Connective tissue from this outer sheath extends into the body of the muscle, subdividing it into columns, or fascicles (these are the strings in stringy meat). Each of these fascicles is thus surrounded by its own connective tissue sheath, which is known as the perimysium (peri = around).Dissection of a muscle fascicle under a microscope reveals that it, in turn, is composed of many muscle fibers, or myofibers.ach is surrounded by a plasma membrane, or sarcolemma, enveloped by a thin connective tissue layer called an endomy-sium (fig. 12.1). Since the connective tissue of the tendons, epimysium, perimysium, and endomysium is continuous, muscle fibers do not normally pull out of the tendons when they contract.