Pulsing dynamics in randomly wired glider cellular automata

Sustained rhythmic oscillations, pulsing dynamics, emerge spontaneously when the local connection scheme is randomised in 3-value cellular automata that feature"glider" dynamics. Time-plots of pulsing measures maintain a distinct waveform for each glider rule, and scatter plots of entropy/density and the density return-map show unique signatures, which have the characteristics of chaotic strange attractors. We present case studies, possible mechanisms, and implications for oscillatory networks in biology.

💡 Research Summary

The paper investigates a striking dynamical phenomenon that emerges when the local neighbourhood connections of three‑state, k‑totalistic cellular automata (CA) are replaced by random wiring. The authors focus on rules that support “gliders” – mobile, coherent patterns that interact through collisions and are a hallmark of complex CA behaviour. By using the DDLab platform, they systematically randomise the k inputs of each cell on a 2‑D hexagonal lattice (k = 6 or 7) and, in later sections, also on a 3‑D lattice. The randomisation can be completely unrestricted, confined to a local zone, or partially “freed” (one or more wires left truly random while the rest stay in the original neighbourhood).

Main discovery – Pulsing.
When a glider‑supporting rule is subjected to any form of random wiring, the automaton spontaneously settles into a sustained rhythmic oscillation that the authors call “pulsing”. The oscillation is visible in the space‑time diagrams as a global density wave that repeatedly sweeps across the lattice. Quantitatively, two global observables are tracked at every time step: (i) the density of each state (0, 1, 2) and (ii) the input‑entropy, i.e. the Shannon entropy of the histogram of neighbourhood‑type frequencies actually looked up in the rule table. For each rule the time series of these observables exhibits a characteristic waveform: a well‑defined wavelength (number of steps per cycle), wave‑height (amplitude) and shape (phase). Different glider rules produce distinct waveforms, providing a fingerprint for each rule.

Statistical signatures.
The authors plot (a) entropy versus non‑zero density and (b) a density return map (ρ(t) versus ρ(t + 1)). Both scatter plots form compact, non‑trivial structures that resemble chaotic strange attractors: they are sensitive to initial conditions, locally unstable, yet globally bounded. The attractor‑like sets persist as the lattice size grows; larger lattices simply sharpen the patterns, while very small lattices sometimes fall into uniform fixed points (e.g., all zeros).

Robustness tests.

• Network size: Waveforms are invariant from 50 × 50 up to 200 × 200 cells; only the statistical spread shrinks with size.
• Re‑randomisation: Re‑wiring the network at every step (annealed disorder) does not alter the waveform, indicating that the phenomenon depends only on the statistical distribution of inputs, not on a particular wiring.
• Partial localisation: Confining random connections to a limited radius creates local spiral‑like density waves, yet the global pulsing persists.
• Freed wires: Allowing one or more inputs to be truly random while the rest stay in the original neighbourhood changes the waveform’s details but not its existence.
• 3‑D extension: The same behaviour appears on cubic lattices, confirming that the effect is not tied to two‑dimensional geometry.
• Noise and asynchronous updates: Adding random perturbations to cell values or updating cells asynchronously leaves the pulsing essentially unchanged, demonstrating strong dynamical robustness.

Biological relevance.
The authors argue that the CA pulsing model offers a fresh perspective on biological oscillations such as cardiac rhythms, neuronal low‑frequency oscillations, and intracellular calcium waves. In these systems, local excitable elements (analogous to gliders) coexist with a network of connections that are neither perfectly regular nor completely random; synaptic plasticity, gap‑junction remodeling, or vascular coupling can be viewed as a form of dynamic random wiring. The fact that a simple discrete system can generate stable, rule‑specific rhythmic activity while tolerating noise and structural changes suggests that similar mechanisms might underlie the robustness of biological clocks.

Theoretical contribution.
By introducing random wiring as a new axis of CA classification, the paper defines a “pulsing CA” class distinct from the traditional Wolfram classes (ordered, chaotic, complex). It demonstrates that input‑entropy dynamics can serve as an automatic diagnostic for discovering such behaviours, and it bridges concepts from continuous dynamical systems (strange attractors) to discrete, deterministic cellular automata. The work also highlights how breaking spatial locality while preserving a homogeneous update rule can generate global periodicity—a counter‑intuitive result that enriches our understanding of self‑organisation in complex networks.

In summary, the study shows that three‑state, k‑totalistic cellular automata that support gliders inevitably develop a robust, rule‑specific pulsing rhythm when their neighbourhood connections are randomised. The rhythm is captured by global density and entropy measures, exhibits chaotic‑like attractor signatures, and persists under a wide range of perturbations. These findings open avenues for modelling biological oscillators and for extending CA theory to incorporate stochastic connectivity as a source of emergent temporal order.