Improved functional-weight approach to oscillatory patterns in excitable networks

Studies of sustained oscillations on complex networks with excitable node dynamics received much interest in recent years. Although an individual unit is non-oscillatory, they may organize to form various collective oscillatory patterns through networked connections. An excitable network usually possesses a number of oscillatory modes dominated by different Winfree loops and numerous spatiotemporal patterns organized by different propagation path distributions. The traditional approach of the so-called dominant phase-advanced drive method has been well applied to the study of stationary oscillation patterns on a network. In this paper, we develop the functional-weight approach that has been successfully used in studies of sustained oscillations in gene-regulated networks by an extension to the high-dimensional node dynamics. This approach can be well applied to the study of sustained oscillations in coupled excitable units. We tested this scheme for different networks, such as homogeneous random networks, small-world networks, and scale-free networks and found it can accurately dig out the oscillation source and the propagation path. The present approach is believed to have the potential in studies competitive non-stationary dynamics.     

Oscillatory and anti-oscillatory motifs in genetic regulatory networks

Recently,self-sustained oscillatory genetic regulatory networks(GRNs) have attracted significant attention in the biological field.Given a GRN,it is important to anticipate whether the network could generate oscillation with proper parameters,and what the key ingredients for the oscillation are.In this paper the ranges of some function-related parameters which are favorable to sustained oscillations are considered.In particular,some oscillatory motifs appearing with high-frequency in most of the oscillatory GRNs are observed.Moreover,there are some anti-oscillatory motifs which have a strong oscillation repressing effect.Some conclusions analyzing these motif effects and constructing oscillatory GRNs are provided.     

Evolutionary design of oscillatory genetic networks

The present study is devoted to the design and statistical investigations of dynamical gene expression networks. In our model problem, we aim to design genetic networks which would exhibit stable periodic oscillations with a prescribed temporal period. While no rational solution of this problem is available, we show that it can be effectively solved by running a computer evolution of the network models. In this process, structural rewiring mutations are applied to the networks with inhibitory interactions between genes and the evolving networks are selected depending on whether, after a mutation, they closer approach the targeted dynamics. We show that, by using this method, networks with required oscillation periods, varying by up to three orders of magnitude, can be constructed by changing the architecture of regulatory connections between the genes. Statistical properties of designed networks, including motif distributions and Laplacian spectra, are considered.     

Topological origin of global attractors in gene regulatory networks

Fixed-point attractors with global stability manifest themselves in a number of gene regulatory networks. This property indicates the stability of regulatory networks against small state perturbations and is closely related to other complex dynamics. In this paper, we aim to reveal the core modules in regulatory networks that determine their global attractors and the relationship between these core modules and other motifs. This work has been done via three steps. Firstly, inspired by the signal transmission in the regulation process, we extract the model of chain-like network from regulation networks. We propose a module of “ideal transmission chain(ITC)”, which is proved sufficient and necessary(under certain condition) to form a global fixed-point in the context of chain-like network. Secondly, by examining two well-studied regulatory networks(i.e., the cell-cycle regulatory networks of Budding yeast and Fission yeast), we identify the ideal modules in true regulation networks and demonstrate that the modules have a superior contribution to network stability(quantified by the relative size of the biggest attraction basin). Thirdly, in these two regulation networks, we find that the double negative feedback loops, which are the key motifs of forming bistability in regulation, are connected to these core modules with high network stability. These results have shed new light on the connection between the topological feature and the dynamic property of regulatory networks.     

Dominant phase-advanced driving analysis of self-sustained oscillations in biological networks

Oscillatory behaviors can be ubiquitously observed in various systems. Biological rhythms are significant in governing living activities of all units. The emergence of biological rhythms is the consequence of large numbers of units. In this paper we discuss several important examples of sustained oscillations in biological media, where the unit composed in the system does not possess the oscillation behavior. The dominant phase-advanced driving method is applied to study the skeletons and oscillatory organizing motifs in excitable networks and gene regulatory networks.     

少节点基因调控网络的控制

近年来,自组织振荡网络受到越来越多科学家的关注,对生物体的生长、发育起调控作用的基因调控网络即是其中的一种.本文研究了少节点基因调控网络的控制问题.运用多相位超前驱动方法对该种网络进行调控,可以有效地提高对网络的控制效率.通过数值模拟,发现对于少节点基因调控网络,当系统参数确定时,网络的有效控制率可以达到95％以上(10节点网络);当系统参数不确定时,控制的效率也非常高.     

Oscillation sources and wave propagation paths in complex networks consisting of excitable nodes

Self-sustained oscillations in complex networks consisting of nonoscillatory nodes have attracted long-standing interest in diverse natural and social systems. We study the self-sustained periodic oscillations in random networks consisting of excitable nodes. We reveal the underlying dynamic structure by applying a dominant phase-advanced driving method. The oscillation sources and wave propagation paths can be illustrated clearly via the dynamic structure revealed. Then we are able to control the oscillations with surprisingly high efficiency based on our understanding.     

Transforming a complex network to an acyclic one

Acyclic networks are a class of complex networks in which links are directed and do not have closed loops. Here we present an algorithm for transforming an ordinary undirected complex network into an acyclic one. Further analysis of an acyclic network allows one to find the structural properties of the network. With our approach one can find the communities and key nodes in complex networks. Also we propose a new parameter of complex networks which can mark the most vulnerable nodes of the system. The proposed algorithm can be applied to finding communities and bottlenecks in general complex networks.     

Phase Transitions of an Epidemic Spreading Model in Small-World Networks

We propose a modified susceptible-infected-refractory-susceptible (SIRS) model to investigate the global oscillations of the epidemic spreading inWatts-Strogatz (WS) small-world networks. It is found that when an individual immunity does not change or decays slowly in an immune period, the system can exhibit complex transition from an infecting stationary state to a large amplitude sustained oscillation or an absorbing state with no infection. When the immunity decays rapidly in the immune period, the transition to the global oscillation disappears and there is no oscillation. Furthermore, based on thespatio-temporal evolution patterns and the phase diagram, it is disclosed that a long immunity period takes an important role in the emergence of the global oscillation in small-world networks.     

Attractive target wave patterns in complex networks consisting of excitable nodes

This review describes the investigations of oscillatory complex networks consisting of excitable nodes,focusing on the target wave patterns or say the target wave attractors.A method of dominant phase advanced driving(DPAD) is introduced to reveal the dynamic structures in the networks supporting oscillations,such as the oscillation sources and the main excitation propagation paths from the sources to the whole networks.The target center nodes and their drivers are regarded as the key nodes which can completely determine the corresponding target wave patterns.Therefore,the center(say node A) and its driver(say node B) of a target wave can be used as a label,(A,B),of the given target pattern.The label can give a clue to conveniently retrieve,suppress,and control the target waves.Statistical investigations,both theoretically from the label analysis and numerically from direct simulations of network dynamics,show that there exist huge numbers of target wave attractors in excitable complex networks if the system size is large,and all these attractors can be labeled and easily controlled based on the information given by the labels.The possible applications of the physical ideas and the mathematical methods about multiplicity and labelability of attractors to memory problems of neural networks are briefly discussed.     

Finite superconducting square wire-network based on two-dimensional crystalline Mo2C

Superconducting wire-networks are paradigms to study Cooper pairing issues, vortex dynamics and arrangements. Recently, emergent low-dimensional crystalline superconductors were reported in the minimal-disorder limit, providing novel platforms to reveal vortices-related physics. Study on superconducting loops with high-crystallinity is thus currently demanded. Here, we report fabrication and transport measurement of finite square-network based on two-dimensional crystalline superconductor Mo₂C. We observe oscillations in the resistance as a function of the magnetic flux through the loops. Resistance dips at both matching field and fractional fillings are revealed. Temperature and current evolutions are carried out in magnetoresistance to study vortex dynamics. The amplitude of oscillation is enhanced due to the interaction between thermally activated vortices and the currents induced in the loops. The driving current reduces the effective activation energy for vortex, giving rise to stronger vortex interaction. Moreover, by the thermally activated vortex creep model, we derive the effective potential barrier for vortex dissipation, which shows well-defined correspondence with structures in magnetoresistance. Our work shows that low-dimensional crystalline superconducting network based on Mo₂C possesses pronounced potential in studying the modulation of vortex arrangements and dynamics, paving the way for further investigations on crystalline superconducting network with various configurations.     

The birhythmicity increases the diversity of p53 oscillation induced by DNA damage

The tumor suppressor p53 mediates the cellular response to various stresses. It was experimentally shown that the concentration of p53 can show oscillations with short or long periods upon DNA damage. The underlying mechanism for this phenomenon is still not fully understood. Here, we construct a network model comprising the ATM-p53-Wip1 and p53-Mdm2 negative feedback loops and ATM autoactivation. We recapitulate the typical features of p53 oscillations including p53 birhythmicity. We show the dependence of p53 birhythmicity on various factors such as the phosphorylation status of ATM. We also perform stochastic simulation and find the noise-induced transitions between two modes of p53 oscillation,which increases the p53 variability in both the amplitude and period. These results suggest that p53 birhythmicity enhances the responsiveness of p53 network, which may facilitate its tumor suppressive function.     

Effect of the nonmagnetic layer in a Co/Cu/CoO trilayer structure on the exchange coupling in it

The dependence of the exchange bias of epitaxial single-crystal Co/Cu/CoO trilayer films on the copper layer thickness and temperature is studied. The exchange bias of the hysteresis loops of the ferromagnetic cobalt layer as a function of the copper layer thickness is found to have a well-pronounced oscillating character. The oscillations manifest themselves over the entire temperature range in which an exchange bias takes place (77–220 K). The complex variation of the oscillation amplitude with the nonmagnetic layer thickness can be explained by the superposition of two interlayer exchange coupling oscillation periods (λ₁ ≈ 10–11 Å, λ₂ ≈ 20 Å) having differentamplitudes and temperature dependences.     

Modeling “preattention” and “attention” information processing by synchronization of neural activity

Oscillatory neural-network preattention and attention models are examined. A two-layer network of Wilson-Cowan oscillators is used to show that two-frequency oscillations can appear in response to a compound stimulus. It is shown that these oscillations can be synchronized at the low frequency, which can be interpreted as feature binding. Partial synchronization is studied in a model of a network of phased oscillators with a central element. Formulas are given for approximation of the average frequency of the central oscillator for small and large networks. The results describe the effect of a distracting stimulus on attention focus.Institute of Mathematical Problems of Biology, Russian Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 37, No. 8, pp. 933–944, August, 1994.     

Complex Networks: from Graph Theory to Biology

The aim of this text is to show the central role played by networks in complex system science. A remarkable feature of network studies is to lie at the crossroads of different disciplines, from mathematics (graph theory, combinatorics, probability theory) to physics (statistical physics of networks) to computer science (network generating algorithms, combinatorial optimization) to biological issues (regulatory networks). New paradigms recently appeared, like that of ‘scale-free networks’ providing an alternative to the random graph model introduced long ago by Erdös and Renyi. With the notion of statistical ensemble and methods originally introduced for percolation networks, statistical physics is of high relevance to get a deep account of topological and statistical properties of a network. Then their consequences on the dynamics taking place in the network should be investigated. Impact of network theory is huge in all natural sciences, especially in biology with gene networks, metabolic networks, neural networks or food webs. I illustrate this brief overview with a recent work on the influence of network topology on the dynamics of coupled excitable units, and the insights it provides about network emerging features, robustness of network behaviors, and the notion of static or dynamic motif.     

Nonlocal Mechanism for Synchronization of Time Delay Networks

We present the interplay between synchronization of networks with heterogeneous delays and the greatest common divisor (GCD) of loops composing the network. We distinguish between two types of networks; (I) chaotic networks and (II) population dynamic networks with periodic activity driven by external stimuli. For type (I), in the weak chaos region, the units of a chaotic network characterized by GCD=1 are in a chaotic zero-lag synchronization, whereas for GCD>1, the network splits into GCD-clusters in which clustered units are in zero-lag synchronization. These results are supported by simulations of chaotic systems, self-consistent and mixing arguments, as well as analytical solutions of Bernoulli maps. Type (II) is exemplified by simulations of Hodgkin Huxley population dynamic networks with unidirectional connectivity, synaptic noise and distribution of delays within neurons belonging to a node and between connecting nodes. For a stimulus to one node, the network splits into GCD-clusters in which cluster neurons are in zero-lag synchronization. For complex external stimuli, the network splits into clusters equal to the greatest common divisor of loops composing the network (spatial) and the periodicity of the external stimuli (temporal). The results suggest that neural information processing may take place in the transient to synchronization and imply a much shorter time scale for the inference of a perceptual entity.     

Bifurcation of synchronous oscillations into torus in a system of two reciprocally inhibitory silicon neurons: experimental observation and modeling

Oscillatory activity in the central nervous system is associated with various functions, like motor control, memory formation, binding, and attention. Quasiperiodic oscillations are rarely discussed in the neurophysiological literature yet they may play a role in the nervous system both during normal function and disease. Here we use a physical system and a model to explore scenarios for how quasiperiodic oscillations might arise in neuronal networks. An oscillatory system of two mutually inhibitory neuronal units is a ubiquitous network module found in nervous systems and is called a half-center oscillator. Previously we created a half-center oscillator of two identical oscillatory silicon (analog Very Large Scale Integration) neurons and developed a mathematical model describing its dynamics. In the mathematical model, we have shown that an in-phase limit cycle becomes unstable through a subcritical torus bifurcation. However, the existence of this torus bifurcation in experimental silicon two-neuron system was not rigorously demonstrated or investigated. Here we demonstrate the torus predicted by the model for the silicon implementation of a half-center oscillator using complex time series analysis, including bifurcation diagrams, mapping techniques, correlation functions, amplitude spectra, and correlation dimensions, and we investigate how the properties of the quasiperiodic oscillations depend on the strengths of coupling between the silicon neurons. The potential advantages and disadvantages of quasiperiodic oscillations (torus) for biological neural systems and artificial neural networks are discussed.     

Coupled Feedback Loops Involving PAGE4, EMT and Notch Signaling Can Give Rise to Non-Genetic Heterogeneity in Prostate Cancer Cells

Non-genetic heterogeneity is emerging as a crucial factor underlying therapy resistance in multiple cancers. However, the design principles of regulatory networks underlying non-genetic heterogeneity in cancer remain poorly understood. Here, we investigate the coupled dynamics of feedback loops involving (a) oscillations in androgen receptor (AR) signaling mediated through an intrinsically disordered protein PAGE4, (b) multistability in epithelial–mesenchymal transition (EMT), and (c) Notch–Delta–Jagged signaling mediated cell-cell communication, each of which can generate non-genetic heterogeneity through multistability and/or oscillations. Our results show how different coupling strengths between AR and EMT signaling can lead to monostability, bistability, or oscillations in the levels of AR, as well as propagation of oscillations to EMT dynamics. These results reveal the emergent dynamics of coupled oscillatory and multi-stable systems and unravel mechanisms by which non-genetic heterogeneity in AR levels can be generated, which can act as a barrier to most existing therapies for prostate cancer patients.     

Designer gene networks: Towards fundamental cellular control

The engineered control of cellular function through the design of synthetic genetic networks is becoming plausible. Here we show how a naturally occurring network can be used as a parts list for artificial network design, and how model formulation leads to computational and analytical approaches relevant to nonlinear dynamics and statistical physics. We first review the relevant work on synthetic gene networks, highlighting the important experimental findings with regard to genetic switches and oscillators. We then present the derivation of a deterministic model describing the temporal evolution of the concentration of protein in a single-gene network. Bistability in the steady-state protein concentration arises naturally as a consequence of autoregulatory feedback, and we focus on the hysteretic properties of the protein concentration as a function of the degradation rate. We then formulate the effect of an external noise source which interacts with the protein degradation rate. We demonstrate the utility of such a formulation by constructing a protein switch, whereby external noise pulses are used to switch the protein concentration between two values. Following the lead of earlier work, we show how the addition of a second network component can be used to construct a relaxation oscillator, whereby the system is driven around the hysteresis loop. We highlight the frequency dependence on the tunable parameter values, and discuss design plausibility. We emphasize how the model equations can be used to develop design criteria for robust oscillations, and illustrate this point with parameter plots illuminating the oscillatory regions for given parameter values. We then turn to the utilization of an intrinsic cellular process as a means of controlling the oscillations. We consider a network design which exhibits self-sustained oscillations, and discuss the driving of the oscillator in the context of synchronization. Then, as a second design, we consider a synthetic network with parameter values near, but outside, the oscillatory boundary. In this case, we show how resonance can lead to the induction of oscillations and amplification of a cellular signal. Finally, we construct a toggle switch from positive regulatory elements, and compare the switching properties for this network with those of a network constructed using negative regulation. Our results demonstrate the utility of model analysis in the construction of synthetic gene regulatory networks. (c) 2001 American Institute of Physics.     

Charged string loops in Reissner–Nordström black hole background

We study the motion of current carrying charged string loops in the Reissner–Nordström black hole background combining the gravitational and electromagnetic field. Introducing new electromagnetic interaction between central charge and charged string loop makes the string loop equations of motion to be non-integrable even in the flat spacetime limit, but it can be governed by an effective potential even in the black hole background. We classify different types of the string loop trajectories using effective potential approach, and we compare the innermost stable string loop positions with loci of the charged particle innermost stable orbits. We examine string loop small oscillations around minima of the string loop effective potential, and we plot radial profiles of the string loop oscillation frequencies for both the radial and vertical modes. We construct charged string loop quasi-periodic oscillations model and we compare it with observed data from microquasars GRO 1655-40, XTE 1550-564, and GRS 1915+105. We also study the acceleration of current carrying string loops along the vertical axis and the string loop ejection from RN black hole neighbourhood, taking also into account the electromagnetic interaction.