Self-Replicating
Neural Cellular Automata

1One network per pixel

Most neural cellular automata train a single update rule and share it across every cell. This substrate does the opposite: each pixel owns its own small network, and the only way to stay alive is to copy a living neighbour's.

The grid is W × H with two depth channels. The first is an alpha channel that says whether a cell is alive; the second holds hidden “chemistry”. Every pixel additionally owns an agent — a two-layer network that reads the depth-wise Moore neighbourhood and emits the cell's next state. A cell is alive only if it has both an above-threshold alpha value and a non-empty agent; an alpha value on its own cannot bring a cell to life, and an agent left in a cell whose alpha has fallen is zeroed along with the state.

A small fraction of founder cells is seeded uniformly at random and given Xavier-initialised weights. Everything else starts empty. Nothing is trained: there is no objective, no gradient and no fitness function. The only selective filter is viability — a genome whose output falls below the alpha threshold deletes its own cell.

Phenotype is never mutated directly. Cell states are produced entirely by the agents, so every observable change is a read-out of a change in the weights.

The agent, all 44 parameters of it

The input is the flattened depth-wise neighbourhood: 3 × 3 positions × 2 channels = 18 values. A hidden width of 2 and an output of 2 gives 44 parameters — 36 + 2 weights and biases in the first layer, 4 + 2 in the second. Only the 40 weights mutate; the biases are left untouched. Two hidden units is a deliberate lower bound rather than a tuned choice: it is the smallest layer that still admits a non-linear, non-separable map, so any structure that appears cannot be attributed to spare capacity. Capacity is also expensive here — a 200 × 200 grid holds 40,000 independent agents, about 1.8 million parameters, each deep-copied and mutated across 1000 generations and five folds.

2Clone and mutate

One generation is a single scan over every cell. Four things can happen to a cell, and which one happens decides whether its lineage continues.

Mutation only happens at birth

One detail of the released implementation is worth stating plainly, because it is easy to miss from the four-step summary. When a cell enters the inheritance branch, the reference code checks whether the cell's own network is still all-zero. If it is — the cell was dead, and this is a birth — the neighbour's genome is copied and mutated. If the cell was already alive, the neighbour's genome is copied unchanged. Only dead cells acquire new weights. The consequence is that the mutation rate is coupled to the death rate: a substrate with no turnover produces almost no new genomes, which is exactly what the unbudgeted runs in section 6 show.

Because inheritance requires a living neighbour, living cells create a local positive feedback: if my neighbour is alive, I have a better chance of living longer. That feedback drives the growth phase. Repeated clone-and-mutate steps then let ancestor–descendant lineages drift until a descendant is, by the coarse-graining used here, a different species.

4Six ways to measure diversity

All six tools are the same object: a coarse-graining φ that labels each living cell, and a functional D of the resulting label distribution. They differ only in what φ looks at and what D computes.

Phenotypic tools take φ to be the observable cell state rounded to one decimal place. Genotypic tools take φ to be a colour code of the agent's weight vector. For D the paper uses either richness — the size of the support — or normalised Shannon entropy, the q = 0 and q → 1 members of the Hill-number family, plus two dispersion statistics that summarise the same field by its spread.

“Species” here means one thing only: a label class of φ. A rounded cell-state type for phenotypic diversity, a genome colour class for genotypic diversity. There is no claim of reproductive isolation, and every count of species below is a count of occupied labels at a stated coarse-graining.

The six tools on common axes

This is the paper's Table 3. It is also the argument for treating GEP as the headline phenotypic signal: it is the only phenotypic tool here that is simultaneously bounded, normalised by a fixed |Σ|, and responsive to the whole type distribution rather than to one summary statistic of it.

5A full genome beats three loci

Prior work visualises a few random gene values directly as an RGB triple. An RGB encoding admits three, so three is exactly the practice under test — and three of forty turns out to be far too few.

GHC hashes an agent's entire weight vector to 24 bits and splits it into an (R, G, B) triple. RWSP fixes three weight indices for the whole run and colours each cell by those three values only — a genetic microscope pointed at three loci. Because the fixed loci are rarely the mutated ones, RWSP tends toward large single-colour colonies while GHC reflects the real turnover.

The asymmetry is quantifiable rather than incidental. A single inheritance event perturbs at least one probed locus with probability 1 − (1 − ppp)³, against 1 − (1 − ppp)⁴⁰ for the whole genome.

How much does hashing 44 values into 24 bits cost?

Two different questions sit behind that figure. The empirical test measures how often a genome and its own mutated descendant hash to the same colour — that is what decides whether GHC can see a mutation at all.

The combinatorial bound measures how often two unrelated genomes coincide, which is what decides whether the unique-colour count under-reports the population. With 2²⁴ = 16,777,216 colours, the expected number of colliding pairs among N living agents is about N²/2²⁵. At full occupancy of a 200 × 200 grid, N = 40,000 gives roughly 48 colliding pairs — about 0.12% of agents. On the 50 × 50 small runs it is 0.19 pairs. Both are two orders of magnitude below the GHC–RWSP gaps.

7The phenotype–genotype trade-off

Drive phenotypic diversity up and genotypic diversity collapses. Drive genotypic diversity up and the phenotype collapses to a single dominant type. One parameter sets where on that frontier a run lands.

At low ppp many types coexist in CTFP but few unique genomes appear in the genotypic count. At high ppp many unique genomes appear but the phenotype collapses onto one type. This mirrors the resource-allocation trade-off reported for bacteria, where investing in fast growth — few, specialised genes — sacrifices trait flexibility and the other way round.

Why the two scales should trade off

The most plausible mechanism is saturation of the agent nonlinearity, and it follows from the update rule rather than from any tuning. Mutation is additive and never reset, so a weight travelling down a lineage accumulates perturbations at a rate proportional to ppp × ip, each drawn uniformly on [−1, 1] and hence of variance 1/3. The weight vector performs an unbiased random walk whose variance grows linearly in the number of inheritance events. A hidden unit sums 18 such weights against neighbourhood values, so for unit-scale inputs its pre-activation standard deviation grows like √(6 · ppp · ip · g) after g generations.

Saturated units emit states at the extremes of the output range, and those extremes round into a single CTFP bin. The phenotype collapses onto one dominant type while the underlying weight vectors keep diverging — which is exactly what the genotypic count reports. Viability selection sharpens the effect: a lineage that saturates below the alpha threshold removes its own cells, so the survivors are the ones saturating on the live side of the threshold. At low ppp the converse holds. Weights stay in the near-linear regime, small genotypic differences map to different rounded states and several types coexist, while few genomes drift far enough for the hash to register novelty.

8Same phenotype, different genotype

Because phenotype is a read-out of genotype, two runs with identical settings can converge to the same observable colonies while carrying different lineages.

We are deliberately careful about what this shows. Two runs whose grids agree step for step are behaviourally equivalent at the resolution measured here; the two genome sets were not evaluated on any external task, so “same phenotype” means observational equivalence rather than demonstrated functional identity. The clusters below likewise arise from drift under viability selection, with no fitness function beyond liveness — which is why the lineages are described as diverging rather than as adapting, and why open-endedness is used in the descriptive sense of continuing production of novel genotypes without a fixed objective, rather than as a claim that any formal open-endedness criterion has been met.

The released animations

Two small demonstration runs from the repository, in all three views. These are 30 × 30 grids, chosen for legibility — the full-resolution 1000-generation animations for every long run are on the project's video channel.

Frames are sampled and palettised for the web. Click a panel to play it.

9What this does not settle

The claims above are bounded by the design in specific ways. These are the paper's own, set out so they can be weighed directly.

What would settle them

Three of the extensions follow directly from the limits above and can be specified now. The saturation account can be tested by re-instrumenting the simulator to record weight-magnitude histograms and pre-activation variance, then re-running the ppp sweep under that instrumentation. Lineage ground truth can be obtained by logging the parent index at every inheritance event, which converts GHC and CNWA from descriptive tools into estimators that can be scored against a known genealogy. And the adversarial annihilation kernel — the wound tool in section 3 — becomes a measurement once the recovery time of the living-cell fraction and of the global entropy, and the genotypic composition of the refilled region, are logged. That is the untrained analogue of the regeneration experiments in Growing NCA.

10Cite this work

@article{Jain2026SelfReplicatingNCA,
  author    = {Jain, Sanyam and Reimers, Felix Simon and Nichele, Stefano},
  title     = {Self-Replicating Neural Cellular Automata: Quantifying Emergent
               Phenotypic and Genotypic Diversity in an Open-Ended Substrate},
  journal   = {Journal of Imaging},
  publisher = {MDPI},
  year      = {2026},
  note      = {Accepted; volume, pages and DOI to follow}
}

@mastersthesis{Jain2024AIGA,
  author    = {Jain, Sanyam},
  title     = {AI Generating Algorithms with Self-Organizing Neural Cellular
               Automata: Quantifying Genetic and Phenotypic Diversity,
               Emerging Growth, and Speciation},
  school    = {{\O}stfold University College},
  address   = {Halden, Norway},
  year      = {2024}
}