Epidural electrical stimulation (EES) of the lumbosacral spinal cord can re-enable stepping below a spinal cord injury by exciting the intrinsic locomotor circuitry when descending drive from the brain is lost. This 2D version runs the exact same threshold dynamics as the 3D model, plotted as live oscillator traces instead of an anatomical scene. Two mechanisms combine:
Recruitment (Henneman's size principle):
I_pool(x) = A · exp(-(x - x_electrode)² / 2σ²), σ ≈ 0.5 + 0.05·A
unit i fires once I_pool exceeds its threshold Tᵢ (small, low-threshold units recruited first)
Current amplitude A and electrode position x set how much depolarizing current reaches the flexor pool (rostral, ≈L2–L4) versus the extensor pool (caudal, ≈L5–S1).
Half-center CPG (Matsuoka oscillator), one pair {flexor, extensor}:
τ·ẋᵢ = -xᵢ - β·vᵢ - w·yⱼ + uᵢ
τ'·v̇ᵢ = -vᵢ + yᵢ
yᵢ = max(0, xᵢ) (i,j = flexor,extensor; j is the other neuron)
Each side's tonic drive uᵢ is the sum of a small residual descending drive (set by Injury completeness — 100% = no spared supraspinal input) and the stimulation-evoked drive from recruitment above. Reciprocal inhibition (the −w·yⱼ term) and self-adaptation (vᵢ) make the pair burst in antiphase — the classic half-center explanation (Brown, 1911) for alternating flexor/extensor rhythm. Below its own threshold drive the oscillator's fixed point is stable and both traces sit flat at zero — a genuine threshold, not a scripted cutoff; above it, the fixed point loses stability and a limit cycle (alternating bursting) emerges from the same equations.
- Amplitude / position — control how much of each motor pool is recruited (size principle) and where along the cord the current is centered.
- Frequency — below ≈15 Hz the CPG drive is delivered as discrete pulses (choppy stepping); the 20–50 Hz band used in real EES protocols drives smooth, well-fused alternation; above ≈60 Hz reciprocal inhibition breaks down and both pools co-contract instead of stepping.
- Injury completeness — near 100% (motor-complete injury) the legs stay still without stimulation; lower values leave enough spared drive for some rhythm even at zero amplitude.
This mirrors the mechanism behind published human locomotor-EES trials (Harkema et al. 2011; Angeli et al. 2018; Wagner et al. 2018): the spinal circuitry below the lesion still contains a working step-generator — stimulation supplies the missing excitability to switch it on.