Neurone - Transmission of Action Potential.

I'm having trouble getting my head around this, it doesn't really make any sense.

I understand the Na-K pump, and its role in the resting potential. I get the Action potential, how the stimulus changes permeability of the axon membrane to Na+ ions, thus increasing the PD from -65 mV to +40 mV, and how when the channel closes and K+ ions flood out to stabilise the electrochemical balance to reach the original resting potential.

But what I don't understand is how this whole sequence of action potential, can be transmitted across a neurone because the movement of ions is happening perpendicular to the supposed direction.

This is what I think happens but I'm not 100% sure and if anyone could correct me, please do and explain.

So the influx of Na+ ions inside the axon from the action potential, moves across the axon to a more negative potential area. This depolarises the surrounding negative area to positive due to the surge of Na+ down an electrochemical gradient. This then skips the stage of the action potential mechanism, where the permeability of Na+ ions is made available to diffuse them inside the axon, and goes straight into the stage where the K+ channels open to stabilise the electrochemical balance. So the same Na+ ions that flooded in from the first action potential move across the whole axon to the end.

Any help, much appreciated.