Cognition as least action: the Physarum Lagrangian

TL;DR

The paper recasts Physarum polycephalum's adaptive transport as a graph-based least-action problem, introducing a Lagrangian $\mathcal{L}_{\phi}$ whose stationary conditions reproduce Poiseuille–Kirchhoff flow and a dual energy $\mathcal{E}(p;D)$, while coupling to a gradient-flow free-energy $\mathcal{F}(D)$ that governs conductance adaptation. On ring, binary-branch, and square-lattice topologies, the framework predicts steady-state networks that minimize energy dissipation under prescribed boundary conditions, reproducing the organism's tendency to prune inefficient paths and concentrate flow along shortest routes. This variational perspective argues that Physarum's problem-solving emerges from fundamental physical principles rather than explicit computation, offering a unifying view of morphogenesis and constrained navigation in aneural systems. The approach may inform understanding of other pre-neural cognitive processes and inspire energy-aware design in transport-like networks within living matter.

Abstract

The slime mould Physarum polycephalum displays adaptive transport dynamics and network formation that have inspired its use as a model of biological computation. We develop a Lagrangian formulation of Physarum's adaptive dynamics on predefined graphs, showing that steady states arise as extrema of a least-action functional balancing metabolic dissipation and transport efficiency. The organism's apparent ability to find optimal paths between nutrient sources and sinks emerges from minimizing global energy dissipation under predefined boundary conditions that specify the problem to be solved. Applied to ring, tree, and lattice geometries, the framework accurately reproduces the optimal conductance and flux configurations observed experimentally. These results show that Physarum's problem-solving on constrained topologies follows a physics-based variational principle, revealing least-action dynamics as the foundation of its adaptive organization.

Figures (4)

• Figure 1: (a) Physarum polycephalum forming a dynamic transport network. The plasmodium organizes its protoplasmic veins into an adaptive, spatially efficient network that optimizes nutrient distribution and environmental exploration (image courtesy of Audrey Dussutour). (b) Navigating a maze. Starting from a uniform inoculation throughout the labyrinth, the plasmodium retracts from dead ends, maintaining only the optimal path connecting food sources, thereby illustrating decentralized problem-solving through adaptive network reorganization. This problem has been mapped into a graph (c) and numerically solved Tero2007, showing that the problem can be represented in terms of many coupled ordinary differential equations on a graph.
• Figure 2: Shortest paths by Physarum on simple graphs. Here, (a) and (b) indicate the initial and final state, respectively, of the ring-shaped network experiment. The lengths of longer and shorter paths are $L_1 = 42$ and $L_2 = 13$ mm, respectively. In (c) and (d), we display the initial and final states, respectively, of the T-shaped graph experiment. Images a--d redrawn using ChatGPT (GPT-5, OpenAI, 2025) based on the original photographs from Tero2007. The insets show, respectively, the initial and the final state of the simulation, which consisted of four nodes and three edges. The width of the black lines reflects the conductivity of each path.
• Figure 3: Schematic representation of the binary tree topology used in the Physarum variational model. In (a), the root node (black circle on the left) represents the nutrient source, and each internal node branches into a left ($L$) and a right ($R$) descendant. At depth $\Lambda$, some terminal nodes (black circles on the right) act as sinks, while others remain inactive. The network fluxes $Q_{iL}$ and $Q_{iR}$ evolve according to the Poiseuille relations, and the conductances $D_{ij}$ adapt dynamically, leading to the pruning of all branches that do not connect the source to at least one active sink. In (b), the notation used for the left/right nodes is indicated.
• Figure 4: Formation and decay of a self-connected loop in Physarum polycephalum. Panels (a–c) show a time sequence of the organism expanding from the central food source through two opposite branches, which eventually merge to form a closed loop (c). After contact, the entire network retracts and disappears, indicating loss of flow and nutrient transport. Panel (d) represents the corresponding idealized topology, where the source and sink coincide ($s = t$), forming a single closed cycle.