Cerebellum to muscle — the motor output, quantified

The §8 motor output, made quantitative: units recruit small-first and force rises monotonically; firing rate fuses twitches into a smooth tetanus along a saturating curve derived here from the measured twitch shape, with the tetanus amplitude set by a locked measured ratio (~3.9×); the stretch reflex divides a disturbance by (1+gain) ≈ 1/10.4; and the cerebellum's supervised error decays to zero with an opposite-sign after-effect. These structures are forced; absolute gains, ratios and counts stay open.

This chapter refines §8 by locking its structure and measuring it, while leaving every absolute number open. Recruitment follows the size principle, so force over drive is monotone; the force-frequency relation is twitch fusion — a saturating curve derived here from the measured twitch shape, with the tetanus amplitude set by the locked measured ratio (~3.9×); the dual code shows low force is recruitment-limited and high force rate-limited; the stretch reflex rejects a disturbance by the factor (1+gain), here ×10.4; and the cerebellar delta rule drives error from 0.400 to 1.6×10⁻⁵ with a per-trial factor 0.880 ≈ 1−lr, leaving an opposite-sign after-effect when the perturbation is removed. In the closed loop the reflex rejects a disturbance instantly and the cerebellum cancels the residual over trials, from 0.058 to 0.0001. The orders, shapes and signs are forced [F] and verified [V]; absolute gains, latencies, unit counts and force-per-Hz remain [O].

Recruitment: the size principle from Ohm's law

Force is built by recruiting motor units in ascending size order, and that order is derived, not assumed. A motor neuron fires when its synaptic current reaches its rheobase — the current Ohm's law sets as the threshold depolarisation divided by the cell's input resistance. Input resistance falls as a motor neuron grows, so a larger neuron needs more current to reach threshold — a higher rheobase — and is recruited later; small units come first, and the size principle follows from the biophysics. The constants here are not representative guesses but two cited measured datasets. Fleshman et al. (1981), recording cat medial-gastrocnemius motoneurons, report an input-resistance range of 0.8–5.1 MΩ and a rheobase range of 0.8–17.1 nA. From the resistance span alone, constant-threshold Ohm's law predicts a rheobase span of ×6.375; a single threshold depolarisation of 7.471 mV — the pool's geometric-mean cell, derived not tuned — places the predicted rheobases at 1.465 nA for the smallest cell and 9.339 nA for the largest, both inside the measured 0.8–17.1 nA range, so the magnitude is validated by containment rather than by fitting a target. The order and that containment are forced and verified. The same data also mark where pure Ohm's law stops: the measured rheobase span is ×21.375, exceeding the conductance-only prediction by ×3.353. Gustafsson & Pinter (1984) saw the same signature in paired cells — rheobase tracks input conductance, yet its range exceeds the conductance range by roughly twofold, because the threshold depolarisation itself rises with rheobase instead of staying constant. That drift, and the absolute force in newtons, stay open [O]; what this chapter locks is the recruitment order and the containment of the derived rheobases inside the cited measured range.

Force-frequency: twitch fusion into tetanus

Within a recruited unit, raising the firing rate fuses successive twitches into a progressively smoother contraction, and that saturating force-frequency relation is derived here with no free parameter. Each twitch is the analytical waveform of Fuglevand, Winter & Patla (1993) — the impulse response of a critically-damped second-order system, rising to a single peak and decaying — and summing that twitch train at a fixed firing rate gives a steady-state force whose fusion index, the ratio of its trough to its peak over one cycle, measures how fused the contraction is. Because the twitch carries a single timescale, its contraction time, the whole curve is universal in the dimensionless rate ν = f · T (firing rate times contraction time): the fusion index climbs from 0.001 at ν = 0.1, through 0.622 at ν = 0.5 and 0.884 at ν = 1, to 0.969 at ν = 2 and 0.998 at ν = 8, rising monotonically and saturating toward a fully fused tetanus. It crosses a stated well-fused criterion of 0.90 — a readable landmark on the curve, not a fitted target — at ν = 1.083. This reproduces the measured sigmoidal tension-frequency relation of Rack & Westbury (1969), where individual twitches at low rates fuse into a smooth tetanus as rate rises. What stays open is everything dimensional: the amplitude of a fully fused tetanus is the locked measured twitch-to-tetanus ratio (~3.9×, muscle- and species-dependent); the absolute fusion frequency is ν/T and depends on the contraction time, which ranges from tens of milliseconds to ~150 ms across muscles; and the sigmoidal plateau of the mean force needs nonlinear calcium summation, which this linear-summation model does not include. The dimensionless fusion shape is forced and verified; the absolute rate, the tetanus amplitude and the mean-force plateau remain open [O].

The reflex rejects disturbances

The stretch reflex is negative feedback, so it divides a length disturbance by one plus its loop gain. A unit disturbance with a loop gain of 9.4 is reduced to about 0.096 — a rejection of ×10.4 — exactly the setpoint control of §7 at the spinal scale. The sign (negative feedback) and the rejection form 1/(1+gain) are forced; the absolute loop gain is open.

The cerebellum learns and shows an after-effect

The cerebellum is supervised, so its error decays geometrically and leaves an after-effect. A climbing-fibre error drives a delta-rule correction, and the motor error falls from 0.400 to 1.6×10⁻⁵ with a per-trial factor 0.880 close to 1−lr; when the perturbation is then removed, the learned correction makes the output undershoot the target by the perturbation, an after-effect opposite in sign to what was learned. The geometric decay and the after-effect sign are forced; the absolute learning rate is open.

The closed loop and what stays open

Put together, the reflex and the cerebellum reject a disturbance on two timescales. The reflex gives instant partial rejection while the cerebellum learns a feedforward cancellation over trials, so the residual falls from 0.058 to 0.0001 — fast feedback plus slow feedforward. What this chapter derives and verifies is the recruitment order (from Ohm's law on the cited measured motoneuron data of Fleshman 1981 and Gustafsson & Pinter 1984), the monotone force-recruitment and force-frequency rises, the dual-code crossover, the reflex feedback sign and form, and the cerebellar decay and after-effect sign; what stays open, in the ledger, is the absolute newton force scale and every absolute gain, latency, unit count and force-per-hertz, none asserted as evidence.