Francis A. Sullivan, SJ, in his The Christology of Theodore of Mopsuestia, published in 1956, has a nice way of getting a sense of the terrain of a large amount of patristic Christology. A major argument by the Arians against the Nicene position was to challenge the supporters of the latter to explain how the one subject to passion and death could be consubstantial with God, who is impassible. Sullivan puts the core of this challenge into syllogistic form (p. 162):
MAJOR: The Word is the subject even of the human operations and sufferings of Christ.
MINOR: Whatever is predicated of the Word must be predicated of Him according to nature.
CONCLUSION: The nature of the Word is limited and affected by the human operations and sufferings of Christ.
The Arians then took this to indicate that the Word could not be divine, since the divine nature could not be so limited and affected. There were sects, less popular and influential (and in general regarded by all other parties as raving loons) that rejected this additional assumption by claiming that passibility could in fact be attributed to God.
The Nicene position, which accepts that the divine nature is impassible but rejects Arianism, requires rejecting the syllogism. The syllogism, interpreted as pretty much anyone would regard the natural interpretation, is valid. So the Nicene position requires rejecting either the Major or the Minor. This eventually -- not immediately, but eventually -- created a significant rift between, on the one side, Antioch and (a bit later) Constantinople, and, on the other, Rome and Alexandria, all of whom were firm supporters of the Nicene orthodoxy, but who split on the question of why the Arian argument was wrong.
The Antiochenes, beginning apparently with Eustathius of Antioch, rejected the Major, i.e., the claim that the Word is subject even of the human operations and sufferings of Christ. This effectively disposes of the Arian argument. It runs into some initial difficulty in terms of how to interpret certain Scriptural expressions, but the Antiochenes did take the trouble to address the matter, and the whole position finally achieves its strongest defense in the works of Theodore of Mopsuestia -- and it is a fairly strong defense. This is the reason for the peculiarity of Theodore's career -- while not completely avoiding all controversy on the matter, he effectively becomes the major Antiochene theologian, and one of the greatest theologians of the day; he lives a life widely respected and dies in communion with the Church and no aspersion at all on his reputation as orthodox; his writings are later condemned in harsh terms. There is no doubt of Theodore's commitment to Nicene doctrine or his opposition to Arianism, the major heresy of his day; there is no doubt that he put considerable thought both into the rejection of the Major nor -- what is perhaps more important for his generally good reputation at his death -- is there doubt about his sincerity and honesty in trying to take into account the objections people were occasionally already raising against it (to the extent that he sometimes sounds very much like later Alexandrian orthodoxy, although always giving the expressions an Antiochene interpretation). But there becomes no way to defend the Antiochene position against those objections which are worked out with increasing force, without the development of Nestorianism.
Athanasius, on the other side, clearly addresses the Minor, i.e., the claim that whatever is predicated of the Word must be predicated of Him according to nature, and the Alexandrians preserve this tradition. In particular, he takes the Minor to be equivocating, and distinguishes between what is predicated of the Word according to human nature and what is predicated of Him according to divine nature. This lets Athanasius preserve a robust doctrine of the Incarnation -- the Word actually does become flesh and dwells among us, since Athanasius can keep the Major -- but without muddling together the divine nature and the human nature and making the divine nature passible. It did have some difficulties to work out, such as how we predicate human attributes to the Divine Word, but Athanasius did so at least roughly, and later Alexandrians up to and including Cyril with slowly increasing sophistication.
The opposition between the Antiochenes and the Alexandrians is muddled a bit by the rise of the Apollinarian heresy, which vehemently opposed the disunifying of Christ that they saw in the Antiochene approach, and in fact anticipates some of Athanasius's response to it, but gets sidetracked with the assumption that a human being is a mind or spirit with a human body, and that therefore Jesus is the Divine Word because He is just the Divine Word with a human body. The Antiochenes opposed this for a number of reasons that were entirely right; and therefore it is because of opposition to Apollinarianism, which was on serious examination obviously untenable, that the Antiochene theologians came to be so certain that they were right. Some of Athanasius's most serious difficulties in argument were concerned with overcoming this by-then entrenched certainty that the options were either the Antiochene position or Apollinarianism; he had to argue that one could reject the Minor without being committed to the latter. This was not actually difficult to do in itself, but the assumption became so entrenched so quickly among those who were already inclined to the Antiochene position that making them see the point was often quite difficult.
These are not by any means the only issues or lines of influence, nor even the only important ones -- there are theological reasons why it eventually became so clear that the Alexandrian position had to be the right one that only become apparent when you consider how the two positions interact with other aspects of Christian doctrine or prayer. Famously, the question of Mary as the God-bearer, Theotokos, or Mother of God was the one that eventually did in the Antiochene position, but it was only an especially eminent case. The Antiochene position is very plausible considered only in itself, but it creates a remarkable number of problems when it comes to how it relates to other theological doctrines. Nonetheless, Sullivan's account of the period in terms of responses to the Arian syllogism gives a nice handle on a large portion of the theology of the Church Fathers.
