Fast Fourier Transforms (FFT) are widely known as useful tools for the evaluation of spectral data. This article discusses the applicability of FFT methods to crystallographic problems. Formulae are derived which make it possible to use fast Fourier transforms for the general Fourier summation of crystallographic data in all space groups and for the computation of slant planes at arbitrary positions in the unit cell. As even moderate resolutions would produce arrays of data too large to fit within internal computer memory, they must be kept to an external storage device. The organization of data and the resulting time requirements are thoroughly discussed. An ALGOL 60 program has been developed with as many redundancies eliminated as possible. An example shows that for a problem of moderate size this algorithm is faster by an order of magnitude than those which have traditionally been used.
Network neuroscience is a promising approach to explore cognitive processes in neurological disorders. Alzheimer's disease (AD) and frontotemporal dementia (FTD) show network dysfunctions linked with cognitive deficits. Within this framework, network abnormalities between AD and FTD show both convergent and divergent patterns. However, these functional patterns are far from being established and their relevance to cognitive processes remains to be elucidated. In this study, we aimed to investigate the relationship between cognition and functional connectivity of major cognitive networks in these diseases. Twenty-three bvFTD, 22 AD and 20 controls underwent cognitive evaluation and resting-state functional MRI. Principal component analysis was used to describe cognitive variance across participants. Brain network connectivity was estimated with connectome analysis. Connectivity matrices were created assessing correlations between parcels within each functional network. The following cognitive networks were considered: default mode (DMN), dorsal attention (DAN), ventral attention (VAN) and frontoparietal (FPN) networks. The relationship between cognition and connectivity was assessed using a robust convergent correlation-wise and interaction analyses. Three principal cognitive components explained more than 80% of the cognitive variance: the first component (cogPC1) loaded on memory, the second component (cogPC2) loaded on emotion and language, the third component (cogPC3) loaded on the visuo-spatial and attentional domains. Compared to HC, AD and bvFTD showed impairment in all cogPCs (p<0.002), and bvFTD scored worse than AD in cogPC2 (p=0.031). At the network level, the DMN showed a robust association in the whole group with cogPC1 and cogPC2, and the VAN with cogPC2. By contrast, DAN and FPN showed a divergent pattern between diagnosis and connectivity for cogPC2. We confirmed these results by means of a multivariate analysis (canonical correlation). These results suggest that a low-dimensional representation can account for a large variance in cognitive scores in the continuum from normal to pathological aging. Moreover, cognitive components showed both convergent and divergent patterns with connectivity across AD and bvFTD. The convergent pattern was observed across the networks primarily involved in these diseases (i.e., the DMN and VAN), while a divergent FC-cognitive pattern was mainly observed between attention/executive networks and the language/emotion cognitive component, suggesting the co-existence of compensatory and detrimental mechanisms underlying these components.
Background Divergent functional connectivity (FC) abnormalities between networks linked with core cognitive deficits have been reported in Alzheimer’s disease (AD) and behavioural variant frontotemporal dementia (bvFTD), the default mode (DMN) and ventral‐attention (VAN), respectively. As cognition relies on the balance between brain networks, we investigated the coupling patterns between different cognitive domains and networks. Method Twenty‐one bvFTD (age: 71±10), 22 AD (age: 72±6) and 20 healthy controls (HC, age: 72±6) underwent cognitive evaluation and 3T resting‐state functional MRI (rsfMRI). We investigated the relationship between memory, language, executive, visuospatial and emotion recognition with FC of cognitive networks: DMN, VAN, frontoparietal (FPN), and dorsal‐attention (DAN). The visual network (VIS) was included as a control network. FC was assessed with a connectome analysis. Cortical parcels were defined using a high‐resolution structural MRI parcellation. Connectivity matrices were created assessing correlations between parcels within each functional network (Yeo 2011 parcellation). Correlation values were thresholded to retain positive values. FC and cognitive differences between groups were assessed with Kruskal‐Wallis test, Spearman’s correlation was used to assess relationships between FC and cognition. Result Compared to HC, AD and bvFTD showed impairment in all cognitive domains (p<0.001), and bvFTD scored worse than AD in emotion recognition (p=0.013). At the network level, AD showed reduced FC in the DMN, FPN and DAN (p<0.05) and bvFTD in the VAN (p<0.05) compared to HC. In bvFTD, reduced VAN connectivity was associated with executive deficits (r=0.444; p=0.038). In AD, reduced FPN connectivity was associated with language deficits (r=0.441; p=0.04), while the opposite pattern was observed in bvFTD (r=‐0.468; p=0.028). Moreover, in AD patients DMN connectivity was negatively associated with executive deficits (r=‐0.56; p=0.007). A significant interaction was detected between AD and bvFTD for the following cognitive‐network associations: VAN‐executive: p=0.027; FPN‐language: p=0.015. No significant association was detected between DMN‐memory in AD or VAN‐emotion recognition in bvFTD (p>0.06), nor between cognition and VIS (p>0.09 for all associations). Conclusion These results suggest a divergent FC‐cognitive pattern between AD and bvFTD in attention/executive networks/functions. We speculate that, as pathology spreads, disease‐specific symptom‐network coupling might weaken, and aberrant patterns may emerge in other domains.
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