Half-Centre Gait Oscillator

What is modelled

Walking, swimming and breathing are rhythmic, yet a spinal cord cut off from the brain and from sensory feedback can still produce the rhythm when it receives a steady chemical drive. The simplest circuit that does it is a half-centre: two neurons that inhibit each other, each slowly tiring while it is active. One fires and suppresses the other until it tires; then the other takes over. Two such pairs, one per leg, with the flexors inhibiting each other, alternate left and right.

Equations

• Neuron τ dui/dt = −ui − β vi − w yj − c yk + s, adaptation τ′ dvi/dt = −vi + yi, output y = max(0, u) (Matsuoka 1985).
• Rhythm exists when 1 + τ/τ′ < w < 1 + β (Hopf condition of the linearised pair).
• The equations are homogeneous of degree one in (u, v, s): output amplitude is proportional to the drive s while the period is independent of it, and multiplying both time constants multiplies the period by the same factor.
• Without a rhythm both neurons settle at y = s/(1 + β + w); hip angle = 0.3 rad × (flexor − extensor).

Checks run in the page

• Doubling the drive doubles every state.
• Doubling both time constants replays the motion at half speed and doubles the period.
• Step halving of the solver.
• Rhythm condition against the simulated behaviour, and the resting output without rhythm.

Honest simplifications

This is a four-neuron caricature; real spinal circuits have many interneurons, sensory feedback that shapes the phase of each step, and different rhythm generators for different gaits. The stick figure moves its ground line at a speed proportional to the drive for effect, which is not derived from the model, and the knee bend is a simple function of the flexor output.

• Press Walking rhythm: left and right legs alternate. Then press Strong drive, same rhythm: bigger strides, same step rate.
• Press Fast limbs (short time constants): the rhythm speeds up in proportion to tau.
• Press Too little inhibition, no rhythm: the legs freeze at a steady output.