The plasma membrane of the skeletal muscle fiber is called a sarcolemma. The muscle fiber contains long cylindrical structures, the myofibrils. The myofibrils almost entirely fill the cell and push the nuclei to the outer edges of the cell under the sarcolemma.he connective tissue also attaches skeletal muscle to the skeleton and other tissues and transmits the force of a contraction to the moving part. Connective tissue binds skeletal muscle fibers in a hierarchical pattern. Individual muscle cell fibers are surrounded by delicate connective tissue called endomysium.