On Connectivity Spaces

This paper presents some basic facts about the so-called connectivity spaces. In particular, it studies the generation of connectivity structures, the existence of limits and colimits in the main categories of connectivity spaces, the closed monoidal category structure given by the so-called tensor product on integral connectivity spaces; it defines homotopy for connectivity spaces and mention briefly related difficulties; it defines smash product of pointed integral connectivity spaces and shows that this operation results in a closed monoidal category with such spaces as objects. Then, it studies finite connectivity spaces, associating a directed acyclic graph with each such space and then defining a new numerical invariant for links: the connectivity order. Finally, it mentions the not very wellknown Brunn-Debrunner-Kanenobu theorem which asserts that every finite integral connectivity space can be represented by a link.Comment: 30 pages, 2 figure

Short Cycles Connectivity

Short cycles connectivity is a generalization of ordinary connectivity. Instead by a path (sequence of edges), two vertices have to be connected by a sequence of short cycles, in which two adjacent cycles have at least one common vertex. If all adjacent cycles in the sequence share at least one edge, we talk about edge short cycles connectivity. It is shown that the short cycles connectivity is an equivalence relation on the set of vertices, while the edge short cycles connectivity components determine an equivalence relation on the set of edges. Efficient algorithms for determining equivalence classes are presented. Short cycles connectivity can be extended to directed graphs (cyclic and transitive connectivity). For further generalization we can also consider connectivity by small cliques or other families of graphs

The Role of Landscape Connectivity in Planning and Implementing Conservation and Restoration Priorities. Issues in Ecology

Landscape connectivity, the extent to which a landscape facilitates the movements of organisms and their genes, faces critical threats from both fragmentation and habitat loss. Many conservation efforts focus on protecting and enhancing connectivity to offset the impacts of habitat loss and fragmentation on biodiversity conservation, and to increase the resilience of reserve networks to potential threats associated with climate change. Loss of connectivity can reduce the size and quality of available habitat, impede and disrupt movement (including dispersal) to new habitats, and affect seasonal migration patterns. These changes can lead, in turn, to detrimental effects for populations and species, including decreased carrying capacity, population declines, loss of genetic variation, and ultimately species extinction. Measuring and mapping connectivity is facilitated by a growing number of quantitative approaches that can integrate large amounts of information about organisms’ life histories, habitat quality, and other features essential to evaluating connectivity for a given population or species. However, identifying effective approaches for maintaining and restoring connectivity poses several challenges, and our understanding of how connectivity should be designed to mitigate the impacts of climate change is, as yet, in its infancy. Scientists and managers must confront and overcome several challenges inherent in evaluating and planning for connectivity, including: •characterizing the biology of focal species; •understanding the strengths and the limitations of the models used to evaluate connectivity; •considering spatial and temporal extent in connectivity planning; •using caution in extrapolating results outside of observed conditions; •considering non-linear relationships that can complicate assumed or expected ecological responses; •accounting and planning for anthropogenic change in the landscape; •using well-defined goals and objectives to drive the selection of methods used for evaluating and planning for connectivity; •and communicating to the general public in clear and meaningful language the importance of connectivity to improve awareness and strengthen policies for ensuring conservation. Several aspects of connectivity science deserve additional attention in order to improve the effectiveness of design and implementation. Research on species persistence, behavioral ecology, and community structure is needed to reduce the uncertainty associated with connectivity models. Evaluating and testing connectivity responses to climate change will be critical to achieving conservation goals in the face of the rapid changes that will confront many communities and ecosystems. All of these potential areas of advancement will fall short of conservation goals if we do not effectively incorporate human activities into connectivity planning. While this Issue identifies substantial uncertainties in mapping connectivity and evaluating resilience to climate change, it is also clear that integrating human and natural landscape conservation planning to enhance habitat connectivity is essential for biodiversity conservation

Caffeine-Induced Global Reductions in Resting-State BOLD Connectivity Reflect Widespread Decreases in MEG Connectivity.

In resting-state functional magnetic resonance imaging (fMRI), the temporal correlation between spontaneous fluctuations of the blood oxygenation level dependent (BOLD) signal from different brain regions is used to assess functional connectivity. However, because the BOLD signal is an indirect measure of neuronal activity, its complex hemodynamic nature can complicate the interpretation of differences in connectivity that are observed across conditions or subjects. For example, prior studies have shown that caffeine leads to widespread reductions in BOLD connectivity but were not able to determine if neural or vascular factors were primarily responsible for the observed decrease. In this study, we used source-localized magnetoencephalography (MEG) in conjunction with fMRI to further examine the origins of the caffeine-induced changes in BOLD connectivity. We observed widespread and significant (p < 0.01) reductions in both MEG and fMRI connectivity measures, suggesting that decreases in the connectivity of resting-state neuro-electric power fluctuations were primarily responsible for the observed BOLD connectivity changes. The MEG connectivity decreases were most pronounced in the beta band. By demonstrating the similarity in MEG and fMRI based connectivity changes, these results provide evidence for the neural basis of resting-state fMRI networks and further support the potential of MEG as a tool to characterize resting-state connectivity

Functional connectivity changes and their relationship with clinical disability and white matter integrity in patients with relapsing-remitting multiple sclerosis

Background and objective: To define the pathological substrate underlying disability in multiple sclerosis by evaluating the relationship of resting-state functional connectivity with microstructural brain damage, as assessed by diffusion tensor maging, and clinical impairments. Methods: Thirty relapsing–remitting patients and 24 controls underwent 3T-MRI; motor abilities were evaluated by using measures of walking speed, hand dexterity and balance capability, while information processing speed was evaluated by a paced auditory serial addiction task. Independent component analysis and tract-based spatial statistics were applied to RS-fMRI and diffusion tensor imaging data using FSL software. Group differences, after dual regression, and clinical correlations were modelled with GeneralLinear-Model and corrected for multiple comparisons. Results: Patients showed decreased functional connectivity in 5 of 11 resting-state-networks (cerebellar, executive-control, medial-visual, basal ganglia and sensorimotor), changes in inter-network correlations and widespread white matter microstructural damage. In multiple sclerosis, corpus callosum microstructural damage positively correlated with functional connectivity in cerebellar and auditory networks. Moreover, functional connectivity within the medial-visual network inversely correlated with information processing speed. White matter widespread microstructural damage inversely correlated with both the paced auditory serial addiction task and hand dexterity. Conclusions: Despite the within-network functional connectivity decrease and the widespread microstructural damage, the inter-network functional connectivity changes suggest a global brain functional rearrangement in multiple sclerosis. The correlation between functional connectivity alterations and callosal damage uncovers a link between functional and structural connectivity. Finally, functional connectivity abnormalities affect information processing speed rather than motor abilities