The idea that the brain is electrically powered is sort of a misconception. Neurons can be very long, and they do conduct signals, so the confusion is understandable.
The axons of neurons are more like a tube than a wire. On the outside of the axon there is a high concentration of potassium, and on the inside, a high concentration of sodium, and the imbalance of the charges of the Na and K ions sets up a voltage difference across the membrane of the axon.
When the nerve fires, small channels in the membrane open up, and diffusion carries the K ions in, and the Na ions out, which decreases the voltage difference across the membrane. This change in voltage causes the channels a little further down the axon to open up, which in turn triggers the next set of channels, and so on. So instead of a current running through a wire, the nerve impulse is more like a set of falling dominoes.
Gribbling and Purple are exactly right. Our brains simply don't operate like electrical devices, so we would be unaffected by an EMP. A regular electric shock, on the other hand, will pass through the entire body, and can mess with the voltages in neurons and muscles enough to cause problems, but that's something different.