The dog is clearly described as being a product of nature and a creature familiar with the conditions, one that knows it shouldn't be traveling at this point but willing to put up with the stupidity of the man, at least for now. It had instinct that makes it "slink along at the man's heels" for now, despite the fact that it knows it should be close to a fire or curled up under the snow right now.
The dog does not communicate with the man with any more than his tail drooping down to signify his submission and perhaps a little bit of apprehension as well.
The dog is forced to go in front of the man in a couple of occasions and demonstrates the instinct that preserves him despite the ice forming on his legs, etc. The man helps the dog to clear the ice from between its toes, but doesn't realize the idiocy of his trek in the incredible cold.
Their relationship is further described as a "keen intimacy," where the dog was only used by the man so it didn't try to communicate its apprehension at the journey.