Time-delayed Interactions Research Articles

Nash equilibrium of time-delay interaction complex networks subject to persistent disturbances

In this study, the authors consider the problem of Nash equilibrium solutions of multi-agent game of information warfare in complex networks, of which the inner dynamics are nonlinear, the interactions among vertex states are with time delays, and vertex states are also with persistent disturbances. A successive approximation approach is used to obtain an optimal control law for different agents in the network systems. An illustration example is presented to demonstrate the characteristic of multi-agent game in networks.

Engineering the synchronization of neuron action potentials using global time-delayed feedback stimulation.

We experimentally demonstrate the use of continuous, time-delayed, feedback stimulation for controlling the synchronization of neuron action potentials. Phase-based models were experimentally constructed from a single synaptically isolated cultured hippocampal neuron. These models were used to determine the stimulation parameters necessary to produce the desired synchronization behavior in the action potentials of a pair of neurons coupled through a global time-delayed interaction. Measurements made using a dynamic clamp system confirm the generation of the synchronized states predicted by the experimentally constructed phase model. This model was then utilized to extrapolate the feedback stimulation parameters necessary to disrupt the action potential synchronization of a large population of globally interacting neurons.

Neural network reconstruction using kinetic Ising models with memory

Ising models with simple Markov chain kinetics have been recently introduced as a tool for inferring asymmetric interactions in neuronal networks [1]. In such an approach, one discretizes time and uses the spike pattern at time step t to predict the pattern at time step t+1 and infer the effective interaction between neurons J(i,j) which influences these dynamics. It is however a priori hard to justify that the effect of spikes in only one time bin from the temporal discretization determines the future state of the system. What happens if we use shorter/longer time bins than the characteristic time steps of the network? How do the inferred couplings change if we allow for interactions with memory of multiple past time steps? To answer these questions, we extend the kinetic Ising approach to higher-order Markov chains by introducing time-delayed interactions using (1) a set of couplings derived from scaling the original J(i,j) and (2) an auxiliary set of couplings K(i,j). A model of this sort is closely related to the Generalized Linear Model and its simplicity allows for detailed analysis of the model parameters. We apply this extended kinetic Ising model to two types of data sets: (1) a realizable case of randomly-connected Ising network with memory of past states and (2) a realistic cortical network simulation with cross-correlated Hodgkin-Huxley dynamics [2]. In the first case, we test the accuracy from assuming simple Markov chain kinetics to reconstruct higher-order networks using an exact iterative algorithm and mean-field approximations, such as in [1]. In the second case, we aim to identify true synaptic connections in a sparse network. The results show that, not surprisingly, the quality of inferring connections sharply decays at short and long time bins. The effect of adding memory is different depending on the time bin size and there is an optimal combination of time bin and memory, indicating strong interactions at specific time delays. The left panel in Figure ​Figure11 illustrates connections between 30 neurons out of the original network composed of 500 excitatory neurons and 500 inhibitory neurons. The inferred network using memory is shown in the middle panel. To have stable balanced states, the excitatory connections are somewhat weaker than the inhibitory ones and are harder to detect using a one-step Markov chain. The time-delayed interactions K(i,j) particularly improve the identification of these weak excitatory connections; see the right panel of Figure ​Figure1.1. Furthermore, K(i,j) is in general linearly dependent on J(i,j) when this connection in the original network is excitatory, but they are less correlated when the connection is inhibitory. This demonstrates that the network has multiple characteristic dynamics which cannot be explained by simple kinetic Ising models. Figure 1 Left: Original network with excitatory (red) and inhibitory connections (blue). Middle: Inferred connectivity using kinetic Ising models with memory. Colors mark their connection strengths. Right: Radio operating characteristics (ROC) curves of inferred ... The authors would like to thank John Hertz for providing the key simulations for the cortical column [2].

Role of local network oscillations in resting-state functional connectivity

Spatio-temporally organized low-frequency fluctuations (<0.1 Hz), observed in BOLD fMRI signal during rest, suggest the existence of underlying network dynamics that emerge spontaneously from intrinsic brain processes. Furthermore, significant correlations between distinct anatomical regions-or functional connectivity (FC)-have led to the identification of several widely distributed resting-state networks (RSNs). This slow dynamics seems to be highly structured by anatomical connectivity but the mechanism behind it and its relationship with neural activity, particularly in the gamma frequency range, remains largely unknown. Indeed, direct measurements of neuronal activity have revealed similar large-scale correlations, particularly in slow power fluctuations of local field potential gamma frequency range oscillations. To address these questions, we investigated neural dynamics in a large-scale model of the human brain's neural activity. A key ingredient of the model was a structural brain network defined by empirically derived long-range brain connectivity together with the corresponding conduction delays. A neural population, assumed to spontaneously oscillate in the gamma frequency range, was placed at each network node. When these oscillatory units are integrated in the network, they behave as weakly coupled oscillators. The time-delayed interaction between nodes is described by the Kuramoto model of phase oscillators, a biologically-based model of coupled oscillatory systems. For a realistic setting of axonal conduction speed, we show that time-delayed network interaction leads to the emergence of slow neural activity fluctuations, whose patterns correlate significantly with the empirically measured FC. The best agreement of the simulated FC with the empirically measured FC is found for a set of parameters where subsets of nodes tend to synchronize although the network is not globally synchronized. Inside such clusters, the simulated BOLD signal between nodes is found to be correlated, instantiating the empirically observed RSNs. Between clusters, patterns of positive and negative correlations are observed, as described in experimental studies. These results are found to be robust with respect to a biologically plausible range of model parameters. In conclusion, our model suggests how resting-state neural activity can originate from the interplay between the local neural dynamics and the large-scale structure of the brain.

Stochastic dynamics ofNbistable elements with global time-delayed interactions: Towards an exact solution of the master equations for finiteN

We consider a network of N noisy bistable elements with global time-delayed couplings. In a two-state description, where elements are represented by Ising spins, the collective dynamics is described by an infinite hierarchy of coupled master equations which was solved at the mean-field level in the thermodynamic limit. When the number of elements is finite, as is the case in actual laser networks, an analytical description was deemed so far intractable and numerical studies seemed to be necessary. In this paper we consider the case of two interacting elements and show that a partial analytical description of the stationary state is possible if the stochastic process is time symmetric. This requires some relationship between the transition rates to be satisfied.

Effect of time delay on the onset of synchronization of the stochastic Kuramotomodel

We consider the Kuramoto model of globally coupled phase oscillatorswith time-delayed interactions, that is subject to Ornstein–Uhlenbeck(Gaussian) colored or non-Gaussian colored noise. We investigate numericallythe interplay between the influences of the finite correlation time of noiseτ and thetime delay τd on the onset of the synchronization process. The cases for identical and nonidentical oscillatorshave both been considered. Among the obtained results for identical oscillators is a large increaseof the synchronization threshold as a function of time delay for the colored non-Gaussiannoise compared to the case of the colored Gaussian noise at low noise correlation timeτ. However, the difference reduces remarkably for large noise correlation times. For the caseof nonidentical oscillators, the incoherent state may become unstable around the maximumvalue of the threshold (as a function of time delay) even at lower coupling strength valuesin the presence of colored noise as compared to the noiseless case. We have studiedthe dependence of the critical value of the coupling strength (the threshold ofsynchronization) on given parameters of the stochastic Kuramoto model in great detailand presented results for possible cases of colored Gaussian and non-Gaussiannoises.

Amplitude death in nonlinear oscillators with nonlinear coupling

Amplitude death is the cessation of oscillations that occurs in coupled nonlinear systems when fixed points are stabilized as a consequence of the interaction. We show here that this phenomenon is very general: it occurs in nonlinearly coupled systems in the absence of parameter mismatch or time delay although time-delayed interactions can enhance the effect. Application is made to synaptically coupled model neurons, nonlinearly coupled Rössler oscillators, as well as to networks of nonlinear oscillators with nonlinear coupling. By suitably designing the nonlinear coupling, arbitrary steady states can be stabilized.

Efficient transfer and concentration of energy between explosive dual bubbles via time-delayed interactions

The dynamics of a high heat flux thermal bubble is constrained by the thermal energy carried on the bubble surface right after the bubble formation because of thermal isolation of vapor. This article proposes a way by assigning time delays between dual bubbles to transfer effectively energy from one bubble into the other, thus, breaks energy limitation that one single bubble can usually carry. Experiment result has demonstrated that the useful work as large as 40% can be transferred from one bubble into the other for the ignition time delay set between 2 and 3 μs in a dual bubble system. At the same time, the total extractable useful work in a dual bubble system is 20% higher than twice that of a single-bubble system with the same input heat energy. This phenomenon opens up a new way to transfer or concentrate energies from distributed energy sources with limit energy density into a much higher one for higher power application.

시간 지연 상호 연계를 가진 비선형 시스템의 분산 적응 제어: 지능적인 접근법

A decentralized adaptive control method is proposed for large-scale systems with unknown time-delayed nonlinear interconnections unmatched in control inputs. It is assumed that the time-delayed interaction terms are bounded by unknown nonlinear bounding functions. The nonlinear bounding functions and uncertain nonlinear functions of large-scale systems are compensated by the function approximation technique using neural networks. The dynamic surface control method is extended to design the proposed memoryless local controller for each subsystem of uncertain nonlinear large-scale time delay systems. Therefore, although the interconnected systems consist of a large number of subsystems, the proposed controller can be designed simply. We prove that all the signals in the total closed-loop system are semiglobally uniformly bounded and the control errors converge to an adjustable neighborhood of the origin. Finally, an example is given to demonstrate the effectiveness and applicability of the proposed scheme.

Detecting robust time-delayed regulation in Mycobacterium tuberculosis

BackgroundTime delays are often found in gene regulation though most techniques of building gene regulatory networks are not capable of capturing such phenomena. Here we look at the delays in the DNA repair system of Mycobacterium tuberculosis which is unusually slow in the bacteria. We propose a method based on a skip-chain model to study this phenomena in gene networks. The Viterbi paths of the underlying Markov chains find the most likely regulatory interactions among genes, taking care of very long delays. Using the derived networks, we discuss the delayed regulations and robustness of the DNA damage seen in the bacterium.ResultsWe evaluated our method on time-course gene expressions after DNA damage with Mitocyin C. Several time-delayed interactions were observed with our analysis. The presence of hubs in the networks indicates that a small number of transcriptional factors regulate the rest of the system. We demonstrate the use of priors to overcome over-fitting problem in the generation of networks. We compare our results with the gene networks derived with dynamic Bayesian networks (DBN).ConclusionDifferent transcription networks are active at different stages, and constant feedback and regulation is maintained throughout the activities of a biological pathway. Skip-chain models are capable of capturing, long distant and the time-delayed regulations. Use of a Dirichlet prior over parameters and Gibbs prior over structure can greatly reduce the over-fitting in the new model.

Looking at map networks with time delay interactions from a local perspective

The evolution of networks of coupled chaotic maps with delayed interactions can be studied in the usual way by analyzing the evolution of the state of elements at each iteration time (the “Simulator” point of view), or it can be analyzed from the point of view of a single element (the “Observer” perspective) that is receiving delayed information from the other elements in the system. In the usual “Simulator” timeframe, an absolute time (i.e., the number of iterations t) is adopted to define the system state at each time t. In the “Observer” framework, the state of an element in the system is given by its state at the “Simulator” time t - τ , where τ is the information travel time between that element and the Observer. We emphasize the convenience of the analysis of the system dynamics in the “Observer” timeframe by showing that the “Observer” dynamics differs substantially from the one in the “Simulator” timeframe, and that the system dynamics in the “Observer” timeframe reflects the proper causality of the interactions among the elements.

Unequal autonegative feedback by GH models the sexual dimorphism in GH secretory dynamics.

Growth hormone (GH) secretion, controlled principally by a GH-releasing hormone (GHRH) and GH release-inhibiting hormone [somatostatin (SRIF)] displays vivid sexual dimorphism in many species. We hypothesized that relatively small differences within a dynamic core GH network driven by regulatory interactions among GH, GHRH, and SRIF explain the gender contrast. To investigate this notion, we implemented a minimal biomathematical model based on two coupled oscillators: time-delayed reciprocal interactions between GH and GHRH, which endow high-frequency (40-60 min) GH oscillations, and time-lagged bidirectional GH-SRIF interactions, which mediate low-frequency (occurring every 3.3 h) GH volleys. We show that this basic formulation, sufficient to explain GH dynamics in the male rat [Farhy LS, Straume M, Johnson ML, Kovatchev BP, and Veldhuis JD. Am J Physiol Regulatory Integrative Comp Physiol 281: R38-R51, 2001], emulates the female pattern of GH release, if autofeedback of GH on SRIF is relaxed. Relief of GH-stimulated SRIF release damps the slower volleylike oscillator, allowing emergence of the underlying high-frequency oscillations that are sustained by the GH-GHRH interactions. Concurrently, increasing variability of basal somatostatin outflow introduces quantifiable, sex-specific disorderliness of the release process typical of female GH dynamics. Accordingly, modulation of GH autofeedback on SRIF within the interactive GH-GHRH-SRIF ensemble and heightened basal SRIF variability are sufficient to transform the well-ordered, 3.3-h-interval, multiphasic, volleylike male GH pattern into a femalelike profile with irregular pulses of higher frequency.

Neural networks to model dynamic systems with time delays

An algorithm is proposed to identify a neural network model that represents a nonlinear dynamic system with a multivariate time delay response. The algorithm consists of two major parts. The first one identifies the time delay vector for a given neural network structure. This task is accomplished by using an exhaustive integer enumeration algorithm that minimizes a statistical parameter to assess the performance of the neural network model. The second part uses a cross-validation strategy to identify the best neural network model. Since the structure that models a nonlinear system is usually unknown, the identification strategy consists of selecting several neural network structures and identifying the best time delay vector for each network. The modeling process starts with the simplest structure and progressively the complexity of the network is increased to end up with a complex structure. Finally, the network that offers the simplest structure with the best network performance is the one that exhibits the appropriate neural network structure with the corresponding optimal time delay vector. The Monte Carlo simulation technique was used to test the performance of the algorithm under the presence of linear and nonlinear relationships among several variables of dynamic systems and with a different time delay applied to each input variable. The introduced algorithm is used to detect a chemical reaction delay among enriched amyl acetate, acetic acid, water, and the pH of erythromycin sail. An appropriate neural network model was designed to model the pH of the erythromycin during a continuous extraction process. To the best of the authors knowledge the proposed algorithm is the only one currently available to identify time delay interactions in the multivariate input output variables of a system. The major drawback of the introduced algorithm is that it becomes very slow as the number of system inputs increases. This algorithm works efficiently in a system that involves five inputs or less.

Spontaneous phase oscillation induced by inertia and time delay

We consider a system of coupled oscillators with finite inertia and time-delayed interaction, and investigate the interplay between inertia and delay both analytically and numerically. The phase velocity of the system is examined; revealed in numerical simulations is the emergence of spontaneous phase oscillation without external driving, which turns out to be in good agreement with analytical results derived in the strong-coupling limit. Such self-oscillation is found to suppress synchronization, and its frequency is observed to decrease with inertia and delay. We obtain the phase diagram, which displays oscillatory and stationary phases in the appropriate regions of the parameters.

Pulse dynamics in a reaction–diffusion system

We formulate a theory for pulse dynamics in an excitable reaction–diffusion system not only in one dimension but also in higher dimensions. In the singular limit where the width of pulse boundaries is infinitesimally thin, we derive the equation of motion for a pair of interacting pulses (spots in higher dimensions). This equation exhibits a bifurcation that a motionless pulse loses stability and begins to propagate. An inertia term appears which originates from a time-delayed interaction between pulses mediated by slow diffusion of the inhibitor. By employing the analogy of particle motion in a conserved system, we can clarify the mechanism that the pulses undergo an elastic-like collision in a special parameter regime as observed by computer simulations even though the system is purely dissipative.

Globally coupled maps with time delay interactions

We analyze the dynamical behavior of globally coupled logistic maps with time-delayed interaction. It is found that certain properties of time-delay coupling could lead to the clustering at low couplings and the break-up of clusters (desynchronization) in the strong coupling regimes. It is also demonstrated that the system of two coupled logistic maps may exhibit multistability and on–off intermittency for properly chosen delay times.

Effects of time-delayed interactions on dynamic patterns in a coupled phase oscillator system.

We investigate the effects of time delayed interactions in the network of neural oscillators. We perform the stability analysis in the vicinity of a synchronized state at vanishing time delay and present a related phase diagram. In the simulations it is shown that time delay induces various phenomena such as clustering where the system is spontaneously split into two phase locked groups, synchronization, and multistability. Time delay effects should be considered both in the natural and artificial neural systems whose information processing is based on the spatiotemporal dynamic patterns.

Time Delay in the Kuramoto Model of Coupled Oscillators

We generalize the Kuramoto model of coupled oscillators to allow time-delayed interactions. New phenomena include bistability between synchronized and incoherent states, and unsteady solutions with time-dependent order parameters. We derive exact formulas for the stability boundaries of the incoherent and synchronized states, as a function of the delay, in the special case where the oscillators are identical. The experimental implications of the model are discussed for populations of chirping crickets, where the finite speed of sound causes communication delays, and for physical systems such as coupled phase-locked loops or lasers.

Associative memory based on synchronized firing of spiking neurons with time-delayed interactions

We study associative memory of a neural network of spiking neurons with time-delayed synaptic interactions incorporating the time taken by an action potential to propagate along the axon. Individual spiking neurons are described by a set of nonlinear differential equations capable of exhibiting excitability such as that of Hodgkin-Huxley and FitzHugh neurons. When a simple learning rule of the autocorrelation type based on random patterns is assumed, memory retrieval is shown to be accompanied by synchronized firing of neurons. The reduced dynamics with a few degrees of freedom of the network with a finite number of stored patterns is analytically derived in the limit of infinitely many neurons. The dependence of the appearance of retrieval states on the distribution of time delay and on the size of refractory period given implicitly in the model is obtained, showing good agreement between the result of numerical simulations and that obtained from the reduced dynamics. The behavior of the network with an extensive number of patterns is also investigated and an approximate analysis is presented to discuss the storage capacity.

Dynamics of Globally Coupled Rotators with Multiplicative Noise

We study the dynamics of globally coupled rotator model with multiplicative noise concentrating on time-delay effect. It is shown that at a critical noise intensity the system undergoes a noise-induced transition and is split into clusters. Time-delayed interaction in the system affects the picture of the transition suppressing the clustering of the rotators and generates the switching phenomenon between the clusters. For some values of delay time the system has two stable steady states which show different dynamics. According to the initial condition the system evolves into one of the stable steady states. We discuss the nature of the transition and the switching phenomenon in the system.