I don't really have a dog in that fight. I leave it to the neuroscientist. I imagine a lot of this depends on what level of cognition we're talking about. Certainly sitting around and reasoning to yourself involves stimuli and responses in your brain - neurons firing, activating other neurons, awakening recognitions that call other thoughts to the fore.
Or something like that.
That sentence probably made actual neuroscientists cringe.
But the point is this seems like a semantic question in a lot of ways. I don't see why there has to be an inherent difference. If you want to highlight certain elements of the problem, perhaps you're interested in drawing a demarcation somewhere.
The point is that "reason" or "logic" is just the word we use to talk about cognitive processes that follow the rules of specific word-games that we pre-establish because when our cognitive processes abide (or even just remain semi-faithful to! first order approximations are great!) by those rules we get good results. "Reasoning" (I don't think) isn't different on a neurological level than other sorts of brain activity.
I think Current has it exactly right when he writes: "It may be true that a human and some other creature both respond in fundamentally mechanical ways to inputs, but come to different conclusions. That doesn't affect types of reason or logic that apply to a problem regardless of the actor. "Logic" (or "Reason" or whatever) is an aspect of a proposed solution to a carefully formed problem, it's nothing to do with that solver. It doesn't require believing anything incredible about existing lifeforms."