Muscle cells, also known as muscle fibers or myocytes, are the fundamental units of your muscles. Humans have three types of muscle: skeletal, smooth and cardiac. Your skeletal muscles are under conscious control, while your smooth muscle -- found in the walls of your blood vessels and your hollow organs -- and cardiac muscle are not. All muscle cells share four primary properties that distinguish them from other cells.
For a muscle to contract and do work, its cells must be stimulated, most often by the nerves supplying them. Nervous impulses cause the release of the neurotransmitter acetylcholine at the nerve-muscle junction, and the acetylcholine activates receptors on the surface of the muscle cell. This results in an influx of positively charged sodium ions into the muscle cell and a depolarization of the muscle cell membrane, which in the resting state is quite negatively charged. If the membrane becomes sufficiently depolarized, an action potential results; the muscle cell is then "excited" from an electrochemical standpoint.
In the case of skeletal muscles, muscle cells contract when stimulated by neural input; smooth and cardiac muscles do not require this input. When a muscle cell is excited, the impulse travels along various membranes of the cell to its interior, where it leads to the opening of calcium channels. Calcium ions flow toward and bind to a protein molecule called troponin, leading to sequential changes in shape and position of the associated proteins tropomyosin, myosin and actin. The upshot is that myosin binds to small strands within the cell called myofilaments and pulls them along, causing the cell to shorten, or contract. Since this is going on simultaneously and in a coordinated fashion in many thousands of myocytes at the same time, the muscle as a whole contracts.
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Most of your body's cells lack the capacity to stretch; attempting to do so only damages or destroys them. Your long, cylindrical muscle cells, however, are a different story. Muscle cells contract, and in order for them to retain this ability, they must accordingly possess extensibility, or the capacity to lengthen. Your muscle cells can be stretched to about three times their contracted length without rupturing. This is important because in a lot of coordinated movements, so-called antagonistic muscles operate such that one is lengthening while the other is contracting. For example, when you run, the hamstring in the back of your thigh contracts while your quadriceps are extended and conversely.
When something is described as elastic, this is simply a statement that it can be stretched or contracted by some amount above or below its resting or default length without damaging it, and that it will return to this resting length once the stimulus for stretching or contraction is removed. Your muscles require the property of elastic recoil for them to be able to do their jobs. If, say, your biceps muscles failed to recoil to their resting length after being stretched during a series of curling exercises, they would become slack, and slack muscles with no tension are unable to generate any force and are therefore useless as levers.