Limit Cycle Of Length Research Articles

Reconstruction of a gene regulatory network of the induced systemic resistance defense response in Arabidopsis using boolean networks

BackgroundAn important process for plant survival is the immune system. The induced systemic resistance (ISR) triggered by beneficial microbes is an important cost-effective defense mechanism by which plants are primed to an eventual pathogen attack. Defense mechanisms such as ISR depend on an accurate and context-specific regulation of gene expression. Interactions between genes and their products give rise to complex circuits known as gene regulatory networks (GRNs). Here, we explore the regulatory mechanism of the ISR defense response triggered by the beneficial bacterium Paraburkholderia phytofirmans PsJN in Arabidopsis thaliana plants infected with Pseudomonas syringae DC3000. To achieve this, a GRN underlying the ISR response was inferred using gene expression time-series data of certain defense-related genes, differential evolution, and threshold Boolean networks.ResultsOne thousand threshold Boolean networks were inferred that met the restriction of the desired dynamics. From these networks, a consensus network was obtained that helped to find plausible interactions between the genes. A representative network was selected from the consensus network and biological restrictions were applied to it. The dynamics of the selected network showed that the largest attractor, a limit cycle of length 3, represents the final stage of the defense response (12, 18, and 24 h). Also, the structural robustness of the GRN was studied through the networks’ attractors.ConclusionsA computational intelligence approach was designed to reconstruct a GRN underlying the ISR defense response in plants using gene expression time-series data of A. thaliana colonized by P. phytofirmans PsJN and subsequently infected with P. syringae DC3000. Using differential evolution, 1000 GRNs from time-series data were successfully inferred. Through the study of the network dynamics of the selected GRN, it can be concluded that it is structurally robust since three mutations were necessary to completely disarm the Boolean trajectory that represents the biological data. The proposed method to reconstruct GRNs is general and can be used to infer other biologically relevant networks to formulate new biological hypotheses.

Stability Proof of Iterative Interference Cancellation for OFDM Signals With Blanking Nonlinearity in Impulsive Noise Channels

Blanking nonlinearity is a widely applied technique for suppressing impulsive noise. However, for orthogonal frequency-division multiplexing signals, the blanking nonlinearity induces intercarrier interference, which must be removed in order to fully exploit the advantage of the blanking nonlinearity and to avoid performance degradation. For this purpose, interference cancellation techniques are attracting a considerable attention. Particularly, the nonlinear iterative ones show a good complexity-performance behavior. Nevertheless, one of the most important features of nonlinear iterative methods in the context of interference cancellation often remains disregarded: The stability. In this correspondence, we prove the asymptotical stability in the sense of Lyapunov of a nonlinear iterative interference cancellation technique for OFDM signals with blanking nonlinearity in impulsive noise channels. This is achieved by reformulating the considered technique as a multithreshold recurrent neural network and by taking advantage of the extensive research on the Lyapunov stability of recurrent neural networks. We show that the considered nonlinear iterative interference cancellation technique either reaches a fixed point or a limit cycle of length two. The obtained results can be utilized to find an upper bound on the required number of iterations to settle on a fixed point.

Convergence of higher-order two-state neural networks with modified updating

The Hopfield network is a standard tool for maximizing aquadratic objective function over the discrete set {− 1,1} n . It is well-known that if a Hopfield network is operated in anasynchronous mode, then the state vector of the network converges to a local maximum of the objective function; if the network is operated in asynchronous mode, then the state vector either converges to a local maximum, or else goes into a limit cycle of length two. In this paper, we examine the behaviour ofhigher-order neural networks, that is, networks used for maximizing objective functions that are not necessarily quadratic. It is shown that one can assume, without loss of generality, that the objective function to be maximized ismultilinear. Three methods are given for updating the state vector of the neural network, called the asynchronous, the best neighbour and the gradient rules, respectively. For Hopfield networks with a quadratic objective function, the asynchronous rule proposed here reduces to the standard asynchronous updating, while the gradient rule reduces to synchronous updating; the best neighbour rule does not appear to have been considered previously. It is shown that both the asynchronous updating rule and the best neighbour rule converge to a local maximum of the objective function within a finite number of time steps. Moreover, under certain conditions, under the best neighbour rule, each global maximum has a nonzero radius of direct attraction; in general, this may not be true of the asynchronous rule. However, the behaviour of the gradient updating rule is not well-understood. For this purpose, amodified gradient updating rule is presented, that incorporates bothtemporal as well as spatial correlations among the neurons. For the modified updating rule, it is shown that, after a finite number of time steps, the network state vector goes into a limit cycle of lengthm, wherem is the degree of the objective function. Ifm = 2, i.e., for quadratic objective functions, the modified updating rule reduces to the synchronous updating rule for Hopfield networks. Hence the results presented here are “true” generalizations of previously known results.

Energy and attractors in parallel Potts dynamics

The authors derive an energy function for multibit threshold automata networks (e.g. the Potts model) updated in parallel, and show that their dynamics admit as attractors fixed points and limit cycles of length 2. They compare this result with similar known results on networks of binary automata (e.g. the Ising model).

Exact results for the average dynamic behavior of some non-linear neural networks.

We have studied the global dynamic behavior of neural-like networks of synchronous threshold elements by writing a master equation as a function of parameter values using statistical methods. Exact results for highly connected networks and no correlation are obtained, showing that in this case (contrary to previous results) the average activity can only display simple stable behaviour, the sole exception being special cases of a slow passage through a tangent bifurcation, and a limit cycle of length two. By introducing an appropriate probabilistic hypothesis, we also study the average activity and correlation for highly connected networks with the topology of a (Cayley) tree. In this case the dynamic is ruled by a pair of coupled equations linking activity and correlation, and the tendency is for the correlation to disappear over time. However, under reasonable biological conditions, this tendency will be extremely slow, giving rise to a region of pseudo-stability.