Which system runs perpendicular to the myofibril and functions with the triad as a microtransportation network to spread the action potential from the fiber's outer membrane inward to deeper regions of the cell?

The preferable options will be -

T-tubule system

Explanation:

T-tubules or transverse tubules are a part of the skeletal and cardiac muscle membrane systems. They are the cell membrane extensions that can penetrate into the center of those muscles. It contains a large amount of on channels, transporters, and pumps. It allows the transmission of the action potential, frequently. T-tubules also regulates the cellular calcium concentration.