The A10 nerve clips are small objects worn by Eva pilots on their heads which aid in synchronization with the Evas. The Evangelion Death & Rebirth Theatrical Program booklet states:
The Evangelion are controlled mainly by linking with the cranial nerves of the pilot. Of these, the most important is this A^10 nerve. ... It is associated with higher order brain functions such as memory, cognizance and the execution of movement, and with emotions such as anxiety, fear, happiness and pleasure. In addition, it is also said to play an important role in affection between parent and child, and between lovers.
In Yoshiyuki Sadamoto's introduction to the bonus materials at the end of vol. 1 of the manga, he writes:
On the floor of the midbrain is the ventral tegmental system, that neurobiologists call region A10. Cells soaked in dopamine, certain emotions are processed here: such as the thoughts of two lovers - or a parent and a child. And it is the synchronization of the threads and bundles of A10 that splice pilot and Eva together; to become one entity, to fight. In other words, the power of love drives this weapon of mass destruction.
Asuka near-constantly wears hers, during her daily life and at school.