Action potentials are mediated by electric currents and can be modeled by electronic circuits.
An electric current is the flow of charge. Therefore, action potentials are mediated by current flow. However, action potentials are mediated by the flow of ions across the membrane (Fig. 1), whereas current flow in electric circuits is typically mediated by the flow of electrons (Fig. 2).
Fig. 1. Action potential propagation. Source: Chen, Carnegie Mellon.
The action potential propagates in the direction of the dendrite. However, the underlying current flow is perpendicular to the axonal membrane and is mediated by ion channels. These ion channels are activated through the sensing of the electrical field. For example, voltage-operated Na+ channels are activated by a small depolarizing potential, while voltage-gated potassium channels open during the later phase of the action potential by a large depolarization.
Whatever the differences, an axon can be well modeled by a parallel resistor-capacitor circuit powered by the membrane potential. The electrochemical properties of the cell membrane makes it equivalent to a circuit with a set of resistors (which are voltage-dependent), batteries (whose voltage depends on ion concentration differences), and a capacitor all connected in parallel (see Fig. 2).
Fig. 2. Electronic circuit representation of the axon. Source: Yale University.
As to your question on charge balance; neutrality is maintained, because Na+ and K+ flow are going in opposite directions and neutralize the net flux of charge.