Randomize the Update Schedule

Most cellular automata in this book have used synchronous updates:

all cells observe state_t
all cells update
state_t+1 appears

Neural cellular automata are typically trained with randomized update schedules: there is no global clock, and cells cannot rely on exact phase relationships. The aim is robustness to timing, which we will measure rather than assume.

Stochastic firing

Instead of applying every predicted update, sample a binary mask:

def stochastic_update(x, dx, fire_rate=0.5):
    mask = (
        torch.rand(
            x.shape[0], 1, x.shape[2], x.shape[3],
            device=x.device,
        ) <= fire_rate
    )
    return x + dx * mask

Half the cells update on average.

Which half changes every step.

(Verified: fire_rate=1.0 reproduces the full update exactly; 0.0 freezes the state exactly; fixed seeds reproduce identical trajectories; mean fire fraction ≈ 0.5 over repeated draws.)

Put it inside the NCA

Chapter 38’s NeuralCA stays the owner of the architecture; here we extend it with scheduling rather than redefining it silently:

class NeuralCA(nn.Module):
    def __init__(self, channels=16, hidden=128, fire_rate=0.5):
        super().__init__()
        self.rule = LocalRule(channels, hidden)
        self.fire_rate = fire_rate

    def forward(self, x):
        dx = self.rule(x)

        mask = (
            torch.rand(
                x.shape[0], 1, x.shape[2], x.shape[3],
                device=x.device,
            ) <= self.fire_rate
        )

        y = x + dx * mask
        return apply_life_mask(x, y)

The rule itself remains deterministic for a given local state.

The execution schedule is stochastic.

That is a different claim from Chapter 9’s stochastic automata, where probability lived inside the transition law itself. Here the transition is deterministic and only its application is sampled — stochastic scheduling of a deterministic learned update, not a stochastic rule. The two chapters share the seed discipline, not the semantics.

Why this matters

A synchronized model can accidentally rely on exact phase relationships:

step 20: everybody emits signal A
step 21: everybody interprets signal A

Randomized updates make that fragile strategy unreliable.

The learned process must tolerate cells being slightly out of phase.

That pushes the system toward more local, self-correcting coordination.

Measure update-rate robustness

Do not train with fire_rate=0.5 and assume the model works everywhere.

Test, with an explicitly defined evaluation (rollout at a fixed rate, final visible loss):

def evaluate(model, fire_rate, initial, target, steps=64):
    model.fire_rate = fire_rate
    final = rollout(model, initial, steps)
    return float(F.mse_loss(final[:, :4], target))


for rate in [0.25, 0.4, 0.5, 0.6, 0.75, 1.0]:
    print(rate, evaluate(model, rate, initial, target))

This creates an update-schedule robustness curve — with the scope stated honestly: robustness to random skipped updates at these rates, not to arbitrary scheduler adversaries or fully asynchronous orderings. One update opportunity, three scheduling regimes:

    flowchart LR
    O[update opportunity] --> M{mask draw}
    M -->|fires| A[apply learned delta]
    M -->|skipped| S[state unchanged]
    A --> N[next state]
    S --> N
  

stochastic scheduling
  ≠ stochastic transition semantics

robust to random skipped updates
  ≠ robust to arbitrary scheduler adversaries

asynchronous-looking execution
  ≠ fully asynchronous model

Stochastic does not mean nondeterministic experiments

For reproducible evaluation, seed the generator:

torch.manual_seed(42)

For stronger experiment isolation, use explicit torch.Generator instances where practical and store the seed with the run metadata. (Verified: generator-seeded masks reproduce bit-exactly.)

The book’s recurring rule still applies:

randomness should be part of the experiment definition, not an invisible source of variance.

Local synchronization without a clock

The deeper idea is that coordination does not require a global scheduler.

Repeated local interactions can create enough effective synchronization for a global pattern to emerge.

This connects back to everything we have studied:

local information
local state
local update
        ↓
global organization

But now the organization has to survive timing noise too.

The next problem is harder still.

Once the target has grown, can the same local dynamics keep it there? The next chapter trains for persistence.

Research

• Mordvintsev, A., Randazzo, E., Niklasson, E. & Levin, M. — Growing Neural Cellular Automata (Distill, 2020). The canonical specification used here: per-cell stochastic update masks (p = 0.5 in training, framed as removing the global clock) applied to a deterministic residual update. Verify any scheduling claim against this mechanism before generalizing to “asynchronous.” https://doi.org/10.23915/distill.00023

• Berto, F. & Tagliabue, J. — Cellular Automata (Stanford Encyclopedia of Philosophy). Covers asynchronous updating as a legitimate modeling relaxation (Ingerson & Buvel) — the lineage that keeps this chapter’s Bernoulli skipping distinct from Chapter 9’s probabilistic rules. https://plato.stanford.edu/entries/cellular-automata/