The globe artichoke is a variety of a species of thistle cultivated as a food.
The edible portion of the plant consists of the flower buds before the flowers come into bloom.
The budding artichoke flower-head is a cluster of many budding small flowers (an inflorescence) together with many bracts, on an edible base.
Once the buds bloom, the structure changes to a coarse, barely edible form.
Another variety of the same species is the cardoon, a perennial plant native to the Mediterranean region. Both wild forms and cultivated varieties (cultivars) exist. (source)
So can rabbits eat artichoke plants?
They can eat the leaves which are fine for them, but they should avoid the stems and other parts of the plant.