Effect of small-world topology on wave propagation on networks of excitable elements

Isele, Thomas; Schöll, Eckehard

doi:10.1088/1367-2630/17/2/023058

Cited by 23 publications

(11 citation statements)

References 44 publications

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“…As another typical solution in reaction-diffusion equations on continuous media, a traveling wave solution is well known, which moves with constant speed without changing the profile. In [23,24,25,26], wave-like phenomena were also observed in reaction-diffusion models on networks. Interestingly, the occurrence of propagation failure of waves and pinned waves in network organized reaction-diffusion models were reported.…”

Section: Discussionmentioning

confidence: 88%

Turing instability in reaction–diffusion models on complex networks

Ide

Izuhara

Machida

2016

Physica A: Statistical Mechanics and its Applications

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In this paper, the Turing instability in reaction-diffusion models defined on complex networks is studied. Here, we focus on three types of models which generate complex networks, i.e. the Erdős-Rényi, the Watts-Strogatz, and the threshold network models. From analysis of the Laplacian matrices of graphs generated by these models, we numerically reveal that stable and unstable regions of a homogeneous steady state on the parameter space of two diffusion coefficients completely differ, depending on the network architecture. In addition, we theoretically discuss the stable and unstable regions in the cases of regular enhanced ring lattices which include regular circles, and networks generated by the threshold network model when the number of vertices is large enough.

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Section: Discussionmentioning

confidence: 88%

Turing instability in reaction–diffusion models on complex networks

Ide

Izuhara

Machida

2016

Physica A: Statistical Mechanics and its Applications

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“…The interplays between local reaction kinetics (nodes), the physical processes that create coupling (link), and the architecture of the network in such systems can lead to a wealth of self-organized phenomena, including synchronization, [4,6,8] stationary Turing and oscillatory patterns, [9,10,11,12,13] or excitation waves. [14,15,16] Stationary patterns generated via the Turing[17] mechanism have been observed in experiments for both continuous [18] and networked [19] systems. Here, an alternative mechanism for emergence of stationary patterns in networks is experimentally explored.…”

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confidence: 99%

Self‐Organized Stationary Patterns in Networks of Bistable Chemical Reactions

et al. 2016

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Experiments with a network of bistable electrochemical reactions, organized in regular and non-regular tree networks, are presented to confirm an alternative to Turing mechanism for formation selforganized stationary patterns. The results show that the pattern formation can be described by identification of domains that can be activated individually or in combinations. The method was also demonstrated to localization of chemical reactions to network substructures and identification of critical sites whose activation results in complete activation of the system. While the experiments were performed with a specific nickel electrodissolution system, they reproduce all the salient dynamical behavior of a general network model with a single nonlinearity parameter. This indicates that the considered pattern formation mechanism is very robust and similar behavior can thus be expected in other natural or engineered networked systems, which exhibit, at least locally, a tree-like structure.In biological context, many chemical reactions take place in discrete units, e.g., in cells, that form a complex network. The interplays between local reaction kinetics (nodes), the physical processes that create coupling (link), and the architecture of the network in such systems can lead to a wealth of self-organized phenomena, including synchronization, [4,6,8] stationary Turing and oscillatory patterns, [9,10,11,12,13] or excitation waves. [14,15,16] Stationary patterns generated via the Turing[17] mechanism have been observed in experiments for both continuous [18] and networked [19] systems. Here, an alternative mechanism for emergence of stationary patterns in networks is experimentally explored. We focus on network-organized systems of bistable elements with diffusive connections between them. Bistable elements can be found in a broad class of chemical reactions (e.g., with autocatalysis [20,21]) but also in cellular [22] and engineered systems.[23] Such elements can have local or diffusive connections between them. For regular lattices and linear chains (i.e., for relatively simple networks), it is known that, under sufficiently weak coupling, the fronts fail to propagate, and thus stationary domains can be formed, [24,25,26,27] whereas at strong coupling the fronts spread [27,28] and a uniform state is eventually established. Recently, analogues phenomena were theoretically investigated for complex 1

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“…In addition, when → 0 eqs. (13,14) are simplified to those of the homogenized continuum model CM. This is expected, since by → 0 we mean the discrete nature of the phenomena is not relevant and the classical continuum hypothesis is valid.…”

Section: B Discretization-dependent Homogenized Modelmentioning

confidence: 99%

“…In the simple case of 1D model this summation may involve only first neighbors (to represent, for instance, the discrete Laplacian operator). Nevertheless, it is possible to use complex networks in this summation term [6,13]. In summary, the continuum space is replaced by a discrete set of components whereas time is still continuous, leading us to a system of coupled ordinary differential equations.…”

Section: Introductionmentioning

confidence: 99%

Discretization-dependent model for weakly connected excitable media

2018

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Pattern formation has been widely observed in extended chemical and biological processes. Although the biochemical systems are highly heterogeneous, homogenized continuum approaches formed by partial differential equations have been employed frequently. Such approaches are usually justified by the difference of scales between the heterogeneities and the characteristic spatial size of the patterns. Under different conditions, for example, under weak coupling, discrete models are more adequate. However, discrete models may be less manageable, for instance, in terms of numerical implementation and mesh generation, than the associated continuum models. Here we study a model to approach discreteness which permits the computer implementation on general unstructured meshes. The model is cast as a partial differential equation but with a parameter that depends not only on heterogeneities sizes, as in the case of quasicontinuum models, but also on the discretization mesh. Therefore, we refer to it as a discretization-dependent model. We validate the approach in a generic excitable media that simulates three different phenomena: the propagation of action membrane potential in cardiac tissue, in myelinated axons of neurons, and concentration waves in chemical microemulsions.

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Effect of small-world topology on wave propagation on networks of excitable elements

Cited by 23 publications

References 44 publications

Turing instability in reaction–diffusion models on complex networks

Turing instability in reaction–diffusion models on complex networks

Self‐Organized Stationary Patterns in Networks of Bistable Chemical Reactions

Discretization-dependent model for weakly connected excitable media

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