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

On the role of the corpus callosum in interhemispheric functional connectivity in humans

Resting state functional connectivity is defined in terms of temporal correlations between physiologic signals, most commonly studied using functional magnetic resonance imaging. Major features of functional connectivity correspond to structural (axonal) connectivity. However, this relation is not one-to-one. Interhemispheric functional connectivity in relation to the corpus callosum presents a case in point. Specifically, several reports have documented nearly intact interhemispheric functional connectivity in individuals in whom the corpus callosum (the major commissure between the hemispheres) never develops. To investigate this question, we assessed functional connectivity before and after surgical section of the corpus callosum in 22 patients with medically refractory epilepsy. Section of the corpus callosum markedly reduced interhemispheric functional connectivity. This effect was more profound in multimodal associative areas in the frontal and parietal lobe than primary regions of sensorimotor and visual function. Moreover, no evidence of recovery was observed in a limited sample in which multiyear, longitudinal follow-up was obtained. Comparison of partial vs. complete callosotomy revealed several effects implying the existence of polysynaptic functional connectivity between remote brain regions. Thus, our results demonstrate that callosal as well as extracallosal anatomical connections play a role in the maintenance of interhemispheric functional connectivity

Dynamic BOLD functional connectivity in humans and its electrophysiological correlates

Neural oscillations subserve many human perceptual and cognitive operations. Accordingly, brain functional connectivity is not static in time, but fluctuates dynamically following the synchronization and desynchronization of neural populations. This dynamic functional connectivity has recently been demonstrated in spontaneous fluctuations of the Blood Oxygen Level-Dependent (BOLD) signal, measured with functional Magnetic Resonance Imaging (fMRI). We analyzed temporal fluctuations in BOLD connectivity and their electrophysiological correlates, by means of long (≈50 min) joint electroencephalographic (EEG) and fMRI recordings obtained from two populations: 15 awake subjects and 13 subjects undergoing vigilance transitions. We identified positive and negative correlations between EEG spectral power (extracted from electrodes covering different scalp regions) and fMRI BOLD connectivity in a network of 90 cortical and subcortical regions (with millimeter spatial resolution). In particular, increased alpha (8-12 Hz) and beta (15-30 Hz) power were related to decreased functional connectivity, whereas gamma (30-60 Hz) power correlated positively with BOLD connectivity between specific brain regions. These patterns were altered for subjects undergoing vigilance changes, with slower oscillations being correlated with functional connectivity increases. Dynamic BOLD functional connectivity was reflected in the fluctuations of graph theoretical indices of network structure, with changes in frontal and central alpha power correlating with average path length. Our results strongly suggest that fluctuations of BOLD functional connectivity have a neurophysiological origin. Positive correlations with gamma can be interpreted as facilitating increased BOLD connectivity needed to integrate brain regions for cognitive performance. Negative correlations with alpha suggest a temporary functional weakening of local and long-range connectivity, associated with an idling state

The Connectivity Order of Links

We associate at each link a connectivity space which describes its splittability properties. Then, the notion of order for finite connectivity spaces results in the definition of a new numerical invariant for links, their connectivity order. A section of this short paper presents a theorem which asserts that every finite connectivity structure can be realized by a link : the Brunn-Debrunner-Kanenobu Theorem.Comment: 4 pages, 2 figure

Connectivity Properties of Factorization Posets in Generated Groups

We consider three notions of connectivity and their interactions in partially ordered sets coming from reduced factorizations of an element in a generated group. While one form of connectivity essentially reflects the connectivity of the poset diagram, the other two are a bit more involved: Hurwitz-connectivity has its origins in algebraic geometry, and shellability in topology. We propose a framework to study these connectivity properties in a uniform way. Our main tool is a certain linear order of the generators that is compatible with the chosen element.Comment: 35 pages, 17 figures. Comments are very welcome. Final versio

Early affective changes and increased connectivity in preclinical Alzheimer's disease.

IntroductionAffective changes precede cognitive decline in mild Alzheimer's disease and may relate to increased connectivity in a "salience network" attuned to emotionally significant stimuli. The trajectory of affective changes in preclinical Alzheimer's disease, and its relationship to this network, is unknown.MethodsOne hundred one cognitively normal older adults received longitudinal assessments of affective symptoms, then amyloid-PET. We hypothesized amyloid-positive individuals would show enhanced emotional reactivity associated with salience network connectivity. We tested whether increased global connectivity in key regions significantly related to affective changes.ResultsIn participants later found to be amyloid positive, emotional reactivity increased with age, and interpersonal warmth declined in women. These individuals showed higher global connectivity within the right insula and superior temporal sulcus; higher superior temporal sulcus connectivity predicted increasing emotional reactivity and decreasing interpersonal warmth.ConclusionsAffective changes should be considered an early preclinical feature of Alzheimer's disease. These changes may relate to higher functional connectivity in regions critical for social-emotional processing

Connectivity of tropicalizations

We show that the tropicalization of an irreducible variety over a complete or algebraically closed valued field is connected through codimension 1, giving an affirmative answer in all characteristics to a question posed by Einsiedler, Lind, and Thomas in 2003.Comment: 7 page