The Mishnah Berura (note 7) cites the custom of reciting at least the first pasuk (verse) of the Shema right after the ‘birchos hashachar’. (Ed: See footnote, pp. 28-9 in the Artscroll Siddur,for a fascinating history behind this practice!)
Regarding ‘Elokai, neshama’, one immediately notices a departure from the standard format of a bracha: it has no opening (Heb: ‘p’sicha’) of ‘Baruch Atah…’! Why not? Hold that question for a moment while we address another: Why does the siddur place this bracha on page 18, after ‘Birchos HaTorah’, but writes a footnote on page 14 that some people recite it immediately following ‘asher yatzar’? What difference does it make where it is recited?
If you guessed that these two questions are inter-related, you’re exactly right! Basically, there are two reasons that a bracha would not start with an opening of ‘Baruch…’:
1) It was instituted as a ‘bracha has’mucha l’chaverta’ – i.e. to follow on the heels of a preceding bracha (such as brachos #2-19 of the Shemoneh Esrei and brachos #2-3 of ‘Birkas Hamazon’); OR
2) It is a bracha of thanksgiving (‘hoda’ah’) – (Ed: I was not real familiar with this reason.)
So here’s the thing about the placement of ‘Elokai, neshama’: if you subscribe to reason #2, then ‘Elokai, neshama’, which is indeed a bracha of ‘hoda’ah’ (i.e. as distinct from a bracha of ‘shevach’, praise, of which there are many more than the former) can really be placed anywhere – even after the paragraphs of Torah study which follow the ‘Birchos HaTorah’ (see p.16). However, in case you want to “play it safe” and satisfy reason #1, then you place it directly after a preceding bracha, namely ‘asher yatzar’! That is why some say it on page 18 and some say it on page 14!