Spatially dependent network model

Overall it is considered that the spatially independent network models, whilst simpler, should strictly be used for lightly crosslinked, homogeneous networks, whereas the spahally dependent models, although computationally intensive and limited by pre-defined lattice structure, provide a better understanding of network heterogeneities in highly crosslinked systems. 

Several extensions of this statistical formalism are possible and would allow taking into account more physiological and neuroanatomic features. For example, most of neural network models consider that every neuron receives exactly the same input current from the electrode, however in the case of DBS neurons do not receive the same current depending of their position with respect to the electrode [12]. This consideration is the first element in favor of adding a spatial state variable to the population density. Another example that goes in favor of this idea is the fact that peculiar features of small-world networks should be included. This class of complex networks was recently formalized [66] and some recent studies... 

Dusek (1986a) characterizes network-formation models into the following categories spatially independent and spatially dependent models. Of the spatially independent models, there are statistical models (in which network structure is developed from various interacting monomer units) and kinetic models (in which each concentration of species is modelled by a kinetic differential equation). 

Currently, the landslide hazard spatial prediction methods can be divided into qualitative methods and quantitative methods. As we all know qualitative forecasting method mainly depends on the subjective experience and the predicted accuracy of qualitative methods is lower than it of quantitative methods. So the qualitative methods have been gradually replaced by the quantitative methods. Quantitative models can be divided into statistic analysis models, deterministic models, probabilistic model, fuzzy information optimization processing and neurd network models. 

The pore radius assigned to each component of the network model is generated random numbers from 0 to 1 according to the probability density given from the pore-size measurement. The numerical assumption of the occurrence of the pore size is important in this model. The water saturation and hydraulic conductivity computed depend on the spatial distribution of the pore radius. The pore radii larger than 30 Ha, which can not be measured by the mercury intrusion method, are approximated to appear at the same probability as 3CTJm given by the PSD curve of Toyoura sand. 

From a fit of Equation (10) to spatially resolved relaxation curves, images of the parameters A, B, T2, q M2 have been obtained [3- - 32]. Here A/(A + B) can be interpreted as the concentration of cross-links and B/(A + B) as the concentration of dangling chains. In addition to A/(A + B) also q M2 is related to the cross-link density in this model. In practice also T2 has been found to depend on cross-link density and subsequently strain, an effect which has been exploited in calibration of the image in Figure 7.6. Interestingly, carbon-black as an active filler has little effect on the relaxation times, but silicate filler has. Consequently the chemical cross-link density of carbon-black filled elastomers can be determined by NMR. The apparent insensitivity of NMR to the interaction of the network chains with carbon black filler particles is explained with paramagnetic impurities of carbon black, which lead to rapid relaxation of the NMR signal in the vicinity of the filler particles. 

An important role in the present model is played by the strongly non-linear elastic response of the rubber matrix that transmits the stress between the filler clusters. We refer here to an extended tube model of rubber elasticity, which is based on the following fundamental assumptions. The network chains in a highly entangled polymer network are heavily restricted in their fluctuations due to packing effects. This restriction is described by virtual tubes around the network chains that hinder the fluctuation. When the network elongates, these tubes deform non-affinely with a deformation exponent v=l/2. The tube radius in spatial direction p of the main axis system depends on the deformation ratio as follows ... 

Our initial studies of dynamics in biochemical networks included spatially localized components [32]. As a consequence, there will be delays involved in the transport between the nuclear and cytoplasmic compartments. Depending on the spatial structure, different dynamical behaviors could be faciliated, but the theoretical methods are useful to help understand the qualitative features. In other (unpublished) work, computations were carried out in feedback loops with cyclic attractors in which a delay was introduced in one of the interactions. Although the delay led to an increase of the period, the patterns of oscillation remained the same. However, delays in differential equations that model neural networks and biological control systems can introduce novel dynamics that are not present without a delay (for example, see Refs. 57 and 58). 

The connectivity and spatial arrangement of objects within a network stmcture and the resulting macroscopic effects can be described by the percolation theory. In all its variations the percolatimi theory focuses on critical phenomena that originate from the spatial formation of a network and result in sharp transitions in the behaviour of the system of interest (Kirkpatrick 1973). Percolation models have been applied with various degrees of success to the description of the electrical behaviour of polymer nanocomposites. In these systems the insulating polymer matrix is loaded with cmiductive filler whose network formation leads to a sharp insulation-conductor transition (Lux 1993). Experimental work and theoretical predictions have established that the system s conductivity o follows a power-law dependence in accordance with percolation theory... 

For the network structure, let us consider regular networks. When such networks deform, the front factor of Eq. (20) differs by two orders of magnitude. This depends on whether it is assumed that the crosslink point moves proportionately to the deformation of the whole body, although they are fixed spatially (affine model) [38], or whether the networks are fixed to the frame while the internal crosslink points can move freely (ghost model) [39]. In any case, if Eq. (20) is rewritten as... 

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