Muscle Twitch and Tetanus Simulator

What is a muscle twitch?

A muscle twitch is the single brief contraction a skeletal muscle makes in response to one stimulus: after a short latent period the force rises to a peak and then falls back to rest. In this simulator each twitch has the shape a(t) = (t/T) e^(1 − t/T), which rises from zero, peaks at the contraction time T and relaxes smoothly, and the twitches of a train of stimuli add together. That adding is the whole of summation and tetanus: a twitch that begins before the previous one has relaxed starts from the force already there.

The shape is the impulse response of a critically damped system, which Milner-Brown, Stein and Yemm used for human motor unit twitches in 1973, and on which Fuglevand, Winter and Patla built their widely used model of a motor unit pool in 1993. Force cannot rise without limit, so the summed activation A, counted in single twitches, passes through a ceiling: F = Fmax[1 − (1 − ρ)^A], where ρ is the twitch-to-tetanus ratio. One twitch reaches ρ of the maximum, and every further twitch’s worth of activation closes the same fraction of the gap that is left. That ceiling is this model’s stated choice rather than a law of muscle.

The three phases: latent period, contraction and relaxation

The latent period lasts a few milliseconds. The stimulus has fired an action potential, which spreads along the muscle fibre and down its T tubules and releases calcium from the sarcoplasmic reticulum, but the force has not yet begun to rise. The simulator opens at 2 ms.

The contraction phase runs from the first rise in force to the peak. Calcium binds troponin, tropomyosin moves off the binding sites on actin, and myosin cross-bridges cycle and pull. Its length, the contraction time, is what makes a muscle fast or slow. Cooper and Eccles measured it in the cat in 1930 at about 40 ms for gastrocnemius, a fast muscle, about 100 ms for soleus, a slow one, and under 10 ms for the medial rectus that turns the eye inwards.

The relaxation phase runs from the peak back to rest, as calcium is pumped back into the sarcoplasmic reticulum and the cross-bridges let go. In most muscles it is slower than the rise. In this model it is set by the contraction time: a single twitch falls to half its peak 1.769 contraction times after it, 70.75 ms for a 40 ms contraction time, and it has no sharp end, so the phase strip under the stage stops where the twitch has all but relaxed.

Using the simulator

The stage is a force recording, like one taken from a muscle held at a fixed length. The marks under the time axis are the stimuli, and the coloured strip under the first one is a single twitch’s three phases, to scale. The dashed curve is a single twitch on its own, so you can see how far the train climbs above it. Play draws the run with a pen at a quarter of real time, the scrubber moves through it by hand, and the arrow keys change the frequency when the stage is selected.

The sliders set the stimulation frequency, the number of pulses, the share of motor units the stimulus recruits, the contraction time, the latent period and the twitch-to-tetanus ratio. The readouts give the peak force against a single twitch, the peak as a percentage of the whole muscle’s maximum, the largest dip between pulses, the fusion frequency and the twitch’s half-relaxation time. The plot underneath is the force-frequency curve: the peak and the lowest force of a long train at every frequency, which close up and meet at the fusion frequency.

Worked example: ten pulses at 20 Hz

The simulator opens on a fast muscle, with a 40 ms contraction time, a 2 ms latent period and a twitch-to-tetanus ratio of 0.2, given ten maximal pulses at 20 Hz.

• A single twitch peaks at F = 1 − 0.8^1 = 0.2, 20 percent of the maximum, 42 ms after its stimulus.
• The pulses are Δ = 1000/20 = 50 ms apart, which is 1.25 contraction times, so in the sums each earlier twitch is weighted by a further factor of q = e^(−1.25) = 0.2865.
• After the tenth pulse the two sums are S₀ = 1 + q + … + q⁹ = 1.4015 and S₁ = q + 2q² + … + 9q⁹ = 0.56274.
• The activation peaks T − ΔS₁/S₀ = 40 − 20.08 = 19.92 ms after the last twitch begins, at 450 + 2 + 19.92 = 471.9 ms.
• There it is A = S₀ e^(ΔS₁/(T S₀)) = 1.4015 × e^0.5019 = 1.4015 × 1.6519 = 2.3151 twitches’ worth.
• The force is F = 1 − 0.8^2.3151 = 1 − 0.5965 = 0.4035, which is 40.35 percent of the maximum and 40.346/20 = 2.017 times a single twitch.
• Just before the tenth pulse the force is down to 34.73 percent, from the 40.34 percent it reached after the ninth: a dip of 1 − 34.73/40.34 = 13.91%, so the trace is an unfused tetanus.

Those are the readouts at the opening settings. The sums have nearly reached their limits for a train that never ends, 1/(1 − q) = 1.4016 and q/(1 − q)² = 0.56279, which is why ten pulses at this frequency already sit on the force-frequency curve.

Wave summation: why a second twitch climbs higher

If a second stimulus arrives before the first twitch has relaxed, the second twitch starts from the force that is still there and climbs above a single twitch. This is wave summation, also called temporal summation. It is not a bigger action potential: a muscle fibre is refractory for only a few milliseconds, far less than its twitch lasts, so the second action potential is an ordinary one. What has changed is that calcium and force have not yet fallen back.

Set two pulses at 20 Hz to see it on its own. The second twitch begins 50 ms after the first, when the first is still at 0.9735 of its peak activation, and the pair peaks at 1.699 twitches of activation: 31.56 percent of the maximum, 1.578 times a single twitch. Spread the pulses out and the gain shrinks. At 5 Hz the peak of a ten-pulse train is only 3.69 percent above a single twitch, and the readout calls them separate twitches.

A twitch on its own never reaches the force the fibres could make. Calcium is released and taken back up so quickly that the cross-bridges stop pulling before they have taken up the slack in the elastic tissue between them and the tendon. In a train the calcium stays high, the elastic tissue stays stretched, and the force can climb towards the maximum, which is why a tetanus is several times stronger than a twitch.

Unfused and fused tetanus

A train of stimuli keeps summation going. At moderate frequencies the force rises and falls with every pulse, which is unfused or incomplete tetanus, the sawtooth on the stage at the opening settings. Raise the frequency and the dips shrink, partly because the pulses close up and partly because the ceiling flattens the top: near the maximum, an extra twitch’s worth of activation adds little force, so losing some between pulses costs little. When the dips are too small to see, the trace is fused or complete tetanus, one smooth contraction.

At 100 Hz the ten pulses of the opening train fuse and climb to 83.23 percent of the maximum, 4.161 times a twitch. That is short of the 91.22 percent a long train at 100 Hz reaches, because ten pulses 10 ms apart are over in 90 ms, a little over two contraction times, and the force has not finished climbing. Raise the number of pulses and the dot on the force-frequency plot rises to its curve. No train can pass 1/ρ times a twitch, which is five times at ρ = 0.2.

The fusion frequency, and why slow muscles fuse sooner

Fusion has no sharp edge, so the simulator draws one: a train counts as fused when its force falls by less than 1 percent between pulses, and the fusion frequency is the lowest frequency at which a long train manages that. Because the whole twitch scales with T, the fusion frequency is a fixed number divided by the contraction time. At the default ratio that number is 2.3637, which gives 2.3637/0.040 s = 59.09 Hz for the opening muscle and 2.3637/0.100 s = 23.64 Hz for a muscle as slow as soleus.

So the same train can be unfused for one muscle and nearly fused for another. Set the contraction time to 100 ms and the opening train of ten pulses at 20 Hz sums to 3.459 times a twitch, with dips of only 1.129 percent, just short of fusion, because each pulse now lands half a contraction time after the one before rather than 1.25. Real muscles keep the same order: slow postural muscles such as soleus fuse at much lower frequencies than fast ones, and the eye muscles, the fastest of all, need the highest.

A short train can look fused below the fusion frequency, because its dips grow as the train goes on and the force nears its plateau. The trace readout says when that is the reason.

Recruitment: grading force by stimulus intensity

Each muscle fibre, and each motor unit, obeys the all-or-none law: a stimulus that reaches its threshold makes it twitch fully, and one that does not leaves it at rest. A whole muscle still gives a graded response to a graded stimulus, because its motor units have different thresholds. A weak stimulus to the nerve recruits a few, a stronger one more, and a maximal stimulus all of them. The motor units recruited slider stands for that: half the units give half the force with exactly the same timing, so the gain and the fusion frequency do not change.

The threshold belongs to the membrane. The resting membrane potential simulator shows where the resting potential comes from, and the action potential simulator shows the all-or-none spike that each stimulus here starts, in a nerve rather than a muscle fibre.

In the body the nervous system grades force both ways at once, by recruiting more motor units and by firing each one faster, which is called rate coding. In a voluntary contraction many motor units fire at rates that would give an unfused tetanus on their own, but out of step with each other, so their ripples cancel and the total force is smooth. Voluntary recruitment runs from the smallest motor units to the largest, the size principle, which is the reverse of an electrical stimulus to a nerve: there the largest axons have the lowest thresholds and are recruited first. The upper limb muscles explorer shows in 3D the arm and forearm muscles whose motor units work this way.

Why the heart cannot be tetanised

Cardiac muscle cannot sum its twitches into a tetanus. Its action potential has a long plateau, and ventricular muscle stays refractory for about 0.25 to 0.30 s, nearly the whole of its contraction, so a second contraction cannot begin until the first is almost over. That is what lets the heart relax and refill between beats. The ECG rhythm simulator shows the electrical side of that rhythm, beat by beat.

What this model leaves out

• Different twitches in one muscle. Every motor unit here has the same twitch. Real muscles mix fast and slow units, so a whole muscle’s twitch is a blend and its force-frequency curve is broader.
• Relaxation set on its own. Relaxation here follows from the contraction time, and the twitch never quite ends. Real muscles vary the two separately, and many relax faster than this shape does, which would raise their fusion frequency above the model’s.
• Treppe and potentiation. A rested muscle stimulated repeatedly gives twitches that grow over the first few, the staircase or treppe effect, and a tetanus leaves the twitches after it stronger for a while. Here every twitch is the same.
• Fatigue. A long tetanus fades as the fibres tire. Here the force holds for as long as the stimuli last.
• Length and shortening. The recording is isometric, at one length. Real force depends on the overlap of actin and myosin, and a muscle allowed to shorten makes less force the faster it shortens.
• The true ceiling. The saturating curve captures that force cannot pass what all the cross-bridges can hold, not the details of calcium binding or of the elastic tissue in series.
• Temperature. Twitches speed up as a muscle warms, so a cold muscle fuses at a lower frequency.

Common mistakes

• Thinking a stronger stimulus makes each fibre contract harder. Each fibre gives an all-or-none twitch. A stronger stimulus recruits more motor units; it does not make the ones already firing pull harder.
• Thinking action potentials summate. Twitches sum; action potentials do not. Each is over in a few milliseconds, long before the twitch it triggers has ended.
• Adding twitch peaks to predict a tetanus. Ten twitches do not give ten times the force. At the opening settings ten pulses give 2.017 times a twitch, and no train passes five times.
• Confusing the two meanings of tetanus. Physiological tetanus is a normal, sustained contraction. The disease tetanus is caused by a bacterial toxin that blocks the inhibition of motor neurons, so muscles go into spasm.
• Reading the latent period as idle time. The force is flat, but the action potential is spreading and calcium is being released: it is excitation-contraction coupling, not a pause.
• Taking a fused tetanus as the strongest. A smooth trace is not the most force. At the fusion frequency of 59.09 Hz a long train reaches 76.41 percent of the maximum, and higher frequencies still add force.