Appendix — Cellular Gaits

§ THE MATH

Everything we hand-set or evolved, written out. Each rule is one equation and one plain-language line saying what it means. Nothing here is a magic number — every constant comes straight from the source.

1 · The update rule (evolved)

For every cell p and its 3×3 Moore neighborhood (zero-padded at the grid border), the same tiny two-layer network runs everywhere. gis the gain — the playground's knob, normally 1.

a_{o} (p) = b_{o}^{1} + i = 1 \sum 4 Δ \in 3 \times 3 \sum W_{o, i, Δ}^{1} s_{i} (p + Δ)

h_{o} (p) = tanh (g \cdot a_{o} (p))

s_{c}^{'} (p) = clip (b_{c}^{2} + o = 1 \sum 16 W_{c, o}^{2} h_{o} (p), - 1, 1)

In words: gather a cell and its eight neighbors (conv1: 4 → 16 channels, 3×3, pad 1), squash through tanh, then mix back down to the new cell state (conv2: 16 → 4, 1×1) and clamp to $[- 1, 1]$ . The whole rule is $16 \cdot 4 \cdot 9 + 16 + 4 \cdot 16 + 4 = 660$ numbers — the vector CMA-ES evolves.

2 · Motor mapping (hand-set)

u = vec (s_{0} [0 : 7, 0 : 6]) \in [- 1, 1]^{42}, q^{*} = clip (u, - 1, 1) \cdot 3.14 rad

Channel 0 of the top-left 7×6 sub-grid, read row by row, gives 42 numbers. Each becomes a joint target for one of FlyGym's 42 position actuators (kp = 50). Control runs at 250 Hz (dt = 0.004 s), with 40 physics steps at 10 kHz per control step and one CA tick per control step — so a 3-second rollout is 750 steps.

3 · Fitness (hand-set objective)

F = (x_{thorax}^{end} - x_{thorax}^{start}) - 0.05 N_{below}, z_{thr} = 0.5 z_{thorax}

Reward forward distance walked; subtract a small penalty for every control step the thorax sags below half its standing height ( $N_{below}$ = number of such steps). Distance is measured after a short warm-up so it reflects control, not the initial settle.

4 · Evolution (CMA-ES)

θ^{*} = ar g θ \in R^{660} max F (θ) \equiv ar g θ min (- F (θ))

Because the score comes from a contact-rich physics simulator it is not differentiable, so we cannot take gradients. CMA-ES only needs to score candidates: population 32, initial step size $σ_{0} = 0.3$ , 50 generations. Best v1 reached $F \approx 86.6 mm$ .

5 · The criticality instruments (this analysis)

These were added to probe the controller — they are not part of evolving or running it. g is the gain from rule 1.

r (t) = \frac{1}{N} k = 1 \sum N (s_{t + 1, k} - s_{t, k})^{2}, N = 256

State-change rate: how far the whole grid moves in one tick (root mean square per cell). This is the number that stays pinned across the sweep.

\tilde{s}_{0} = s_{0} + ε e, d_{t} = ∥ \tilde{s}_{t} - s_{t} ∥, λ \approx \frac{1}{T} t \sum ln \frac{d _{t}}{ε}

\tilde{s}_{t} \leftarrow s_{t} + \frac{ε}{d _{t}} (\tilde{s}_{t} - s_{t}), ε = 1 0^{- 3}

Poor-man's Lyapunov exponent: start a twin a hair $ε$ away, watch the gap $d_{t}$ each tick, average the log growth — then rescale the twin back to length $ε$ every tick so the gap stays in the linear regime (and discard the first ~12 ticks). $λ < 0$ ordered, $λ \approx 0$ edge, $λ > 0$ chaotic.

§ SYSTEM DESIGN

The walking and training system at a glance — two coupled loops. The runtime control loop (green, every 250 Hz step) runs the NCA controller through the motor mapping into the MuJoCo fly body; the proprioceptive feedback arc is now closed — it runs across the perturbation, chemotaxis, and escape behaviors (the v1 walking shown across the other tabs stays open-loop). The training loop (amber, every 3-second rollout) scores the body with the fitness function, hands it to CMA-ES, and feeds the evolved parameters θ back into the controller. This diagram is the walking system; the embodied brain↔body loop — the real FlyWire connectome run as a spiking brain driving the body — is a separate system, walked out on the Embodied tab. Hover, tap, or focus any block to see the real model behind it.