Force-Frequency: Twitch Summation to Fused Tetanus

Twitch summation reproduces the muscle force-frequency relationship: individual twitches (2nd-order critically-damped activation impulses) superpose into a smooth fused tetanus at 25 Hz, inside the cited fusion band 20–100 Hz. The fusion frequency tracks 1/τ across fast↔slow fibres — a verified, anchor-respecting result (code output).

Mean force rises monotonically with stimulation frequency and the fusion index falls below 0.10 at 25 Hz (band 20–100 Hz). Shortening the cited contraction time raises the fusion frequency in step (25 ms→50 Hz, 50 ms→25 Hz, 90 ms→13 Hz), confirming the 1/τ law with no tuning. The tetanus:twitch AMPLITUDE ratio is left uncalibrated [O]: linear superposition has no force ceiling.

Twitches that add up

A single motor spike drives a unit twitch modelled as the impulse response of a 2nd-order critically-damped activation→force cascade k(u) = (u/τ)·e^1−u/τ, peaking at the cited contraction time τ. Repetitive stimulation superposes these twitches; as the inter-spike interval falls below τ the ripples merge into a fused plateau.

Sweeping stimulation frequency, the steady mean force increases monotonically and the fusion index (ripple/peak) drops below 0.10 at 25 Hz, within the cited whole-muscle fusion band 20–100 Hz. The 25 Hz value is a code output that depends on its inputs: the 50 ms contraction time, the 0.10 ripple criterion and the frequency grid (…20, 25, 30… Hz). Rerun on a 1 Hz grid the same code gives 22 Hz; with a 0.05 or 0.20 criterion it gives 32 or 15 Hz. This is the discriminant: a non-summating actuator would not fuse.

Fusion tracks 1/τ (no tuning)

Across the physiological contraction-time range the fusion frequency moves inversely with τ: 25 ms→50 Hz, 50 ms→25 Hz, 90 ms→13 Hz. Faster fibres fuse at higher frequency, exactly as the 1/τ law requires, and the anchor was never adjusted to hit the band. Because the twitch kernel depends on time only through u/τ, fusion frequency × τ is fixed by construction (1250 ms·Hz here, up to grid rounding): the 1/τ tracking is a consistency check, not an independent test.

What is NOT claimed

The tetanus:twitch amplitude ratio from pure linear superposition has no contractile force ceiling, so it is uncalibrated and graded [O]. A bounded ratio needs Ca²⁺/cross-bridge activation-saturation kinetics whose half-activation constant the jamming substrate does not fix; tuning it to the cited ~3–5 would violate No-Tuning.