The Purusha medha

purush sukta itself give detail explanation of how purush medha is performed. but basic idea is the animal tied as offering in purush medha is purusha itself. and this purusha doesn't mean a person but the aatma as the purush sukta is regarding the purusha which pervades entire universe. 

Verse VI:
When the devas (gods) performed a sacrifice with the Purusha as the offering, the season of spring was the molten butter (used in all sacrifices), summer the sacred fuel and autumn the oblation.

Verse VII:
When the devas performed the sacrifice, in which they bound the Purusha as the sacrificial animal, there were seven poles in the sacrificial enclosure and three times seven logs of sacred firewood.

Verse VIII:
They placed Him, the Sacrifice, Purusha, the firstborn, on the sacrificial grass and besprinkled Him with the sacred water. Then the devas, the sadhyas and the rishis performed the sacrifice with Him as the offering.

Verse IX:
From that sacrifice, in which everything was offered as an oblation, was collected fresh butter. Creatures living in air (birds) and those living in forests and villages came in to being.

Verse X:
From that sacrifice, in which everything was offered as oblation, came the Riks (Vedic Hymns) and the Samans (Hymns which are sung - the Saama Gaana). The Chandases (Vedic Metres) came from it; and from it the Yajus (mantra for the ritual) was born.

Verse XI:
Horses were born of it, and animals with two rows of teeth. Cows were born of it, and so were goats and sheep.

though it appears from first reading that some animal is tied in yajna, but it's context itself says that the purusha is aatma of universe. and if one has to perform purush medha, it does in it's aatma. some other terms like names of animals may be confusing in this context, so the meanings of asva ( horse ), aja ( goat), avi ( sheep ) and gau ( cow ) is as follows.

 real meanings of ashva are:
1)'ashnute adhvanam' ( one with high speed).
2)'ashnute vyapnoti' ( the which spread rapidly)
3)'bahu ashnaniti ashvah' ( one which eats too much is also called ashva).
in vedas, it's meaning have actually double significance, anything that moves with high speed is portrayed as horse animal. brahmana scriptures has called rays, fire, sun and even god as ashva. for example:
'sauyaryo va ashvah' (gopath brahman 2.3.19) - rays of sun is ashva.
'agnirva ashvah' ( shatpath brahman 2.6.2.5) - agni is ashva.
'ashvo it ishwaro wa ashvah' (shatpath brahman 13.3.3.5) - being in all over the world, god is ashva.
other synonym for ashva used in vedas are 'arva', 'vaaji', which mean restless and strong respectively.

similarly, aja do not mean goat, but voice or something that is born of agni, as sound is also said to be born from agni.
'vaak va ajah' ( shatpath brahman 7.5.2.21) - sound is aja.
'aagneyo va ajah' (shatpath brahman 6.4.4.15) - that born from agni is aja.

word 'avi' is used for sheep as well as protecting feature. 
shatpath brahman 6.1.2.33 says earth is 'avi' as it protects the praja. 

word 'gau' is used for that which moves. mostly it is used for rays, but portrayed as animal cow, as that which nourish the body. other meanings of 'gau' are;
a) worlds- 'ime loka gauh' ( S. Br. 6.5.2.17)
b) antariksh - 'antariksham gauh' ( A. Br. 4.15)
c) sun - 'gavo va aaditya' ( A. Br. 4.15)
d) food- 'annam vai gauh' ( T. Br. 3.18.3)
e) life - 'prano hi gauh' ( S. Br. 4. 3.4.25)
shatpath brahman 8.5.2.11 says, "gau is born of agni" ( aagneyo vai gauh )

similar meanings of animals are applied while performing animal sacrifices like ashvamedha etc.