An astika is someone who believes in the existence of God and a nastika is one who does not. This is the common man’s understanding about the terms astika and nastika. If this basis were to be applied to the दर्शन- darshanas (systems of philosophy) some of the astika darshanas like Sankhya would not qualify as astika darshana as they do not conceive of a principle called Isvara or God. They are able to account for the existence of the manifested world comprising of Sentient and non Sentient beings with merely two principles – प्रकृति and पुरुष (Prakrti and Purusha). So from the view point of philosophy the mere acceptance of existence of a permanent atma and non refutation of the Vedas can said to be basis for the classification of astika and nastika darshanas. This article, which was presented in a seminar at Madurantakam Patashala sometime in 2014, briefly discusses how each astika school (Nyaya, Vaiseshika, Sankhya, Yoga, Mimamsa and Vedanta) defines the Isvara tattva.
The link to the article is is https://drive.google.com/open?id=0B5TNRFElGMV1UFVvajNpYkRsLXZWMjFXbDNia05GMnBDdjVr