonic Response to Monotonic Stimulus: Regulation of Glyoxylate Shunt Gene-Expression Dynamics in Mycobacterium tuberculosis

Understanding how dynamical responses of biological networks are constrained by underlying network topology is one of the fundamental goals of systems biology. Here we employ monotone systems theory to formulate a theorem stating necessary conditions for non-monotonic time-response of a biochemical network to a monotonic stimulus. We apply this theorem to analyze the non-monotonic dynamics of the σ^B-regulated glyoxylate shunt gene expression in Mycobacterium tuberculosis cells exposed to hypoxia. We first demonstrate that the known network structure is inconsistent with observed dynamics. To resolve this inconsistency we employ the formulated theorem, modeling simulations and optimization along with follow-up dynamic experimental measurements. We show a requirement for post-translational modulation of σ^B activity in order to reconcile the network dynamics with its topology. The results of this analysis make testable experimental predictions and demonstrate wider applicability of the developed methodology to a wide class of biological systems.     

Perspective: network-guided pattern formation of neural dynamics

The understanding of neural activity patterns is fundamentally linked to an understanding of how the brain''s network architecture shapes dynamical processes. Established approaches rely mostly on deviations of a given network from certain classes of random graphs. Hypotheses about the supposed role of prominent topological features (for instance, the roles of modularity, network motifs or hierarchical network organization) are derived from these deviations. An alternative strategy could be to study deviations of network architectures from regular graphs (rings and lattices) and consider the implications of such deviations for self-organized dynamic patterns on the network. Following this strategy, we draw on the theory of spatio-temporal pattern formation and propose a novel perspective for analysing dynamics on networks, by evaluating how the self-organized dynamics are confined by network architecture to a small set of permissible collective states. In particular, we discuss the role of prominent topological features of brain connectivity, such as hubs, modules and hierarchy, in shaping activity patterns. We illustrate the notion of network-guided pattern formation with numerical simulations and outline how it can facilitate the understanding of neural dynamics.     

Cooperative differentiation through clustering in multicellular populations

The coordinated development of multicellular organisms is driven by intercellular communication. Differentiation into diverse cell types is usually associated with the existence of distinct attractors of gene regulatory networks, but how these attractors emerge from cell-cell coupling is still an open question. In order to understand and characterize the mechanisms through which coexisting attractors arise in multicellular systems, here we systematically investigate the dynamical behavior of a population of synthetic genetic oscillators coupled by chemical means. Using bifurcation analysis and numerical simulations, we identify various attractors and attempt to deduce from these findings a way to predict the organized collective behavior of growing populations. Our results show that dynamical clustering is a generic property of multicellular systems. We argue that such clustering might provide a basis for functional differentiation and variability in biological systems.     

Mutual Information Rate and Bounds for It

The amount of information exchanged per unit of time between two nodes in a dynamical network or between two data sets is a powerful concept for analysing complex systems. This quantity, known as the mutual information rate (MIR), is calculated from the mutual information, which is rigorously defined only for random systems. Moreover, the definition of mutual information is based on probabilities of significant events. This work offers a simple alternative way to calculate the MIR in dynamical (deterministic) networks or between two time series (not fully deterministic), and to calculate its upper and lower bounds without having to calculate probabilities, but rather in terms of well known and well defined quantities in dynamical systems. As possible applications of our bounds, we study the relationship between synchronisation and the exchange of information in a system of two coupled maps and in experimental networks of coupled oscillators.     

Spatially Organized Dynamical States in Chemical Oscillator Networks: Synchronization,Dynamical Differentiation,and Chimera Patterns

Dynamical processes in many engineered and living systems take place on complex networks of discrete dynamical units. We present laboratory experiments with a networked chemical system of nickel electrodissolution in which synchronization patterns are recorded in systems with smooth periodic, relaxation periodic, and chaotic oscillators organized in networks composed of up to twenty dynamical units and 140 connections. The reaction system formed domains of synchronization patterns that are strongly affected by the architecture of the network. Spatially organized partial synchronization could be observed either due to densely connected network nodes or through the ‘chimera’ symmetry breaking mechanism. Relaxation periodic and chaotic oscillators formed structures by dynamical differentiation. We have identified effects of network structure on pattern selection (through permutation symmetry and coupling directness) and on formation of hierarchical and ‘fuzzy’ clusters. With chaotic oscillators we provide experimental evidence that critical coupling strengths at which transition to identical synchronization occurs can be interpreted by experiments with a pair of oscillators and analysis of the eigenvalues of the Laplacian connectivity matrix. The experiments thus provide an insight into the extent of the impact of the architecture of a network on self-organized synchronization patterns.     

Diversity of temporal correlations between genes in models of noisy and noiseless gene networks

Gene regulatory networks (GRNs) are parallel information processing systems, binding past events to future actions. Since cell types stably remain in restricted subsets of the possible states of the GRN, they are likely the dynamical attractors of the GRN. These attractors differ in which genes are active and in the amount of information propagating within the network. Using mutual information (I) as a measure of information propagation between genes in a GRN, modeled as finite-sized Random Boolean Networks (RBN), we study how the dynamical regime of the GRN affects I within attractors (I(A)). The spectra of I(A) of individual RBNs are found to be scattered and diverse, and distributions of I(A) of ensembles are non-trivial and change shape with mean connectivity. Mean and diversity of I(A) values maximize in the chaotic near-critical regime, whereas ordered near-critical networks are the best at retaining the distinctiveness of each attractor's I(A) with noise. The results suggest that selection likely favors near-critical GRNs as these both maximize mean and diversity of I(A), and are the most robust to noise. We find similar I(A) distributions in delayed stochastic models of GRNs. For a particular stochastic GRN, we show that both mean and variance of I(A) have local maxima as its connectivity and noise levels are varied, suggesting that the conclusions for the Boolean network models may be generalizable to more realistic models of GRNs.     

Stimulus-induced transitions between spike-wave discharges and spindles with the modulation of thalamic reticular nucleus

It is believed that thalamic reticular nucleus (TRN) controls spindles and spike-wave discharges (SWD) in seizure or sleeping processes. The dynamical mechanisms of spatiotemporal evolutions between these two types of activity, however, are not well understood. In light of this, we first use a single-compartment thalamocortical neural field model to investigate the effects of TRN on occurrence of SWD and its transition. Results show that the increasing inhibition from TRN to specific relay nuclei (SRN) can lead to the transition of system from SWD to slow-wave oscillation. Specially, it is shown that stimulations applied in the cortical neuronal populations can also initiate the SWD and slow-wave oscillation from the resting states under the typical inhibitory intensity from TRN to SRN. Then, we expand into a 3-compartment coupled thalamocortical model network in linear and circular structures, respectively, to explore the spatiotemporal evolutions of wave states in different compartments. The main results are: (i) for the open-ended model network, SWD induced by stimulus in the first compartment can be transformed into sleep-like slow UP-DOWN and spindle states as it propagates into the downstream compartments; (ii) for the close-ended model network, weak stimulations performed in the first compartment can result in the consistent experimentally observed spindle oscillations in all three compartments; in contrast, stronger periodic single-pulse stimulations applied in the first compartment can induce periodic transitions between SWD and spindle oscillations. Detailed investigations reveal that multi-attractor coexistence mechanism composed of SWD, spindles and background state underlies these state evolutions. What’s more, in order to demonstrate the state evolution stability with respect to the topological structures of neural network, we further expand the 3-compartment coupled network into 10-compartment coupled one, with linear and circular structures, and nearest-neighbor (NN) coupled network as well as its realization of small-world (SW) topology via random rewiring, respectively. Interestingly, for the cases of linear and circular connetivities, qualitatively similar results were obtained in addition to the more irregularity of firings. However, SWD can be eventually transformed into the consistent low-amplitude oscillations for both NN and SW networks. In particular, SWD evolves into the slow spindling oscillations and background tonic oscillations within the NN and SW network, respectively. Our modeling and simulation studies highlight the effect of network topology in the evolutions of SWD and spindling oscillations, which provides new insights into the mechanisms of cortical seizures development.     

Impact of the topology of global macroeconomic network on the spreading of economic crises

Throughout economic history, the global economy has experienced recurring crises. The persistent recurrence of such economic crises calls for an understanding of their generic features rather than treating them as singular events. The global economic system is a highly complex system and can best be viewed in terms of a network of interacting macroeconomic agents. In this regard, from the perspective of collective network dynamics, here we explore how the topology of the global macroeconomic network affects the patterns of spreading of economic crises. Using a simple toy model of crisis spreading, we demonstrate that an individual country's role in crisis spreading is not only dependent on its gross macroeconomic capacities, but also on its local and global connectivity profile in the context of the world economic network. We find that on one hand clustering of weak links at the regional scale can significantly aggravate the spread of crises, but on the other hand the current network structure at the global scale harbors higher tolerance of extreme crises compared to more "globalized" random networks. These results suggest that there can be a potential hidden cost in the ongoing globalization movement towards establishing less-constrained, trans-regional economic links between countries, by increasing vulnerability of the global economic system to extreme crises.     

Computational models of signalling networks for non-linear control

Artificial signalling networks (ASNs) are a computational approach inspired by the signalling processes inside cells that decode outside environmental information. Using evolutionary algorithms to induce complex behaviours, we show how chaotic dynamics in a conservative dynamical system can be controlled. Such dynamics are of particular interest as they mimic the inherent complexity of non-linear physical systems in the real world. Considering the main biological interpretations of cellular signalling, in which complex behaviours and robust cellular responses emerge from the interaction of multiple pathways, we introduce two ASN representations: a stand-alone ASN and a coupled ASN. In particular we note how sophisticated cellular communication mechanisms can lead to effective controllers, where complicated problems can be divided into smaller and independent tasks.     

Territorial spatial vulnerability assessment based on PSO-BP neural network: A case study in Shenzhen,China

Studies on vulnerability, a fairly new research paradigm inspired by global environmental change, are expanding to support scientific decision-making for developing sustainable, resilient territorial space at the regional level. The goal of this research is to reveal the vulnerability of a human-environment coupled system and its interaction mechanisms amidst a changing environment. This study started with the vulnerability scoping diagram (VSD) conceptual model and built up a research framework of territorial spatial vulnerability according to the three structural elements of ecological protection, agricultural production and urban construction. Following the requirements for intelligent computing, we introduced particle swarm optimization and BP neural network learning algorithms to evaluate the vulnerability of Shenzhen's territory. We also introduced flow cytometry to analyze the mechanism of Shenzhen's territorial spatial vulnerability driven by human and natural forces. The results showed that the territorial spatial vulnerability of Shenzhen was low under various functional orientations. Among them, ecological protection-oriented territorial spatial vulnerability was high in the west and low in the east and followed an inverted U-shaped trend in the north-south direction. Agricultural production-oriented territorial spatial vulnerability was high in the east and low in the west, high in the north and low in the south. Urban construction-oriented territorial spatial vulnerability was high in the west and low in the east, high in the north and low in the south. Under the complex orientation of the gigantic human-environment coupled system, the vulnerability of territory was symmetrical and balanced in the north-south direction and fluctuated with an overall declining trend in the east-west direction. In vulnerability, ecological protection, agricultural production and urban construction systems independently accounted for 35.642%, 38.209%, and 26.149% of the territorial spatial vulnerability, respectively.