In all likelyhood atkins will result in sub-optimal muscle gains, as the insulin will be continuously suppressed. Atkins is good for fatloss though.
The purpose of the keto diets WITH carbs is to get insulin spikes and nutrient loading that comes with it, which results in a very anabolic state inside the muscle cell. Insulin is very potent that way. There are reasons to believe that these diets also train the muscle to increase its glycogen storage capacity, which of course results in a larger muscle.