Asymptotic Lenia: Continuous Cellular Automata

• Asymptotic Lenia is a continuum-limit framework replacing discrete clipping with natural boundedness, enabling rigorous analysis using PDEs.
• It bridges continuous cellular automata with reaction–diffusion and kernel-based Turing models through innovative mathematical formulations.
• The model facilitates quantitative assessments of emergent complexity and evolutionary dynamics using metrics like Lyapunov exponents and entropy.

Asymptotic Lenia is the mathematically principled, continuum-limit variant of the Lenia family of continuous cellular automata (CCAs), formulated as a system of partial differential equations (PDEs) or integro-differential equations that eliminate discretization artifacts present in the original, clipped-update models. This formulation enables rigorous analytical investigation using dynamical systems theory, connection with reaction–diffusion models, and quantitative characterization of emergent complexity and evolutionary activity in artificial life systems. Asymptotic Lenia is central to the paper of open-ended evolution, autopoiesis, and complexity in computational media, providing a theoretical and computational bridge between artificial and biological patterns.

1. Mathematical Formulation and Continuum Limit

Asymptotic Lenia is defined by a continuous-time evolution equation of the following form: $\frac{\partial u(x, t)}{\partial t} = T(K * u) - u$ where $u(x, t)$ is the continuous state of the system over space and time, $K$ is a normalized, radially symmetric kernel ($K(0) = 0, \int_{\Omega} K(x)\,dx = 1$), $*$ denotes spatial convolution, and $T$ is a Lipschitz continuous, bounded “target” or growth function, typically an unstretched Gaussian; $0 \leq T(\cdot) \leq 1$ (Kojima et al., 2023, Yevenko et al., 4 Aug 2025).

Unlike classical Lenia, which applies an explicit clip operator to confine values to $[0,1]$, Asymptotic Lenia’s target function naturally keeps states bounded. This continuous, non-clipped setting permits the derivation of the model’s dynamics from the limit of the discrete Euler method as $\Delta t \to 0$, i.e.,

$$A(x, t+\Delta t) = A(x, t) + \Delta t \left(T(K * A(x, t)) - A(x, t)\right)$$

Time-step tick independence is achieved in this limit, ensuring that solutions are robust to numerical integration parameters (Kojima et al., 2023).

The model is mathematically equivalent to a kernel-based Turing (KT) model when $T$ is chosen piecewise-linear, connecting Lenia’s space of behaviors with classical reaction–diffusion systems (Kojima et al., 2023).

2. Existence, Uniqueness, and Entropy

Standard ODE existence theorems (e.g., Picard–Lindelöf) fail for Lenia due to discontinuities introduced by clipping in the discrete variant. However, in the Asymptotic Lenia setting, or when employing arc field theory, one establishes the existence and uniqueness of the flow $u(x, t)$ in the forward direction, as the mapping is continuous and satisfies relevant Lipschitz bounds (Calcaterra et al., 2022).

The removal of explicit clipping renders Asymptotic Lenia time-reversible and entropy-neutral; there is no fundamental increase in entropy or irreversible information loss. This contrasts with original Lenia, where clipping induces an arrow of time and allows finite-time death—a property modeling biological irreversibility (Calcaterra et al., 2022). The difference is significant: Asymptotic Lenia is more mathematically tractable, but certain biological phenomena (death, extinction) are suppressed or fundamentally altered.

3. Dynamical Systems Theory and Complexity Measures

The PDE formulation of Asymptotic Lenia supports a rigorous dynamical systems analysis (Yevenko et al., 4 Aug 2025). Four main solution classes arise:

1. Solitons: Purely translating, stable wave packets, satisfying an ansatz $A(x, t) = e^{t\gamma}A_0$.
2. Rotators: Solutions invariant under spatial rotations, defined via a generator $\gamma$ from the Lie algebra of SE(2).
3. Periodic Solutions: Translating solutions with added time-periodic oscillatory modulation.
4. Chaotic Patterns: Solutions exhibiting internal oscillations and translation, with small positive Lyapunov exponents.

Existence of a global attractor is guaranteed by dissipativity. The approach bounds minimum $m(t)$ and maximum $M(t)$ values of $A[i, j](t)$ to the compact interval $[–\varepsilon, 1+\varepsilon]^{N^2}$ after transient time. The attractor may be fractal, typically with dimension $>4$ for chaotic cases.

Complexity is effectively quantified by the Kaplan–Yorke dimension,

$$D_{\text{KY}} = j + \frac{\sum_{i=1}^j \lambda_i}{|\lambda_{j+1}|}$$

where $\{\lambda_i\}$ are ordered Lyapunov exponents and $j$ is maximal such that $\sum_{i=1}^j \lambda_i \geq 0$ (Yevenko et al., 4 Aug 2025). Complementary measures include computation of covariant Lyapunov vectors (CLVs) for local instability direction analysis. An open-source implementation of LE/CLV computation is available, facilitating practical assessment of complexity in large-scale CCA simulations (Yevenko et al., 4 Aug 2025).

4. Emergence, Evolution, and Open-Ended Dynamics

Asymptotic Lenia’s analytical continuity supports the systematic emergence of diverse pattern classes—including solitons, complex oscillators, rotors, and chaotic "organisms". Genetic and quality-diversity algorithms have been shown to drive the sustained discovery of self-organizing patterns, a phenomenon interpreted as open-ended evolution (Faldor et al., 6 Jun 2024, Lorantos et al., 3 Jun 2025). Adaptive evolutionary methodologies (MAP-Elites, AURORA) maximize not only homeostasis but also distinctiveness and population sparsity, rewarding the continual colonization of new behavioral niches.

Intrinsic multi-objective ranking strategies have empirically led to increases in pattern repertoire variance (+0.91%), sustainable mass, and modular complexity, supporting the claim that Asymptotic Lenia can realize open-ended evolutionary regimes without prematurity or stagnation (Lorantos et al., 3 Jun 2025). These findings are supported by monitoring census statistics, entropy in VAE-derived latent spaces, and the prevalence of new "elites" over long simulation periods.

5. Structural Stability, Autopoiesis, and Discretization Effects

In Asymptotic Lenia, autopoietic competence—the ability of a structure to maintain itself—is closely tied to simulation coarseness (Davis, 30 Jul 2024). Glider persistence requires a suitable range of temporal ($\Delta t$) and spatial (kernel scale $k_r$) discretization; beyond a certain refinement threshold, patterns lose their homeostasis and may dissolve. The self-maintenance of these patterns is thus non-Platonic, depending on finite resolution rather than idealized, infinitely precise dynamics.

Instability maps in parameter space reveal fractal boundaries between stable and unstable regimes, with intricate detail persisting to floating point limits. This fractal retention is characteristic of a chaotic, multi-scale system and implies undecidability in the fate of specific initial conditions (Davis, 30 Jul 2024). The removal of hard clipping, via choice of a naturally bounded $T$, alters but does not eliminate non-Platonic autopoietic phenomena.

6. Extensions: Reaction–Diffusion and Mass Conservation Models

Asymptotic Lenia is mathematically equivalent to generalized reaction–diffusion (RD) systems, specifically kernel-based Turing (KT) models (Kojima et al., 2023). By emulating the non-local kernel convolution with a sum of local auxiliary diffusion fields, reaction–diffusion PDEs can reproduce Lenia's continuum-limit behavior, albeit with a caveat: the required reaction terms (e.g., Gaussian $T$) are non-polynomial and violate mass-action constraints, disallowing interpretation as a conventional chemical kinetics system.

Flow-Lenia introduces mass conservation into the Lenia framework by treating the state not as normalized activation but as “matter” transported by a divergence-free flow field (Plantec et al., 10 Jun 2025). The flow $\mathbf{F}$ is computed as a balance of affinity gradients (for pattern formation) and total mass gradients (for regulatory feedback), ensuring exact conservation at each timestep: $$\mathbf{F}^t_i = (1 - \alpha^t) \nabla U^t_i - \alpha^t \nabla A_\Sigma^t,\quad \alpha^t(x) = [(A_\Sigma^t(x)/\beta_A)^n]_0^1$$ Parameter embedding and reintegration tracking schemes allow for simulation of multispecies populations and evolutionary dynamics in mass-conserved artificial life ecosystems.

7. Metrics, Implementation, and Broader Implications

Quantitative analysis of complexity and evolutionary diversification in Asymptotic Lenia leverages both traditional statistics (mass, variance) and dynamical system invariants (Lyapunov exponents, entropy, attractor dimensions). Evolutionary activity frameworks, count-based and non-neutral activity metrics, and diversity distances in parameter space provide fine-grained insight into asymptotic dynamics and regime transitions (Plantec et al., 10 Jun 2025).

Implementation of these analyses (e.g., for computing full Lyapunov spectra and CLVs on CCAs of up to $128 \times 128$ size) is computationally efficient, supporting routine large-scale simulations.

In summary, Asymptotic Lenia supplies a unifying, analytically tractable framework for the paper of artificial life, pattern formation, and open-ended evolution. Its dynamical equations, structural stability properties, connection with RD systems, and quantitative metrics for complexity provide a foundation for both theoretical inquiry and practical exploration in artificial and natural complex systems (Yevenko et al., 4 Aug 2025, Kojima et al., 2023, Plantec et al., 10 Jun 2025, Davis, 30 Jul 2024, Lorantos et al., 3 Jun 2025, Faldor et al., 6 Jun 2024, Chan, 2018, Jain et al., 2023, Chan, 2020, Calcaterra et al., 2022, Davis et al., 2022).