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Spiral waves on excitable media strongly influence the functions of living systems in both a positive and negative way. The spiral formation mechanism has thus been one of the major themes in the field of reaction-diffusion systems. Although the widely believed origin of spiral waves is the interaction of traveling waves, the heterogeneity of an excitable medium has recently been suggested as a probable cause. We suggest one possible origin of spiral waves using a Belousov-Zhabotinsky reaction and a discretized FitzHugh-Nagumo model. The heterogeneity of the reaction field is shown to stochastically generate unidirectional sites, which can induce spiral waves. Furthermore, we found that the spiral wave vanished with only a small reduction in the excitability of the reaction field. These results reveal a gentle approach for controlling the appearance of a spiral wave on an excitable medium.

Biological network systems such as gene regulatory networks, metabolic networks and protein interaction networks depend on their network structures and dynamics. In this contribution, we report on attractors and their dynamical stability to rewiring in the core of the gene regulatory network of Escherichia coli and compare them with those of artificial networks with scale-free topology, using both threshold dynamics and Kauffman dynamics. ©2010 IEEE.

We study intrinsic properties of attractor in Boolean dynamics of complex networks with scale-free topology, comparing with those of the so-called Kauffman's random Boolean networks. We numerically study both frozen and relevant nodes in each attractor in the dynamics of relatively small networks (20 <= N <= 200). We investigate numerically robustness of an attractor to a perturbation. An attractor with cycle length of epsilon(c) in a network of size N consists of epsilon(c) states in the state space of 2(N) states; each attractor has the arrangement of N nodes, where the cycle of attractor sweeps epsilon(c) states. We define a perturbation as a flip of the state on a single node in the attractor state at a given time step. We show that the rate between unfrozen and relevant nodes in the dynamics of a complex network with scale-free topology is larger than that in Kauffman's random Boolean network model. Furthermore, we find that in a complex scale-free network with fluctuation of the in-degree number, attractors are more sensitive to a state flip for a highly connected node (i.e. input-hub node) than to that for a less connected node. By some numerical examples, we show that the number of relevant nodes increases, when an input. hub node is coincident with and/or connected with an output-hub node (i.e. a node with large output-degree) one another. (c) 2008 Elsevier Ltd. All rights reserved.

We study the intrinsic properties of attractors in the Boolean dynamics in complex network with scale-free topology, comparing with those of the so-called random Kauffman networks. We have numerically investigated the frozen and relevant nodes for each attractor, and the robustness of the attractors to the perturbation that flips the state of a single node of attractors in the relatively small network (N = 30 similar to 200). It is shown that the rate of frozen nodes in the complex networks with scale-free topology is larger than that in the random Kauffman model. Furthermore, we have found that in the complex scale-free networks with fluctuations of in-degree number the attractors are more sensitive to the state flip of a highly connected node than to the state flip of a less connected node.

We present some intrinsic properties of attractor states in Boolean dynamics in complex networks, comparing with that of random Kauffman networks. The frozen nodes and robustness of each attractor have been investigated in the relatively small network (N similar to 200).

We study the Boolean dynamics of the "quenched" Kauffman models with a directed scale-free network, comparing with that of the original directed random Kauffman networks and that of the directed exponential-fluctuation networks. We have numerically investigated the distributions of the state cycle lengths and its changes as the network size N and the average degree (k) of nodes increase. In the relatively small network (N similar to 150), the median, the mean value and the standard deviation grow exponentially with N in the directed scale-free and the directed exponential-fluctuation networks with (k) = 2, where the function forms of the distributions are given as an almost exponential. We have found that for the relatively large N similar to 10(3) the growth of the median of the distribution over the attractor lengths asymptotically changes from algebraic type to exponential one as the average degree (k) goes to (k) = 2. The result supports the existence of the transition at (k)(c) = 2 derived in the annealed model. (C) 2007 Elsevier Ltd. All rights reserved.

We study the nature of the fitness landscapes of a "quenched" Kauffman's Boolean model with a scale-free network. We have numerically calculated the rugged fitness landscapes, the distributions, their tails, and the correlation between the fitness of local optima and their Hamming distance from the highest optimum found, respectively. We have found that (a) there is an interesting difference between random and scale-free networks such that the statistics of the rugged fitness landscapes is Gaussian for the random network while it is non-Gaussian with a tail for the scale-free network (b) as the average degree increases, there is a phase transition at the critical value of c=2, below which there is a global order and above which the order goes away. © 2005 The American Physical Society.