The muscle in question is the masseter, a muscle that, in humans, runs from the forward part of the cheekbone down to the back of the jaw, and which is important in chewing - it is often particularly large in herbivores, for example. It has two parts, one deep and one superficial, and in rodents, one or both of these parts extend much further forward than normal, reaching up onto the side of the snout. Generally speaking, in squirrel-like rodents, its the superficial part of the muscle that does this, in guinea pigs and their relatives, its the deep part, and in mouse-like rodents, it's both. Because the shape of the skull has to be modified to allow for these attachments, it is possible to tell which of the three possible arrangements even a fossil species had.
But there is one rodent species alive today that doesn't have any of these features. (Actually, mole rats don't have the full muscle arrangement, either, but we can tell from the shape of their skulls that their ancestors once did). That animal is the mountain beaver (Aplodontia rufa), and it's only found along the Pacific Coast of North America. Despite the name, it is not particularly closely related to real beavers, and it doesn't look much like them, either, looking more like some kind of giant vole. Indeed, its a burrowing animal that doesn't like water, and usually only ventures above ground at night.
Although there is no question that it is a rodent, the complete lack of the muscle arrangements found in any of its relatives often result in it being described as the "most primitive living rodent." Although it seems to be more closely related to squirrels than anything else, it doesn't seem to have any particularly close relatives, and is the sole living member of its family, a rather strange animal not quite like any other rodent.
Or, at least, not like any other rodent that's alive today. Because this odd creature has to have come from somewhere, and it turns out that some of its extinct relatives were even stranger. While there are a number of fossils belonging to the mountain beaver family itself, there was also once another family of rodents belonging to the same evolutionary line, one that split from the squirrels very early on. These animals were the mylagaulids, and some of them had a feature that no other rodent has had before or since: they had horns.
Mylagaulines Other Mountain
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These were, like mountain beavers, burrowing animals. We can tell this from their powerful fore-claws, and because they had the sort of wedge-shaped head that is often found in animals that habitually push earth out of the way with their snouts (the golden moles being a particularly extreme example). They were quite large for rodents, and had small eyes, as one might expect for an animal that spends much of its time underground. Indeed, for the most part, thwy would have looked like over-sized gophers.
Except, of course, that gophers do not have horns. The mylagaulids' horns were not placed on the top of their head, as in cattle or antelopes, but rather on the tips their noses, like a rhinoceros. Unlike the horns of rhinoceroses, however, these had a solid bony core, and there were two of them, side by side, each stout and conical. It's not entirely clear hat the horns were for; it's been suggested in the past that they used them to shovel earth out of the way while digging, although newer research suggests that they may instead have fought with them..
Not all mylogaulids had horns, and there's some dispute as to how many really did. That's partly because most of what we know of the group comes from very fragmentary fossils - many species are known only from their teeth, which obviously makes it difficult to say what the rest of the animal might have looked like. We know for certain that one genus, Ceratogaulus, had horns, because we have a fairly complete skeleton for that (you may also find mention of Epigaulus in older books - it's the same animal), but, beyond that, it's less clear. A fossil recently described by Nicholas Czaplewski of the Oklahoma Museum of Natural History may change that, however.
The horns of this fossil are different from those of any Ceratogaulus species, because of the presence of an unusual groove along the back. On the basis of the shape of its teeth, Czaplewski instead places it in the closely related genus Mylagaulus - one of the ones for which we just never had the relevant part of the skull before. If he's right, horns may be more widespread in this unusual group than we previously thought.
However, some other fossils described in the same issue of the Journal of Vertebrate Paleontology may shed more light on the origins of the group. These do not belong to a new species, but they are far more complete than previously studied specimens. Jonathan Calede of the Unviersity of Washington and Samantha Hopkins of the University of Oregon place the fossils in the species Alphagaulus pristinus, one that has a key place in the evolution of these strange animals.
The more specialised mylagaulids, including all of the horned ones, belong to a more narrowly defined group, the mylagaulines, while those outside this group tend to be earlier forms less well adapted to a burrowing lifestyle. Alphagualus pristinus is significant in that it seems to be the most primitive of the mylagaulines, representing the transition from the less specialised ancestral species to the burrowing - and eventually horned - ones.
So, the animal was well suited for burrowing, although the authors point out that its adaptations to doing so were less extreme than those in the later, horned, forms. It seems to mark a transition to a new lifestyle, and shows that - as if often the case - not all the adaptations appeared at once in the group's evolutionary history.
Significantly, however, some of the fossils belonged not to adults, but to juveniles. We can tell this because the bones in their skulls had yet to fully fuse, and, together with other differences in the skeleton, that enables to learn something about how these animals grew up. In particular, the younger specimens had less extreme adaptations than those found in the older animals, with more slender arms and a rounder head. That means that the animals became more specialised for digging as they aged, something also seen in beavers and marmots. In this way, as they grew older, becoming large enough and independent enough to start digging their own burrows, the animals mirrored the development of their own ancestors over time.
[Picture from Wikimedia Commons. Cladogram adapted from Blanga-Kanfi et al.]