The directed Physarum dynamics is known to solve positive linear programs: minimize @math subject to @math and @math for a positive cost vector @math . The directed Physarum dynamics evolves a positive vector @math according to the dynamics @math . Here @math is the solution to @math that minimizes the "energy" @math . In this paper, we study the non-uniform directed dynamics @math , where @math is a positive diagonal matrix. The non-uniform dynamics is more complex than the uniform dynamics (with @math being the identity matrix), as it allows each component of @math to react with different speed to the differences between @math and @math . Our contribution is to show that the non-uniform directed dynamics solves positive linear programs.
is a slime mold that apparently is able to solve shortest path problems. Nakagaki, Yamada, and T ' o th @cite_0 report about the following experiment; see Figure . They built a maze, covered it by pieces of Physarum (the slime can be cut into pieces which will reunite if brought into vicinity), and then fed the slime with oatmeal at two locations. After a few hours the slime retracted to a path following the shortest path in the maze connecting the food sources. The authors report that they repeated the experiment with different mazes; in all experiments, Physarum retracted to the shortest path.
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2138415668 1964894406 2003810228 2031275178
Physarum Polycephalum is a slime mold that apparently is able to solve shortest path problems. A mathematical model has been proposed by biologists to describe the feedback mechanism used by the slime mold to adapt its tubular channels while foraging two food sources s0 and s1. We prove that, under this model, the mass of the mold will eventually converge to the shortest s0-s1 path of the network that the mold lies on, independently of the structure of the network or of the initial mass distribution. This matches the experimental observations by the biologists and can be seen as an example of a "natural algorithm", that is, an algorithm developed by evolution over millions of years. Physarum polycephalum is a unicellular, multinucleate slime mold in the Mycetozoa group that exhibits remarkable capabilities, including the ability to construct efficient networks when foraging food [1]. The Physarum cell grows, as long as nutrition is abundant. When nutrition is limited, Physarum forms a network of interconnected veins; the veins are gel-like tubes in which the cytoplasm flows. It has been experimentally observed that, with two food sources, Physarum's tubular network often retracts to the shortest path between the food sources. Tero, Kobayashi and Nakagaki [2] proposed a model for the dynamics of the Physarum, which in the computer simulations converged to the shortest path on any initial network. We analytically prove that, under this model, the mass of the mold has to eventually converge to the shortest path in the initial network between the two food sources, independently of the structure of the initial network or of the initial mass distribution. This presentation is based on the work in [3,4] and is supported by the Italian Flagship Initiative "InterOmics" (PB.P05). References [1] Toshiyuki Nakagaki, Hiroyasu Yamada, Agota Toth: Maze-solving by an amoeboid organism. Nature 407:470 (2000) [2] Atsushi Tero, Ryo Kobayashi, Toshiyuki Nakagaki: A mathematical model for adaptive transport network in path finding by true slime mold. Journal of Theoretical Biology 244: 553-564 (2007) [3] Vincenzo Bonifaci, Kurt Mehlhorn, Girish Varma: Physarum can compute shortest paths. Journal of Theoretical Biology 309: 121-133 (2012) [4] Vincenzo Bonifaci: Physarum can compute shortest paths: A short proof. Information Processing Letters 113(1-2): 4-7 (2013) We describe here a mathematical model of the adaptive dynamics of a transport network of the true slime mold Physarum polycephalum, an amoeboid organism that exhibits path-finding behavior in a maze. This organism possesses a network of tubular elements, by means of which nutrients and signals circulate through the plasmodium. When the organism is put in a maze, the network changes its shape to connect two exits by the shortest path. This process of path-finding is attributed to an underlying physiological mechanism: a tube thickens as the flux through it increases. The experimental evidence for this is, however, only qualitative. We constructed a mathematical model of the general form of the tube dynamics. Our model contains a key parameter corresponding to the extent of the feedback regulation between the thickness of a tube and the flux through it. We demonstrate the dependence of the behavior of the model on this parameter. When two food sources are presented to the slime mold Physarum in the dark, a thick tube for absorbing nutrients is formed that connects the food sources through the shortest route. When the light-avoiding organism is partially illuminated, however, the tube connecting the food sources follows a different route. Defining risk as the experimentally measurable rate of light-avoiding movement, the minimum-risk path is exhibited by the organism, determined by integrating along the path. A model for an adaptive-tube network is presented that is in good agreement with the experimental observations.