Does anyone know the deeper reason to why we place the straps of the tefillin shel rosh in a spot where the amygdala is?
The Alter Rebbe in his Shulchan Aruch HaRav (Chapter 25: Hilchos Tefillin) writes that the reason we put the tefillin shel rosh in a place near the brain, has to do with the fact that we need to subjugate our soul, which abides in the brain, according to the Alter Rebbe (this is also explained in Tanya, Likutei Amarim chapter 9).
When putting on tefillin, one should keep in mind that the Holy One, blessed be He, commanded us to write these four passages which speak of the Unity of [G‑d] and the Exodus from Egypt, and to place them on the arm, opposite the heart, and on the head, over the brain.
This we should do in order to remember the miracles and wonders which He wrought on our behalf and which reflect His Unity, [demonstrating] that He has the power and the sovereignty to do as He desires in the upper realms and in the lower realms. One should [therefore] subjugate his soul, which abides in his brain, to the Holy One, blessed be He, and the desires and thoughts of his heart to His service. Thus, by putting on tefillin, he will remember the Creator, blessed be He, and minimize his pleasure-seeking.
The Alter Rebbe writes in Tanya (Likutei Amarim, chapter 9):
But the abode of the divine soul is in the brains that are in the head, and from there it extends to all the limbs; and also in the heart, in the right ventricle wherein there is no blood, as is written, "The heart of the wise man is on his right." It is [the source of] man's fervent love towards G‑d which, like flaming coals, flares up in the heart of discerning men who understand and reflect, with the [faculty of] knowledge of their brain, on matters that arouse this love
From there, e.g. from the place in the brains where the abode of the divine soul is, is where the source of man's fervent love towards G-d is.
A similair explanation can be found in the commentary of the Kav HaYashar (68:1):
There is also a third aspect of the soul called the nefesh, which is a lower level than the ruach. The seat of the neshamah is in the brain, near where the head tefillin are placed. The seat of the ruach is in the heart, across from the tefillin that are worn on the left arm.
In Derech Hashem it says:
Moreover, there are two main organs in a person from which the soul derives great strength. And they are the brain and the heart. And the Creator, may He be blessed, commanded that this light first be brought down into the brain through the head tefillin and that the brain, and the soul in it, be refined through it. Then it expands afterwards to the heart through the arm tefillin that is across from it, so that it too is refined by it. And through this, it comes out that man is completely included in the drawing down of this holiness to all of his characteristics; he is crowned by it and attains a great holiness.
This all might be the reason why the evil kingdom of Rome once issued a decree that one who was wearing the tefillin shel rosh, was going to be pierced through the brains, as the Gemara notes (Shabbos 49a, cited by the Mei HaShiloach, volume 1, Devarim, Ki Teitzei)
The Gemara asks: And why did they call Elisha Man of Wings? Because on one occasion the evil kingdom of Rome issued a decree against Israel that, as punishment, they would pierce the brain of anyone who dons phylacteries.
The Malbim, in his commentary to Yechezkel 13:18 writes:
הוי למתפרות כסתות, שכמו שבקדושה מניחים תפילין על היד כנגד הלב לשעבד תאות ומחשבת הלב לה', ומניחים על הראש כנגד המוח לשעבד כח השכל והמחשבה אליו יתברך
In his commentary, the Malbim notes two reasons why we put the tefillin set in the places we put them. 1) arm-tefillin; because of our desires that are coming from our hearts, 2) head-tefillin; because chochmah (wisdom) comes from the brain and this way we make it subservient to G-d.
The same idea is brought down by the Maharal in his Netzach Yisrael (23:8)