Apollinarianism was discussed in the video lecture below:
Grenz, Guretzki and Nordling explain that this view was the teaching of fourth century bishop of Laodicia, Apollinarius, that reasoned in the incarnation of Christ, God the Son, took on a human body but not a human mind or human spirit. (13).
The divine logos took the place of the human mind or spirit (nous νοῦς) (13).
In other words, the incarnate God-man, had a divine mind and spirit alone. (13).
Erickson explains that apollinarianism was an overreaction to Arianism. (714).
Apollinarius taught the dualism of the incarnate Jesus Christ having two complete natures, absurd. (714). Instead Jesus Christ was part human, but mostly divine. (714).
Therefore, the incarnate God-man was different than every other human being. (715). Christ did not have a human mind or spirit (nous νοῦς) (715). In this view, the centre of Christ's consciousnesses is the divine. (715). This serves as explanation of how Jesus Christ would remain sinless. (715).
Erickson correctly points out that with this view, the divine nature of Jesus Christ would swallow up the human nature. (715). It also would have Christ lacking human will, human reason, and human mind and this doctrine was condemned at the Council of Constantinople in 381. (716).
Being incarnate implies, fully divine, as in fully God, and fully human, I am in agreement with Erickson and Bruce Gore from the video that in one person, Jesus Christ possessed and still possesses one human nature, spirit/mind and one divine nature, spirit/mind.
The spirit/mind of God, although triune, is infinite and eternal. Jesus Christ's human spirit/mind is finite and everlasting, but not eternal as it was created in the incarnation. This explains, in part, why as God the Son, Jesus Christ in his humanity depended on God and God the Father for guidance. Jesus Christ was still fully God, but humbled himself as fully human.
Philippians 2: 8 New American Standard Bible
Being found in appearance as a man, He humbled Himself by becoming obedient to the point of death, even death on a cross.
There is a theological debate whether or not Jesus Christ had his sinful human nature purged from his humanity with the incarnation; or that Jesus Christ had the sinful human nature from the fall, and never accessed it.
Thiessen quotes Strong who states that the incarnation purged depravity from Christ. p.305. This view would be contrasted by scholars such as Mounce and Cranfield:
Cranfield in his Romans commentary p.176, comments on the likeness of sinful flesh. Cranfield states with what he thinks is the best explanation, that being that the Greek word for likeness is not to water down Christ's fallen human nature, as in being fully human, but is to draw attention that the fallen nature was assumed but Christ did not become a fallen human being.
So, unlike some views that reason Christ's sinful human nature was purged out at the incarnation, this view reasons it was there but because of his perfect obedience and I would reason deity, he did not become a fallen human being that sinned. So, in a sense in the likeness of sinful flesh, he had fallen human nature. In another sense in the likeness of sinful flesh, he did not have an active fallen human nature that would have led to sinful thoughts and choices.
Mounce in his Romans commentary sees it the same way on p.175-176, Christ took upon a fallen nature but did not become completely like us, as in sinners.
CRANFIELD, C.E.B. (1992) Romans: A Shorter Commentary, Grand Rapids, William B. Eerdmans Publishing Company.
ERICKSON, MILLARD J. (1994). Christian Theology, Grand Rapids, Baker Book House.
GRENZ, STANLEY J. DAVID GURETZKI and CHERITH FEE NORDLING (1999) Pocket Dictionary of Theological Terms, Downers Grove, Ill., InterVarsity Press.
MOUNCE, R.H. (1995) The New American Commentary: Romans, Nashville, Broadman & Holman Publishers.
THEISSEN, HENRY, CLARENCE (1956) Introductory Lectures in Systematic Theology, Grand Rapids, Eerdmans.