To give a direct example from your own body: the heat is capable of beating on its own without any signals from the brain. A region of the heart called the pacemaker generates a rhythmic signal which starts a wave of muscle contraction which runs across the heart and causes it to beat in a sequence which pumps blood (random or disorganized contraction of cardiac muscle is called fibrillation, pumps little to no blood, and is what those paddles you see in ever medical show are for). The heart is only involved in sending signals which tell the pacemaker to speed up or slow down. But the basic function of the heart itself runs on its own.

I'm a high schooler so I don't really know anything, but I'm pretty sure the way our brains work isn't much like a computer or robot at all (a comparison I feel is implied by words like "run" and "system"). It may perform computations, but at the end of the day, a computer is called a computer when it can run software to complete essentially any computation. If it can't do that, it's typically reclassified as a "calculator" or a "machine", or it has some specialized name. However, evolution is not concerned with the computations it doesn't need to make, so the brain is more like specialized "hardware" to perform specifically its tasks. To make a visual analogy, if "completing a task" is compared to "connecting point A to point B", a computer will take a "rectangular" path to leave room for further infrastructure and to make things more readable. However, readability and future iterations are no concern for a "machine" optimized purely by the force of killing ones that are not. It takes a direct approach, connecting A and B with a diagonal line. I don't know if that makes sense, but it's how I visualize the difference between a computer and the brain.

So simply "scaling it down" in the same way you can a computer, by putting in processors with less power but that do the same thing, doesn't make so much sense to me. A "smaller brain" would have to somehow do the exact same things, and that means it has to be intelligently built because simply "making it smaller" wouldn't accomplish nerve function in any way other than a seizure. The brain accomplishes its exact task. If nature could make it smaller and still do the same thing, it would. In fact, the reason humans are born so underdeveloped is that if we developed any further, our heads would be too big and kill the mother every time. Maybe human innovation could build something smaller out of neurons that functions the same, but I doubt it to be frank. Nature has been perfecting this thing for about as long as the brain has existed. You could probably put something that has an analogous function into a human skull (like a baby's brain in an adult's body or an engineered "lite" version of an adult' brain), but obviously we can see that the function would be at a lower level. Let's, of course, assume that we know how to "wire" the correct neuron outputs to the nervous system to make things connect properly.

When it comes to putting a smaller species' brain into a human's head, I think it's out of the question. Like I've mused, the human brain does exactly what a human brain should do. The function of, say, a monkey is completely foreign. It's almost like if you tried to take two people who speak different languages, and write code in different programming languages, to accomplish different tasks, and try to mash their code together and make it work together. The inputs and outputs just won't align in a way that makes coherent sense. It's possible we could map out every input and output in a monkey brain, and find every entry signal in our nervous system, and map connections so that the outputs more or less make sense, but that really only strikes me as possible because we're so closely related to monkeys. I don't see it as concievable for something like, say, an octopus.

I sound a lot more confident above than I really am. I should stress again that I am a high schooler, and I don't know much at all about neuroscience. The above is nothing more than musings derived from common sense, and lacks the nuance of work composed by someone truly educated in the field.

tl;dr, a "smaller" brain simply won't do the same thing without basically starting from scratch to engineer a new brain, if my musings are correct.