In computer science, there are various classes of "machine", which are basically constructs that can solve problems, and they are classed based on what set of problems they can solve. However, some machines are more powerful than others, as the more powerful machines can compute the results of the lesser machines, but not vice-versa.
However, of particular note is the machine at the top of the hierarchy,
Alan Turing's Universal machine. The reason this is important is that any Turing machine is
itself calculable by a Turing machine. Thus, any construction that can simulate a Turing machine (such as the computer you're viewing this on) can also simulate
every other Turing-machine-like construct.
The universe, and everything in it, must either be equivalent to a Turing machine, or be even greater than it. This becomes obvious when you consider that, if it weren't, we wouldn't be able to build Turing machines.
However, now we've ran into a problem. The only theoretical devices that are greater than a Turing machine is a
hypercomputer, and, as you can see from the page, all of them involve computing an infinite number of things in a finite time.
Since it's generally assumed that this isn't possible, we must conclude that the universe is a Turing machine-equivalent, and thus
simulatable by a Turing machine. Since simulating the universe involves simulating everything in it, the brain is thus simultable by a Turing machine with sufficiently large memory and/or a sufficiently long time. As mentioned earlier, your computer's processor is Turing-equivalent. As is
Conway's Game Of Life.