Mathematical analysis to an adaptive network of the Plasmodium system

Abstract

Adaptive networks are emerging a lot of fields of science, recently. In this paper, we consider about a mathematical model for adaptive network made by the plasmodium. The organism contains a tube network by means of which nutrients and signals circulate through the body. The tube network changes its shape to connect two exits through the shortest path when the organism is put in a maze and food is placed at two exits. Recently, a mathematical model for this adaptation process of the plasmodium has been proposed. Here we analyze it mathematically rigorously. In ring-shaped network and Wheatstone bridge-shaped network, we mainly show that the globally asymptotically stable equilibrium point of the model corresponds to the shortest path connecting two special points on the network in the case where the shortest path is determined uniquely. From the viewpoint of mathematical technique, especially in the case of Wheatstone bridge-shaped network, we show that there is a simple but novel device used here by which we prove the global asymptotic stability, even when Lyapunov function cannot be constructed.

Physarum Can Compute Shortest Paths

“…Miyaji and Ohnishi [MO07,MO08] initiated the analytical investigation of the model. They argued convergence against the shortest path if G is a planar graph and s 0 and s 1 lie on the same face in some embedding of G. Ito et al [IJNT11] study the dynamics (1) in a directed graph G = (V, E); they do not claim that the model is justified on biological grounds.…”

“…Theorem 1 (Miyashi-Ohnishi [MO07]) For a network of parallel links, the dynamics converge against D 1 = 1 and D i = 0 for i ≥ 2.…”

“…For a path P , let W (P ) := e∈P L e ln D e be its weighted sum of log diameters, and let ∆(P ) = p a − p b be the potential difference between its endpoints. The function W (P ) was introduced by Miyaji and Ohnishi [MO07,MO08].…”

“…This holds true for any initial condition and assumes the uniqueness of the shortest path. Miyaji and Ohnishi [MO07,MO08] initiated the analytical investigation of the model. They argued convergence against the shortest path if G is a planar graph and s 0 and s 1 lie on the same face in some embedding of G.…”

Numerical approximation of nonhomogeneous boundary conditions on networks for a hyperbolic system of chemotaxis modeling the Physarum dynamics

“…In that paper, it was shown numerically that, in the asymptotic steady state of the model, the solution converges to the minimum-length solution between a source and a sink on any input graph. The theoretical analysis of such ODE model was carried out in successive works such as [18,19,1]. In particular, in [18] there was the proof of convergence for two parallel links, while the convergence of this model for any network to the shortest path connecting the source and the sink was analytically proved in [1].…”