Behavioral Architecture in Lenia

• Behavioral architecture in Lenia is defined by local convolution and bell-shaped growth maps that generate self-organizing, lifelike agent behaviors.
• Occlusion experiments reveal distinct attractor basin regions, with metrics like recovery time and morphological distortion tracking sensitivity.
• Emergent traits such as agnosiophobia arise naturally from dynamic constraints, offering insights into robust, adaptive artificial agency.

Behavioral architecture in Lenia refers to the set of dynamical, informational, and morphological principles underlying the emergence and maintenance of complex, life-like behaviors in Lenia-based cellular automata and their extensions. Lenia provides a deterministic, continuous-valued, spatially extended substrate in which localized, self-organizing patterns (“creatures”) arise and demonstrate a rich behavioral repertoire—including locomotion, morphology preservation, sensitivity to environmental topology, and higher-level adaptive traits. Its architecture links local, kernel-based update rules to global attractor structure and can be interpreted both as a model system for artificial agency and as a dynamical blueprint underlying phenomena such as “agnosiophobia”—the emergent avoidance of sensory voids. This article develops the mathematical and conceptual framework for Lenia’s behavioral architecture, with an emphasis on recent advances in the study of informational topography, dynamical basins, and sensitivity under occlusion (Cool et al., 29 May 2026).

1. Mathematical and Dynamical Foundations

The canonical Lenia system is a discrete-time, continuous-state cellular automaton defined over a toroidal lattice of size $N \times N$. Each cell’s state $A_t(\mathbf{x}) \in [0,1]$ is updated in parallel through three steps (Cool et al., 29 May 2026):

1. Neighborhood Convolution: For each cell $\mathbf{x}$, compute the local potential as a normalized, radially symmetric convolution:

$$U_t(\mathbf{x}) = (K * A_t)(\mathbf{x}) = \sum_{\mathbf{y}} K(\mathbf{y})\,A_t(\mathbf{x} + \mathbf{y}),$$

with $\sum_{\mathbf{y}} K(\mathbf{y}) = 1$.

1. Growth Mapping: Pass $U_t(\mathbf{x})$ through a rule-specific unimodal function $G: [0,1] \to \mathbb{R}$, typically bell-shaped.
2. State Update and Clipping:

$$A_{t+1}(\mathbf{x}) = \mathrm{clip}_{[0,1]}\Bigl( A_t(\mathbf{x}) + \Delta t\, G(U_t(\mathbf{x})) \Bigr),$$

where $\mathrm{clip}_{[0,1]}(z) = \min\{\max\{z,0\},1\}$, and $\Delta t$ is the time step.

The choice of kernel $A_t(\mathbf{x}) \in [0,1]$0, growth map $A_t(\mathbf{x}) \in [0,1]$1, and time step $A_t(\mathbf{x}) \in [0,1]$2 defines the agent’s “genotype.” Self-organization, motility, and robustness to perturbation arise as global attractors of the deterministic mapping $A_t(\mathbf{x}) \in [0,1]$3 (Chan, 2018, Chan, 2020, Hudcová et al., 5 Jan 2026).

2. Informational Topography and Sensory Occlusion

To interrogate the informational architecture of Lenia agents, recent work introduces environmental regions from which sensory information is withheld—simulating “blind spots” or occluded perceptual fields (Cool et al., 29 May 2026). The convolution is locally modified by a binary mask $A_t(\mathbf{x}) \in [0,1]$4 (1=occluded, 0=visible): $A_t(\mathbf{x}) \in [0,1]$5 This renormalization causes asymmetric signals as the kernel overlaps masked and unmasked regions, affecting the local growth/decay field and shaping agent-environment interaction.

Targeted “pixelwise occlusion sweeps” reveal the sensitivity landscape of specific body sites: occlusions may drive the trajectory toward the boundary of the attractor basin, with three possible outcomes—recovery (return to canonical morphology), catastrophic failure (death/explosion), or slow, distorted reorientation (Cool et al., 29 May 2026).

3. Attractor-Basin Geometry and Dynamical Interpretation

Each agent is associated with a high-dimensional attractor manifold $A_t(\mathbf{x}) \in [0,1]$6, whose internal symmetries (position, heading) reflect translation and rotation invariances of Lenia’s dynamics. The corresponding basin of attraction $A_t(\mathbf{x}) \in [0,1]$7 encompasses all states that re-converge to $A_t(\mathbf{x}) \in [0,1]$8. Environmental occlusions act as structured perturbations partitioning $A_t(\mathbf{x}) \in [0,1]$9 into:

• Deep basin zones—rapid, minimal recovery after perturbation (stable operation);
• Near-basin-boundary—slow, highly distorted recoveries associated with pronounced heading change;
• Extrinsic states—trajectories irreversibly leave $\mathbf{x}$0 (death, metamorphosis).

Sensitivity maps derived from body-occlusion sweeps empirically chart these regions, with maximal heading reorientation localized to neighborhoods immediately adjacent to basin-lethal zones. There is an absence of rapid, undistorted reorientations, indicating critical slowing near the basin boundary (Cool et al., 29 May 2026).

4. Emergent Behavioral Propensities: Agnosiophobia

A central emergent trait, “agnosiophobia,” is the agent’s systematic avoidance of informationally occluded regions in its environment. Notably, this avoidance is not explicitly programmed; rather, it arises from the interplay of morphological self-maintenance with informational gradients induced by the masked convolution:

• As the agent approaches an occlusion, a spatial gradient in the renormalized potential $\mathbf{x}$1 develops across its structure.
• This gradient generates an asymmetric $\mathbf{x}$2-driven growth field, introducing a torque on the agent that steers it away from the blind zone.
• The underlying dynamics enforce heading adjustment as a byproduct of shape preservation, not because of any explicit “danger” computation.

Empirical experiments confirm robust avoidance across several agent genotypes; survival correlates with the spatial distribution of sensitivity in the agent’s morphology. Agents with “buffer zones” are more resilient, while those lacking such zones (e.g., S1s) fail catastrophically under occlusion (Cool et al., 29 May 2026).

5. Quantitative Analysis and Metrics

The response of Lenia agents to targeted occlusions is characterized by the following metrics (Cool et al., 29 May 2026):

• Recovery Time ($\mathbf{x}$3): Number of frames until the pattern returns to its canonical attractor.
• Maximum Morphological Distortion ($\mathbf{x}$4): Max Wasserstein-1 distance between current morphology and median canonical profile during recovery.
• Heading Change ($\mathbf{x}$5): Angle by which the centroid trajectory is deflected pre- versus post-recovery.
• Sensitivity Mapping: For each body site, systematically masked, the trio ($\mathbf{x}$6) is recorded and mapped.

Correlation analyses show that large heading changes occur only in conjunction with extended, morphologically distorted recoveries—i.e., only near basin boundaries.

6. Broader Implications for Artificial Life and Cognitive Agency

The architecture revealed in Lenia—kernel convolution, bell-shaped growth, and attractor-basin geometry—underpins a set of computationally robust, lifelike behaviors that are emergent rather than hard-coded. Crucially:

• Self-preserving dynamics produce navigation-like responses by exploiting local asymmetries in information, even in the absence of explicit sensorimotor coupling.
• Morphological sensitivity mapping provides a concrete methodology for assessing the “cognitive” structure of embodied virtual agents, operationalizing concepts such as the cognitive domain (Maturana & Varela).
• Informational topography (the arrangement of sensory occlusions) is shown to be as significant as tangible obstacles in shaping agent behavior; emergent avoidance is rooted in the architecture’s dynamical properties, not explicit representation or information-theoretic inference.

These results extend the conceptual toolkit for analyzing autonomous, self-maintaining agents in excitable media and strengthen the paradigm in which behavior emerges from the interplay of local update rules, global attractor geometry, and informational features of the environment (Cool et al., 29 May 2026).