Towards a stationary Monge-Kantorovich dynamics: the Physarum Polycephalum experience

TL;DR

This paper extends a model of Physarum Polycephalum's pathfinding ability from a graph-based to a continuous domain, demonstrating convergence to solutions of the Monge-Kantorovich optimal transport problem.

Contribution

It introduces a continuous domain extension of the Physarum model and shows its asymptotic behavior aligns with the Monge-Kantorovich equations, linking biological dynamics to optimal transport theory.

Findings

Model converges to an equilibrium resembling MK solutions

Numerical methods successfully simulate long-term Physarum behavior

Extension bridges biological modeling with optimal transport mathematics

Abstract

In this work we study and expand a model describing the dynamics of a unicellular slime mold, Physarum Polycephalum (PP), which was proposed to simulate the ability of PP to find the shortest path connecting two food sources in a maze. The original model describes the dynamics of the slime mold on a finite dimensional planar graph using a pipe-flow analogy whereby mass transfer occurs because of pressure differences with a conductivity coefficient that varies with the flow intensity. We propose an extension of this model that abandons the graph structure and moves to a continuous domain. Numerical evidence, shows that the model is capable of describing the slime mold dynamics also for large times, accurately reproducing the PP behavior. A notable result related to the original model is that it is equivalent to an optimal transportation problem over the graph as time tends to infinity. In our case, we can only conjecture that our extension presents a time-asymptotic equilibrium. This equilibrium point is precisely the solution of the Monge-Kantorovich (MK) equations at the basis of the PDE formulation of optimal transportation problems. Numerical results obtained with our approach, which combines P1 Finite Elements with forward Euler time stepping, show that the approximate solution converges at large times to an equilibrium configuration that well compares with the numerical solution of the MK-equations.