|MadSci Network: Cell Biology|
Good question, and the answer may surprise you.
First, it's important to think about where on the neuron you are looking, because the parts of a neuron are specialized for how they deal with current flow/voltage differences (that's actually part of what makes neurons so special). Second, it's also important to remember how small neurons are (I'll get to why in a minute).
In the early 1950s, Hodgkin and Huxley characterized the currents flowing through the squid giant axon (a big axon, almost 1 mm in diameter). In that system, they showed that the current flow during a voltage step was approximately 1 mA/cm2. That is on the high end for a neuron.
In fact, at a synapse, a "typical" response might only be a few pA (that's pico, or 10-12). You might wonder why that number is so small. You probably already know that a "normal" neuron will fire an action potential when it reaches a certain threshold of depolarization (above where the neuron "rests"). The action potential can result in a brief depolarization of up to 150 mV above that resting point. Here is where the small size of the neuron comes into play. If you think about what it would take to change the potential difference across a membrane enclosing a very small space (a typical neuron might be 15 um (micrometer) in diameter or 7.5 um in radius), you will see that it doesn't take a whole lot of charge difference (provided by ions) to change the potential (or voltage) in that volume. If changing the potential by so much is easy with the movement of few ions, then you won't need a lot of amps to do it.
I highly recommend Neuroscience for Kids as a place to pick up more of this kind of information. You might especially be interested in the section on action potentials.
I also used the following books in my quest:
Those are specialists books, but a good book for a person interested in neurons and neuroscience is The Amazing Brain by Ornstein and Thompson.
I hope this answered your question, or at least put you on the right path to find the answer. Good luck.
Try the links in the MadSci Library for more information on Cell Biology.