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How can "aesthetic" meaning "I breathe in" and "I gasp" have any relation to what the writer has explained below?

Context:

Indeed, the solving of a perceptual conundrum brings its own reward, and we experience what feels like a little stab of pleasure when we unscramble a confusing image to make sense of it, the internal top-down/bottom-up synthesis involved in visualisation making direct links to the limbic system, the brain regions concerned with physical processes, emotions and memories. The root meaning of ‘aesthetic’ in early Greek is ‘I breathe in’, ‘I gasp’ and it acknowledges this frisson of emotion.

(Art and Science, Siân Ede)

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    The author is just saying something about the etymology of the word aesthetic. – James Kingsbery Sep 18 '15 at 14:52
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I would say the origin of 'gasp' comes from the Indo-European root *gheis-d- which is also the root of geist - ghost/spirit. A very interesting and multifaceted origin, as befits its complex conceptuality

Geist. m. Old and Middle High German geist, Old Saxon gêst from West Germanic *gaista- m.: "supernatural character or being, emotional state;" also in Old English gâst, Old Frisian jêst. From the Indo-European root *gheis-d- : "to be out of one's mind" [au er sich sein], also in Sanskrit hîd-: "to be angry" [zürnen]; unexpanded in Avestan zaêsa-: "terrible, dreadful" [schauderhaft], Gothic usgeinan: "to be frightened" and usgaisjan: "to frighten" [erschrecken], Old Norse geiskafullr: "completely frightened." To the extensive family of *ghe/ghei- : "yawn" [gähnen], belongs also the sense "to open wide one's mouth", hence probably a derived root *gheies-: "Wide opening of the mouth"; the -d- is probably a short grade of *dô-: "to give." Therefore *gheis-d- would mean: "to bring about a wide opening of the mouth" [Mundaufsperrung herbeiführen]; "to make someone open his mouth wide."

http://www.albany.edu/offcourse/july99/hache.html

also https://philosophy.stackexchange.com/a/11396/5154

gasp (v.) late 14c., gaspen, "open the mouth wide; exhale," of uncertain origin, perhaps from Old Norse geispa "to yawn," or its Danish cognate gispe "gasp," which probably are related to Old Norse gapa "open the mouth wide" (see gap (n.)). Related: Gasped; gasping.

http://www.etymonline.com/index.php?term=gasp

It is certainly interesting to see it linked to aisthetikos. Of course, spirit well suits the OP's expression about emotions. I would have also looked to pneuma in this analysis.

I will endeavour to develop this answer further in due course.

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I am more familiar with the Eastern approaches to breath, so there will admittedly be speculation here regarding potential Greek mindsets. In many Eastern philosophies, such as those at the root of Kung Fu or Yoga, everything gets tied to the breath as part of the flow of life. In such systems, breathing in is often strongly associated with an acceptance of the world around you, so from that vantage point, this "breathing in" or "gasping" might be associated with suddenly letting a deep unresolved truth resolve itself, and inviting it in.

This reading would be validated if, in later paragraphs, Siân Ede chooses to describe a process of sharing that which you have resolved with the world as an exhaling activity. The Eastern approach to breath very rarely includes only one half of the process; it almost always covers both inhale and exhale.

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One of Greek root words she's referring to is aisthanesthai [1]

which means "to perceive (by the senses or by the mind), to feel,",

By top-down/bottom-up synthesis she means that there is a conscious process of perception, the top-down, in which you observe, as well as a bottom-up process, in which objects exert affects [2] on you. These two processes link cognitive aspects of experience (even in abstract) to emotional/sensory aspects.

When we discover one of these links between the cognitive and the sensory, there is a stab of pleasure, an 'aha!' moment of understanding, when we recognise some pattern in our experience.

the easiest way to understand what that means is to consider music. We can immediately recognise when a song is sad or happy, because those songs imitate the way we feel when we are sad or happy. If you're excited your pulse starts to race, similarly up tempo songs start to make us feel excited. so there is a sense in which, not only do we understand a song cognitively, we also understand it physically, in terms of the affect it produces.

in terms of your question, could you give a bit more context? I feel like I need to read the paragraph that came before this one in order to answer what relation 'aesthetic' has to the argument, but from what is there she seems to be making some argument related to, maybe why we seek knowledge and understanding? at the very least she's talking about the pleasure we feel when we solve or understand something. The paper you're quoting from is also called 'Art and Science' so maybe she's trying to show that science has aesthetic qualities just like art?

So as far as 'Aesthetic' and 'I breath in', 'I gasp' acknowledging this frisson of emotion I think she is just referring to the fact that it's root words mean both to perceive and to feel. In other words, mind and body. So 'aesthetic' is word for understanding with both mind and body. Its important to understand that this does not mean understanding in the mundane sense of what we already know, but the kind that happens very infrequently in moments that seem like intuitive leaps, moments of spontaneous understanding that produce intense emotional shivers of excitement (think Archimedes in the bathtub), and are in some sense the aesthetic side of science.

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