Internalized Morphogenesis: A Self-Organizing Model for Growth, Replication, and Regeneration via Local Token Exchange in Modular Systems

This study presents an internalized morphogenesis model for autonomous systems, such as swarm robotics and micro-nanomachines, that eliminates the need for external spatial computation. Traditional self-organizing models often require calculations across the entire coordinate space, including empty areas, which is impractical for resource-constrained physical modules. Our proposed model achieves complex morphogenesis through strictly local interactions between adjacent modules within the “body.” By extending the “Ishida token model,” modules exchange integer values using an RD-inspired discrete analogue without solving differential equations. The internal potential, derived from token accumulation and aging, guides autonomous growth, shrinkage, and replication. Simulations on a hexagonal grid demonstrated the emergence of limb-like extensions, self-division, and robust regeneration capabilities following structural amputation. A key feature is the use of the body boundary as a natural sink for information entropy (tokens) to maintain a dynamic equilibrium. These results indicate that sophisticated morphological behaviors can emerge from minimal, internal-only rules. This framework offers a computationally efficient and biologically plausible approach to developing self-repairing, adaptive, and autonomous hardware.

💡 Research Summary

The paper introduces an “internalized morphogenesis” framework that eliminates any reliance on external spatial computation, enabling autonomous growth, self‑division, and regeneration through strictly local token exchange among adjacent modules. Traditional reaction‑diffusion (RD) and cellular automata (CA) approaches require updating the entire coordinate space, including empty regions, which is infeasible for resource‑constrained swarm robots or micro‑nanomachines. Building on the earlier Ishida token model, the authors devise a discrete, integer‑based token system that mimics diffusion and decay without solving partial differential equations.

The computational domain is a hexagonal lattice (6‑neighbour connectivity) where each cell is either occupied (state 1) or empty (state 0). Only occupied cells participate in any computation; empty cells are never updated, guaranteeing that all dynamics are confined to the “body”. At each time step every module generates a single token with an aging label v = 1. Tokens propagate up to Z = 20 hops, each hop dividing the token amount equally among the six neighbours. Two token classes emerge: propagating tokens (the moving “wavefront”) whose label increments each hop, and accumulated tokens, which each module stores per label. The accumulated tokens constitute the internal potential.

Potential is compared against two thresholds: a growth threshold T_g and a shrink threshold T_s. If a boundary module’s potential exceeds T_g, a new module is created on the adjacent empty cell, extending the body; if it falls below T_s, the boundary module is removed, causing contraction. Because tokens are never allowed to leave the occupied region, the body boundary acts as a natural sink, preventing uncontrolled token accumulation and maintaining a dynamic equilibrium.

Simulation experiments on a 150 × 150 hexagonal grid start from a central circular seed. The system spontaneously generates limb‑like protrusions, demonstrating that spatial heterogeneity in potential can drive anisotropic growth. When the body reaches a critical size, the internal potential distribution triggers a bifurcation: the structure splits into two roughly equal daughter bodies, illustrating self‑division without any explicit replication protocol. In an amputation scenario where a portion of the body is removed, the remaining modules quickly rebuild the missing region; the loss of neighbours causes a local surge in potential, which drives rapid token accumulation and new module creation at the wound edge.

The authors analyze computational complexity, showing it scales linearly with the number of occupied cells (O(N)), and each module performs only five simple operations per step. This makes the algorithm suitable for low‑power microcontrollers typical of swarm robots. The paper also discusses limitations: the current implementation is two‑dimensional, parameter sensitivity (token decay rate, thresholds) can affect pattern stability, and real‑world communication delays or packet loss have not been modeled. Future work includes extending the model to three dimensions, automated parameter tuning via reinforcement learning, and hardware validation on platforms such as Kilobots or SMORES.

In summary, the study provides a minimal yet powerful set of local rules that achieve complex morphogenetic behaviours—growth, division, and regeneration—entirely within the occupied body. By treating token exchange as a discrete analogue of diffusion and using the body boundary as an entropy sink, the framework offers a computationally lightweight, biologically plausible route toward self‑repairing, adaptive, and autonomous modular systems.