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Therapeutic agents acting on RNA, including RNA modification, RNA editing and RNA interference (RNAi), play a vital role in gene function study, signaling pathway, drug discovery, disease treatment, vaccine development and so on. Therein, RNAi as an emerging gene therapy has been widely applied in many cancer studies by silencing oncogenes or specific mRNA in malignant tumor cells. Mechanism and efficiency of RNAi are the key issues of RNAi technology. RNA silencing involves dynamic modeling, analyses and optimal control of RNAi gene system. Physiological delay and Hill response describing off-target are considerable elements involving RNAi efficiency. In this paper, we first formulate a four-dimensional RNAi model with time delays and Hill functions, and then investigate the complex dynamic behaviors including the number, existence and stability of internal equilibria and Hopf bifurcations of single delay and two delays. Furthermore, based on the specific mRNA degradation adopted in impulsive patterns, we build an optimal problem by adding exogenous dsRNA at alterable time points in variable dosages in a treatment session. By the method of gradient formula, we can find the optimal impulsive time and proportion of dsRNA. Finally, simulation indicates that (1) physiological lags not only raise the oscillations of mRNA but also cut down the levels of cost; (2) smaller delays and larger rates of siRNA–mRNA complex formation and dsRNA synthesis imply the rapid composition of RISC and fast synthesis of dsRNA leading to more desirable therapeutic schedule, which affords evidence for gene regulation and RNAi; (3) a larger half-saturation coefficient characterizes a unique and stable higher targeted mRNA, whereas a smaller half-saturation coefficient generates bistability in which the higher and lower targeted mRNAs simultaneously emerge; and (4) the bistability will provide a good guidance to control, suppress and degrade targeted mRNA.

In this paper, without assuming the boundedness, strict monotonicity and differentiability of the activation functions, we utilize a new Lyapunov function to analyze the global convergence of a class of neural networks models with time delays. A new sufficient condition guaranteeing the existence, uniqueness and global exponential stability of the equilibrium point is derived. This stability criterion imposes constraints on the feedback matrices independently of the delay parameters. The result is compared with some previous works. Furthermore, the condition may be less restrictive in the case that the activation functions are hyperbolic tangent.

In this paper a hopping robot motion with offset mass is discussed. A mathematical model has been considered and an efficient single layered neural network has been developed to suit to the dynamics of the hopping robot, which ensures guaranteed tracking performance leading to the stability of the otherwise unstable system. The neural network takes advantage of the robot regressor dynamics that expresses the highly nonlinear robot dynamics in a linear form in terms of the known and unknown robot parameters. Time delays in the control mechanism play a vital role in the motion of hopping robots. The present work also enables us to estimate the maximum time delay admissible with out losing the guaranteed tracking performance. Further this neural network does not require offline training procedures. The salient features are highlighted by appropriate simulations.

Understanding the timing requirements for evacuation of people has focused primarily on independent pedestrians rather than pedestrians emotionally connected. However, the main statistical effects observed in crowds, the so-called “faster is slower”, “clever is not always better” and the “low visibility enhancement”, cannot explain the overall behavior of a crowd during an evacuation process when correlated pedestrians due to, for example feelings, are present. Our research addresses this issue and examines the statistical behavior of a mixture of individuals and couples during a (panic) escaping process. We found that the attractive feeling among couples plays an important role in the time delays during the evacuation of a single exit room.

In this article, synchronization inside complex networks with double time-delays and nonlinear inner-coupling functions are studied. Here double time-delays mean that each node vector field and every coupling node have retard time, while nonlinear inner-coupling functions refer to all the components of every node that are nonlinearly coupled. The theoretical criterion respecting synchronization is derived. And illustrative numerical examples are also given.

Since time delays are very vital and inevitable in the transmission of neurons, this paper is mainly concerned with the dynamic analysis of the magnetic flux e-HR neuron model with time delays. Specifically, the stability of the system at the equilibrium point and the conditions for bifurcations are analyzed to explore the impact of time delays on dynamics behavior. Results show that the system exhibits Hopf bifurcation within the specific range of time delay. Moreover, the increase of time delay can induce the tip discharge behavior. In addition, we investigate the stability and bifurcation behavior of the model with dual time delays. Our results indicate that the system displays rich dynamics performance under certain conditions. Furthermore, numerical simulations show time delays play a crucial role in the dynamics process for the magnetic flux e-HR neuron model.

In this paper, we study the relaxation time of a bistable system driven by correlated noises when there are two different kinds of time delays in the deterministic (τ) and random (θ) forces, respectively. The expression for the relaxation time T_c is derived by the projection operator method, in which the effects of the memory kernels are taken into account. After introducing a dimensionless parameter R (R = D/α, where D is the strength of multiplicative noise and α is the strength of additive noise), and then performing numerical computations, we find the following: (1) for the case of R ≤ 1, the relaxation time T_c increases as τ increases, i.e. τ slows down the fluctuation decay of the dynamical variable for the case of R ≤ 1; (2) however, for the case of R > 1, T_c decreases as τ increases, i.e. τ enhances the fluctuation decay of the dynamical variable for the case of R > 1; (3) T_c decreases as θ increases, i.e. θ enhances the fluctuation decay of the dynamical variable for the three cases of R (R > 1, R=1, and R < 1).

In this paper, some criteria are derived for global robust asymptotic stability of a class of interval neural networks with multiple constant or time-varying delays via the Lyapunov–Krasovskii stability theory for functional differential equations and the linear matrix inequality (LMI) approach. The main results are applied to the uncertain systems with single delay, and several new criteria to determine the asymptotical convergence of these systems are obtained. Numerical examples are also given to show the effectiveness of our results.

In this paper, several simple and new conditions are presented to ensure existence, uniqueness and global exponential robust stability of the equilibrium point of neural networks with delays by using contraction mapping principle, homeomorphism property and Lyapunov method. The conditions given in this paper are easy to verify and apply, and the obtained results improve and extend to those in the literature. In addition, two examples are given to show the effectiveness of the proposed results.

In this paper, a two-neuron network with different time delays is investigated. By analyzing the associated characteristic equation, we obtain the conditions for delay-dependent and delay-independent asymptotic stability, respectively. Furthermore, we find that if the delay is used as a bifurcation parameter, Hopf bifurcation would occur. The direction and stability of the bifurcating periodic solutions are determined by using the Nyquist criterion and the graphical Hopf bifurcation theorem. Some examples are included to illustrate our results.

This paper concerns the computation and local stability analysis of periodic solutions to semi-explicit differential algebraic equations with time delays (delay DAEs) of index 1 and index 2. By presenting different formulations of delay DAEs, we motivate our choice of a direct treatment of these equations. Periodic solutions are computed by solving a periodic two-point boundary value problem, which is an infinite-dimensional problem for delay DAEs. We investigate two collocation methods based on piecewise polynomials: collocation at Radau IIA and Gauss–Legendre nodes. Using the obtained collocation equations, we compute an approximation to the Floquet multipliers which determine the local asymptotic stability of a periodic solution. Based on numerical experiments, we present orders of convergence for the computed solutions and Floquet multipliers and compare our results with known theoretical convergence results for initial value problems for delay DAEs. We end with examples on bifurcation analysis of delay DAEs.

There is great interest in methods for computing that do not involve digital machines. Many computational paradigms were inspired by brain research, such as Boolean neuronal logic [McCulloch & Pitts, 1943], the perceptron [Rosenblatt, 1958], attractor neural networks [Hopfield, 1982] and cellular neural nets [Chua & Yang, 1988]. All these paradigms abstract biological circuits to artificial neural networks, i.e. interconnected units (neurons) that perform computations based on the connections between the units (synapses). Here we present a novel computational framework based on polychronous wavefront dynamics. It is entirely different from an artificial neural network paradigm, rather it is based on temporal and spatial patterns of activity in pulse-propagating media and their interaction with transponders, which create pulses in response to receiving appropriate inputs, e.g. two coincident input pulses. A pulse propagates as a circular wave from its source to other transponders. Computations result from interactions between transponders, and they are encoded by the exact physical locations of transponders and by precise timings of pulses. We illustrate temporal pattern recognition, reverberating memory, temporal signal analysis and basic logical operations using polychronous wavefront computations. This work reveals novel principles for designing nanoscale computational devices.

In this paper, we discuss in detail the local stability, Hopf bifurcation and pitchfork bifurcation of an inertial two-neuron system with time delays by applying the second-order approximation approach and analyzing the associated transcendental equation, respectively. Comparison has been made between the two methods. Numerical results have been presented to verify the analytical predictions. Specially, numerical examples show the time-varying control and the impulsive control which can both improve the stability of the system effectively.

The paper presents a detailed analysis on the collective dynamics and delayed state feedback control of a three-dimensional delayed small-world network. The trivial equilibrium of the model is first investigated, showing that the uncontrolled model exhibits complicated unbounded behavior. Then three control strategies, namely a position feedback control, a velocity feedback control, and a hybrid control combined velocity with acceleration feedback, are then introduced to stabilize this unstable system. It is shown in these three control schemes that only the hybrid control can easily stabilize the 3-D network system. And with properly chosen delay and gain in the delayed feedback path, the hybrid controlled model may have stable equilibrium, or periodic solutions resulting from the Hopf bifurcation, or complex stranger attractor from the period-doubling bifurcation. Moreover, the direction of Hopf bifurcation and stability of the bifurcation periodic solutions are analyzed. The results are further extended to any "d" dimensional network. It shows that to stabilize a "d" dimensional delayed small-world network, at least a "d – 1" order completed differential feedback is needed. This work provides a constructive suggestion for the high dimensional delayed systems.

This paper aims to study price dynamics in two different continuous time cobweb models with delays close to [Hommes, 1994]. In both cases, the stationary equilibrium may be not representative of the long-term dynamics of the model, since it is possible to observe endogenous and persistent fluctuations (supercritical Hopf bifurcations) even if a deterministic context without external shocks is considered. In the model in which markets are in equilibrium every time, we show that the existence of time delays in the expectations formation mechanism may cause chaotic dynamics similar to those obtained in [Hommes, 1994] in a discrete time context. From a mathematical point of view, we apply the Poincaré–Lindstedt perturbation method to study the local dynamic properties of the models. In addition, several numerical experiments are used to investigate global properties of the systems.

In this paper, a novel version of nonlinear model, i.e. a complex-valued love model with two time delays between two individuals in a love affair, has been proposed. A notable feature in this model is that we separate the emotion of one individual into real and imaginary parts to represent the variation and complexity of psychophysiological emotion in romantic relationship instead of just real domain, and make our model much closer to reality. This is because love is a complicated cognitive and social phenomenon, full of complexity, diversity and unpredictability, which refers to the coexistence of different aspects of feelings, states and attitudes ranging from joy and trust to sadness and disgust. By analyzing associated characteristic equation of linearized equations for our model, it is found that the Hopf bifurcation occurs when the sum of time delays passes through a sequence of critical value. Stability of bifurcating cyclic love dynamics is also derived by applying the normal form theory and the center manifold theorem. In addition, it is also shown that, for some appropriate chosen parameters, chaotic behaviors can appear even without time delay.

This paper focuses on the dynamic behaviors of delay-coupled networks consisting of an arbitrary number of nonidentical neurons with unidirectional connections and an electrical synapse from one neuron onto itself. The stability criteria and different types of bifurcations are discussed by decomposing and analyzing the associated characteristic equation. Then, the study turns to the validation of theoretical results through numerical simulations and various interesting neural activities are observed, such as nontrivial equilibria, different patterns of oscillations, multistability coexistence, multiple switches between the quiescent and different periodic states, and transitions between the coexisting equilibria and the coexistence of equilibria and periodic oscillations. It is shown that time delays play important roles in the system performance and can be used to control neural activities.

We consider the neutrino fluxes and photon lensing from Gamma Ray Bursts and galaxy shocks, in order to predict measurable quantum corrections at late times. In particular, changes of the Einstein shell condition are expected to modify the time delay of neutrinos and photons at the Planck scale. Hence, we study the consequences of the Lorentz invariance violation through the use of such photons and neutrinos. The Lorentz invariance violation is expected to depend on the cosmological model, considered a priori in the Einstein equations; therefore, we focus on the so-called Hořava model which has recently attracted great interest. In addition, we compare our results with the ΛCDM and modified Chaplygin gas models, respectively. We infer several theoretical constraints which could reveal such quantum gravity effects.

A 2-dimensional neural network with time delayed connections between neurons is considered. Based on the construction of Liapunov functionals, we obtain sufficient criteria to ensure local and global asymptotic stability of the equilibrium of the neural network.

In an organism, genes encode proteins, some of which in turn regulate other genes. Such interactions work in highly structured but incredibly complex ways, and make up a genetic regulatory network. Recently, nonlinear delay differential equations have been proposed for describing genetic regulatory networks in the state-space form. In this paper, we study stability properties of genetic regulatory networks with time delays, by the notion of delay-independent stability. We first present necessary and sufficient conditions for delay-independent local stability of genetic regulatory networks with a single time delay, and then extend the main result to genetic regulatory networks with multiple time delays. To illustrate the main theory, we analyze delay-independent stability of three genetic regulatory networks in E. coli or zebra fish. For E. coli, an autoregulatory network and a repressilatory network are analyzed. The results show that these two genetic regulatory networks with parameters in the physiological range are delay-independently robustly stable. For zebra fish, an autoregulatory network for the gene her1 is analyzed. The result shows that delay-independent stability of this network depends on the initial number of protein molecules, which is in agreement with the existing biological knowledge. The theories presented in this paper provide a very useful complement to the previous work and a framework for further studying the stability of more complex genetic regulatory networks.