Final answer:
The Sarcoplasmic Reticulum (SR) is the network of membrane channels that runs parallel to the myofibrils and stores calcium ions, which are released during muscle contraction.
Step-by-step explanation:
The network of membrane channels that surround and run parallel to the myofibrils is the Sarcoplasmic Reticulum (SR). This specialized type of endoplasmic reticulum is critical in muscle physiology as it stores calcium ions (Ca++). During muscle contraction, the propagation of an action potential along the sarcolemma enters the T-tubules, which are invaginations of the sarcolemma.
These T-tubules ensure that the action potential can reach the SR membrane even deep within the muscle fiber. The close proximity of T-tubules to the SR facilitates the opening of calcium channels in the SR, releasing Ca++ into the sarcoplasm, thus initiating the contraction of the muscle fiber by affecting the sarcomeres, the contractile units of the muscle.