Consider this. Could a pigeon walk if you put his neck in a brace? I say no.
-- Birdman, San Diego
Of course the elves couldn't resist this one. They headed out with Grandma's new pillowcases and stalked flying rats for a day or so. They had an empty toilet paper roll ready just in case they caught one. Stick his head through it and see what transpires. I know this is the kind of direct, hands-on science you've all come to expect from Worldwide Alice LLC, and we were ready with plenty of data-recording gear to back up our conclusions. Turns out those suckers can move when they need to. So we're forced back on the work of others, in this case, two biologists in Canada who finally got tired of being asked this stupid question at cocktail parties and finally did something about it.
Why do those annoying flying rats bob-bob-bob-bob their heads? Hypothesis: Has something to do with vision; many other birds have stereotyped head movements related to vision. Pigeons have eyes set at the sides of their heads, giving them a visual field of nearly 340 degrees. The back of his own neck is about the only thing a pigeon can't see when he's standing still. When he's motoring along the ground, all that scenery would slide through his visual field, right and left eye "seeing" independently. Too much information! Turns out, pigeons see much as we do, in a series of "snapshots" (saccade) rather than in continuous motion. To keep our pigeon oriented, there's a place in his little pigeon brain where motor signals from his little pigeon legs meet visual signals from his little red pigeon eyes, then his little pigeon neck muscles zap into action. Head thrusts forward, head (and eyes) stay in place, he orients himself visually, body follows; head thrust, orient, body follows; repeat; repeat; repeat. The clever Canadians did their initial pigeon studies by putting the birds on little treadmills. The birds kept walking, but since the visual signals didn't change, they didn't bob their heads as they moved. So yes, birdman, a pigeon could walk if he was in a neck brace, but apparently only if he is on a treadmill.