pattern - Find the missing elephants

Determine color and direction of the missing elephants in the picture.

Here's a textual representation of the above.

G< R< R< G< R< M> G< R< R> M<

G< G< G> G< G> G< G< G< G> G>
G< M> M> G< G< ?? M> M> G< G<
Y< Y> G> G< Y< Y> Y< G> G< Y>
G< G< G> G> M< M> G> G> M< M<
R> M> ?? G< R> M< Y< Y> G> R>
R> R< R< R< R> R> R< ?? R> R<
R< G< R< G< R< G> G> R> G> R<
Y> Y> Y< Y< Y> Y< Y> Y< Y< Y<
Y> Y< Y< Y< Y< Y> Y> Y< Y> Y<

• R: red


• Y: yellow


• G: green


• M: magenta


• >: facing right


• <: facing left


• ?: gap

Hint 1:

The elephant herd has no boundaries. The right side joins the left side and the bottom joins the top.

Hint 2:

Elephants are social animals: they count on themselves and on their neighbours.

Hint 3:

With more fellows around, an elephant must see further.

Hint 4:

Eight are the neighbours. One is the self. Sight overflows. Where is the match?

Hint 5:

Almost everything in this puzzle is random.

Answer

OK. I finally figured out this accurs slightly frustrating puzzle. I shall forever curse the name of GOTO for more than an hour spent convolving kernels with elephant herds in vain attempts to find some sort of linear invariant.

The missing cells, in standard reading order, are

green elephant facing right
yellow elephant facing left
red elephant facing left

The rationale is as follows:

An elephant's "gaze" must be fixed on an elephant of identical colour. Let $D$ be the number of elephants in the 3x3 cell block surrounding an elephant that share the elephant's direction (this includes the elephant him/herself). The grid is toroidal, hence the grid "wraps around" at the edges, from right to left, top to bottom. An elephant's gaze is fixed on the cell $D$ cells ahead of him/her in the direction (s)he's looking (respecting wrapping). We shall call this property "elephanticity".

The three colours and directions listed in the above spoiler are the only three combinations that yield elephanticity for all cells in the rows in which the empty cells appear as well as the rows immediately above and below them.