I am currently reading Hyvarinen, A. Juha karhunen & Erkki Oja (2001), Independent Component Analysis. to get a deeper understanding of ICA. Is there a subtype of ICA that is usually used in FC analysis and a particular algorithm used to find the independent components? Is there a good publication concerning modern ICA methods used in resting state analysis written at about the same level of mathematical detail as the text I cite above?
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I am not a regular ICA user, but as there hasn’t been a response yet I’ll go ahead.
Classic group ICA decomposition include tensorial ICA, implemented in MELODIC (https://www.fmrib.ox.ac.uk/datasets/techrep/tr04cb1/tr04cb1.pdf), and temporal concatenation in the GIFT software of Calhoun and coll (see this review A review of group ICA for fMRI data and ICA for joint inference of imaging, genetic, and ERP data). Another classic technique often used in combination with MELODIC to get back individualized ICA results is dual regression (DualRegression - FslWiki).
In terms of modern variants, there is a large number of them. But if I were to pick two from the same influential groups above, you may want to have a look at PROFUMO (Modelling subject variability in the spatial and temporal characteristics of functional modes - ScienceDirect) and NeuroMark (NeuroMark: An automated and adaptive ICA based pipeline to identify reproducible fMRI markers of brain disorders - ScienceDirect).
Note that other space-time decomposition models are explored in the literature. Stochastic online matrix factorization (https://hal.archives-ouvertes.fr/hal-01431618v3/document) in particular has been shown to lead to high-quality compression of fMRI data Fine-grain atlases of functional modes for fMRI analysis - ScienceDirect
What those three recent papers have in common is to scale to very large datasets, and also better capture inter-individual (and temporal) variability. I am not aware of a didactic comprehensive text which would review different recent models of space-time decomposition of fMRI data. The 2009 review of Calhoun I linked above is nice, but it’s now over a decade old. Maybe others will have recommendations, which I’ll be happy to hear about 
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Sorry for the delay in my replay thank you very very much for giving me these reviews. I really appreciate it. If I’m not mistaken the tensorial ICA from melodic is the most commonly used correct?
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I don’t have statistics, but tensorial ICA has indeed been very influential and is the ICA flavor implemented in FSL, which is a very popular package. So it’s safe to say it’s up there, likely with the ICA implementation of the GIFT toolbox.
Good luck with the reading!