Basic idea:

Sodium ions go into the cell and the electrical potential becomes more positive .
Potassium ions move out of the cell and the electrical potential becomes more negative.

Because these processes influence themselves and one another, and these changes happen with an associated time constant (fast, slow, etc.), the result is a transient change in electrical potential in the cell, like a "blip" or a "spike" of work potential. The current is created in a way that allows it to move away from the place that it originated, and therefore allows the cell to influence other cells that are far away.
More Precise Questions that Math Can Help Answer:

What will happen if you increase/decrease extracellular ion concentrations?
Why are action potentials all-or-none?
What balances of parameters will create an action potential?

It wouldn't be completely honest to say that you can't get some understanding of these questions without completely understanding the mathematics involved, but by understanding the mathematics, you can gain a deeper, more reliable understanding. After all, mathematics is just the process of making your understandings of the relationships between things more precise and self-consistent.

References:
Kandel, E., Schwartz J., and Jessel, T., (eds.) "Principles of Neural Science. Third Edition," New York: Elsevier, 1991.

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