What are the limits of a Muslim's ties with his non-Muslim relatives, especially a mother who shuns him, and is he sinful for not attempting to maintain ties with her? And how should his relationship with his non-Muslim siblings be?

It is obligatory to be dutiful to a non-Muslim mother, to treat her kindly, and not to sever ties with her. However, it is not permissible to obey her in disobedience to Allah, as Allah the Almighty says: (And if they endeavor to make you associate with Me that of which you have no knowledge, do not obey them, but accompany them in [this] world with appropriate kindness and follow the way of those who turn back to Me). Similarly, it is permissible to maintain ties with non-Muslim relatives, treat them kindly, visit them, and offer them gifts, as Allah the Almighty says: (Allah does not forbid you from those who do not fight you because of religion and do not expel you from your homes - from being righteous toward them and acting justly toward them). And among your duties to them is inviting them to Islam, for indeed, that Allah guides one man through you is better for you than red camels.