We present observational evidence from studies on primary cortical cultures from AD transgenic mice, APPSwe/PS1ΔE9 (APP/PS1) mice, for significant decrease in total spine density at DIV-15 and onward. This indicates reduction in potential healthy synapses and strength of connections among neurons. Based on this, a network model of neurons is developed, that explains the consequent loss of coordinated activity and transmission efficiency among neurons that manifests over time. The critical time when structural connectivity in the brain undergoes a phase-transition, from initial robustness to irreparable breakdown, is estimated from this model. We also show how the global efficiency of signal transmission in the network decreases over time. Moreover, the number of multiple paths of high efficiency decreases rapidly as the disease progresses, indicating loss of structural plasticity and inefficiency in choosing alternate paths or desired paths for any pattern of activity. Thus loss of spines caused by β -Amyloid (A β ) peptide results in disintegration of the neuronal network over time with consequent cognitive dysfunctions in Alzheimer’s Disease (AD).
We present a systematic and detailed study of the robustness of directed networks under random and targeted removal of links. We work with a set of network models of random and scale free type, generated with specific features of clustering and assortativity. Various strategies like random deletion of links, or deletions based on betweenness centrality and degrees of source and target nodes, are used to breakdown the networks. The robustness of the networks to the sustained loss of links is studied in terms of the sizes of the connected components and the inverse path lengths. The effects of clustering and 2-node degree correlations, on the robustness to attack, are also explored. We provide specific illustrations of our study on three real-world networks constructed from protein-protein interactions and from transport data.
With complex networks emerging as an effective tool to tackle multidisciplinary problems, models of network generation have gained an importance of their own. These models allow us to extensively analyze the data obtained from real-world networks, study their relevance and corroborate theoretical results. In this work, we introduce methods, based on degree preserving rewiring, that can be used to tune the clustering and degree-correlations in directed networks with random and scale-free topologies. They provide null-models to investigate the role of the mentioned properties along with their strengths and limitations. We find that in the case of clustering, structural relationships, that are independent of topology and rewiring schemes are revealed, while in the case of degree-correlations, the network topology is found to play an important role in the working of the mechanisms. We also study the effects of link-density on the efficiency of these rewiring mechanisms and find that in the case of clustering, the topology of the network plays an important role in determining how link-density affects the rewiring process, while in the case of degree-correlations, the link-density and topology, play no role for sufficiently large number of rewiring steps. Besides the intended purpose of tuning network properties, the proposed mechanisms can also be used as a tool to reveal structural relationships and topological constraints.
In year 2013, L. Thivagar et al. introduced nano topological space and he analysed some properties of weak open sets. In this paper we shall introduce Kasaj-topological space. We shall introduce some new classes of weak open sets namely Kasaj-pre-open sets and Kasaj-semi-open sets in Kasaj topological spaces and analyze their basic properties. We shall also define new types of continuous functions namely Kasaj-continuous function, Kasaj-pre-continuous function, Kasaj-semi-continuous function in Kasaj topological space.
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