Is the first independent component of independent component analysis always important?

Question

I was looking at a neuroscience paper that used ICA to reduce dimensionality of calcium signaling profiles in 20 randomly selected neurons of a zebrafish brain.

I presume that in Figure 2, ICA was used to remove shared signals/artifacts between neurons, which should produce 20 independent components.

However, it appeared that the researchers only focused on the first independent component for analysis. I was wondering how or why the 1st IC has was chosen over the other 19 ICs?

Answer 1

Just to close the loop in this question:

1. Strictly speaking, Independent Component Analysis (ICA) extracts the corresponding independent components (ICs) (i.e. source signals) from the observed data (i.e. mixed signals) simultaneously. As such there is no ordering involved. (See final point.)
2. These ICs are not sorted by construction, on that matter, their amplitude is defined only up to a multiplicative constant. We might wish to normalise them in a particular way (e.g. unit norm) and then treat them corresponding scores (i.e. mixing coefficients) as weights and then look at that weight variance but that is a somewhat arbitrary decision.
3. There has been worked on ranking ICs; e.g. Determining the optimal number of independent components for reproducible transcriptomic data analysis (2017) by Kairov et al., Ranking and averaging independent component analysis by reproducibility (2008) by Yang et al. This work is strongly focused on the reproducibility of the relevant decomposition. I think that is the best approach.
4. Particular for the article in question: I suspect the author informally followed a reproducibility approach (they do comment that they did independent runs of this analysis to check it reproducibility) but did not fully explore a particular approach. Also some slightly lax reviewing was involved as the statement "Independent component analysis and PCA (...) were performed using built-in MatLab functions." is inaccurate; MATLAB does not have an built-in ICA function only a PCA one. For ICA a toolbox of their choice was used (Hyvärinen's FastICA which, granted, it is as close as one can get to a reference implementation but it is not the only ICA implementation). For that matter: FastICA does use an incremental approach in finding ICs and that implemenation does make it relevant to suggest there is a first component. That said, that it implementation dependent and other ICA implementations (e.g. JADE from High-Order Contrasts for Independent Component Analysis (1999) by Cardoso CuBICA: independent component analysis by simultaneous third- and fourth-order cumulant diagonalization (2004) by Blaschke & Wiskott) do not have such an approach and make the notion of "first IC" arbitrary (as it should - and therefore we have point 3 :) ).